source
stringlengths 31
168
| text
stringlengths 51
3k
|
|---|---|
https://en.wikipedia.org/wiki/Mertens%20conjecture
|
In mathematics, the Mertens conjecture is the statement that the Mertens function is bounded by . Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermite (reprinted in ), and again in print by , and disproved by .
It is a striking example of a mathematical conjecture proven false despite a large amount of computational evidence in its favor.
Definition
In number theory, we define the Mertens function as
where μ(k) is the Möbius function; the Mertens conjecture is that for all n > 1,
Disproof of the conjecture
Stieltjes claimed in 1885 to have proven a weaker result, namely that was bounded, but did not publish a proof. (In terms of , the Mertens conjecture is that .)
In 1985, Andrew Odlyzko and Herman te Riele proved the Mertens conjecture false using the Lenstra–Lenstra–Lovász lattice basis reduction algorithm:
and
It was later shown that the first counterexample appears below but above 1016. The upper bound has since been lowered to or approximately but no explicit counterexample is known.
The law of the iterated logarithm states that if is replaced by a random sequence of +1s and −1s then the order of growth of the partial sum of the first terms is (with probability 1) about which suggests that the order of growth of might be somewhere around . The actual order of growth may be somewhat smaller; in the early 1990s Steve Gonek conjectured that the order of growth of was which was affirmed by Ng (2004), based on a heuristic argument, that assumed the Riemann hypothesis and certain conjectures about the averaged behavior of zeros of the Riemann zeta function.
In 1979, Cohen and Dress found the largest known value of for and in 2011, Kuznetsov found the largest known negative value for In 2016, Hurst computed for every but did not find larger values of .
In 2006, Kotnik and te Riele improved the upper bound and showed that there are infinitely many values of for which but without giving any specific value for such an . In 2016, Hurst made further improvements by showing
and
Connection to the Riemann hypothesis
The connection to the Riemann hypothesis is based on the Dirichlet series
for the reciprocal of the Riemann zeta function,
valid in the region . We can rewrite this as a
Stieltjes integral
and after integrating by parts, obtain the reciprocal of the zeta function
as a Mellin transform
Using the Mellin inversion theorem we now can express in terms of as
which is valid for , and valid for on the Riemann hypothesis.
From this, the Mellin transform integral must be convergent, and hence
must be for every exponent e greater than . From this it follows that
for all positive is equivalent to the Riemann hypothesis, which therefore would have followed from the stronger Mertens hypothesis, and follows from the hypothesis of Stieltjes that
References
Further reading
External links
Ana
|
https://en.wikipedia.org/wiki/Linear%20differential%20equation
|
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable .
Such an equation is an ordinary differential equation (ODE). A linear differential equation may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives.
Types of solution
A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any.
The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.
Basic terminology
The highest order of derivation that appears in a (linear) differential equation is the order of the equation. The term , which does not depend on the unknown function and its derivatives, is sometimes called the constant term of the equation (by analogy with algebraic equations), even when this term is a non-constant function. If the constant term is the zero function, then the differential equation is said to be homogeneous, as it is a homogeneous polynomial in the unknown function and its derivatives. The equation obtained by replacing, in a linear differential equation, the constant term by the zero function is the . A differential equation has constant coefficients if only constant functions appear as coefficients in the associated homogeneous equation.
A of a differential equation is a function that satisfies the equation.
The solutions of a homogeneous linear differential equation form a vector space. In the ordinary case, this vector space has a finite dimension, equal to the order of the equation. All solutions of a linear dif
|
https://en.wikipedia.org/wiki/Spline
|
Spline may refer to:
Mathematics
Spline (mathematics), a mathematical function used for interpolation or smoothing
Smoothing spline, a method of smoothing using a spline function
Devices
Spline (mechanical), a mating feature for rotating elements
Flat spline, a device to draw curves
Spline drive, a type of screw drive
Spline cord, a type of thin rubber cord used to secure a window screen to its frame
Spline (or star filler), a type of plastic cable filler for CAT cable
Other
Spline (alien beings), in Stephen Baxter's Xeelee Sequence novels
See also
|
https://en.wikipedia.org/wiki/Janko%20group
|
In the area of modern algebra known as group theory, the Janko groups are the four sporadic simple groups J1, J2, J3 and J4 introduced by Zvonimir Janko. Unlike the Mathieu groups, Conway groups, or Fischer groups, the Janko groups do not form a series, and the relation among the four groups is mainly historical rather than mathematical.
History
Janko constructed the first of these groups, J1, in 1965 and predicted the existence of J2 and J3. In 1976, he suggested the existence of J4. Later, J2, J3 and J4 were all shown to exist.
J1 was the first sporadic simple group discovered in nearly a century: until then only the Mathieu groups were known, M11 and M12 having been found in 1861, and M22, M23 and M24 in 1873. The discovery of J1 caused a great "sensation" and "surprise" among group theory specialists. This began the modern theory of sporadic groups.
And in a sense, J4 ended it. It would be the last sporadic group (and, since the non-sporadic families had already been found, the last finite simple group) predicted and discovered, though this could only be said in hindsight when the Classification theorem was completed.
References
Sporadic groups
|
https://en.wikipedia.org/wiki/Mathieu%20group
|
In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups M11, M12, M22, M23 and M24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They are the first sporadic groups to be discovered.
Sometimes the notation M8 M9, M10, M20 and M21 is used for related groups (which act on sets of 9, 10, 20, and 21 points, respectively), namely the stabilizers of points in the larger groups. While these are not sporadic simple groups, they are subgroups of the larger groups and can be used to construct the larger ones. John Conway has shown that one can also extend this sequence up, obtaining the Mathieu groupoid M13 acting on 13 points. M21 is simple, but is not a sporadic group, being isomorphic to PSL(3,4).
History
introduced the group M12 as part of an investigation of multiply transitive permutation groups, and briefly mentioned (on page 274) the group M24, giving its order. In he gave further details, including explicit generating sets for his groups, but it was not easy to see from his arguments that the groups generated are not just alternating groups, and for several years the existence of his groups was controversial. even published a paper mistakenly claiming to prove that M24 does not exist, though shortly afterwards in he pointed out that his proof was wrong, and gave a proof that the Mathieu groups are simple. finally removed the doubts about the existence of these groups, by constructing them as successive transitive extensions of permutation groups, as well as automorphism groups of Steiner systems.
After the Mathieu groups no new sporadic groups were found until 1965, when the group J1 was discovered.
Multiply transitive groups
Mathieu was interested in finding multiply transitive permutation groups, which will now be defined. For a natural number k, a permutation group G acting on n points is k-transitive if, given two sets of points a1, ... ak and b1, ... bk with the property that all the ai are distinct and all the bi are distinct, there is a group element g in G which maps ai to bi for each i between 1 and k. Such a group is called sharply k-transitive if the element g is unique (i.e. the action on k-tuples is regular, rather than just transitive).
M24 is 5-transitive, and M12 is sharply 5-transitive, with the other Mathieu groups (simple or not) being the subgroups corresponding to stabilizers of m points, and accordingly of lower transitivity (M23 is 4-transitive, etc.). These are the only two 5-transitive groups that are neither symmetric groups nor alternating groups .
The only 4-transitive groups are the symmetric groups Sk for k at least 4, the alternating groups Ak for k at least 6, and the Mathieu groups M24, M23, M12, and M11. The full proof requires the classification of finite simple groups, but some special cases have been known for much longer.
It is a classical result of Jordan that the symmetric and alternating gr
|
https://en.wikipedia.org/wiki/Stone%20space
|
In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in the 1930s in the course of his investigation of Boolean algebras, which culminated in his representation theorem for Boolean algebras.
Equivalent conditions
The following conditions on the topological space are equivalent:
is a Stone space;
is homeomorphic to the projective limit (in the category of topological spaces) of an inverse system of finite discrete spaces;
is compact and totally separated;
is compact, T0 , and zero-dimensional (in the sense of the small inductive dimension);
is coherent and Hausdorff.
Examples
Important examples of Stone spaces include finite discrete spaces, the Cantor set and the space of -adic integers, where is any prime number. Generalizing these examples, any product of finite discrete spaces is a Stone space, and the topological space underlying any profinite group is a Stone space. The Stone–Čech compactification of the natural numbers with the discrete topology, or indeed of any discrete space, is a Stone space.
Stone's representation theorem for Boolean algebras
To every Boolean algebra we can associate a Stone space as follows: the elements of are the ultrafilters on and the topology on called , is generated by the sets of the form where
Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to the Boolean algebra of clopen sets of the Stone space ; and furthermore, every Stone space is homeomorphic to the Stone space belonging to the Boolean algebra of clopen sets of These assignments are functorial, and we obtain a category-theoretic duality between the category of Boolean algebras (with homomorphisms as morphisms) and the category of Stone spaces (with continuous maps as morphisms).
Stone's theorem gave rise to a number of similar dualities, now collectively known as Stone dualities.
Condensed mathematics
The category of Stone spaces with continuous maps is equivalent to the pro-category of the category of finite sets, which explains the term "profinite sets". The profinite sets are at the heart of the project of condensed mathematics, which aims to replace topological spaces with "condensed sets", where a topological space X is replaced by the functor that takes a profinite set S to the set of continuous maps from S to X.
See also
References
Further reading
Boolean algebra
Categorical logic
General topology
|
https://en.wikipedia.org/wiki/Walsh%20function
|
In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous function in Fourier analysis. They can thus be viewed as a discrete, digital counterpart of the continuous, analog system of trigonometric functions on the unit interval. But unlike the sine and cosine functions, which are continuous, Walsh functions are piecewise constant. They take the values −1 and +1 only, on sub-intervals defined by dyadic fractions.
The system of Walsh functions is known as the Walsh system. It is an extension of the Rademacher system of orthogonal functions.
Walsh functions, the Walsh system, the Walsh series, and the fast Walsh–Hadamard transform are all named after the American mathematician Joseph L. Walsh. They find various applications in physics and engineering when analyzing digital signals.
Historically, various numerations of Walsh functions have been used; none of them is particularly superior to another. This articles uses the Walsh–Paley numeration.
Definition
We define the sequence of Walsh functions , as follows.
For any natural number k, and real number , let
be the jth bit in the binary representation of k, starting with as the least significant bit, and
be the jth bit in the fractional binary representation of , starting with as the most significant fractional bit.
Then, by definition
In particular, everywhere on the interval, since all bits of k are zero.
Notice that is precisely the Rademacher function rm.
Thus, the Rademacher system is a subsystem of the Walsh system. Moreover, every Walsh function is a product of Rademacher functions:
Comparison between Walsh functions and trigonometric functions
Walsh functions and trigonometric functions are both systems that form a complete, orthonormal set of functions, an orthonormal basis in Hilbert space of the square-integrable functions on the unit interval. Both are systems of bounded functions, unlike, say, the Haar system or the Franklin system.
Both trigonometric and Walsh systems admit natural extension by periodicity from the unit interval to the real line . Furthermore, both Fourier analysis on the unit interval (Fourier series) and on the real line (Fourier transform) have their digital counterparts defined via Walsh system, the Walsh series analogous to the Fourier series, and the Hadamard transform analogous to the Fourier transform.
Properties
The Walsh system is a commutative multiplicative discrete group isomorphic to , the Pontryagin dual of Cantor group . Its identity is , and every element is of order two (that is, self-inverse).
The Walsh system is an orthonormal basis of Hilbert space . Orthonormality means
,
and being a basis means that if, for every , we set then
It turns out that for every , the series converge to for almost every .
The Walsh system (in Walsh-Paley nu
|
https://en.wikipedia.org/wiki/Group%20algebra%20of%20a%20locally%20compact%20group
|
In functional analysis and related areas of mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra (or more generally a Banach algebra), such that representations of the algebra are related to representations of the group. As such, they are similar to the group ring associated to a discrete group.
The algebra Cc(G) of continuous functions with compact support
If G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called a Haar measure. Using the Haar measure, one can define a convolution operation on the space Cc(G) of complex-valued continuous functions on G with compact support; Cc(G) can then be given any of various norms and the completion will be a group algebra.
To define the convolution operation, let f and g be two functions in Cc(G). For t in G, define
The fact that is continuous is immediate from the dominated convergence theorem. Also
where the dot stands for the product in G. Cc(G) also has a natural involution defined by:
where Δ is the modular function on G. With this involution, it is a *-algebra.
Theorem. With the norm:
Cc(G) becomes an involutive normed algebra with an approximate identity.
The approximate identity can be indexed on a neighborhood basis of the identity consisting of compact sets. Indeed, if V is a compact neighborhood of the identity, let fV be a non-negative continuous function supported in V such that
Then {fV}V is an approximate identity. A group algebra has an identity, as opposed to just an approximate identity, if and only if the topology on the group is the discrete topology.
Note that for discrete groups, Cc(G) is the same thing as the complex group ring C[G].
The importance of the group algebra is that it captures the unitary representation theory of G as shown in the following
Theorem. Let G be a locally compact group. If U is a strongly continuous unitary representation of G on a Hilbert space H, then
is a non-degenerate bounded *-representation of the normed algebra Cc(G). The map
is a bijection between the set of strongly continuous unitary representations of G and non-degenerate bounded *-representations of Cc(G). This bijection respects unitary equivalence and strong containment. In particular, U is irreducible if and only if U is irreducible.
Non-degeneracy of a representation of Cc(G) on a Hilbert space H means that
is dense in H.
The convolution algebra L1(G)
It is a standard theorem of measure theory that the completion of Cc(G) in the L1(G) norm is isomorphic to the space L1(G) of equivalence classes of functions which are integrable with respect to the Haar measure, where, as usual, two functions are regarded as equivalent if and only if they differ only on a set of Haar measure zero.
Theorem. L1(G) is a Banach *-algebra with the convolution product and involution defined above and with the L1 norm. L1(G) also has a bounde
|
https://en.wikipedia.org/wiki/Support%20%28mathematics%29
|
In mathematics, the support of a real-valued function is the subset of the function domain containing the elements which are not mapped to zero. If the domain of is a topological space, then the support of is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis.
Formulation
Suppose that is a real-valued function whose domain is an arbitrary set The of written is the set of points in where is non-zero:
The support of is the smallest subset of with the property that is zero on the subset's complement. If for all but a finite number of points then is said to have .
If the set has an additional structure (for example, a topology), then the support of is defined in an analogous way as the smallest subset of of an appropriate type such that vanishes in an appropriate sense on its complement. The notion of support also extends in a natural way to functions taking values in more general sets than and to other objects, such as measures or distributions.
Closed support
The most common situation occurs when is a topological space (such as the real line or -dimensional Euclidean space) and is a continuous real- (or complex-) valued function. In this case, the of , , or the of , is defined topologically as the closure (taken in ) of the subset of where is non-zero that is,
Since the intersection of closed sets is closed, is the intersection of all closed sets that contain the set-theoretic support of
For example, if is the function defined by
then , the support of , or the closed support of , is the closed interval since is non-zero on the open interval and the closure of this set is
The notion of closed support is usually applied to continuous functions, but the definition makes sense for arbitrary real or complex-valued functions on a topological space, and some authors do not require that (or ) be continuous.
Compact support
Functions with on a topological space are those whose closed support is a compact subset of If is the real line, or -dimensional Euclidean space, then a function has compact support if and only if it has , since a subset of is compact if and only if it is closed and bounded.
For example, the function defined above is a continuous function with compact support If is a smooth function then because is identically on the open subset all of 's partial derivatives of all orders are also identically on
The condition of compact support is stronger than the condition of vanishing at infinity. For example, the function defined by
vanishes at infinity, since as but its support is not compact.
Real-valued compactly supported smooth functions on a Euclidean space are called bump functions. Mollifiers are an important special case of bump functions as they can be used in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution
|
https://en.wikipedia.org/wiki/Function%20field%20%28scheme%20theory%29
|
The sheaf of rational functions KX of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of varieties, such a sheaf associates to each open set U the ring of all rational functions on that open set; in other words, KX(U) is the set of fractions of regular functions on U. Despite its name, KX does not always give a field for a general scheme X.
Simple cases
In the simplest cases, the definition of KX is straightforward. If X is an (irreducible) affine algebraic variety, and if U is an open subset of X, then KX(U) will be the field of fractions of the ring of regular functions on U. Because X is affine, the ring of regular functions on U will be a localization of the global sections of X, and consequently KX will be the constant sheaf whose value is the fraction field of the global sections of X.
If X is integral but not affine, then any non-empty affine open set will be dense in X. This means there is not enough room for a regular function to do anything interesting outside of U, and consequently the behavior of the rational functions on U should determine the behavior of the rational functions on X. In fact, the fraction fields of the rings of regular functions on any affine open set will be the same, so we define, for any U, KX(U) to be the common fraction field of any ring of regular functions on any open affine subset of X. Alternatively, one can define the function field in this case to be the local ring of the generic point.
General case
The trouble starts when X is no longer integral. Then it is possible to have zero divisors in the ring of regular functions, and consequently the fraction field no longer exists. The naive solution is to replace the fraction field by the total quotient ring, that is, to invert every element that is not a zero divisor. Unfortunately, in general, the total quotient ring does not produce a presheaf much less a sheaf. The well-known article of Kleiman, listed in the bibliography, gives such an example.
The correct solution is to proceed as follows:
For each open set U, let SU be the set of all elements in Γ(U, OX) that are not zero divisors in any stalk OX,x. Let KXpre be the presheaf whose sections on U are localizations SU−1Γ(U, OX) and whose restriction maps are induced from the restriction maps of OX by the universal property of localization. Then KX is the sheaf associated to the presheaf KXpre.
Further issues
Once KX is defined, it is possible to study properties of X which depend only on KX. This is the subject of birational geometry.
If X is an algebraic variety over a field k, then over each open set U we have a field extension KX(U) of k. The dimension of U will be equal to the transcendence degree of this field extension. All finite transcendence degree field extensions of k correspond to the rational function field of some variety.
In the particular case of an algebraic curve C, that is, d
|
https://en.wikipedia.org/wiki/Hilbert%27s%20sixteenth%20problem
|
Hilbert's 16th problem was posed by David Hilbert at the Paris conference of the International Congress of Mathematicians in 1900, as part of his list of 23 problems in mathematics.
The original problem was posed as the Problem of the topology of algebraic curves and surfaces (Problem der Topologie algebraischer Kurven und Flächen).
Actually the problem consists of two similar problems in different branches of mathematics:
An investigation of the relative positions of the branches of real algebraic curves of degree n (and similarly for algebraic surfaces).
The determination of the upper bound for the number of limit cycles in two-dimensional polynomial vector fields of degree n and an investigation of their relative positions.
The first problem is yet unsolved for n = 8. Therefore, this problem is what usually is meant when talking about Hilbert's sixteenth problem in real algebraic geometry. The second problem also remains unsolved: no upper bound for the number of limit cycles is known for any n > 1, and this is what usually is meant by Hilbert's sixteenth problem in the field of dynamical systems.
The Spanish Royal Society for Mathematics published an explanation of Hilbert's sixteenth problem.
The first part of Hilbert's 16th problem
In 1876, Harnack investigated algebraic curves in the real projective plane and found that curves of degree n could have no more than
separate connected components. Furthermore, he showed how to construct curves that attained that upper bound, and thus that it was the best possible bound. Curves with that number of components are called M-curves.
Hilbert had investigated the M-curves of degree 6, and found that the 11 components always were grouped in a certain way. His challenge to the mathematical community now was to completely investigate the possible configurations of the components of the M-curves.
Furthermore, he requested a generalization of Harnack's curve theorem to algebraic surfaces and a similar investigation of surfaces with the maximum number of components.
The second part of Hilbert's 16th problem
Here we are going to consider polynomial vector fields in the real plane, that is a system of differential equations of the form:
where both P and Q are real polynomials of degree n.
These polynomial vector fields were studied by Poincaré, who had the idea of abandoning the search for finding exact solutions to the system, and instead attempted to study the qualitative features of the collection of all possible solutions.
Among many important discoveries, he found that the limit sets of such solutions need not be a stationary point, but could rather be a periodic solution. Such solutions are called limit cycles.
The second part of Hilbert's 16th problem is to decide an upper bound for the number of limit cycles in polynomial vector fields of degree n and, similar to the first part, investigate their relative positions.
Results
It was shown in 1991/1992 by Yulii Ilyashenko and Jean Écal
|
https://en.wikipedia.org/wiki/Singleton%20%28mathematics%29
|
In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set is a singleton whose single element is .
Properties
Within the framework of Zermelo–Fraenkel set theory, the axiom of regularity guarantees that no set is an element of itself. This implies that a singleton is necessarily distinct from the element it contains, thus 1 and {1} are not the same thing, and the empty set is distinct from the set containing only the empty set. A set such as is a singleton as it contains a single element (which itself is a set, however, not a singleton).
A set is a singleton if and only if its cardinality is . In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton
In axiomatic set theory, the existence of singletons is a consequence of the axiom of pairing: for any set A, the axiom applied to A and A asserts the existence of which is the same as the singleton (since it contains A, and no other set, as an element).
If A is any set and S is any singleton, then there exists precisely one function from A to S, the function sending every element of A to the single element of S. Thus every singleton is a terminal object in the category of sets.
A singleton has the property that every function from it to any arbitrary set is injective. The only non-singleton set with this property is the empty set.
Every singleton set is an ultra prefilter. If is a set and then the upward of in which is the set is a principal ultrafilter on Moreover, every principal ultrafilter on is necessarily of this form. The ultrafilter lemma implies that non-principal ultrafilters exist on every infinite set (these are called ).
Every net valued in a singleton subset of is an ultranet in
The Bell number integer sequence counts the number of partitions of a set (), if singletons are excluded then the numbers are smaller ().
In category theory
Structures built on singletons often serve as terminal objects or zero objects of various categories:
The statement above shows that the singleton sets are precisely the terminal objects in the category Set of sets. No other sets are terminal.
Any singleton admits a unique topological space structure (both subsets are open). These singleton topological spaces are terminal objects in the category of topological spaces and continuous functions. No other spaces are terminal in that category.
Any singleton admits a unique group structure (the unique element serving as identity element). These singleton groups are zero objects in the category of groups and group homomorphisms. No other groups are terminal in that category.
Definition by indicator functions
Let be a class defined by an indicator function
Then is called a singleton if and only if there is some such that for all
Definition in Principia Mathematica
The following definition was introduced by Whitehead and Russell
‘ Df.
The symbol ‘ denotes the sin
|
https://en.wikipedia.org/wiki/Lexicostatistics
|
Lexicostatistics is a method of comparative linguistics that involves comparing the percentage of lexical cognates between languages to determine their relationship. Lexicostatistics is related to the comparative method but does not reconstruct a proto-language. It is to be distinguished from glottochronology, which attempts to use lexicostatistical methods to estimate the length of time since two or more languages diverged from a common earlier proto-language. This is merely one application of lexicostatistics, however; other applications of it may not share the assumption of a constant rate of change for basic lexical items.
The term "lexicostatistics" is misleading in that mathematical equations are used but not statistics. Other features of a language may be used other than the lexicon, though this is unusual. Whereas the comparative method used shared identified innovations to determine sub-groups, lexicostatistics does not identify these. Lexicostatistics is a distance-based method, whereas the comparative method considers language characters directly. The lexicostatistics method is a simple and fast technique relative to the comparative method but has limitations (discussed below). It can be validated by cross-checking the trees produced by both methods.
History
Lexicostatistics was developed by Morris Swadesh in a series of articles in the 1950s, based on earlier ideas. The concept's first known use was by Dumont d'Urville in 1834 who compared various "Oceanic" languages and proposed a method for calculating a coefficient of relationship. Hymes (1960) and Embleton (1986) both review the history of lexicostatistics.
Method
Create word list
The aim is to generate a list of universally used meanings (hand, mouth, sky, I). Words are then collected for these meaning slots for each language being considered. Swadesh reduced a larger set of meanings down to 200 originally. He later found that it was necessary to reduce it further but that he could include some meanings that were not in his original list, giving his later 100-item list. The Swadesh list in Wiktionary gives the total 207 meanings in a number of languages. Alternative lists that apply more rigorous criteria have been generated, e.g. the Dolgopolsky list and the Leipzig–Jakarta list, as well as lists with a more specific scope; for example, Dyen, Kruskal and Black have 200 meanings for 84 Indo-European languages in digital form.
Determine cognacies
A trained and experienced linguist is needed to make cognacy decisions. However, the decisions may need to be refined as the state of knowledge increases. However, lexicostatistics does not rely on all the decisions being correct. For each pair of words (in different languages) in this list, the cognacy of a form could be positive, negative or indeterminate. Sometimes a language has multiple words for one meaning, e.g. small and little for not big.
Calculate lexicostatistic percentages
This percentage is related to the proportion of
|
https://en.wikipedia.org/wiki/Fourier%20inversion%20theorem
|
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency and phase information about a wave then we may reconstruct the original wave precisely.
The theorem says that if we have a function satisfying certain conditions, and we use the convention for the Fourier transform that
then
In other words, the theorem says that
This last equation is called the Fourier integral theorem.
Another way to state the theorem is that if is the flip operator i.e. , then
The theorem holds if both and its Fourier transform are absolutely integrable (in the Lebesgue sense) and is continuous at the point . However, even under more general conditions versions of the Fourier inversion theorem hold. In these cases the integrals above may not converge in an ordinary sense.
Statement
In this section we assume that is an integrable continuous function. Use the convention for the Fourier transform that
Furthermore, we assume that the Fourier transform is also integrable.
Inverse Fourier transform as an integral
The most common statement of the Fourier inversion theorem is to state the inverse transform as an integral. For any integrable function and all set
Then for all we have
Fourier integral theorem
The theorem can be restated as
By taking the real part of each side of the above we obtain
Inverse transform in terms of flip operator
For any function define the flip operator by
Then we may instead define
It is immediate from the definition of the Fourier transform and the flip operator that both and match the integral definition of , and in particular are equal to each other and satisfy .
Since we have and
Two-sided inverse
The form of the Fourier inversion theorem stated above, as is common, is that
In other words, is a left inverse for the Fourier transform. However it is also a right inverse for the Fourier transform i.e.
Since is so similar to , this follows very easily from the Fourier inversion theorem (changing variables ):
Alternatively, this can be seen from the relation between and the flip operator and the associativity of function composition, since
Conditions on the function
When used in physics and engineering, the Fourier inversion theorem is often used under the assumption that everything "behaves nicely". In mathematics such heuristic arguments are not permitted, and the Fourier inversion theorem includes an explicit specification of what class of functions is being allowed. However, there is no "best" class of functions to consider so several variants of the Fourier inversion theorem exist, albeit with compatible conclusions.
Schwartz functions
The Fourier inversion theorem holds for all Schwartz functions (roughly speaking, smooth functions that decay quickly and whose derivatives all decay quickly). This condition has the benefit that it i
|
https://en.wikipedia.org/wiki/Trivial%20topology
|
In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the consequence that all points of the space are "lumped together" and cannot be distinguished by topological means. Every indiscrete space is a pseudometric space in which the distance between any two points is zero.
Details
The trivial topology is the topology with the least possible number of open sets, namely the empty set and the entire space, since the definition of a topology requires these two sets to be open. Despite its simplicity, a space X with more than one element and the trivial topology lacks a key desirable property: it is not a T0 space.
Other properties of an indiscrete space X—many of which are quite unusual—include:
The only closed sets are the empty set and X.
The only possible basis of X is {X}.
If X has more than one point, then since it is not T0, it does not satisfy any of the higher T axioms either. In particular, it is not a Hausdorff space. Not being Hausdorff, X is not an order topology, nor is it metrizable.
X is, however, regular, completely regular, normal, and completely normal; all in a rather vacuous way though, since the only closed sets are ∅ and X.
X is compact and therefore paracompact, Lindelöf, and locally compact.
Every function whose domain is a topological space and codomain X is continuous.
X is path-connected and so connected.
X is second-countable, and therefore is first-countable, separable and Lindelöf.
All subspaces of X have the trivial topology.
All quotient spaces of X have the trivial topology
Arbitrary products of trivial topological spaces, with either the product topology or box topology, have the trivial topology.
All sequences in X converge to every point of X. In particular, every sequence has a convergent subsequence (the whole sequence or any other subsequence), thus X is sequentially compact.
The interior of every set except X is empty.
The closure of every non-empty subset of X is X. Put another way: every non-empty subset of X is dense, a property that characterizes trivial topological spaces.
As a result of this, the closure of every open subset U of X is either ∅ (if U = ∅) or X (otherwise). In particular, the closure of every open subset of X is again an open set, and therefore X is extremally disconnected.
If S is any subset of X with more than one element, then all elements of X are limit points of S. If S is a singleton, then every point of X \ S is still a limit point of S.
X is a Baire space.
Two topological spaces carrying the trivial topology are homeomorphic iff they have the same cardinality.
In some sense the opposite of the trivial topology is the discrete topology, in which every subset is open.
The trivial topology belongs to a uniform space in which the whole cartesian product X × X is the only entourage.
Le
|
https://en.wikipedia.org/wiki/Projective%20linear%20group
|
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space V on the associated projective space P(V). Explicitly, the projective linear group is the quotient group
PGL(V) = GL(V)/Z(V)
where GL(V) is the general linear group of V and Z(V) is the subgroup of all nonzero scalar transformations of V; these are quotiented out because they act trivially on the projective space and they form the kernel of the action, and the notation "Z" reflects that the scalar transformations form the center of the general linear group.
The projective special linear group, PSL, is defined analogously, as the induced action of the special linear group on the associated projective space. Explicitly:
PSL(V) = SL(V)/SZ(V)
where SL(V) is the special linear group over V and SZ(V) is the subgroup of scalar transformations with unit determinant. Here SZ is the center of SL, and is naturally identified with the group of nth roots of unity in F (where n is the dimension of V and F is the base field).
PGL and PSL are some of the fundamental groups of study, part of the so-called classical groups, and an element of PGL is called projective linear transformation, projective transformation or homography. If V is the n-dimensional vector space over a field F, namely the alternate notations and are also used.
Note that and are isomorphic if and only if every element of F has an nth root in F. As an example, note that , but that ; this corresponds to the real projective line being orientable, and the projective special linear group only being the orientation-preserving transformations.
PGL and PSL can also be defined over a ring, with an important example being the modular group, .
Name
The name comes from projective geometry, where the projective group acting on homogeneous coordinates (x0:x1: ... :xn) is the underlying group of the geometry. Stated differently, the natural action of GL(V) on V descends to an action of PGL(V) on the projective space P(V).
The projective linear groups therefore generalise the case PGL(2, C) of Möbius transformations (sometimes called the Möbius group), which acts on the projective line.
Note that unlike the general linear group, which is generally defined axiomatically as "invertible functions preserving the linear (vector space) structure", the projective linear group is defined constructively, as a quotient of the general linear group of the associated vector space, rather than axiomatically as "invertible functions preserving the projective linear structure". This is reflected in the notation: PGL(n, F) is the group associated to GL(n, F), and is the projective linear group of (n−1)-dimensional projective space, not n-dimensional projective space.
Collineations
A related group is the collineation group, which is defined axiomatically. A collineation is an invertible (or mo
|
https://en.wikipedia.org/wiki/Congruence%20subgroup
|
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible 2 × 2 integer matrices of determinant 1 in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.
The existence of congruence subgroups in an arithmetic group provides it with a wealth of subgroups, in particular it shows that the group is residually finite. An important question regarding the algebraic structure of arithmetic groups is the congruence subgroup problem, which asks whether all subgroups of finite index are essentially congruence subgroups.
Congruence subgroups of 2×2 matrices are fundamental objects in the classical theory of modular forms; the modern theory of automorphic forms makes a similar use of congruence subgroups in more general arithmetic groups.
Congruence subgroups of the modular group
The simplest interesting setting in which congruence subgroups can be studied is that of the modular group .
Principal congruence subgroups
If is an integer there is a homomorphism induced by the reduction modulo morphism . The principal congruence subgroup of level in is the kernel of , and it is usually denoted . Explicitly it is described as follows:
This definition immediately implies that is a normal subgroup of finite index in . The strong approximation theorem (in this case an easy consequence of the Chinese remainder theorem) implies that is surjective, so that the quotient is isomorphic to Computing the order of this finite group yields the following formula for the index:
where the product is taken over all prime numbers dividing .
If then the restriction of to any finite subgroup of is injective. This implies the following result:
If then the principal congruence subgroups are torsion-free.
The group contains and is not torsion-free. On the other hand, its image in is torsion-free, and the quotient of the hyperbolic plane by this subgroup is a sphere with three cusps.
Definition of a congruence subgroup
A subgroup in is called a congruence subgroup if there exists such that contains the principal congruence subgroup . The level of is then the smallest such .
From this definition it follows that:
Congruence subgroups are of finite index in ;
The congruence subgroups of level are in one-to-one correspondence with the subgroups of
Examples
The subgroup , sometimes called the Hecke congruence subgroup of level , is defined as the preimage by of the group of upper triangular matrices. That is,
The index is given by the formula:
where the product is taken over all prime numbers dividing . If is prime then is in natural bijection with the projective line over the finite field , and explic
|
https://en.wikipedia.org/wiki/Cofiniteness
|
In mathematics, a cofinite subset of a set is a subset whose complement in is a finite set. In other words, contains all but finitely many elements of If the complement is not finite, but is countable, then one says the set is cocountable.
These arise naturally when generalizing structures on finite sets to infinite sets, particularly on infinite products, as in the product topology or direct sum.
This use of the prefix "" to describe a property possessed by a set's mplement is consistent with its use in other terms such as "meagre set".
Boolean algebras
The set of all subsets of that are either finite or cofinite forms a Boolean algebra, which means that it is closed under the operations of union, intersection, and complementation. This Boolean algebra is the on A Boolean algebra has a unique non-principal ultrafilter (that is, a maximal filter not generated by a single element of the algebra) if and only if there exists an infinite set such that is isomorphic to the finite–cofinite algebra on In this case, the non-principal ultrafilter is the set of all cofinite sets.
Cofinite topology
The cofinite topology (sometimes called the finite complement topology) is a topology that can be defined on every set It has precisely the empty set and all cofinite subsets of as open sets. As a consequence, in the cofinite topology, the only closed subsets are finite sets, or the whole of Symbolically, one writes the topology as
This topology occurs naturally in the context of the Zariski topology. Since polynomials in one variable over a field are zero on finite sets, or the whole of the Zariski topology on (considered as affine line) is the cofinite topology. The same is true for any irreducible algebraic curve; it is not true, for example, for in the plane.
Properties
Subspaces: Every subspace topology of the cofinite topology is also a cofinite topology.
Compactness: Since every open set contains all but finitely many points of the space is compact and sequentially compact.
Separation: The cofinite topology is the coarsest topology satisfying the T1 axiom; that is, it is the smallest topology for which every singleton set is closed. In fact, an arbitrary topology on satisfies the T1 axiom if and only if it contains the cofinite topology. If is finite then the cofinite topology is simply the discrete topology. If is not finite then this topology is not Hausdorff (T2), regular or normal because no two nonempty open sets are disjoint (that is, it is hyperconnected).
Double-pointed cofinite topology
The double-pointed cofinite topology is the cofinite topology with every point doubled; that is, it is the topological product of the cofinite topology with the indiscrete topology on a two-element set. It is not T0 or T1, since the points of each doublet are topologically indistinguishable. It is, however, R0 since topologically distinguishable points are separated. The space is compact as the product of two compact space
|
https://en.wikipedia.org/wiki/Birational%20geometry
|
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.
Birational maps
Rational maps
A rational map from one variety (understood to be irreducible) to another variety , written as a dashed arrow , is defined as a morphism from a nonempty open subset to . By definition of the Zariski topology used in algebraic geometry, a nonempty open subset is always dense in , in fact the complement of a lower-dimensional subset. Concretely, a rational map can be written in coordinates using rational functions.
Birational maps
A birational map from X to Y is a rational map such that there is a rational map inverse to f. A birational map induces an isomorphism from a nonempty open subset of X to a nonempty open subset of Y, and vice versa: an isomorphism between nonempty open subsets of X, Y by definition gives a birational map . In this case, X and Y are said to be birational, or birationally equivalent. In algebraic terms, two varieties over a field k are birational if and only if their function fields are isomorphic as extension fields of k.
A special case is a birational morphism , meaning a morphism which is birational. That is, f is defined everywhere, but its inverse may not be. Typically, this happens because a birational morphism contracts some subvarieties of X to points in Y.
Birational equivalence and rationality
A variety X is said to be rational if it is birational to affine space (or equivalently, to projective space) of some dimension. Rationality is a very natural property: it means that X minus some lower-dimensional subset can be identified with affine space minus some lower-dimensional subset.
Birational equivalence of a plane conic
For example, the circle with equation in the affine plane is a rational curve, because there is a rational map given by
which has a rational inverse g: X ⇢ given by
Applying the map f with t a rational number gives a systematic construction of Pythagorean triples.
The rational map is not defined on the locus where . So, on the complex affine line , is a morphism on the open subset , . Likewise, the rational map is not defined at the point (0,−1) in .
Birational equivalence of smooth quadrics and Pn
More generally, a smooth quadric (degree 2) hypersurface X of any dimension n is rational, by stereographic projection. (For X a quadric over a field k, X must be assumed to have a k-rational point; this is automatic if k is algebraically closed.) To define stereographic projection, let p be a point in X. Then a birational map from X to the projective space of lines through p is given by sending a point q in X to the line through p and q. This is a birational equivalence but not an isomorphism of variet
|
https://en.wikipedia.org/wiki/Polydivisible%20number
|
In mathematics a polydivisible number (or magic number) is a number in a given number base with digits abcde... that has the following properties:
Its first digit a is not 0.
The number formed by its first two digits ab is a multiple of 2.
The number formed by its first three digits abc is a multiple of 3.
The number formed by its first four digits abcd is a multiple of 4.
etc.
Definition
Let be a positive integer, and let be the number of digits in n written in base b. The number n is a polydivisible number if for all ,
.
Example
For example, 10801 is a seven-digit polydivisible number in base 4, as
Enumeration
For any given base , there are only a finite number of polydivisible numbers.
Maximum polydivisible number
The following table lists maximum polydivisible numbers for some bases b, where represent digit values 10 to 35.
Estimate for Fb(n) and Σ(b)
Let be the number of digits. The function determines the number of polydivisible numbers that has digits in base , and the function is the total number of polydivisible numbers in base .
If is a polydivisible number in base with digits, then it can be extended to create a polydivisible number with digits if there is a number between and that is divisible by . If is less or equal to , then it is always possible to extend an digit polydivisible number to an -digit polydivisible number in this way, and indeed there may be more than one possible extension. If is greater than , it is not always possible to extend a polydivisible number in this way, and as becomes larger, the chances of being able to extend a given polydivisible number become smaller. On average, each polydivisible number with digits can be extended to a polydivisible number with digits in different ways. This leads to the following estimate for :
Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately
Specific bases
All numbers are represented in base , using A−Z to represent digit values 10 to 35.
Base 2
Base 3
Base 4
Base 5
The polydivisible numbers in base 5 are
1, 2, 3, 4, 11, 13, 20, 22, 24, 31, 33, 40, 42, 44, 110, 113, 132, 201, 204, 220, 223, 242, 311, 314, 330, 333, 402, 421, 424, 440, 443, 1102, 1133, 1322, 2011, 2042, 2200, 2204, 2231, 2420, 2424, 3113, 3140, 3144, 3302, 3333, 4022, 4211, 4242, 4400, 4404, 4431, 11020, 11330, 13220, 20110, 20420, 22000, 22040, 22310, 24200, 24240, 31130, 31400, 31440, 33020, 33330, 40220, 42110, 42420, 44000, 44040, 44310, 110204, 113300, 132204, 201102, 204204, 220000, 220402, 223102, 242000, 242402, 311300, 314000, 314402, 330204, 333300, 402204, 421102, 424204, 440000, 440402, 443102, 1133000, 1322043, 2011021, 2042040, 2204020, 2420003, 2424024, 3113002, 3140000, 3144021, 4022042, 4211020, 4431024, 11330000, 13220431, 20110211, 20420404, 24200031, 31400004, 31440211, 40220422, 42110202, 44310242, 132204314, 201102110, 242000311, 314000044, 402204220, 443102421, 1322043140, 2011
|
https://en.wikipedia.org/wiki/Seifert%E2%80%93Van%20Kampen%20theorem
|
In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space in terms of the fundamental groups of two open, path-connected subspaces that cover . It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.
Van Kampen's theorem for fundamental groups
Let X be a topological space which is the union of two open and path connected subspaces U1, U2. Suppose U1 ∩ U2 is path connected and nonempty, and let x0 be a point in U1 ∩ U2 that will be used as the base of all fundamental groups. The inclusion maps of U1 and U2 into X induce group homomorphisms and . Then X is path connected and and form a commutative pushout diagram:
The natural morphism k is an isomorphism. That is, the fundamental group of X is the free product of the fundamental groups of U1 and U2 with amalgamation of .
Usually the morphisms induced by inclusion in this theorem are not themselves injective, and the more precise version of the statement is in terms of pushouts of groups.
Van Kampen's theorem for fundamental groupoids
Unfortunately, the theorem as given above does not compute the fundamental group of the circle – which is the most important basic example in algebraic topology – because the circle cannot be realised as the union of two open sets with connected intersection. This problem can be resolved by working with the fundamental groupoid on a set A of base points, chosen according to the geometry of the situation. Thus for the circle, one uses two base points.
This groupoid consists of homotopy classes relative to the end points of paths in X joining points of A ∩ X. In particular, if X is a contractible space, and A consists of two distinct points of X, then is easily seen to be isomorphic to the groupoid often written with two vertices and exactly one morphism between any two vertices. This groupoid plays a role in the theory of groupoids analogous to that of the group of integers in the theory of groups. The groupoid also allows for groupoids a notion of homotopy: it is a unit interval object in the category of groupoids.
The category of groupoids admits all colimits, and in particular all pushouts.
Theorem. Let the topological space X be covered by the interiors of two subspaces X1, X2 and let A be a set which meets each path component of X1, X2 and X0 = X1 ∩ X2. Then A meets each path component of X and the diagram P of morphisms induced by inclusion
is a pushout diagram in the category of groupoids.
This theorem gives the transition from topology to algebra, in determining completely the fundamental groupoid ; one then has to use algebra and combinatorics to determine a fundamental group at some basepoint.
One interpretation of the theorem is that it computes homotopy 1-types. To see its utility, one can easily f
|
https://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%20theorem
|
In set theory and order theory, the Cantor–Bernstein theorem states that the cardinality of the second type class, the class of countable order types, equals the cardinality of the continuum. It was used by Felix Hausdorff and named by him after Georg Cantor and Felix Bernstein. Cantor constructed a family of countable order types with the cardinality of the continuum, and in his 1901 inaugural dissertation Bernstein proved that such a family can have no higher cardinality.
References
Order theory
|
https://en.wikipedia.org/wiki/Italian%20school%20of%20algebraic%20geometry
|
In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around Rome roughly from 1885 to 1935. There were 30 to 40 leading mathematicians who made major contributions, about half of those being Italian. The leadership fell to the group in Rome of Guido Castelnuovo, Federigo Enriques and Francesco Severi, who were involved in some of the deepest discoveries, as well as setting the style.
Algebraic surfaces
The emphasis on algebraic surfaces—algebraic varieties of dimension two—followed on from an essentially complete geometric theory of algebraic curves (dimension 1). The position in around 1870 was that the curve theory had incorporated with Brill–Noether theory the Riemann–Roch theorem in all its refinements (via the detailed geometry of the theta-divisor).
The classification of algebraic surfaces was a bold and successful attempt to repeat the division of algebraic curves by their genus g. The division of curves corresponds to the rough classification into the three types: g = 0 (projective line); g = 1 (elliptic curve); and g > 1 (Riemann surfaces with independent holomorphic differentials). In the case of surfaces, the Enriques classification was into five similar big classes, with three of those being analogues of the curve cases, and two more (elliptic fibrations, and K3 surfaces, as they would now be called) being with the case of two-dimension abelian varieties in the 'middle' territory. This was an essentially sound, breakthrough set of insights, recovered in modern complex manifold language by Kunihiko Kodaira in the 1950s, and refined to include mod p phenomena by Zariski, the Shafarevich school and others by around 1960. The form of the Riemann–Roch theorem on a surface was also worked out.
Foundational issues
Some proofs produced by the school are not considered satisfactory because of foundational difficulties. These included frequent use of birational models in dimension three of surfaces that can have non-singular models only when embedded in higher-dimensional projective space. In order to avoid these issues, a sophisticated theory of handling a linear system of divisors was developed (in effect, a line bundle theory for hyperplane sections of putative embeddings in projective space). Many modern techniques were found, in embryonic form, and in some cases the articulation of these ideas exceeded the available technical language.
The geometers
According to Guerraggio & Nastasi (page 9, 2005), Luigi Cremona is "considered the founder of the Italian school of algebraic geometry". Later they explain that in Turin the collaboration of Enrico D'Ovidio and Corrado Segre "would bring, either by their own efforts or those of their students, Italian algebraic geometry to full maturity". A one-time student of Segre, H.F. Baker wrote that Corrado Segre "may probably be said to be the father of that wond
|
https://en.wikipedia.org/wiki/Erich%20K%C3%A4hler
|
Erich Kähler (; 16 January 1906 – 31 May 2000) was a German mathematician with wide-ranging interests in geometry and mathematical physics, who laid important mathematical groundwork for algebraic geometry and for string theory.
Education and life
Erich Kähler was born in Leipzig, the son of a telegraph inspector Ernst Kähler. Inspired as a boy to be an explorer after reading books about Sven Hedin that his mother Elsa Götsch had given to him, the young Kähler soon focused his passion for exploration on astronomy. He is said to have written a 50-page thesis on fractional differentiation while still in high school, hoping that it would earn him a PhD. His teachers replied that he would have to attend university courses first.
Kähler enrolled in the University of Leipzig in 1924. He read Galois theory, met the mathematician Emil Artin, and did research under the supervision of Leon Lichtenstein. Still fascinated by celestial mechanics, Kähler wrote a dissertation entitled On the existence of equilibrium solutions of rotating liquids, which are derived from certain solutions of the n-body problem, and received his doctorate in 1928. He continued his studies at Leipzig for the following year, supported by fellowship from the Notgemeinschaft der Deutschen Wissenschaften, except for a research assistantship at the University of Königsberg in 1929. In 1930 Kähler joined the Department of Mathematics at the University of Hamburg to work under the direction of Wilhelm Blaschke, writing a habilitation thesis entitled, "About the integrals of algebraic equations". He took a year in Rome to work with Italian geometers including Enriques, Castelnuovo, Levi-Civita, Severi, and Segre in 1931-1932, which led him to publish his acclaimed work on what are now called Kähler metrics in 1932. Kähler returned to Hamburg after his year in Rome, where he continued to work until going to the University of Konigsberg in 1935, and was offered an ordinary professorship a year later. In 1938 he married his first wife Luise Günther.
In the years leading up to World War II Kähler was a supporter of Hitler and of German nationalism, and reported that he volunteered for the German military in 1935, joined the navy in 1937, and the army on 24 August 1939 before the invasion of Poland. After being stationed at the Saint-Nazaire submarine base in German Occupied France towards the end of the war, Kähler was captured by the Allies and taken to the prisoner of war camp at Ile de Ré, and then to another camp in Mulsanne. Thanks to the French physicist Frederic Joliot-Curie and mathematician Élie Cartan, Kähler was able to study mathematics during this time, receiving books and mathematics papers and working during his imprisonment. He was released in 1947. He reported that his oath to Hitler (as a civil servant) was important to him, and remained an apologist for the Third Reich decades later, in a 1988 interview with Sanford Segal. A former student reported in 1988 that he kept a
|
https://en.wikipedia.org/wiki/Large%20cardinal
|
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least α such that α=ωα). The proposition that such cardinals exist cannot be proved in the most common axiomatization of set theory, namely ZFC, and such propositions can be viewed as ways of measuring how "much", beyond ZFC, one needs to assume to be able to prove certain desired results. In other words, they can be seen, in Dana Scott's phrase, as quantifying the fact "that if you want more you have to assume more".
There is a rough convention that results provable from ZFC alone may be stated without hypotheses, but that if the proof requires other assumptions (such as the existence of large cardinals), these should be stated. Whether this is simply a linguistic convention, or something more, is a controversial point among distinct philosophical schools (see Motivations and epistemic status below).
A is an axiom stating that there exists a cardinal (or perhaps many of them) with some specified large cardinal property.
Most working set theorists believe that the large cardinal axioms that are currently being considered are consistent with ZFC. These axioms are strong enough to imply the consistency of ZFC. This has the consequence (via Gödel's second incompleteness theorem) that their consistency with ZFC cannot be proven in ZFC (assuming ZFC is consistent).
There is no generally agreed precise definition of what a large cardinal property is, though essentially everyone agrees that those in the list of large cardinal properties are large cardinal properties.
Partial definition
A necessary condition for a property of cardinal numbers to be a large cardinal property is that the existence of such a cardinal is not known to be inconsistent with ZF and that such a cardinal Κ would be an uncountable initial ordinal for which LΚ is a model of ZFC. If ZFC is consistent, then ZFC does not imply that any such large cardinals exist.
Hierarchy of consistency strength
A remarkable observation about large cardinal axioms is that they appear to occur in strict linear order by consistency strength. That is, no exception is known to the following: Given two large cardinal axioms A1 and A2, exactly one of three things happens:
Unless ZFC is inconsistent, ZFC+A1 is consistent if and only if ZFC+A2 is consistent;
ZFC+A1 proves that ZFC+A2 is consistent; or
ZFC+A2 proves that ZFC+A1 is consistent.
These are mutually exclusive, unless one of the theories in question is actually inconsistent.
In case 1, we say that A1 and A2 are equiconsistent. In case 2, we say that A1 is consistency-wise stronger than A2 (vice versa for case 3). If A2 is stronger than A1, then ZFC+A1 cannot prove ZFC+A2 is consistent, even with the additional hypothesis that ZFC+A1 is itself consistent (provided of course that it really is). This follow
|
https://en.wikipedia.org/wiki/General%20position
|
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible, which is referred to as special position. Its precise meaning differs in different settings.
For example, generically, two lines in the plane intersect in a single point (they are not parallel or coincident). One also says "two generic lines intersect in a point", which is formalized by the notion of a generic point. Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case.
This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise statements thereof, and when writing computer programs (see generic complexity).
General linear position
A set of points in a -dimensional affine space (-dimensional Euclidean space is a common example) is in general linear position (or just general position) if no of them lie in a -dimensional flat for . These conditions contain considerable redundancy since, if the condition holds for some value then it also must hold for all with . Thus, for a set containing at least points in -dimensional affine space to be in general position, it suffices that no hyperplane contains more than points – i.e. the points do not satisfy any more linear relations than they must.
A set of at most points in general linear position is also said to be affinely independent (this is the affine analog of linear independence of vectors, or more precisely of maximal rank), and points in general linear position in affine d-space are an affine basis. See affine transformation for more.
Similarly, n vectors in an n-dimensional vector space are linearly independent if and only if the points they define in projective space (of dimension ) are in general linear position.
If a set of points is not in general linear position, it is called a degenerate case or degenerate configuration, which implies that they satisfy a linear relation that need not always hold.
A fundamental application is that, in the plane, five points determine a conic, as long as the points are in general linear position (no three are collinear).
More generally
This definition can be generalized further: one may speak of points in general position with respect to a fixed class of algebraic relations (e.g. conic sections). In algebraic geometry this kind of condition is frequently encountered, in that points should impose independent conditions on curves passing through them.
For example, five points determine a conic, but in general six points do not lie on a conic, so being in general position with respect to conics requires that no six points lie on a conic.
General position is preserved under bi
|
https://en.wikipedia.org/wiki/Intersection%20number
|
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency. One needs a definition of intersection number in order to state results like Bézout's theorem.
The intersection number is obvious in certain cases, such as the intersection of the x- and y-axes in a plane, which should be one. The complexity enters when calculating intersections at points of tangency, and intersections which are not just points, but have higher dimension. For example, if a plane is tangent to a surface along a line, the intersection number along the line should be at least two. These questions are discussed systematically in intersection theory.
Definition for Riemann surfaces
Let X be a Riemann surface. Then the intersection number of two closed curves on X has a simple definition in terms of an integral. For every closed curve c on X (i.e., smooth function ), we can associate a differential form of compact support, the Poincaré dual of c, with the property that integrals along c can be calculated by integrals over X:
, for every closed (1-)differential on X,
where is the wedge product of differentials, and is the Hodge star. Then the intersection number of two closed curves, a and b, on X is defined as
.
The have an intuitive definition as follows. They are a sort of dirac delta along the curve c, accomplished by taking the differential of a unit step function that drops from 1 to 0 across c. More formally, we begin by defining for a simple closed curve c on X, a function fc by letting be a small strip around c in the shape of an annulus. Name the left and right parts of as and . Then take a smaller sub-strip around c, , with left and right parts and . Then define fc by
.
The definition is then expanded to arbitrary closed curves. Every closed curve c on X is homologous to for some simple closed curves ci, that is,
, for every differential .
Define the by
.
Definition for algebraic varieties
The usual constructive definition in the case of algebraic varieties proceeds in steps. The definition given below is for the intersection number of divisors on a nonsingular variety X.
1. The only intersection number that can be calculated directly from the definition is the intersection of hypersurfaces (subvarieties of X of codimension one) that are in general position at x. Specifically, assume we have a nonsingular variety X, and n hypersurfaces Z1, ..., Zn which have local equations f1, ..., fn near x for polynomials fi(t1, ..., tn), such that the following hold:
.
for all i. (i.e., x is in the intersection of the hypersurfaces.)
(i.e., the divisors are in general position.)
The are nonsingular at x.
Then the intersection number at the point x (called the intersection multiplicity at x) is
,
where is the local ring of X at x, and the dimension i
|
https://en.wikipedia.org/wiki/Solomon%20Lefschetz
|
Solomon Lefschetz (; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.
Life
He was born in Moscow, the son of Alexander Lefschetz and his wife Sarah or Vera Lifschitz, Jewish traders who used to travel around Europe and the Middle East (they held Ottoman passports). Shortly thereafter, the family moved to Paris. He was educated there in engineering at the École Centrale Paris, but emigrated to the US in 1905.
He was badly injured in an industrial accident in 1907, losing both hands. He moved towards mathematics, receiving a Ph.D. in algebraic geometry from Clark University in Worcester, Massachusetts in 1911. He then took positions in University of Nebraska and University of Kansas, moving to Princeton University in 1924, where he was soon given a permanent position. He remained there until 1953.
In the application of topology to algebraic geometry, he followed the work of Charles Émile Picard, whom he had heard lecture in Paris at the École Centrale Paris. He proved theorems on the topology of hyperplane sections of algebraic varieties, which provide a basic inductive tool (these are now seen as allied to Morse theory, though a Lefschetz pencil of hyperplane sections is a more subtle system than a Morse function because hyperplanes intersect each other). The Picard–Lefschetz formula in the theory of vanishing cycles is a basic tool relating the degeneration of families of varieties with 'loss' of topology, to monodromy. He was an Invited Speaker of the ICM in 1920 in Strasbourg. His book L'analysis situs et la géométrie algébrique from 1924, though opaque foundationally given the current technical state of homology theory, was in the long term very influential (one could say that it was one of the sources for the eventual proof of the Weil conjectures, through SGA 7 also for the study of Picard groups of Zariski surface). In 1924 he was awarded the Bôcher Memorial Prize for his work in mathematical analysis. He was elected to the United States National Academy of Sciences in 1925 and the American Philosophical Society in 1929.
The Lefschetz fixed-point theorem, now a basic result of topology, was developed by him in papers from 1923 to 1927, initially for manifolds. Later, with the rise of cohomology theory in the 1930s, he contributed to the intersection number approach (that is, in cohomological terms, the ring structure) via the cup product and duality on manifolds. His work on topology was summed up in his monograph Algebraic Topology (1942). From 1944 he worked on differential equations.
He was editor of the Annals of Mathematics from 1928 to 1958. During this time, the Annals became an increasingly well-known and respected journal, and Lefschetz played an important role in this.
In 1945 he travelled to Mexico for the first time, where he joined the Institute
|
https://en.wikipedia.org/wiki/W.%20V.%20D.%20Hodge
|
Sir William Vallance Douglas Hodge (; 17 June 1903 – 7 July 1975) was a British mathematician, specifically a geometer.
His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major influence on subsequent work in geometry.
Life and career
Hodge was born in Edinburgh in 1903, the younger son and second of three children of Archibald James Hodge (1869-1938), a searcher of records in the property market and a partner in the firm of Douglas and Company, and his wife, Jane (born 1875), daughter of confectionery business owner William Vallance. They lived at 1 Church Hill Place in the Morningside district.
He attended George Watson's College, and studied at Edinburgh University, graduating MA in 1923. With help from E. T. Whittaker, whose son J. M. Whittaker was a college friend, he then took the Cambridge Mathematical Tripos. At Cambridge he fell under the influence of the geometer H. F. Baker. He gained a Cambridge BA degree in 1925, receiving the MA in 1930 and the Doctor of Science (ScD) degree in 1950.
In 1926 he took up a teaching position at the University of Bristol, and began work on the interface between the Italian school of algebraic geometry, particularly problems posed by Francesco Severi, and the topological methods of Solomon Lefschetz. This made his reputation, but led to some initial scepticism on the part of Lefschetz. According to Atiyah's memoir, Lefschetz and Hodge in 1931 had a meeting in Max Newman's rooms in Cambridge, to try to resolve issues. In the end Lefschetz was convinced.
In 1928 he was elected a Fellow of the Royal Society of Edinburgh. His proposers were Sir Edmund Taylor Whittaker, Ralph Allan Sampson, Charles Glover Barkla, and Sir Charles Galton Darwin. He was awarded the Society's Gunning Victoria Jubilee Prize for the period 1964 to 1968.
In 1930 Hodge was awarded a Research Fellowship at St. John's College, Cambridge. He spent the year 1931–2 at Princeton University, where Lefschetz was, visiting also Oscar Zariski at Johns Hopkins University. At this time he was also assimilating de Rham's theorem, and defining the Hodge star operation. It would allow him to define harmonic forms and so refine the de Rham theory.
On his return to Cambridge, he was offered a University Lecturer position in 1933. He became the Lowndean Professor of Astronomy and Geometry at Cambridge, a position he held from 1936 to 1970. He was the first head of DPMMS.
He was the Master of Pembroke College, Cambridge from 1958 to 1970, and vice-president of the Royal Society from 1959 to 1965. He was knighted in 1959. Amongst other honours, he received the Adams Prize in 1937 and the Copley Medal of the Royal Society in 1974.
He died in Cambridge on 7 July 1975.
Work
The Hodge index theorem was a result on the intersection number theory for curves on an algebraic surface: it determines the sign
|
https://en.wikipedia.org/wiki/Arithmetic%20of%20abelian%20varieties
|
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields).
Integer points on abelian varieties
There is some tension here between concepts: integer point belongs in a sense to affine geometry, while abelian variety is inherently defined in projective geometry. The basic results, such as Siegel's theorem on integral points, come from the theory of diophantine approximation.
Rational points on abelian varieties
The basic result, the Mordell–Weil theorem in Diophantine geometry, says that A(K), the group of points on A over K, is a finitely-generated abelian group. A great deal of information about its possible torsion subgroups is known, at least when A is an elliptic curve. The question of the rank is thought to be bound up with L-functions (see below).
The torsor theory here leads to the Selmer group and Tate–Shafarevich group, the latter (conjecturally finite) being difficult to study.
Heights
The theory of heights plays a prominent role in the arithmetic of abelian varieties. For instance, the canonical Néron–Tate height is a quadratic form with remarkable properties that appear in the statement of the Birch and Swinnerton-Dyer conjecture.
Reduction mod p
Reduction of an abelian variety A modulo a prime ideal of (the integers of) K — say, a prime number p — to get an abelian variety Ap over a finite field, is possible for almost all p. The 'bad' primes, for which the reduction degenerates by acquiring singular points, are known to reveal very interesting information. As often happens in number theory, the 'bad' primes play a rather active role in the theory.
Here a refined theory of (in effect) a right adjoint to reduction mod p — the Néron model — cannot always be avoided. In the case of an elliptic curve there is an algorithm of John Tate describing it.
L-functions
For abelian varieties such as Ap, there is a definition of local zeta-function available. To get an L-function for A itself, one takes a suitable Euler product of such local functions; to understand the finite number of factors for the 'bad' primes one has to refer to the Tate module of A, which is (dual to) the étale cohomology group H1(A), and the Galois group action on it. In this way one gets a respectable definition of Hasse–Weil L-function for A. In general its properties, such as functional equation, are still conjectural – the Taniyama–Shimura conjecture (which was proven in 2001) was just a special case, so that's hardly surprising.
It is in terms of this L-function that the conjecture of Birch and Swinnerton-Dyer is posed
|
https://en.wikipedia.org/wiki/Ramification%20%28mathematics%29
|
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping.
In complex analysis
In complex analysis, the basic model can be taken as the z → zn mapping in the complex plane, near z = 0. This is the standard local picture in Riemann surface theory, of ramification of order n. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus.
In algebraic topology
In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The z → zn mapping shows this as a local pattern: if we exclude 0, looking at 0 < |z| < 1 say, we have (from the homotopy point of view) the circle mapped to itself by the n-th power map (Euler–Poincaré characteristic 0), but with the whole disk the Euler–Poincaré characteristic is 1, n – 1 being the 'lost' points as the n sheets come together at z = 0.
In geometric terms, ramification is something that happens in codimension two (like knot theory, and monodromy); since real codimension two is complex codimension one, the local complex example sets the pattern for higher-dimensional complex manifolds. In complex analysis, sheets can't simply fold over along a line (one variable), or codimension one subspace in the general case. The ramification set (branch locus on the base, double point set above) will be two real dimensions lower than the ambient manifold, and so will not separate it into two 'sides', locally―there will be paths that trace round the branch locus, just as in the example. In algebraic geometry over any field, by analogy, it also happens in algebraic codimension one.
In algebraic number theory
In algebraic extensions of the rational numbers
Ramification in algebraic number theory means a prime ideal factoring in an extension so as to give some repeated prime ideal factors. Namely, let be the ring of integers of an algebraic number field , and a prime ideal of . For a field extension we can consider the
ring of integers (which is the integral closure of in ), and the ideal of . This ideal may or may not be prime, but for finite , it has a factorization into prime ideals:
where the are distinct prime ideals of . Then is said to ramify in if for some ; otherwise it is . In other words, ramifies in if the ramification index is greater than one for some . An equivalent condition is that has a non-zero nilpotent element: it is not a product of finite fields. The analogy with the Riemann surface case was already pointed out by Richard Dedekind and Heinrich M. Weber in the nineteenth century.
The ramification is encoded in by the relative discriminant and in by the relative different.
|
https://en.wikipedia.org/wiki/Generalized%20flag%20variety
|
In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space V over a field F. When F is the real or complex numbers, a generalized flag variety is a smooth or complex manifold, called a real or complex flag manifold. Flag varieties are naturally projective varieties.
Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space V over a field F, which is a flag variety for the special linear group over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the symplectic group. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags.
In the most general sense, a generalized flag variety is defined to mean a projective homogeneous variety, that is, a smooth projective variety X over a field F with a transitive action of a reductive group G (and smooth stabilizer subgroup; that is no restriction for F of characteristic zero). If X has an F-rational point, then it is isomorphic to G/P for some parabolic subgroup P of G. A projective homogeneous variety may also be realised as the orbit of a highest weight vector in a projectivized representation of G. The complex projective homogeneous varieties are the compact flat model spaces for Cartan geometries of parabolic type. They are homogeneous Riemannian manifolds under any maximal compact subgroup of G, and they are precisely the coadjoint orbits of compact Lie groups.
Flag manifolds can be symmetric spaces. Over the complex numbers, the corresponding flag manifolds are the Hermitian symmetric spaces. Over the real numbers, an R-space is a synonym for a real flag manifold and the corresponding symmetric spaces are called symmetric R-spaces.
Flags in a vector space
A flag in a finite dimensional vector space V over a field F is an increasing sequence of subspaces, where "increasing" means each is a proper subspace of the next (see filtration):
If we write the dim Vi = di then we have
where n is the dimension of V. Hence, we must have k ≤ n. A flag is called a complete flag if di = i for all i, otherwise it is called a partial flag. The signature of the flag is the sequence (d1, ..., dk).
A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.
Prototype: the complete flag variety
According to basic results of linear algebra, any two complete flags in an n-dimensional vector space V over a field F are no different from each other from a geometric point of view. That is to say, the general linear group acts transitively on the set of all complete flags.
Fix an ordered basis for V, identifying it with Fn, whos
|
https://en.wikipedia.org/wiki/Finsler%20manifold
|
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski norm is provided on each tangent space , that enables one to define the length of any smooth curve as
Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products.
Every Finsler manifold becomes an intrinsic quasimetric space when the distance between two points is defined as the infimum length of the curves that join them.
named Finsler manifolds after Paul Finsler, who studied this geometry in his dissertation .
Definition
A Finsler manifold is a differentiable manifold together with a Finsler metric, which is a continuous nonnegative function defined on the tangent bundle so that for each point of ,
for every two vectors tangent to at (subadditivity).
for all (but not necessarily for (positive homogeneity).
unless (positive definiteness).
In other words, is an asymmetric norm on each tangent space . The Finsler metric is also required to be smooth, more precisely:
is smooth on the complement of the zero section of .
The subadditivity axiom may then be replaced by the following strong convexity condition:
For each tangent vector , the Hessian matrix of at is positive definite.
Here the Hessian of at is the symmetric bilinear form
also known as the fundamental tensor of at . Strong convexity of implies the subadditivity with a strict inequality if . If is strongly convex, then it is a Minkowski norm on each tangent space.
A Finsler metric is reversible if, in addition,
for all tangent vectors v.
A reversible Finsler metric defines a norm (in the usual sense) on each tangent space.
Examples
Smooth submanifolds (including open subsets) of a normed vector space of finite dimension are Finsler manifolds if the norm of the vector space is smooth outside the origin.
Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds.
Randers manifolds
Let be a Riemannian manifold and b a differential one-form on M with
where is the inverse matrix of and the Einstein notation is used. Then
defines a Randers metric on M and is a Randers manifold, a special case of a non-reversible Finsler manifold.
Smooth quasimetric spaces
Let (M, d) be a quasimetric so that M is also a differentiable manifold and d is compatible with the differential structure of M in the following sense:
Around any point z on M there exists a smooth chart (U, φ) of M and a constant C ≥ 1 such that for every x, y ∈ U
The function d: M × M → [0, ∞] is smooth in some punctured neighborhood of the diagonal.
Then one can define a Finsler function F: TM →[0, ∞] by
where γ is any curve in M with γ(0) = x and γ(0) = v. The Finsler function F obtained in this way restricts to an asymmetric (typically non-Minkowski) norm on each tangent space of M. The induced intrinsic metric of the original quasimet
|
https://en.wikipedia.org/wiki/Inverse%20scattering%20problem
|
In mathematics and physics, the inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters incoming radiation or particles. It is the inverse problem to the direct scattering problem, which is to determine how radiation or particles are scattered based on the properties of the scatterer.
Soliton equations are a class of partial differential equations which can be studied and solved by a method called the inverse scattering transform, which reduces the nonlinear PDEs to a linear inverse scattering problem. The nonlinear Schrödinger equation, the Korteweg–de Vries equation and the KP equation are examples of soliton equations. In one space dimension the inverse scattering problem is equivalent to a Riemann-Hilbert problem. Since its early statement for radiolocation, many applications have been found for inverse scattering techniques, including echolocation, geophysical survey, nondestructive testing, medical imaging, quantum field theory.
References
.
Inverse Acoustic and Electromagnetic Scattering Theory; Colton, David and Kress, Rainer
Scattering theory
Scattering, absorption and radiative transfer (optics)
Inverse problems
|
https://en.wikipedia.org/wiki/Pontryagin%20class
|
In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.
Definition
Given a real vector bundle E over M, its k-th Pontryagin class is defined as
where:
denotes the -th Chern class of the complexification of E,
is the -cohomology group of M with integer coefficients.
The rational Pontryagin class is defined to be the image of in , the -cohomology group of M with rational coefficients.
Properties
The total Pontryagin class
is (modulo 2-torsion) multiplicative with respect to
Whitney sum of vector bundles, i.e.,
for two vector bundles E and F over M. In terms of the individual Pontryagin classes pk,
and so on.
The vanishing of the Pontryagin classes and Stiefel–Whitney classes of a vector bundle does not guarantee that the vector bundle is trivial. For example, up to vector bundle isomorphism, there is a unique nontrivial rank 10 vector bundle over the 9-sphere. (The clutching function for arises from the homotopy group .) The Pontryagin classes and Stiefel-Whitney classes all vanish: the Pontryagin classes don't exist in degree 9, and the Stiefel–Whitney class w9 of E10 vanishes by the Wu formula w9 = w1w8 + Sq1(w8). Moreover, this vector bundle is stably nontrivial, i.e. the Whitney sum of E10 with any trivial bundle remains nontrivial.
Given a 2k-dimensional vector bundle E we have
where e(E) denotes the Euler class of E, and denotes the cup product of cohomology classes.
Pontryagin classes and curvature
As was shown by Shiing-Shen Chern and André Weil around 1948, the rational Pontryagin classes
can be presented as differential forms which depend polynomially on the curvature form of a vector bundle. This Chern–Weil theory revealed a major connection between algebraic topology and global differential geometry.
For a vector bundle E over a n-dimensional differentiable manifold M equipped with a connection, the total Pontryagin class is expressed as
where Ω denotes the curvature form, and H*dR(M) denotes the de Rham cohomology groups.
Pontryagin classes of a manifold
The Pontryagin classes of a smooth manifold are defined to be the Pontryagin classes of its tangent bundle.
Novikov proved in 1966 that if two compact, oriented, smooth manifolds are homeomorphic then their rational Pontryagin classes pk(M, Q) in H4k(M, Q) are the same.
If the dimension is at least five, there are at most finitely many different smooth manifolds with given homotopy type and Pontryagin classes.
Pontryagin classes from Chern classes
The Pontryagin classes of a complex vector bundle is completely determined by its Chern classes. This follows from the fact that , the Whitney sum formula, and properties of Chern classes of its complex conjugate bundle. That is, and . Then, this given the relationfor example, we can apply this formula to find the Pontryagin classes of a complex
|
https://en.wikipedia.org/wiki/Irreducible%20representation
|
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation or irrep of an algebraic structure is a nonzero representation that has no proper nontrivial subrepresentation , with closed under the action of .
Every finite-dimensional unitary representation on a Hilbert space is the direct sum of irreducible representations. Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible.
History
Group representation theory was generalized by Richard Brauer from the 1940s to give modular representation theory, in which the matrix operators act on a vector space over a field of arbitrary characteristic, rather than a vector space over the field of real numbers or over the field of complex numbers. The structure analogous to an irreducible representation in the resulting theory is a simple module.
Overview
Let be a representation i.e. a homomorphism of a group where is a vector space over a field . If we pick a basis for , can be thought of as a function (a homomorphism) from a group into a set of invertible matrices and in this context is called a matrix representation. However, it simplifies things greatly if we think of the space without a basis.
A linear subspace is called -invariant if for all and all . The co-restriction of to the general linear group of a -invariant subspace is known as a subrepresentation. A representation is said to be irreducible if it has only trivial subrepresentations (all representations can form a subrepresentation with the trivial -invariant subspaces, e.g. the whole vector space , and {0}). If there is a proper nontrivial invariant subspace, is said to be reducible.
Notation and terminology of group representations
Group elements can be represented by matrices, although the term "represented" has a specific and precise meaning in this context. A representation of a group is a mapping from the group elements to the general linear group of matrices. As notation, let denote elements of a group with group product signified without any symbol, so is the group product of and and is also an element of , and let representations be indicated by . The representation of a is written as
By definition of group representations, the representation of a group product is translated into matrix multiplication of the representations:
If is the identity element of the group (so that , etc.), then is an identity matrix, or identically a block matrix of identity matrices, since we must have
and similarly for all other group elements. The last two statements correspond to the requirement that is a group homomorphism.
Reducible and irreducible representations
A representation is reducible if it contains a nontrivial G-inva
|
https://en.wikipedia.org/wiki/Eisenstein%27s%20criterion
|
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients.
This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases for irreducibility to be proved with very little effort. It may apply either directly or after transformation of the original polynomial.
This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it.
Criterion
Suppose we have the following polynomial with integer coefficients:
If there exists a prime number such that the following three conditions all apply:
divides each for ,
does not divide , and
does not divide ,
then is irreducible over the rational numbers. It will also be irreducible over the integers, unless all its coefficients have a nontrivial factor in common (in which case as integer polynomial will have some prime number, necessarily distinct from , as an irreducible factor). The latter possibility can be avoided by first making primitive, by dividing it by the greatest common divisor of its coefficients (the content of ). This division does not change whether is reducible or not over the rational numbers (see Primitive part–content factorization for details), and will not invalidate the hypotheses of the criterion for (on the contrary it could make the criterion hold for some prime, even if it did not before the division).
Examples
Eisenstein's criterion may apply either directly (i.e., using the original polynomial) or after transformation of the original polynomial.
Direct (without transformation)
Consider the polynomial . In order for Eisenstein's criterion to apply for a prime number it must divide both non-leading coefficients and , which means only could work, and indeed it does since does not divide the leading coefficient , and its square does not divide the constant coefficient . One may therefore conclude that is irreducible over (and, since it is primitive, over as well). Note that since is of degree 4, this conclusion could not have been established by only checking that has no rational roots (which eliminates possible factors of degree 1), since a decomposition into two quadratic factors could also be possible.
Indirect (after transformation)
Often Eisenstein's criterion does not apply for any prime number. It may however be that it applies (for some prime number) to the polynomial obtained after substitution (for some integer ) of for . The fact that the polynomial after substitution is irreducible then allows concluding that the original polynomial is as well. This procedure is known as applying a shift.
For example consider
|
https://en.wikipedia.org/wiki/Isospectral
|
In mathematics, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity.
The theory of isospectral operators is markedly different depending on whether the space is finite or infinite dimensional. In finite-dimensions, one essentially deals with square matrices.
In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator on a Hilbert space (or Banach space) is still tractable, since the eigenvalues are at most countable with at most a single limit point λ = 0. The most studied isospectral problem in infinite dimensions is that of the Laplace operator on a domain in R2. Two such domains are called isospectral if their Laplacians are isospectral. The problem of inferring the geometrical properties of a domain from the spectrum of its Laplacian is often known as hearing the shape of a drum.
Finite dimensional spaces
In the case of operators on finite-dimensional vector spaces, for complex square matrices, the relation of being isospectral for two diagonalizable matrices is just similarity. This doesn't however reduce completely the interest of the concept, since we can have an isospectral family of matrices of shape A(t) = M(t)−1AM(t) depending on a parameter t in a complicated way. This is an evolution of a matrix that happens inside one similarity class.
A fundamental insight in soliton theory was that the infinitesimal analogue of that equation, namely
A = [A, M] = AM − MA
was behind the conservation laws that were responsible for keeping solitons from dissipating. That is, the preservation of spectrum was an interpretation of the conservation mechanism. The identification of so-called Lax pairs (P,L) giving rise to analogous equations, by Peter Lax, showed how linear machinery could explain the non-linear behaviour.
Isospectral manifolds
Two closed Riemannian manifolds are said to be isospectral if the eigenvalues of their Laplace–Beltrami operator (Laplacians), counted multiplicities, coincide. One of fundamental problems in spectral geometry is to ask to what extent the eigenvalues determine the geometry of a given manifold.
There are many examples of isospectral manifolds which are not isometric. The first example was given in 1964 by John Milnor. He constructed a pair of flat tori of 16 dimension, using arithmetic lattices first studied by Ernst Witt. After this example, many isospectral pairs in dimension two and higher were constructed (for instance, by M. F. Vignéras, A. Ikeda, H. Urakawa, C. Gordon). In particular , based on the Selberg trace formula for PSL(2,R) and PSL(2,C), constructed examples of isospectral, non-isometric closed hyperbolic 2-manifolds and 3-manifolds as quotients of hyperbolic 2-space and 3-space by arithmetic subgroups, constructed using quaternion algebras associated with quadratic exte
|
https://en.wikipedia.org/wiki/Homothety
|
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number called its ratio, which sends point to a point by the rule
for a fixed number .
Using position vectors:
.
In case of (Origin):
,
which is a uniform scaling and shows the meaning of special choices for :
for one gets the identity mapping,
for one gets the reflection at the center,
For one gets the inverse mapping defined by .
In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if ) or reverse (if ) the direction of all vectors. Together with the translations, all homotheties of an affine (or Euclidean) space form a group, the group of dilations or homothety-translations. These are precisely the affine transformations with the property that the image of every line g is a line parallel to g.
In projective geometry, a homothetic transformation is a similarity transformation (i.e., fixes a given elliptic involution) that leaves the line at infinity pointwise invariant.
In Euclidean geometry, a homothety of ratio multiplies distances between points by , areas by and volumes by . Here is the ratio of magnification or dilation factor or scale factor or similitude ratio. Such a transformation can be called an enlargement if the scale factor exceeds 1. The above-mentioned fixed point S is called homothetic center or center of similarity or center of similitude.
The term, coined by French mathematician Michel Chasles, is derived from two Greek elements: the prefix homo- (), meaning "similar", and thesis (), meaning "position". It describes the relationship between two figures of the same shape and orientation. For example, two Russian dolls looking in the same direction can be considered homothetic.
Homotheties are used to scale the contents of computer screens; for example, smartphones, notebooks, and laptops.
Properties
The following properties hold in any dimension.
Mapping lines, line segments and angles
A homothety has the following properties:
A line is mapped onto a parallel line. Hence: angles remain unchanged.
The ratio of two line segments is preserved.
Both properties show:
A homothety is a similarity.
Derivation of the properties:
In order to make calculations easy it is assumed that the center is the origin: . A line with parametric representation is mapped onto the point set with equation , which is a line parallel to .
The distance of two points is and the distance between their images. Hence, the ratio (quotient) of two line segments remains unchanged .
In case of the calculation is analogous but a little extensive.
Consequences: A triangle is mapped on a similar one. The homothetic image of a circle is a circle. The image of an ellipse is a similar one. i.e. the ratio of the two axes is unchanged.
Graphical constructions
using the intercept theorem
If for a homothety with center the image
|
https://en.wikipedia.org/wiki/Spectral%20radius
|
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum. The spectral radius is often denoted by .
Definition
Matrices
Let be the eigenvalues of a matrix . The spectral radius of is defined as
The spectral radius can be thought of as an infimum of all norms of a matrix. Indeed, on the one hand, for every natural matrix norm ; and on the other hand, Gelfand's formula states that . Both of these results are shown below.
However, the spectral radius does not necessarily satisfy for arbitrary vectors . To see why, let be arbitrary and consider the matrix
.
The characteristic polynomial of is , so its eigenvalues are and thus . However, . As a result,
As an illustration of Gelfand's formula, note that as , since if is even and if is odd.
A special case in which for all is when is a Hermitian matrix and is the Euclidean norm. This is because any Hermitian Matrix is diagonalizable by a unitary matrix, and unitary matrices preserve vector length. As a result,
Bounded linear operators
In the context of a bounded linear operator on a Banach space, the eigenvalues need to be replaced with the elements of the spectrum of the operator, i.e. the values for which is not bijective. We denote the spectrum by
The spectral radius is then defined as the supremum of the magnitudes of the elements of the spectrum:
Gelfand's formula, also known as the spectral radius formula, also holds for bounded linear operators: letting denote the operator norm, we have
A bounded operator (on a complex Hilbert space) is called a spectraloid operator if its spectral radius coincides with its numerical radius. An example of such an operator is a normal operator.
Graphs
The spectral radius of a finite graph is defined to be the spectral radius of its adjacency matrix.
This definition extends to the case of infinite graphs with bounded degrees of vertices (i.e. there exists some real number such that the degree of every vertex of the graph is smaller than ). In this case, for the graph define:
Let be the adjacency operator of :
The spectral radius of is defined to be the spectral radius of the bounded linear operator .
Upper bounds
Upper bounds on the spectral radius of a matrix
The following proposition gives simple yet useful upper bounds on the spectral radius of a matrix.
Proposition. Let with spectral radius and a consistent matrix norm . Then for each integer :
Proof
Let be an eigenvector-eigenvalue pair for a matrix A. By the sub-multiplicativity of the matrix norm, we get:
Since , we have
and therefore
concluding the proof.
Upper bounds for spectral radius of a graph
There are many upper bounds for the spectral radius of a graph in terms of its number n of vertices and its number m of edges. For instance, if
where is an integer, the
|
https://en.wikipedia.org/wiki/Cartan
|
Cartan may refer to:
Élie Cartan (1869–1951), French mathematician who worked with Lie groups
Henri Cartan (1904–2008), French mathematician who worked in algebraic topology, son of Élie Cartan
Anna Cartan (1878–1923), French mathematician and teacher, sister of Élie Cartan
Cartan (crater), a lunar crater named for Élie Cartan
Badea Cârțan (1849–1911), Austro-Hungarian Romanian activist
|
https://en.wikipedia.org/wiki/Digital%20geometry
|
Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitized models or images of objects of the 2D or 3D Euclidean space.
Simply put, digitizing is replacing an object by a discrete set of its points. The images we see on the TV screen, the raster display of a computer, or in newspapers are in fact digital images.
Its main application areas are computer graphics and image analysis.
Main aspects of study are:
Constructing digitized representations of objects, with the emphasis on precision and efficiency (either by means of synthesis, see, for example, Bresenham's line algorithm or digital disks, or by means of digitization and subsequent processing of digital images).
Study of properties of digital sets; see, for example, Pick's theorem, digital convexity, digital straightness, or digital planarity.
Transforming digitized representations of objects, for example (A) into simplified shapes such as (i) skeletons, by repeated removal of simple points such that the digital topology of an image does not change, or (ii) medial axis, by calculating local maxima in a distance transform of the given digitized object representation, or (B) into modified shapes using mathematical morphology.
Reconstructing "real" objects or their properties (area, length, curvature, volume, surface area, and so forth) from digital images.
Study of digital curves, digital surfaces, and digital manifolds.
Designing tracking algorithms for digital objects.
Functions on digital space.
Curve sketching, a method of drawing a curve pixel by pixel.
Digital geometry heavily overlaps with discrete geometry and may be considered as a part thereof.
Digital space
A 2D digital space usually means a 2D grid space that only contains integer points in 2D Euclidean space. A 2D image is a function on a 2D digital space (See image processing).
In Rosenfeld and Kak's book, digital connectivity are defined as the relationship among elements in digital space. For example, 4-connectivity and 8-connectivity in 2D. Also see pixel connectivity. A digital space and its (digital-)connectivity determine a digital topology.
In digital space, the digitally continuous function (A. Rosenfeld, 1986) and the gradually varied function (L. Chen, 1989) were proposed, independently.
A digitally continuous function means a function in which the value (an integer) at a digital point is the same or off by at most 1 from its neighbors. In other words, if x and y are two adjacent points in a digital space, |f(x) − f(y)| ≤ 1.
A gradually varied function is a function from a digital space to where and are real numbers. This function possesses the following property: If x and y are two adjacent points in , assume , then , , or . So we can see that the gradually varied function is defined to be more general than the digitally continuous function.
An extension theorem related to above functions was mentioned by A. Rosenfeld (1986) and completed by L. Chen (19
|
https://en.wikipedia.org/wiki/Discrete%20geometry
|
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
Discrete geometry has a large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.
History
Although polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied were: the density of circle packings by Thue, projective configurations by Reye and Steinitz, the geometry of numbers by Minkowski, and map colourings by Tait, Heawood, and Hadwiger.
László Fejes Tóth, H.S.M. Coxeter, and Paul Erdős laid the foundations of discrete geometry.
Topics
Polyhedra and polytopes
A polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions (such as a 4-polytope in four dimensions). Some theories further generalize the idea to include such objects as unbounded polytopes (apeirotopes and tessellations), and abstract polytopes.
The following are some of the aspects of polytopes studied in discrete geometry:
Polyhedral combinatorics
Lattice polytopes
Ehrhart polynomials
Pick's theorem
Hirsch conjecture
Opaque set
Packings, coverings and tilings
Packings, coverings, and tilings are all ways of arranging uniform objects (typically circles, spheres, or tiles) in a regular way on a surface or manifold.
A sphere packing is an arrangement of non-overlapping spheres within a containing space. The spheres considered are usually all of identical size, and the space is usually three-dimensional Euclidean space. However, sphere packing problems can be generalised to consider unequal spheres, n-dimensional Euclidean space (where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions) or to non-Euclidean spaces such as hyperbolic space.
A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions.
Specific topics in this area include:
Circle packings
Sphere packings
Kepler conjecture
Quasicrystals
Aperiodic tilings
Periodic graph
Finite subdivision rules
Structural rigidity and flexibility
Structural rigidity is a combi
|
https://en.wikipedia.org/wiki/Constructive%20solid%20geometry
|
Constructive solid geometry (CSG; formerly called computational binary solid geometry) is a technique used in solid modeling. Constructive solid geometry allows a modeler to create a complex surface or object by using Boolean operators to combine simpler objects, potentially generating visually complex objects by combining a few primitive ones.
In 3D computer graphics and CAD, CSG is often used in procedural modeling. CSG can also be performed on polygonal meshes, and may or may not be procedural and/or parametric.
Contrast CSG with polygon mesh modeling and box modeling.
Workings
The simplest solid objects used for the representation are called geometric primitives. Typically they are the objects of simple shape: cuboids, cylinders, prisms, pyramids, spheres, cones. The set of allowable primitives is limited by each software package. Some software packages allow CSG on curved objects while other packages do not.
An object is constructed from primitives by means of allowable operations, which are typically Boolean operations on sets: union, intersection and difference, as well as geometric transformations of those sets.
A primitive can typically be described by a procedure which accepts some number of parameters; for example, a sphere may be described by the coordinates of its center point, along with a radius value. These primitives can be combined into compound objects using operations like these:
Combining these elementary operations, it is possible to build up objects with high complexity starting from simple ones.
Ray tracing
Rendering of constructive solid geometry is particularly simple when ray tracing. Ray tracers intersect a ray with both primitives that are being operated on, apply the operator to the intersection intervals along the 1D ray, and then take the point closest to the camera along the ray as being the result.
Applications
Constructive solid geometry has a number of practical uses. It is used in cases where simple geometric objects are desired, or where mathematical accuracy is important.
Nearly all engineering CAD packages use CSG (where it may be useful for representing tool cuts, and features where parts must fit together).
The Quake engine and Unreal Engine both use this system, as does Hammer (the native Source engine level editor), and Torque Game Engine/Torque Game Engine Advanced. CSG is popular because a modeler can use a set of relatively simple objects to create very complicated geometry. When CSG is procedural or parametric, the user can revise their complex geometry by changing the position of objects or by changing the Boolean operation used to combine those objects.
One of the advantages of CSG is that it can easily assure that objects are "solid" or water-tight if all of the primitive shapes are water-tight. This can be important for some manufacturing or engineering computation applications. By comparison, when creating geometry based upon boundary representations, additional topological data i
|
https://en.wikipedia.org/wiki/Newport%20Pagnell
|
Newport Pagnell is a town and civil parish in the City of Milton Keynes, Buckinghamshire, England. The Office for National Statistics records Newport Pagnell as part of the Milton Keynes urban area.
The town is separated from the rest of the urban area by the M1 motorway, on which Newport Pagnell Services, the second service station to be opened in the United Kingdom, is located.
The town is more widely known for having the only remaining vellum manufacturer in the United Kingdom, and being the original home of the exclusive sports car manufacturer Aston Martin.
History
The town was first mentioned in the Domesday Book of 1086 as Neuport, Old English for 'New Market Town', but by that time, the old Anglo-Saxon town was dominated by the Norman invaders. The suffix 'Pagnell' came later when the manor passed into the hands of the Pagnell (Paynel) family. It was the principal town of the "Three Hundreds of Newport", a district that had almost the same boundary as the modern City of Milton Keynes UA.
The Grade I listed Tickford Bridge, over the River Ouzel (or Lovat), was built in 1810. It is one of just a few cast iron bridges in Britain that still carry modern road traffic. Near the footbridge at the side, there is a plaque placed by Newport Pagnell Historical Society that gives details of its history and construction. The Ouzel joins the Great Ouse nearby, and a large set of sluice gates, used to control downstream flooding, is located near the bridge.
Between 1817 and 1864, the town was linked to the Grand Junction Canal at Great Linford via the Newport Pagnell Canal. In 1862, the canal owners sold the route to the London and North Western Railway. For a hundred years (1867 to 1967), Newport Pagnell was served by Newport Pagnell railway station, the terminus on the Wolverton to Newport Pagnell branch line. (The route is now a rail trail, part of the Milton Keynes redway system.)
The population of Newport Pagnell and its hinterland at the 1801 Census was 17,576; by 1911 it had grown to 14,428. The population of Newport Pagnell Urban District alone is first recorded at the 1911 Census as 4,238 and had reached 4,743 by 1961.
Industry
From 1954 until 2007, the town was the home to the sports car manufacturer Aston Martin. The Newport Pagnell factory was considered outdated and a new production facility was built near Gaydon in Warwickshire. There is still a service facility in Newport Pagnell, but the factory on the north side of Tickford St has since been demolished apart from the engine shop, board room and offices that are listed buildings. The land behind these has been developed by a housing developer. The buildings at the front, including the house used as a board room, have been restored and will be used as commercial sites. In 2012, Aston Martin completely modernised the service facility and the site also houses a bespoke sales department.
Notable industries in the town include the only remaining vellum manufacturer in the United K
|
https://en.wikipedia.org/wiki/Incidence%20%28geometry%29
|
In geometry, an incidence relation is a heterogeneous relation that captures the idea being expressed when phrases such as "a point lies on a line" or "a line is contained in a plane" are used. The most basic incidence relation is that between a point, , and a line, , sometimes denoted . If the pair is called a flag. There are many expressions used in common language to describe incidence (for example, a line passes through a point, a point lies in a plane, etc.) but the term "incidence" is preferred because it does not have the additional connotations that these other terms have, and it can be used in a symmetric manner. Statements such as "line intersects line " are also statements about incidence relations, but in this case, it is because this is a shorthand way of saying that "there exists a point that is incident with both line and line ". When one type of object can be thought of as a set of the other type of object (viz., a plane is a set of points) then an incidence relation may be viewed as containment.
Statements such as "any two lines in a plane meet" are called incidence propositions. This particular statement is true in a projective plane, though not true in the Euclidean plane where lines may be parallel. Historically, projective geometry was developed in order to make the propositions of incidence true without exceptions, such as those caused by the existence of parallels. From the point of view of synthetic geometry, projective geometry should be developed using such propositions as axioms. This is most significant for projective planes due to the universal validity of Desargues' theorem in higher dimensions.
In contrast, the analytic approach is to define projective space based on linear algebra and utilizing homogeneous co-ordinates. The propositions of incidence are derived from the following basic result on vector spaces: given subspaces and of a (finite-dimensional) vector space , the dimension of their intersection is . Bearing in mind that the geometric dimension of the projective space associated to is and that the geometric dimension of any subspace is positive, the basic proposition of incidence in this setting can take the form: linear subspaces and of projective space meet provided .
The following sections are limited to projective planes defined over fields, often denoted by , where is a field, or . However these computations can be naturally extended to higher-dimensional projective spaces, and the field may be replaced by a division ring (or skewfield) provided that one pays attention to the fact that multiplication is not commutative in that case.
Let be the three-dimensional vector space defined over the field . The projective plane consists of the one-dimensional vector subspaces of , called points, and the two-dimensional vector subspaces of , called lines. Incidence of a point and a line is given by containment of the one-dimensional subspace in the two-dimensional subspace.
Fix a basis f
|
https://en.wikipedia.org/wiki/Mellin%20transform
|
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is
often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions.
The Mellin transform of a function is
The inverse transform is
The notation implies this is a line integral taken over a vertical line in the complex plane, whose real part c need only satisfy a mild lower bound. Conditions under which this inversion is valid are given in the Mellin inversion theorem.
The transform is named after the Finnish mathematician Hjalmar Mellin, who introduced it in a paper published 1897 in Acta Societatis Scientiarum Fennicæ.
Relationship to other transforms
The two-sided Laplace transform may be defined in terms of the Mellin transform by
and conversely we can get the Mellin transform from the two-sided Laplace transform by
The Mellin transform may be thought of as integrating using a kernel xs with respect to the multiplicative Haar measure,
, which is invariant
under dilation , so that
the two-sided Laplace transform integrates with respect to the additive Haar measure , which is translation invariant, so that .
We also may define the Fourier transform in terms of the Mellin transform and vice versa; in terms of the Mellin transform and of the two-sided Laplace transform defined above
We may also reverse the process and obtain
The Mellin transform also connects the Newton series or binomial transform together with the Poisson generating function, by means of the Poisson–Mellin–Newton cycle.
The Mellin transform may also be viewed as the Gelfand transform for the convolution algebra of the locally compact abelian group of positive real numbers with multiplication.
Examples
Cahen–Mellin integral
The Mellin transform of the function is
where is the gamma function. is a meromorphic function with simple poles at . Therefore, is analytic for . Thus, letting and on the principal branch, the inverse transform gives
.
This integral is known as the Cahen–Mellin integral.
Polynomial functions
Since is not convergent for any value of , the Mellin transform is not defined for polynomial functions defined on the whole positive real axis. However, by defining it to be zero on different sections of the real axis, it is possible to take the Mellin transform. For example, if
then
Thus has a simple pole at and is thus defined for . Similarly, if
then
Thus has a simple pole at and is thus defined for .
Exponential functions
For , let . Then
Zeta function
It is possible to use the Mellin transform to produce one of the fundamental formulas for the Riemann zeta function, . Let . Then
Thus,
Generalized Gaussian
For , let (i.
|
https://en.wikipedia.org/wiki/Edward%20Nelson
|
Edward Nelson (May 4, 1932 – September 10, 2014) was an American mathematician. He was professor in the Mathematics Department at Princeton University. He was known for his work on mathematical physics and mathematical logic. In mathematical logic, he was noted especially for his internal set theory, and views on ultrafinitism and the consistency of arithmetic. In philosophy of mathematics he advocated the view of formalism rather than platonism or intuitionism. He also wrote on the relationship between religion and mathematics.
Biography
Edward Nelson was born in Decatur, Georgia, in 1932. He spent his early childhood in Rome where his father worked for the Italian YMCA. At the advent of World War II, Nelson moved with his mother to New York City, where he attended high school at the Bronx High School of Science. His father, who spoke fluent Russian, stayed in St. Petersburg in connection with issues related to prisoners of war. After the war, his family returned to Italy and he attended the Liceo Scientifico Giovanni Verga in Rome.
He received his Ph.D. in 1955 from the University of Chicago, where he worked with Irving Segal. He was a member of the Institute for Advanced Study from 1956 to 1959. He held a position at Princeton University starting in 1959, attaining the rank of professor there in 1964 and retiring in 2013.
In 2012 he became a fellow of the American Mathematical Society. He died in Princeton, New Jersey, on September 10, 2014.
Academic work
Stochastic quantum mechanics
Nelson made contributions to the theory of infinite-dimensional group representations, the mathematical treatment of quantum field theory, the use of stochastic processes in quantum mechanics, and the reformulation of probability theory in terms of non-standard analysis. For many years he worked on mathematical physics and probability theory, and he retained a residual interest in these fields, particularly in connection with possible extensions of stochastic mechanics to field theory.
Four color problem
In 1950, Nelson formulated a popular variant of the four color problem: What is the chromatic number, denoted , of the plane? In more detail, what is the smallest number of colors sufficient for coloring the points of the Euclidean plane such that no two points of the same color are unit distance apart? We know by simple arguments that 4 ≤ χ ≤ 7. The problem was introduced to a wide mathematical audience by Martin Gardner in his October 1960 Mathematical Games column. The chromatic number problem, also now known as the Hadwiger–Nelson problem, was a favorite of Paul Erdős, who mentioned it frequently in his problems lectures. In 2018, Aubrey de Grey showed that χ ≥ 5.
Foundations of mathematics
In the later part of his career, he worked on mathematical logic and the foundations of mathematics. One of his goals was to extend IST (Internal Set Theory—a version of a portion of Abraham Robinson's non-standard analysis) in a natural manner that includes ext
|
https://en.wikipedia.org/wiki/Sierpi%C5%84ski%20space
|
In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed.
It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński.
The Sierpiński space has important relations to the theory of computation and semantics, because it is the classifying space for open sets in the Scott topology.
Definition and fundamental properties
Explicitly, the Sierpiński space is a topological space S whose underlying point set is and whose open sets are
The closed sets are
So the singleton set is closed and the set is open ( is the empty set).
The closure operator on S is determined by
A finite topological space is also uniquely determined by its specialization preorder. For the Sierpiński space this preorder is actually a partial order and given by
Topological properties
The Sierpiński space is a special case of both the finite particular point topology (with particular point 1) and the finite excluded point topology (with excluded point 0). Therefore, has many properties in common with one or both of these families.
Separation
The points 0 and 1 are topologically distinguishable in S since is an open set which contains only one of these points. Therefore, S is a Kolmogorov (T0) space.
However, S is not T1 since the point 1 is not closed. It follows that S is not Hausdorff, or Tn for any
S is not regular (or completely regular) since the point 1 and the disjoint closed set cannot be separated by neighborhoods. (Also regularity in the presence of T0 would imply Hausdorff.)
S is vacuously normal and completely normal since there are no nonempty separated sets.
S is not perfectly normal since the disjoint closed sets and cannot be precisely separated by a function. Indeed, cannot be the zero set of any continuous function since every such function is constant.
Connectedness
The Sierpiński space S is both hyperconnected (since every nonempty open set contains 1) and ultraconnected (since every nonempty closed set contains 0).
It follows that S is both connected and path connected.
A path from 0 to 1 in S is given by the function: and for The function is continuous since which is open in I.
Like all finite topological spaces, S is locally path connected.
The Sierpiński space is contractible, so the fundamental group of S is trivial (as are all the higher homotopy groups).
Compactness
Like all finite topological spaces, the Sierpiński space is both compact and second-countable.
The compact subset of S is not closed showing that compact subsets of T0 spaces need not be closed.
Every open cover of S must contain S itself since S is the only open neighborhood of 0. Therefore, every open cover of S has an open subcover consisting of a single set:
It follows that S is fully normal.
Convergence
Every sequence in S converges to the point 0. This is because the only neighborhood of 0 is S itself.
|
https://en.wikipedia.org/wiki/Separation%20of%20variables
|
In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation.
Ordinary differential equations (ODE)
A differential equation for the unknown will be separable if it can be written in the form
where and are given functions. This is perhaps more transparent when written using as:
So now as long as h(y) ≠ 0, we can rearrange terms to obtain:
where the two variables x and y have been separated. Note dx (and dy) can be viewed, at a simple level, as just a convenient notation, which provides a handy mnemonic aid for assisting with manipulations. A formal definition of dx as a differential (infinitesimal) is somewhat advanced.
Alternative notation
Those who dislike Leibniz's notation may prefer to write this as
but that fails to make it quite as obvious why this is called "separation of variables". Integrating both sides of the equation with respect to , we have
or equivalently,
because of the substitution rule for integrals.
If one can evaluate the two integrals, one can find a solution to the differential equation. Observe that this process effectively allows us to treat the derivative as a fraction which can be separated. This allows us to solve separable differential equations more conveniently, as demonstrated in the example below.
(Note that we do not need to use two constants of integration, in equation () as in
because a single constant is equivalent.)
Example
Population growth is often modeled by the "logistic" differential equation
where is the population with respect to time , is the rate of growth, and is the carrying capacity of the environment.
Separation of variables now leads to
which is readily integrated using partial fractions on the left side yielding
where A is the constant of integration. We can find in terms of at t=0. Noting we get
Generalization of separable ODEs to the nth order
Much like one can speak of a separable first-order ODE, one can speak of a separable second-order, third-order or nth-order ODE. Consider the separable first-order ODE:
The derivative can alternatively be written the following way to underscore that it is an operator working on the unknown function, y:
Thus, when one separates variables for first-order equations, one in fact moves the dx denominator of the operator to the side with the x variable, and the d(y) is left on the side with the y variable. The second-derivative operator, by analogy, breaks down as follows:
The third-, fourth- and nth-derivative operators break down in the same way. Thus, much like a first-order separable ODE is reducible to the form
a separable second-order ODE is reducible to the form
and an nth-order separable ODE is reducible to
Example
Consider the simple nonlinear second-order differential equation:This
|
https://en.wikipedia.org/wiki/Outline%20of%20probability
|
Probability is a measure of the likeliness that an event will occur. Probability is used to quantify an attitude of mind towards some proposition whose truth is not certain. The proposition of interest is usually of the form "A specific event will occur." The attitude of mind is of the form "How certain is it that the event will occur?" The certainty that is adopted can be described in terms of a numerical measure, and this number, between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty) is called the probability. Probability theory is used extensively in statistics, mathematics, science and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
Introduction
Probability and randomness.
Basic probability
(Related topics: set theory, simple theorems in the algebra of sets)
Events
Events in probability theory
Elementary events, sample spaces, Venn diagrams
Mutual exclusivity
Elementary probability
The axioms of probability
Boole's inequality
Meaning of probability
Probability interpretations
Bayesian probability
Frequency probability
Calculating with probabilities
Conditional probability
The law of total probability
Bayes' theorem
Independence
Independence (probability theory)
Probability theory
(Related topics: measure theory)
Measure-theoretic probability
Sample spaces, σ-algebras and probability measures
Probability space
Sample space
Standard probability space
Random element
Random compact set
Dynkin system
Probability axioms
Event (probability theory)
Complementary event
Elementary event
"Almost surely"
Independence
Independence (probability theory)
The Borel–Cantelli lemmas and Kolmogorov's zero–one law
Conditional probability
Conditional probability
Conditioning (probability)
Conditional expectation
Conditional probability distribution
Regular conditional probability
Disintegration theorem
Bayes' theorem
Rule of succession
Conditional independence
Conditional event algebra
Goodman–Nguyen–van Fraassen algebra
Random variables
Discrete and continuous random variables
Discrete random variables: Probability mass functions
Continuous random variables: Probability density functions
Normalizing constants
Cumulative distribution functions
Joint, marginal and conditional distributions
Expectation
Expectation (or mean), variance and covariance
Jensen's inequality
General moments about the mean
Correlated and uncorrelated random variables
Conditional expectation:
law of total expectation, law of total variance
Fatou's lemma and the monotone and dominated convergence theorems
Markov's inequality and Chebyshev's inequality
Independence
Independent random variables
Some common distributions
Discrete:
constant (see also degenerate distribution),
Bernoulli and binomial,
negative binomial,
(discrete) uniform,
geometric,
Poisson, and
hypergeometric.
Continuous:
(continuous) uniform,
exponential,
ga
|
https://en.wikipedia.org/wiki/Cantor%20distribution
|
The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function.
This distribution has neither a probability density function nor a probability mass function, since although its cumulative distribution function is a continuous function, the distribution is not absolutely continuous with respect to Lebesgue measure, nor does it have any point-masses. It is thus neither a discrete nor an absolutely continuous probability distribution, nor is it a mixture of these. Rather it is an example of a singular distribution.
Its cumulative distribution function is continuous everywhere but horizontal almost everywhere, so is sometimes referred to as the Devil's staircase, although that term has a more general meaning.
Characterization
The support of the Cantor distribution is the Cantor set, itself the intersection of the (countably infinitely many) sets:
The Cantor distribution is the unique probability distribution for which for any Ct (t ∈ { 0, 1, 2, 3, ... }), the probability of a particular interval in Ct containing the Cantor-distributed random variable is identically 2−t on each one of the 2t intervals.
Moments
It is easy to see by symmetry and being bounded that for a random variable X having this distribution, its expected value E(X) = 1/2, and that all odd central moments of X are 0.
The law of total variance can be used to find the variance var(X), as follows. For the above set C1, let Y = 0 if X ∈ [0,1/3], and 1 if X ∈ [2/3,1]. Then:
From this we get:
A closed-form expression for any even central moment can be found by first obtaining the even cumulants
where B2n is the 2nth Bernoulli number, and then expressing the moments as functions of the cumulants.
References
Further reading
This, as with other standard texts, has the Cantor function and its one sided derivates.
This is more modern than the other texts in this reference list.
This has more advanced material on fractals.
Continuous distributions
Georg Cantor
|
https://en.wikipedia.org/wiki/Cubic
|
Cubic may refer to:
Science and mathematics
Cube (algebra), "cubic" measurement
Cube, a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex
Cubic crystal system, a crystal system where the unit cell is in the shape of a cube
Cubic function, a polynomial function of degree three
Cubic equation, a polynomial equation (reducible to ax3 + bx2 + cx + d = 0)
Cubic form, a homogeneous polynomial of degree 3
Cubic graph (mathematics - graph theory), a graph where all vertices have degree 3
Cubic plane curve (mathematics), a plane algebraic curve C defined by a cubic equation
Cubic reciprocity (mathematics - number theory), a theorem analogous to quadratic reciprocity
Cubic surface, an algebraic surface in three-dimensional space
Cubic zirconia, in geology, a mineral that is widely synthesized for use as a diamond simulacra
CUBIC, a histology method
Computing
Cubic IDE, a modular development environment
CUBIC TCP, a TCP congestion-avoidance strategy
Media
Cubic (film), a 2002 science-fiction film also known as Equilibrium
Cubic (TV series), a 2014 Thai soap opera
Other
Cubic Corporation, an American company that provides transportation and defense systems
Cubic Transportation Systems, a division of Cubic Corporation
Cubic (river), a tributary of the Ier in northwestern Romania
See also
Cube (disambiguation)
Cubicle, a small area set off by walls for special use, such as a place to work, to shower, or with a toilet
Quadratic, relating to degree 2, as next lower below cubic
Quartic, relating to degree 4, as next higher above cubic
Mathematics disambiguation pages
|
https://en.wikipedia.org/wiki/Bewick%20Bridge
|
Bewick Bridge (1767, Linton, Cambridgeshire – 15 May 1833, Cherry Hinton) was an English vicar and mathematical author.
In 1786, he was admitted as a sizar to study mathematics Peterhouse, Cambridge University, where he graduated as senior wrangler and won the Smith's Prize in 1790.
In October 1790, he was ordained a deacon at Ely, and became a priest in 1792; in the same year he became a Fellow at Peterhouse, during which he spent time as both as college moderator and as proctor. From 1806 until 1816, he was Professor of Mathematics at the East India Company College, Haileybury. He wrote a number of mathematical texts: his Algebra achieved international circulation. He became a Fellow of the Royal Society in 1812.
From 1816 until 1833, he was vicar of Cherry Hinton in Cambridge, where in 1818 he built the vicarage, and he founded the village school in 1832 (now a Church of England PrimarySchool). He died on 15 May 1833, aged 66. In September 2011 the Cherry Hinton Community Junior School was named after Bewick, becoming Bewick Bridge Community Primary School.
References
1767 births
1833 deaths
People from Linton, Cambridgeshire
Alumni of Peterhouse, Cambridge
Fellows of Peterhouse, Cambridge
Fellows of the Royal Society
19th-century English mathematicians
Senior Wranglers
18th-century English Anglican priests
19th-century English Anglican priests
People from Cherry Hinton
|
https://en.wikipedia.org/wiki/Zipf%E2%80%93Mandelbrot%20law
|
In probability theory and statistics, the Zipf–Mandelbrot law is a discrete probability distribution. Also known as the Pareto–Zipf law, it is a power-law distribution on ranked data, named after the linguist George Kingsley Zipf who suggested a simpler distribution called Zipf's law, and the mathematician Benoit Mandelbrot, who subsequently generalized it.
The probability mass function is given by:
where is given by:
which may be thought of as a generalization of a harmonic number. In the formula, is the rank of the data, and and are parameters of the distribution. In the limit as approaches infinity, this becomes the Hurwitz zeta function . For finite and the Zipf–Mandelbrot law becomes Zipf's law. For infinite and it becomes a Zeta distribution.
Applications
The distribution of words ranked by their frequency in a random text corpus is approximated by a power-law distribution, known as Zipf's law.
If one plots the frequency rank of words contained in a moderately sized corpus of text data versus the number of occurrences or actual frequencies, one obtains a power-law distribution, with exponent close to one (but see Powers, 1998 and Gelbukh & Sidorov, 2001). Zipf's law implicitly assumes a fixed vocabulary size, but the Harmonic series with s=1 does not converge, while the Zipf–Mandelbrot generalization with s>1 does. Furthermore, there is evidence that the closed class of functional words that define a language obeys a Zipf–Mandelbrot distribution with different parameters from the open classes of contentive words that vary by topic, field and register.
In ecological field studies, the relative abundance distribution (i.e. the graph of the number of species observed as a function of their abundance) is often found to conform to a Zipf–Mandelbrot law.
Within music, many metrics of measuring "pleasing" music conform to Zipf–Mandelbrot distributions.
Notes
References
Reprinted as
Van Droogenbroeck F.J., 'An essential rephrasing of the Zipf–Mandelbrot law to solve authorship attribution applications by Gaussian statistics' (2019)
External links
Z. K. Silagadze: Citations and the Zipf–Mandelbrot's law
NIST: Zipf's law
W. Li's References on Zipf's law
Gelbukh & Sidorov, 2001: Zipf and Heaps Laws’ Coefficients Depend on Language
C++ Library for generating random Zipf–Mandelbrot deviates.
Discrete distributions
Power laws
Computational linguistics
Quantitative linguistics
Corpus linguistics
|
https://en.wikipedia.org/wiki/Legendre%20transformation
|
In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued functions that are convex on a real variable. Specifically, if a real-valued multivariable function is convex on one of its independent real variables, then the Legendre transform with respect to this variable is applicable to the function. In physical problems, it is used to convert functions of one quantity (such as position, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively). In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism (or vice versa) and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables.
For sufficiently smooth functions on the real line, the Legendre transform of a function can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other. This can be expressed in Euler's derivative notation as
where is an operator of differentiation, represents an argument or input to the associated function, is an inverse function such that ,
or equivalently, as and in Lagrange's notation.
The generalization of the Legendre transformation to affine spaces and non-convex functions is known as the convex conjugate (also called the Legendre–Fenchel transformation), which can be used to construct a function's convex hull.
Definition
Let be an interval, and a convex function; then the Legendre transform of is the function defined by
where denotes the supremum over , e.g., in is chosen such that is maximized at each , or is such that as a bounded value throughout exists (e.g., when is a linear function).
The transform is always well-defined when is convex. This definition requires to be bounded from above in in order for the supremum to exist.
The generalization to convex functions on a convex set is straightforward: has domain
and is defined by
where denotes the dot product of and .
The function is called the convex conjugate function of . For historical reasons (rooted in analytic mechanics), the conjugate variable is often denoted , instead of . If the convex function is defined on the whole line and is everywhere differentiable, then
can be interpreted as the negative of the -intercept of the tangent line to the graph of that has slope .
The Legendre transformation is an application of the duality relationship between points and lines. The functional relationship specified by can be represented equally well as a set of points, or as a set of tangent lines specified by their slope and intercept values.
Understanding the Legendre transform in terms of derivatives
For a differentiable convex function on the real line
|
https://en.wikipedia.org/wiki/Fano%20plane
|
In finite geometry, the Fano plane (after Gino Fano) is a finite projective plane with the smallest possible number of points and lines: 7 points and 7 lines, with 3 points on every line and 3 lines through every point. These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements. The standard notation for this plane, as a member of a family of projective spaces, is . Here stands for "projective geometry", the first parameter is the geometric dimension (it is a plane, of dimension 2) and the second parameter is the order (the number of points per line, minus one).
The Fano plane is an example of a finite incidence structure, so many of its properties can be established using combinatorial techniques and other tools used in the study of incidence geometries. Since it is a projective space, algebraic techniques can also be effective tools in its study.
Homogeneous coordinates
The Fano plane can be constructed via linear algebra as the projective plane over the finite field with two elements. One can similarly construct projective planes over any other finite field, with the Fano plane being the smallest.
Using the standard construction of projective spaces via homogeneous coordinates, the seven points of the Fano plane may be labeled with the seven non-zero ordered triples of binary digits 001, 010, 011, 100, 101, 110, and 111. This can be done in such a way that for every two points p and q, the third point on line pq has the label formed by adding the labels of p and q modulo 2 digit by digit (e.g., 010 and 111 resulting in 101). In other words, the points of the Fano plane correspond to the non-zero points of the finite vector space of dimension 3 over the finite field of order 2.
Due to this construction, the Fano plane is considered to be a Desarguesian plane, even though the plane is too small to contain a non-degenerate Desargues configuration (which requires 10 points and 10 lines).
The lines of the Fano plane may also be given homogeneous coordinates, again using non-zero triples of binary digits. With this system of coordinates, a point is incident to a line if the coordinate for the point and the coordinate for the line have an even number of positions at which they both have nonzero bits: for instance, the point 101 belongs to the line 111, because they have nonzero bits at two common positions. In terms of the underlying linear algebra, a point belongs to a line if the inner product of the vectors representing the point and line is zero.
The lines can be classified into three types.
On three of the lines the binary triples for the points have the 0 in a constant position: the line 100 (containing the points 001, 010, and 011) has 0 in the first position, and the lines 010 and 001 are formed in the same way.
On three of the lines, two of the positions in the binary triples of each point have the same value: in the line 110 (co
|
https://en.wikipedia.org/wiki/PSL%282%2C7%29
|
In mathematics, the projective special linear group , isomorphic to , is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements, PSL(2, 7) is the smallest nonabelian simple group after the alternating group A5 with 60 elements, isomorphic to .
Definition
The general linear group consists of all invertible 2×2 matrices over F7, the finite field with 7 elements. These have nonzero determinant. The subgroup consists of all such matrices with unit determinant. Then is defined to be the quotient group
SL(2, 7) / {I, −I}
obtained by identifying I and −I, where I is the identity matrix. In this article, we let G denote any group isomorphic to .
Properties
G = has 168 elements. This can be seen by counting the possible columns; there are possibilities for the first column, then possibilities for the second column. We must divide by to force the determinant equal to one, and then we must divide by 2 when we identify I and −I. The result is .
It is a general result that is simple for (q being some power of a prime number), unless or . is isomorphic to the symmetric group S3, and is isomorphic to alternating group A4. In fact, is the second smallest nonabelian simple group, after the alternating group .
The number of conjugacy classes and irreducible representations is 6. The sizes of conjugacy classes are 1, 21, 42, 56, 24, 24. The dimensions of irreducible representations 1, 3, 3, 6, 7, 8.
Character table
where:
The following table describes the conjugacy classes in terms of the order of an element in the class, the size of the class, the minimum polynomial of every representative in GL(3, 2), and the function notation for a representative in PSL(2, 7). Note that the classes 7A and 7B are exchanged by an automorphism, so the representatives from GL(3, 2) and PSL(2, 7) can be switched arbitrarily.
The order of group is , this implies existence of Sylow's subgroups of orders 3, 7 and 8. It is easy to describe the first two, they are cyclic, since any group of prime order is cyclic. Any element of conjugacy class 3A56 generates Sylow 3-subgroup. Any element from the conjugacy classes 7A24, 7B24 generates the Sylow 7-subgroup. The Sylow 2-subgroup is a dihedral group of order 8. It can be described as centralizer of any element from the conjugacy class 2A21. In the representation, a Sylow 2-subgroup consists of the upper triangular matrices.
This group and its Sylow 2-subgroup provide a counter-example for various normal p-complement theorems for .
Actions on projective spaces
G = acts via linear fractional transformation on the projective line P1(7) over the field with 7 elements:
Every orientation-preserving automorphism of P1(7) arises in this way, and so can be thought of geometrically as a group of symmetries of the projective line P1(7); the full group of possibly
|
https://en.wikipedia.org/wiki/K%C3%A4hler%20manifold
|
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil. Kähler geometry refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics.
Every smooth complex projective variety is a Kähler manifold. Hodge theory is a central part of algebraic geometry, proved using Kähler metrics.
Definitions
Since Kähler manifolds are equipped with several compatible structures, they can be described from different points of view:
Symplectic viewpoint
A Kähler manifold is a symplectic manifold equipped with an integrable almost-complex structure J which is compatible with the symplectic form ω, meaning that the bilinear form
on the tangent space of X at each point is symmetric and positive definite (and hence a Riemannian metric on X).
Complex viewpoint
A Kähler manifold is a complex manifold X with a Hermitian metric h whose associated 2-form ω is closed. In more detail, h gives a positive definite Hermitian form on the tangent space TX at each point of X, and the 2-form ω is defined by
for tangent vectors u and v (where i is the complex number ). For a Kähler manifold X, the Kähler form ω is a real closed (1,1)-form. A Kähler manifold can also be viewed as a Riemannian manifold, with the Riemannian metric g defined by
Equivalently, a Kähler manifold X is a Hermitian manifold of complex dimension n such that for every point p of X, there is a holomorphic coordinate chart around p in which the metric agrees with the standard metric on Cn to order 2 near p. That is, if the chart takes p to 0 in Cn, and the metric is written in these coordinates as , then
for all a, b in
Since the 2-form ω is closed, it determines an element in de Rham cohomology , known as the Kähler class.
Riemannian viewpoint
A Kähler manifold is a Riemannian manifold X of even dimension 2n whose holonomy group is contained in the unitary group U(n). Equivalently, there is a complex structure J on the tangent space of X at each point (that is, a real linear map from TX to itself with ) such that J preserves the metric g (meaning that ) and J is preserved by parallel transport.
Kähler potential
A smooth real-valued function ρ on a complex manifold is called strictly plurisubharmonic if the real closed (1,1)-form
is positive, that is, a Kähler form. Here are the Dolbeault operators. The function ρ is called a Kähler potential for ω.
Conversely, by the complex version of the Poincaré lemma, known as the local -lemma, every Kähler met
|
https://en.wikipedia.org/wiki/Isolated%20point
|
In mathematics, a point is called an isolated point of a subset (in a topological space ) if is an element of and there exists a neighborhood of that does not contain any other points of . This is equivalent to saying that the singleton is an open set in the topological space (considered as a subspace of ). Another equivalent formulation is: an element of is an isolated point of if and only if it is not a limit point of .
If the space is a metric space, for example a Euclidean space, then an element of is an isolated point of if there exists an open ball around that contains only finitely many elements of .
A point set that is made up only of isolated points is called a discrete set or discrete point set (see also discrete space).
Related notions
Any discrete subset of Euclidean space must be countable, since the isolation of each of its points together with the fact that rationals are dense in the reals means that the points of may be mapped injectively onto a set of points with rational coordinates, of which there are only countably many. However, not every countable set is discrete, of which the rational numbers under the usual Euclidean metric are the canonical example.
A set with no isolated point is said to be dense-in-itself (every neighbourhood of a point contains other points of the set). A closed set with no isolated point is called a perfect set (it contains all its limit points and no isolated points).
The number of isolated points is a topological invariant, i.e. if two topological spaces are homeomorphic, the number of isolated points in each is equal.
Examples
Standard examples
Topological spaces in the following three examples are considered as subspaces of the real line with the standard topology.
For the set the point 0 is an isolated point.
For the set each of the points is an isolated point, but is not an isolated point because there are other points in as close to as desired.
The set of natural numbers is a discrete set.
In the topological space with topology the element is an isolated point, even though belongs to the closure of (and is therefore, in some sense, "close" to ). Such a situation is not possible in a Hausdorff space.
The Morse lemma states that non-degenerate critical points of certain functions are isolated.
Two counter-intuitive examples
Consider the set of points in the real interval such that every digit of their binary representation fulfills the following conditions:
Either or
only for finitely many indices .
If denotes the largest index such that then
If and then exactly one of the following two conditions holds: or
Informally, these conditions means that every digit of the binary representation of that equals 1 belongs to a pair ...0110..., except for ...010... at the very end.
Now, is an explicit set consisting entirely of isolated points but has the counter-intuitive property that its closure is an uncountable set.
Another set with
|
https://en.wikipedia.org/wiki/Flag%20%28linear%20algebra%29
|
In mathematics, particularly in linear algebra, a flag is an increasing sequence of subspaces of a finite-dimensional vector space V. Here "increasing" means each is a proper subspace of the next (see filtration):
The term flag is motivated by a particular example resembling a flag: the zero point, a line, and a plane correspond to a nail, a staff, and a sheet of fabric.
If we write that dimVi = di then we have
where n is the dimension of V (assumed to be finite). Hence, we must have k ≤ n. A flag is called a complete flag if di = i for all i, otherwise it is called a partial flag.
A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.
The signature of the flag is the sequence (d1, ..., dk).
Bases
An ordered basis for V is said to be adapted to a flag V0 ⊂ V1 ⊂ ... ⊂ Vk if the first di basis vectors form a basis for Vi for each 0 ≤ i ≤ k. Standard arguments from linear algebra can show that any flag has an adapted basis.
Any ordered basis gives rise to a complete flag by letting the Vi be the span of the first i basis vectors. For example, the in Rn is induced from the standard basis (e1, ..., en) where ei denotes the vector with a 1 in the ith entry and 0's elsewhere. Concretely, the standard flag is the sequence of subspaces:
An adapted basis is almost never unique (the counterexamples are trivial); see below.
A complete flag on an inner product space has an essentially unique orthonormal basis: it is unique up to multiplying each vector by a unit (scalar of unit length, e.g. 1, −1, i). Such a basis can be constructed using the Gram-Schmidt process. The uniqueness up to units follows inductively, by noting that lies in the one-dimensional space .
More abstractly, it is unique up to an action of the maximal torus: the flag corresponds to the Borel group, and the inner product corresponds to the maximal compact subgroup.
Stabilizer
The stabilizer subgroup of the standard flag is the group of invertible upper triangular matrices.
More generally, the stabilizer of a flag (the linear operators on V such that for all i) is, in matrix terms, the algebra of block upper triangular matrices (with respect to an adapted basis), where the block sizes . The stabilizer subgroup of a complete flag is the set of invertible upper triangular matrices with respect to any basis adapted to the flag. The subgroup of lower triangular matrices with respect to such a basis depends on that basis, and can therefore not be characterized in terms of the flag only.
The stabilizer subgroup of any complete flag is a Borel subgroup (of the general linear group), and the stabilizer of any partial flags is a parabolic subgroup.
The stabilizer subgroup of a flag acts simply transitively on adapted bases for the flag, and thus these are not unique unless the stabilizer is trivial. That is a very exceptional circumstance: it happen
|
https://en.wikipedia.org/wiki/Klein%20quartic
|
In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms if orientation may be reversed. As such, the Klein quartic is the Hurwitz surface of lowest possible genus; see Hurwitz's automorphisms theorem. Its (orientation-preserving) automorphism group is isomorphic to , the second-smallest non-abelian simple group after the alternating group A5. The quartic was first described in .
Klein's quartic occurs in many branches of mathematics, in contexts including representation theory, homology theory, octonion multiplication, Fermat's Last Theorem, and the Stark–Heegner theorem on imaginary quadratic number fields of class number one; see for a survey of properties.
Originally, the "Klein quartic" referred specifically to the subset of the complex projective plane defined by an algebraic equation. This has a specific Riemannian metric (that makes it a minimal surface in ), under which its Gaussian curvature is not constant. But more commonly (as in this article) it is now thought of as any Riemann surface that is conformally equivalent to this algebraic curve, and especially the one that is a quotient of the hyperbolic plane by a certain cocompact group that acts freely on by isometries. This gives the Klein quartic a Riemannian metric of constant curvature that it inherits from . This set of conformally equivalent Riemannian surfaces is precisely the same as all compact Riemannian surfaces of genus 3 whose conformal automorphism group is isomorphic to the unique simple group of order 168. This group is also known as , and also as the isomorphic group . By covering space theory, the group mentioned above is isomorphic to the fundamental group of the compact surface of genus .
Closed and open forms
It is important to distinguish two different forms of the quartic. The closed quartic is what is generally meant in geometry; topologically it has genus 3 and is a compact space. The open or "punctured" quartic is of interest in number theory; topologically it is a genus 3 surface with 24 punctures, and geometrically these punctures are cusps. The open quartic may be obtained (topologically) from the closed quartic by puncturing at the 24 centers of the tiling by regular heptagons, as discussed below. The open and closed quartics have different metrics, though they are both hyperbolic and complete – geometrically, the cusps are "points at infinity", not holes, hence the open quartic is still complete.
As an algebraic curve
The Klein quartic can be viewed as a projective algebraic curve over the complex numbers , defined by the following quartic equation in homogeneous coordinates on :
The locus of this equation in is the original Riemannian surface that Klein described.
Quaternion algebra construction
The compact Klein quartic can be constructed as the quot
|
https://en.wikipedia.org/wiki/Stark%E2%80%93Heegner%20theorem
|
In number theory, the Baker–Heegner–Stark theorem establishes the complete list of the quadratic imaginary number fields whose rings of integers are unique factorization domains. It solves a special case of Gauss's class number problem of determining the number of imaginary quadratic fields that have a given fixed class number.
Let denote the set of rational numbers, and let be a square-free integer. The field is a quadratic extension of . The class number of is one if and only if the ring of integers of is a principal ideal domain (or, equivalently, a unique factorization domain). The Baker–Heegner–Stark theorem can then be stated as follows:
If , then the class number of is one if and only if
These are known as the Heegner numbers.
By replacing with the discriminant of this list is often written as:
History
This result was first conjectured by Gauss in Section 303 of his Disquisitiones Arithmeticae (1798). It was essentially proven by Kurt Heegner in 1952, but Heegner's proof had some minor gaps and the theorem was not accepted until Harold Stark gave a complete proof in 1967, which had many commonalities to Heegner's work, but sufficiently many differences that Stark considers the proofs to be different. Heegner "died before anyone really understood what he had done". Stark formally filled in the gap in Heegner's proof in 1969 (other contemporary papers produced various similar proofs by modular functions, but Stark concentrated on explicitly filling Heegner's gap).
Alan Baker gave a completely different proof slightly earlier (1966) than Stark's work (or more precisely Baker reduced the result to a finite amount of computation, with Stark's work in his 1963/4 thesis already providing this computation), and won the Fields Medal for his methods. Stark later pointed out that Baker's proof, involving linear forms in 3 logarithms, could be reduced to only 2 logarithms, when the result was already known from 1949 by Gelfond and Linnik.
Stark's 1969 paper also cited the 1895 text by Heinrich Martin Weber and noted that if Weber had "only made the observation that the reducibility of [a certain equation] would lead to a Diophantine equation, the class-number one problem would have been solved 60 years ago". Bryan Birch notes that Weber's book, and essentially the whole field of modular functions, dropped out of interest for half a century: "Unhappily, in 1952 there was no one left who was sufficiently expert in Weber's Algebra to appreciate Heegner's achievement."
Deuring, Siegel, and Chowla all gave slightly variant proofs by modular functions in the immediate years after Stark. Other versions in this genre have also cropped up over the years. For instance, in 1985, Monsur Kenku gave a proof using the Klein quartic (though again utilizing modular functions). And again, in 1999, Imin Chen gave another variant proof by modular functions (following Siegel's outline).
The work of Gross and Zagier (1986) combined with that of Goldfe
|
https://en.wikipedia.org/wiki/Fluxion%20%28disambiguation%29
|
A fluxion is a mathematical concept, first formulated by Isaac Newton.
Fluxion may also refer to:
Newton's method for solving an equation
Method of Fluxions, Newton's book on differential calculus
An alternate spelling of fluxon, a quantum of magnetic flux
Fluxion (electronic musician), real name Konstantinos Soublis
Fluxion (album), The Ocean Collective's first LP
|
https://en.wikipedia.org/wiki/Full%20Circle
|
Full Circle may refer to:
Geometry
Full circle (unit), a unit of plane angle
Books
Full Circle, a 1962 novel by Grace Lumpkin
Full Circle, a 1982 memoir by Janet Baker
Full Circle (novel), a 1984 novel by Danielle Steel
Full Circle: The Moral Force of Unified Science, a 1972 book co-written and edited by Edward Haskell
Full Circle (travel book), a 1997 companion book to the travel TV series, by Michael Palin
Batman: Full Circle, a comic book
Full Circle Magazine, a free-distribution on-line Ubuntu PDF magazine launched in 2007
Film, TV and theatre
Full Circle (1935 film), a British film starring Garry Marsh
Full Circle (1977 film) (also known as The Haunting of Julia), starring Mia Farrow
Full Circle (1993 film), a Canadian documentary
Full Circle (1996 film), a Cirque du Soleil documentary
Full Circle (2008 film), a documentary about an Israeli submarine
Full Circle, a 1996 TV film adapted from the novel Full Circle by Danielle Steel
Full Circle (2012 film), a Chinese film directed by Zhang Yang
Television
Full Circle (Doctor Who), a 1980 TV serial
"Full Circle" (Stargate SG-1), a 2002 TV episode
Full Circle (1960 TV series), a short-lived daytime TV serial starring Dyan Cannon and Jean Byron
Full Circle (2013 TV series), a TV series by Audience Network
"Full Circle", a Hercules: The Legendary Journeys episode
"Full Circle", a Princess Gwenevere and the Jewel Riders episode
"The Full Circle", an episode of the TV series Space: 1999
Full Circle with Michael Palin, a 1997 travel TV series
ESPN Full Circle, a televised sports event
Full Circle (miniseries), an American television miniseries
Plays
Full Circle (Melville play), a 1953 play by Alan Melville
Full Circle, a 1956 play by Erich Maria Remarque
Full Circle, a 1998 play by Charles L. Mee
Music
Full circle ringing, a method of hanging (church) bells and ringing them in the "English tradition"
Full Circle (group), a hip hop combined supergroup of Halal Gang and Prime Boys
Albums
Full Circle (Loretta Lynn album), 2016 album by Loretta Lynn
Full Circle (David Benoit album), 2006 album by David Benoit
Full Circle (Boyz II Men album), 2002 album by Boyz II Men
Full Circle (Chillinit album), 2020 album by Chillinit
Full Circle (Creed album), 2009 album by Creed
Full Circle (CTA album), 2007 album by CTA
Full Circle (Holger Czukay, Jah Wobble and Jaki Liebezeit album), 1982 album by Holger Czukay, Jah Wobble and Jaki Liebezeit
Full Circle (Barbara Dickson album), 2004 album by Barbara Dickson
Full Circle (The Doors album), 1972 album by The Doors
Full Circle (Dixie Dregs album), 1994 album by Dixie Dregs
Full Circle (Drowning Pool album), 2007 album by Drowning Pool
Full Circle (Eddie Palmieri album), 2018 album by Eddie Palmieri
Full Circle (FireHouse album), 2011 album by FireHouse
Full Circle (Hieroglyphics album), 2003 album and title track by Hieroglyphics
Full Circle (Icehouse album), 1994 album by Icehouse
Full Circle (Waylon Jennings album)
|
https://en.wikipedia.org/wiki/Parametric
|
Parametric may refer to:
Mathematics
Parametric equation, a representation of a curve through equations, as functions of a variable
Parametric statistics, a branch of statistics that assumes data has come from a type of probability distribution
Parametric derivative, a type of derivative in calculus
Parametric model, a family of distributions that can be described using a finite number of parameters
Parametric oscillator, a harmonic oscillator whose parameters oscillate in time
Parametric surface, a particular type of surface in the Euclidean space R3
Parametric family, a family of objects whose definitions depend on a set of parameters
Science
Parametric process, in optical physics, any process in which an interaction between light and matter does not change the state of the material
Spontaneous parametric down-conversion, in quantum optics, a source of entangled photon pairs and of single photons
Optical parametric amplifier, a type of laser light source that emits light of variable wavelengths
Statistical parametric mapping, a statistical technique for examining differences in brain activity recorded during functional neuroimaging
Parametric search
Financial services
Parametric contract, a financial or investment contract
Parametric insurance, insurance that agrees to make a payment upon the occurrence of a triggering event
Parametric feature based modeler, a modeler using features defined to be parametric shapes associated with attributes
Computing
Parametric polymorphism, a feature of some type systems in computer programming
Parametric animation, a computer-animation technique
Parametric Technology Corporation, an American technology company
Software parametric models, a set of related mathematical equations that incorporates variable parameters
Other uses
Parametric feature based modeler, a modeler using features defined to be parametric shapes associated with attributes
Parametric determinism, a Marxist interpretation of the course of history
Parametric equalizer, a multi-band variable equalizer
Parametric array, a nonlinear transduction mechanism
Parametric design, a design process
See also
Parameter (disambiguation)
|
https://en.wikipedia.org/wiki/Connection%20%28vector%20bundle%29
|
In mathematics, and especially differential geometry and gauge theory, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a linear connection on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a pseudo-Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.
Linear connections are also called Koszul connections after Jean-Louis Koszul, who gave an algebraic framework for describing them .
This article defines the connection on a vector bundle using a common mathematical notation which de-emphasizes coordinates. However, other notations are also regularly used: in general relativity, vector bundle computations are usually written using indexed tensors; in gauge theory, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article on metric connections (the comments made there apply to all vector bundles).
Motivation
Let be a differentiable manifold, such as Euclidean space. A vector-valued function can be viewed as a section of the trivial vector bundle One may consider a section of a general differentiable vector bundle, and it is therefore natural to ask if it is possible to differentiate a section, as a generalization of how one differentiates a function on .
The model case is to differentiate a function on Euclidean space . In this setting the derivative at a point in the direction may be defined by the standard formula
For every , this defines a new vector
When passing to a section of a vector bundle over a manifold , one encounters two key issues with this definition. Firstly, since the manifold has no linear structure, the term makes no sense on . Instead one takes a path such that and computes
However this still does not make sense, because and are elements of the distinct vector spaces and This means that subtraction of these two terms is not naturally defined.
The problem is resolved by introducing the extra structure of a connection to the vector bundle. There are at least three perspectives from which connections can be understood. When formulated precisely, all three perspectives are equivalent.
(Parallel transport) A connection can be viewed as assigning to every differentiable path a linear isomorphism for all Using this isomorphism one can transport to the fibre and then take the difference; explicit
|
https://en.wikipedia.org/wiki/Cobordism
|
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher.
The boundary of an (n + 1)-dimensional manifold W is an n-dimensional manifold ∂W that is closed, i.e., with empty boundary. In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for
piecewise linear and topological manifolds.
A cobordism between manifolds M and N is a compact manifold W whose boundary is the disjoint union of M and N, .
Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right. Cobordism is a much coarser equivalence relation than diffeomorphism or homeomorphism of manifolds, and is significantly easier to study and compute. It is not possible to classify manifolds up to diffeomorphism or homeomorphism in dimensions ≥ 4 – because the word problem for groups cannot be solved – but it is possible to classify manifolds up to cobordism. Cobordisms are central objects of study in geometric topology and algebraic topology. In geometric topology, cobordisms are intimately connected with Morse theory, and h-cobordisms are fundamental in the study of high-dimensional manifolds, namely surgery theory. In algebraic topology, cobordism theories are fundamental extraordinary cohomology theories, and categories of cobordisms are the domains of topological quantum field theories.
Definition
Manifolds
Roughly speaking, an n-dimensional manifold M is a topological space locally (i.e., near each point) homeomorphic to an open subset of Euclidean space A manifold with boundary is similar, except that a point of M is allowed to have a neighborhood that is homeomorphic to an open subset of the half-space
Those points without a neighborhood homeomorphic to an open subset of Euclidean space are the boundary points of ; the boundary of is denoted by . Finally, a closed manifold is, by definition, a compact manifold without boundary (.)
Cobordisms
An -dimensional cobordism is a quintuple consisting of an -dimensional compact differentiable manifold with boundary, ; closed -manifolds , ; and embeddings , with disjoint images such that
The terminology is usually abbreviated to . M and N are called cobordant if such a cobordism exists. All manifolds cobordant to a fixed given manifold M form the cobordism class of M.
Every closed manifold M is the boundary of the non-compact manifold M × [0, 1); for this reason we require W to be compact in the definition of cobordism. Note however that W is not required to be connected;
|
https://en.wikipedia.org/wiki/81%20%28number%29
|
81 (eighty-one) is the natural number following 80 and preceding 82.
In mathematics
81 is:
the square of 9 and the second fourth-power of a prime; 34.
with an aliquot sum of 40; within an aliquot sequence of three composite numbers (81,40,50,43,1,0) to the Prime in the 43-aliquot tree.
a perfect totient number like all powers of three.
a heptagonal number.
a centered octagonal number.
a tribonacci number.
an open meandric number.
the ninth member of the Mian-Chowla sequence.
a palindromic number in bases 8 (1218) and 26 (3326).
a Harshad number in bases 2, 3, 4, 7, 9, 10 and 13.
one of three non-trivial numbers (the other two are 1458 and 1729) which, when its digits (in decimal) are added together, produces a sum which, when multiplied by its reversed self, yields the original number:
8 + 1 = 9
9 × 9 = 81 (although this case is somewhat degenerate, as the sum has only a single digit).
The inverse of 81 is 0. recurring, missing only the digit "8" from the complete set of digits. This is an example of the general rule that, in base b,
omitting only the digit b−2.
In astronomy
Messier object M81, a magnitude 8.5 spiral galaxy in the constellation Ursa Major, also known as Bode's Galaxy, and the first of what is known as the M81 Group of galaxies
The New General Catalogue object NGC 81, a spiral galaxy in the constellation Andromeda
In other fields
Eighty-one is also:
The number of squares on a shogi playing board
The year AD 81, 81 BC, or 1981.
The atomic number of thallium
The symbolic number of the Hells Angels Motorcycle Club. 'H' and 'A' are the 8th and 1st letter of the alphabet, respectively.
The title of a short film by Stephen Burke: 81
The model number of Sinclair ZX81
The number of the department in France called Tarn
The code for international direct dial phone calls to Japan
"+81" is a song by Japanese metalcore band Crystal Lake.
One of two ISBN Group Identifiers for books published in India
The number of stanzas or chapters in the Tao Te Ching (in the most common arrangements).
The number of provinces in Turkey.
The number of prayers said in the Rosary in each night.
"The 81" is a 1965 song by Candy and the Kisses.
Artemis 81 is a 1981 BBC TV science fiction drama.
'The Eighty-One Brothers' is a Japanese fable
The number of possible divinations in the Taixuanjing
In culture
The Arabic characters for the numerals 8 and 1 are visible in the left palm of the human hand.
In China, 81 always reminds people People's Liberation Army as it was founded on August 1.
81 is used to refer to the Hells Angels Motorcycle Club, since H and A are, respectively, the 8th and 1st letters of the alphabet.
See also
List of highways numbered 81
References
Integers
|
https://en.wikipedia.org/wiki/82%20%28number%29
|
82 (eighty-two) is the natural number following 81 and preceding 83.
In mathematics
82 is:
the twenty-third semiprime and the twelfth of the form (2.q).
with an aliquot sum of 44, within an aliquot sequence of four composite numbers (82,44,40,50,43,1,0) to the Prime in the 43-aliquot tree.
a companion Pell number.
a happy number.
palindromic in bases 3 (100013), 9 (1019) and 40 (2240).
In science
The atomic number of lead.
The sixth magic number.
Messier 82, a starburst galaxy in the constellation Ursa Major.
The New General Catalogue object NGC 82, a single star in the constellation Andromeda.
In other fields
Eighty-two is also:
The model number of: Mark 82 bomb, a nonguided general-purpose bomb.
The number of the French department Tarn-et-Garonne.
The code for international direct dial phone calls to South Korea.
The ISBN Group Identifier for books published in Norway.
Title of Dennis Smith's book about firefighters, Report from Engine Co. 82.
The year AD 82, 82 BC, or 1982, stylized as '82.
The number of Trip Murphy's (Matt Dillon) car in Herbie: Fully Loaded (2005).
The second studio album by African electropop outfit Just a Band (2009).
The number (*82) to unblock your caller ID for phones that block anonymous incoming calls.
The very significant number that appears at the end of Kurt Vonnegut's book Hocus Pocus.
In sports
Both the NBA and NHL operate 82-game regular seasons.
In Major League Baseball, the number of games a team must win to secure a winning season.
References
Integers
|
https://en.wikipedia.org/wiki/83%20%28number%29
|
83 (eighty-three) is the natural number following 82 and preceding 84.
In mathematics
83 is:
the sum of three consecutive primes (23 + 29 + 31).
the sum of five consecutive primes (11 + 13 + 17 + 19 + 23).
the 23rd prime number, following 79 (of which it is also a cousin prime) and preceding 89.
a Sophie Germain prime.
a safe prime.
a Chen prime.
an Eisenstein prime with no imaginary part and real part of the form 3n − 1.
a highly cototient number.
the number of primes that are right-truncatable.
a super-prime, because 23 is prime.
In science
Chemistry
The atomic number of bismuth (Bi)
Astronomy
Messier object M83, a magnitude 8.5 spiral galaxy in the constellation Hydra, also known as the Southern Pinwheel Galaxy
The New General Catalogue object NGC 83, a magnitude 12.3 elliptical galaxy in the constellation Andromeda
In religion
Judaism
When someone reaches 83 they may celebrate a second bar mitzvah
In music
M83 is the debut album of the French electronic music group M83
83 is a song written by John Mayer in the Room for Squares album.
83 is a Quebec hip-hop group
83 is a song produced by Maximono
In film and television
Gypsy 83 is 2001 film directed by Todd Stephens
Class of '83 is 2004 film directed by Kurt E. Soderling
83 Hours 'Til Dawn is a 1990 film directed by Donald Wrye
83 is the highest UHF channel on older televisions made before the late 1970s (newer televisions only go up to channel 69, due to the frequency spectrum previously assigned to channels 70–83 in the United States being reassigned to cellular phone service there in the late 1970s-early 1980s). As an example, the television station CIVIC-TV managed by the James Woods character Max Renn in the 1983 film Videodrome was on Channel 83.
In other fields
Eighty-three is also:
The year AD 83, 83 BC, or 1983
The TI-83 series, graphing calculators from Texas Instruments
Konsept83 is a Greek graphic design team.
The model number of Bell XP-83
The number of the French department Var
The ISBN Group Identifier for books published in Poland
The eighth letter of the alphabet is H and the third letter is C, thus 83 stands for "Heil Christ," a greeting sometimes (not always) used by racist organizations that consider themselves also to be Christian. This symbology is also known to be used by many non-racist Christians and non-denominational Churches.
An emoticon based on :3 with wide-open eyes.
References
Integers
|
https://en.wikipedia.org/wiki/84%20%28number%29
|
84 (eighty-four) is the natural number following 83 and preceding 85.
In mathematics
84 is a semiperfect number, being thrice a perfect number, and the sum of the sixth pair of twin primes .
It is the third (or second) dodecahedral number, and the sum of the first seven triangular numbers (1, 3, 6, 10, 15, 21, 28, 36), which makes it the sixth tetrahedral number.
The twenty-second unique prime in decimal, with notably different digits than its preceding (and known following) terms in the same sequence, contains a total of 84 digits.
A hepteract is a seven-dimensional hypercube with 84 penteract 5-faces.
84 is the limit superior of the largest finite subgroup of the mapping class group of a genus surface divided by .
Under Hurwitz's automorphisms theorem, a smooth connected Riemann surface of genus will contain an automorphism group whose order is classically bound to .
There are 84 zero divisors in the 16-dimensional sedenions .
In astronomy
Messier object M84, a magnitude 11.0 lenticular galaxy in the constellation Virgo
The New General Catalogue object NGC 84, a single star in the constellation Andromeda
In other fields
Eighty-four is also:
The year AD 84, 84 BC, or 1984.
The number of years in the , a cycle used in the past by Celtic peoples, equal to 3 cycles of the Julian Calendar and to 4 Metonic cycles and 1 octaeteris
The atomic number of polonium
The model number of Harpoon missile
WGS 84 - The latest revision of the World Geodetic System, a fixed global reference frame for the Earth.
The house number of 84 Avenue Foch
The number of the French department Vaucluse
The code for international direct dial phone calls to Vietnam
The town of Eighty Four, Pennsylvania
The company 84 Lumber
The ISBN Group Identifier for books published in Spain
A variation of the game 42 played with two sets of dominoes.
The film 84 Charing Cross Road (1987) starring Anne Bancroft and Anthony Hopkins
KKNX Radio 84 in Eugene, Oregon
The B-Side to "Up All Night" (Take That song)
British Army term for the 84mm Carl Gustav recoilless rifle.
How many Earth years it takes Uranus to orbit the Sun once
The total number of Vertcoin to be released is 84 million “Vertcoin”
The number of former National Football League player Antonio Brown
See also
List of highways numbered 84
References
Integers
|
https://en.wikipedia.org/wiki/85%20%28number%29
|
85 (eighty-five) is the natural number following 84 and preceding 86.
In mathematics
85 is:
the product of two prime numbers (5 and 17), and is therefore a semiprime of the form (5.q) where q is prime.
specifically, the 24th Semiprime, it being the fourth of the form (5.q).
together with 86 and 87, forms the second cluster of three consecutive semiprimes; the first comprising 33, 34, 35.
with a prime aliquot sum of 23 in the short aliquot sequence (85,23,1,0).
an octahedral number.
a centered triangular number.
a centered square number.
a decagonal number.
the smallest number that can be expressed as a sum of two squares, with all squares greater than 1, in two ways, 85 = 92 + 22 = 72 + 62.
the length of the hypotenuse of four Pythagorean triangles.
a Smith number in decimal.
In astronomy
Messier object M85 is a magnitude 10.5 lenticular galaxy in the constellation Coma Berenices
NGC 85 is a galaxy in the constellation Andromeda
85 Io is a large main belt asteroid
85 Pegasi is a multiple star system in constellation of Pegasus
85 Ceti is a variable star in the constellation of Cetus
85D/Boethin is a periodic comet
In titles and names
Federalist No. 85, by Alexander Hamilton, the last of The Federalist Papers (1788)
The 85 Ways to Tie a Tie, a book by Thomas Fink and Yong Mao
85 Days: The Last Campaign of Robert Kennedy, a book by Jules Witcover
Live/1975–85, an album of live recordings by Bruce Springsteen (1986)
80–85, a compilation album by Bad Religion (1991)
Cupid & Psyche 85, an album by band Scritti Politti (1985)
45/85 was a television documentary on World War II
Minuscule 85, Papyrus 85, Lectionary 85 are early Greek manuscripts of the New Testament
"85", a 2000 rap single by YoungBloodz, from their album Against Da Grain
85°C, a Taiwanese coffee store chain.
In sports
In U.S. college athletics, schools that are members of NCAA Division I are limited to providing athletic scholarships to a maximum of 85 football players in a given season. The specifics vary by the two Division I football subdivisions:
In the top-level FBS, each player provided with a scholarship may, and almost always does, receive a full scholarship.
In the second-level FCS, schools are allowed to provide football-related athletic aid equivalent to 63 full scholarships, but this aid may be divided among up to 85 players as the schools see fit.
In other fields
The year AD 85, 85 BC, or 1985.
The Muslim calendar year 85 AH.
The atomic number of the chemical element astatine
The number of the French department Vendée
The ISBN Group Identifier for books published in Brazil
The radix of the Ascii85 (sometimes called Base85) binary-to-text encoding
The IQ and nickname of Aaron in Alien 3
The number worn by Chad Ochocinco, whose name means "eight five" in Spanish
Arabigere 85 is a village in India
E85 fuel is 85% ethanol and 15% conventional gasoline
MCS-85 was a family of Intel processors including the 8085
TI-85 was a graphing
|
https://en.wikipedia.org/wiki/86%20%28number%29
|
86 (eighty-six) is the natural number following 85 and preceding 87.
In mathematics
86 is:
nontotient and a noncototient.
the 25th distinct semiprime and the 13th of the form (2.q).
together with 85 and 87, forms the middle semiprime in the 2nd cluster of three consecutive semiprimes; the first comprising 33, 34, 35.
an Erdős–Woods number, since it is possible to find sequences of 86 consecutive integers such that each inner member shares a factor with either the first or the last member.
a happy number and a self number in base 10.
with an aliquot sum of 46; itself a semiprime, within an aliquot sequence of seven members (86,46,26,16,15,9,4,3,1,0) in the Prime 3-aliquot tree.
It appears in the Padovan sequence, preceded by the terms 37, 49, 65 (it is the sum of the first two of these).
It is conjectured that 86 is the largest n for which the decimal expansion of 2n contains no 0.
86 = (8 × 6 = 48) + (4 × 8 = 32) + (3 × 2 = 6). That is, 86 is equal to the sum of the numbers formed in calculating its multiplicative persistence.
In science
86 is the atomic number of radon.
There are 86 metals on the modern periodic table.
In other fields
In American English, and particularly in the food service industry, 86 has become a slang term referring to an item being out of stock or discontinued, and by extension to a person no longer welcome on the premises.
The number of the French department Vienne. This number is also reflected in the department's postal code and in the name of a local basketball club, Poitiers Basket 86.
+86 is the code for international direct dial phone calls to China.
An art gallery in Ventura, California, displaying art pieces from such artists Billy Childish, Stacy Lande and Derek Hess, most of which include the number *86 hidden or overtly shown in the art, and some of which fall under the genre of lowbrow.
86 is the device number for a lockout relay function in electrical engineering electrical circuit protection schemes.
86 is often used in Japan as the nickname for the Toyota AE86.
86 is the name of a series of Japanese science fiction light novels written by Asato Asato, later adapted as a manga and an anime.
See also
List of highways numbered 86
Notes
Integers
|
https://en.wikipedia.org/wiki/87%20%28number%29
|
87 (eighty-seven) is the natural number following 86 and preceding 88.
In mathematics
87 is:
the sum of the squares of the first four primes (87 = 22 + 32 + 52 + 72).
the sum of the sums of the divisors of the first 10 positive integers.
the thirtieth semiprime, and the twenty-sixth distinct semiprime and the eighth of the form (3.q).
together with 85 and 86, forms the last semiprime in the 2nd cluster of three consecutive semiprimes; the first comprising 33, 34, 35.
with an aliquot sum of 33; itself a semiprime, within an aliquot sequence of five composite numbers (87,33,15,9,4,3,1,0) to the Prime in the 3-aliquot tree.
5! - 4! - 3! - 2! - 1! = 87
the last two decimal digits of Graham's number.
In sports
Cricket in Australia holds 87 as a superstitiously unlucky score and is referred to as "the devil's number". This originates from the fact that 87 is 13 runs short of a century. 187, 287, and so on are also considered unlucky but are not as common as 87 on its own.
In the National Hockey League, Wayne Gretzky scored a league-high 87 goals with the Edmonton Oilers in the 1983–84 NHL season.
In other fields
Eighty-seven is also:
The atomic number of francium.
An answer to a popular puzzle question states 16, 06, 68, 88, xx, 98. The answer is 87 when looked upside down.
The number of years between the signing of the U.S. Declaration of Independence and the Battle of Gettysburg, immortalized in Abraham Lincoln's Gettysburg Address with the phrase "Four Score and Seven Years ago..."
The model number of Junkers Ju 87.
The number of the French department Haute-Vienne.
The code for international direct dial phone calls to Inmarsat and other services.
The 87 photographic filter blocks visible light, allowing only infrared light to pass.
The ISBN Group Identifier for books published in Denmark.
The opus number of the 24 Preludes and Fugues of Dmitri Shostakovich.
In model railroading, the ratio of the popular H0 scale is 1:87. Proto:87 scale claims to offer precise proportions of wheels and tracks of real railroads.
David Bowie CD (1987) Never Let Me Down includes the song, "'87 and Cry".
The 87th United States Congress met from January 3, 1961, to January 3, 1963, during John F. Kennedy's time in office.
Ed McBain's 87th Precinct: Lightning film starred Randy Quaid (1995).
87 punch includes one bottle of Bacardi rum (eight years aged) and 7-up (two-liter bottle).
Wenger Swiss Army Knife version XXL, listed in the Guinness Book of World Records as the world's most multi-functional penknife with 87 tools.
Sonnet 87 by William Shakespeare.
Vault 87 is a main location in the game Fallout 3.
M87* is the first black hole ever photographed.
Aragorn's age in The Lord of the Rings.
"The Bite of '87" in the Five Nights at Freddy's series.
See also
List of highways numbered 87
References
Integers
|
https://en.wikipedia.org/wiki/89%20%28number%29
|
89 (eighty-nine) is the natural number following 88 and preceding 90.
In mathematics
89 is:
the 24th prime number, following 83 and preceding 97.
a Chen prime.
a Pythagorean prime.
the smallest Sophie Germain prime to start a Cunningham chain of the first kind of six terms, {89, 179, 359, 719, 1439, 2879}.
an Eisenstein prime with no imaginary part and real part of the form .
a Fibonacci number and thus a Fibonacci prime as well. The first few digits of its reciprocal coincide with the Fibonacci sequence due to the identity
a Markov number, appearing in solutions to the Markov Diophantine equation with other odd-indexed Fibonacci numbers.
M89 is the 10th Mersenne prime.
Although 89 is not a Lychrel number in base 10, it is unusual that it takes 24 iterations of the reverse and add process to reach a palindrome. Among the known non-Lychrel numbers in the first 10000 integers, no other number requires that many or more iterations. The palindrome reached is also unusually large.
There are exactly 1000 prime numbers between 1 and 892.
In science
Eighty-nine is:
The atomic number of actinium.
In astronomy
Messier object M89, a magnitude 11.5 elliptical galaxy in the constellation Virgo.
The New General Catalogue object NGC 89, a magnitude 13.5 peculiar spiral galaxy in the constellation Phoenix and a member of Robert's Quartet.
In sports
The Oklahoma Redhawks, an American minor league baseball team, were formerly known as the Oklahoma 89ers (1962–1997). The number alludes to the Land Run of 1889, when the Unassigned Lands of Oklahoma were opened to white settlement. The team's home of Oklahoma City was founded during this event.
In Rugby, an "89" or eight-nine move is a phase following a scrum, in which the number 8 catches the ball and transfers it to number 9 (scrum half).
The Elite 89 Award is presented by the U.S. NCAA to the participant in each of the NCAA's 89 championship finals with the highest grade point average.
89, a 2017 film about a football match, between Liverpool and Arsenal in 1989.
In other fields
Eighty-nine is also:
The designation of Interstate 89, a freeway that runs from New Hampshire to Vermont
The designation of U.S. Route 89, a north–south highway that runs from Montana to Arizona
The ISBN Group Identifier for books published in Korea
"Pop Song 89" by R.E.M.
A model of the Texas Instruments calculator TI-89
California Proposition 89, a 2006 California ballot initiative on campaign finance reform
The title of a currently-unreleased song by Bon Iver
The greatest number of verses in a chapter of a book of the Bible other than the Book of Psalms—specifically Numbers chapter 7.
The number of units of each colour in the board game Blokus
The number of the French department Yonne
Information Is Beautiful cites eighty-nine as one of the words censored on the Chinese internet.
See also
Hellin's law
References
Integers
|
https://en.wikipedia.org/wiki/77%20%28number%29
|
77 (seventy-seven) is the natural number following 76 and preceding 78. Seventy-seven is the smallest positive integer requiring five syllables in English.
In mathematics
77 is:
the 22nd discrete semiprime and the first of the (7.q) family, where q is a higher prime.
with a prime aliquot sum of 19 within an aliquot sequence (77,19,1,0) to the Prime in the 19-aliquot tree. 77 is the second composite member of the 19-aliquot tree with 65
a Blum integer since both 7 and 11 are Gaussian primes.
the sum of three consecutive squares, 42 + 52 + 62.
the sum of the first eight prime numbers.
the number of integer partitions of the number 12.
the largest number that cannot be written as a sum of distinct numbers whose reciprocals sum to 1.
the number of digits of the 12th perfect number.
It is possible for a sudoku puzzle to have as many as 77 givens, yet lack a unique solution.
It and 49 are the only 2-digit numbers whose home primes (in base 10) have not been calculated.
In science
The atomic number of iridium
The boiling point of nitrogen (in kelvins)
The temperature, in Fahrenheit, some characteristics of semiconductors are specifically given in a datasheet (77 °F = 25 °C).
In history
During World War II in Sweden at the border with Norway, "77" was used as a shibboleth (password), because the tricky pronunciation in Swedish made it easy to instantly discern whether the speaker was native Swedish, Norwegian, or German.
In religion
In the Islamic tradition, "77" figures prominently. Muhammad is reported to have explained, "Faith has sixty-odd, or seventy-odd branches, the highest and best of which is to declare and believe that there is no god but Allah without any equals or highers and anyone worthy of worship, and the lowest of which is to remove something harmful from a road. Shyness, too, is a branch of faith." While some scholars refrain from clarifying "sixty-odd or seventy-odd", various numbers have been suggested, 77 being the most common. Some have gone so far as to delineate these branches.
The Gospel of Luke lists 77 generations from Adam to Jesus.
In the Gospel of Matthew Peter asks, "How many times shall I forgive my brother?". Jesus replies, "Seventy-seven times." However this was not intended as literal quantitative instruction. Additionally, depending on the manuscript used for a given New Testament eclectic translation, the result is 77 or 490 (70*7) as it is seen in the King James Version.
In the Book of Genesis, Chapter 4, Lamech says to his wives, "hear my voice ... hearken unto my speech; for I have slain a man for wounding me, and a young man for bruising me; If Cain shall be avenged sevenfold, truly Lamech seventy and sevenfold."
In religious numerology
In certain numerological systems based on the English alphabet, the number 77 is associated with Jesus Christ. CHRIST is C = 3, H = 8, R = 18, I = 9, S = 19, T = 20, which added together equal 77.
'Liber 77' is the gematrian name for Liber OZ- a brief but popular publi
|
https://en.wikipedia.org/wiki/79%20%28number%29
|
79 (seventy-nine) is the natural number following 78 and preceding 80.
In mathematics
79 is:
An odd number.
The smallest number that can not be represented as a sum of fewer than 19 fourth powers.
The 22nd prime number (between and )
An isolated prime without a twin prime, as 77 and 81 are composite.
The smallest prime number p for which the real quadratic field Q[] has class number greater than 1 (namely 3).
A cousin prime with 83.
An emirp in base 10, because the reverse of 79, 97, is also a prime.
A Fortunate prime.
A circular prime.
A prime number that is also a Gaussian prime (since it is of the form ).
A happy prime.
A Higgs prime.
A lucky prime.
A permutable prime, with ninety-seven.
A Pillai prime, because 23! + 1 is divisible by 79, but 79 is not one more than a multiple of 23.
A regular prime.
A right-truncatable prime, because when the last digit (9) is removed, the remaining number (7) is still prime.
A sexy prime (with 73).
The n value of the Wagstaff prime 201487636602438195784363.
Similarly to how the decimal expansion of 1/89 gives Fibonacci numbers, 1/79 gives Pell numbers, that is,
A Leyland number of the second kind.
In science
The atomic number of the chemical element gold (Au) is 79.
In astronomy
Messier object 79 (M79), a magnitude 8.5 globular cluster in the constellation Lepus
New General Catalogue object 79 (NGC 79), a galaxy in the constellation Andromeda
In other fields
Live Seventy Nine, an album by Hawkwind
The years 79 BC, AD 79 or 1979
The number of the French department Deux-Sèvres
The ASCII code of the capital letter O
References
Integers
|
https://en.wikipedia.org/wiki/Shiing-Shen%20Chern
|
Shiing-Shen Chern (; , ; October 28, 1911 – December 3, 2004) was a Chinese-American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geometry" and is widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century, winning numerous awards and recognition including the Wolf Prize and the inaugural Shaw Prize. In memory of Shiing-Shen Chern, the International Mathematical Union established the Chern Medal in 2010 to recognize "an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics".
Chern worked at the Institute for Advanced Study (1943–45), spent about a decade at the University of Chicago (1949-1960), and then moved to University of California, Berkeley, where he co-founded the Mathematical Sciences Research Institute in 1982 and was the institute's founding director. Renowned co-authors with Chern include Jim Simons, an American mathematician and billionaire hedge fund manager. Chern's work, most notably the Chern-Gauss-Bonnet Theorem, Chern–Simons theory, and Chern classes, are still highly influential in current research in mathematics, including geometry, topology, and knot theory; as well as many branches of physics, including string theory, condensed matter physics, general relativity, and quantum field theory.
Name spelling
Chern's surname (陈) is a common Chinese surname which is now usually spelled Chen. The unusual spelling "Chern" is from the old Gwoyeu Romatzyh (GR) romanization for Mandarin Chinese that uses spelling rather than diacritics to indicate tones. The silent r in "Chern" indicates a second-tone, while "Shiing-Shen" indicates a third tone for Shiing and a first tone for Shen. The equivalent pinyin would be "Chén Xǐngshēn".
In English, Chern pronounced his own name as "Churn" (), and this spelling pronunciation is now widely used among English-speaking mathematicians and physicists.
Biography
Early years in China
Chern was born in Xiushui, Jiaxing, China in 1911. He graduated from Xiushui Middle School () and subsequently moved to Tianjin in 1922 to accompany his father. In 1926, after spending four years in Tianjin, Chern graduated from .
At age 15, Chern entered the Faculty of Sciences of the Nankai University in Tianjin and was interested in physics, but not so much the laboratory, so he studied mathematics instead. Chern graduated with a Bachelor of Science degree in 1930. At Nankai, Chern's mentor was mathematician Jiang Lifu, and Chern was also heavily influenced by Chinese physicist Rao Yutai, considered to be one of the founding fathers of modern Chinese informatics.
Chern went to Beijing to work at the Tsinghua University Department of Mathematics as a teaching assistant. At the same time he also registered at Tsinghua Graduate School as a student. He studied projective differential geomet
|
https://en.wikipedia.org/wiki/71%20%28number%29
|
71 (seventy-one) is the natural number following 70 and preceding 72.
In mathematics
Because both rearrangements of its digits (17 and 71) are prime numbers, 71 is an emirp and more generally a permutable prime. It is the largest number which occurs as a prime factor of an order of a sporadic simple group, the largest (15th) supersingular prime.
It is a Pillai prime, since is divisible by 71, but 71 is not one more than a multiple of 9.
It is part of the last known pair (71, 7) of Brown numbers, since .
It is centered heptagonal number.
See also
71 (disambiguation)
References
Integers
|
https://en.wikipedia.org/wiki/73%20%28number%29
|
73 (seventy-three) is the natural number following 72 and preceding 74. In English, it is the smallest natural number with twelve letters in its spelled out name.
In mathematics
73 is the 21st prime number, and emirp with 37, the 12th prime number. It is also the eighth twin prime, with 71. It is the largest minimal primitive root in the first 100,000 primes; in other words, if is one of the first one hundred thousand primes, then at least one of the numbers 2, 3, 4, 5, 6, ..., 73 is a primitive root modulo . 73 is also the smallest factor of the first composite generalized Fermat number in decimal: , and the smallest prime congruent to 1 modulo 24, as well as the only prime repunit in octal (1118). It is the fourth star number.
Sheldon prime
Notably, 73 is the only Sheldon prime to contain both "mirror" and "product" properties:
73, as an emirp, has 37 as its dual permutable prime, a mirroring of its base ten digits, 7 and 3. 73 is the 21st prime number, while 37 is the 12th, which is a second mirroring; and
73 has a prime index of 21 = 7 × 3; a product property where the product of its base-10 digits is precisely its index in the sequence of prime numbers.
Arithmetically, from sums of 73 and 37 with their prime indexes, one obtains:
Meanwhile,
In binary 73 is represented as , while 21 in binary is , with 7 and 3 represented as and respectively; all which are palindromic. Of the seven binary digits representing 73, there are three 1s. In addition to having prime factors 7 and 3, the number represents the ternary (base-3) equivalent of the decimal numeral 7, that is to say: .
Other properties
The row sum of Lah numbers of the form with and is equal to . These numbers represent coefficients expressing rising factorials in terms of falling factorials, and vice-versa; equivalently in this case to the number of partitions of into any number of lists, where a list means an ordered subset.
73 requires 115 steps to return to 1 in the Collatz problem, and 37 requires 21: {37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1}. Collectively, the sum between these steps is 136, the 16th triangular number, where {16, 8, 4, 2, 1} is the only possible step root pathway.
There are 73 three-dimensional arithmetic crystal classes that are part of 230 crystallographic space group types. These 73 groups are specifically symmorphic groups such that all operating lattice symmetries have one common fixed isomorphic point, with the remaining 157 groups nonsymmorphic (the 37th prime is 157).
In five-dimensional space, there are 73 Euclidean solutions of 5-polytopes with uniform symmetry, excluding prismatic forms: 19 from the simplex group, 23 from the demihypercube group, and 31 from the hypercubic group, of which 15 equivalent solutions are shared between and from distinct polytope operations.
In moonshine theory of sporadic groups, 73 is the first non-supersingular prime greater than 71 that does not divide
|
https://en.wikipedia.org/wiki/74%20%28number%29
|
74 (seventy-four) is the natural number following 73 and preceding 75.
In mathematics
74 is:
the twenty-first distinct semiprime and the eleventh of the form (2.q), where q is a higher prime.
with an aliquot sum of 40, within an aliquot sequence of three composite numbers (74,40,50,43,1,0) to the Prime in the 43-aliquot tree.
a palindromic number in bases 6 (2026) and 36 (2236).
a nontotient.
the number of collections of subsets of {1, 2, 3} that are closed under union and intersection.
φ(74) = φ(σ(74)).
There are 74 different non-Hamiltonian polyhedra with a minimum number of vertices.
In science
The atomic number of tungsten
In astronomy
Messier object M74, a magnitude 10.5 spiral galaxy in the constellation Pisces.
The New General Catalogue object NGC 74, a galaxy in the constellation Andromeda.
In music
Seventy-four, one of the Number Pieces by John Cage
"Seventy-Four", a song by the American band Bright from the album The Albatross Guest House
In other fields
Seventy-four is also:
The year AD 74, 74 BC, or 1974
Designates the 7400 series of Integrated Chips. 74xx xx=00-4538
A seventy-four was a third-rate warship with 74 guns.
The registry of the U.S. Navy's nuclear aircraft carrier USS John C. Stennis (CVN-74), named after U.S. Senator John C. Stennis
A hurricane or typhoon is a system with sustained winds of at least 74 mph (64 knots).
The number of the French department Haute-Savoie
In the Bible it is the number of people that were in the presence of God on Mount Sinai and saw God without dying "Exodus 24:9-11"
References
Integers
|
https://en.wikipedia.org/wiki/75%20%28number%29
|
75 (seventy-five) is the natural number following 74 and preceding 76.
In mathematics
75 is a self number because there is no integer that added up to its own digits adds up to 75. It is the sum of the first five pentagonal numbers, and therefore a pentagonal pyramidal number, as well as a nonagonal number.
It is also the fourth ordered Bell number, and a Keith number, because it recurs in a Fibonacci-like sequence started from its base 10 digits: 7, 5, 12, 17, 29, 46, 75...
75 is the count of the number of weak orderings on a set of four items.
Excluding the infinite sets, there are 75 uniform polyhedra in the third dimension, which incorporate star polyhedra as well. Inclusive of 7 families of prisms and antiprisms, there are also 75 uniform compound polyhedra.
In other fields
Seventy-five is:
The atomic number of rhenium
The age limit for Canadian senators
A common name for the Canon de 75 modèle 1897, a French World War I gun
The department number of the city of Paris
The number of balls in a standard game of Bingo in the United States
References
Integers
|
https://en.wikipedia.org/wiki/32%20%28number%29
|
32 (thirty-two) is the natural number following 31 and preceding 33.
In mathematics
32 is the fifth power of two (), making it the first non-unitary fifth-power of the form p5 where p is prime. 32 is the totient summatory function over the first 10 integers, and the smallest number with exactly 7 solutions for . The aliquot sum of a power of two () is always one less than the number itself, therefore the aliquot sum of 32 is 31.
The product between neighbor numbers of 23, the dual permutation of the digits of 32 in decimal, is equal to the sum of the first 32 integers: .
32 is the ninth -happy number, while 23 is the sixth. Their sum is 55, which is the tenth triangular number, while their difference is .
32 is also a Leyland number expressible in the form , where:
On the other hand, a regular 32-sided icosidodecagon contains distinct symmetries.
There are collectively 32 uniform colorings to the 11 regular and semiregular tilings.
The product of the five known Fermat primes is equal to the number of sides of the largest regular constructible polygon with a straightedge and compass that has an odd number of sides, with a total of sides numbering
The first 32 rows of Pascal's triangle in binary represent the thirty-two divisors that belong to this number, which is also the number of sides of all odd-sided constructible polygons with simple tools alone (if the monogon is also included).
There are 32 three-dimensional crystallographic point groups and 32 five-dimensional crystal families, and the maximum determinant in a 7 by 7 matrix of only zeroes and ones is 32. In sixteen dimensions, the sedenions generate a non-commutative loop of order 32, and in thirty-two dimensions, there are at least 1,160,000,000 even unimodular lattices (of determinants 1 or −1); which is a marked increase from the twenty-four such Niemeier lattices that exists in twenty-four dimensions, or the single lattice in eight dimensions (these lattices only exist for dimensions ). Furthermore, the 32nd dimension is the first dimension that holds non-critical even unimodular lattices that do not interact with a Gaussian potential function of the form of root and .
In science
The atomic number of germanium
The freezing point of water at standard atmospheric pressure in degrees Fahrenheit
In the Standard Model of particle physics, there are 32 degrees of freedom among the leptons and all bosons that interact with them (including the graviton, which is generally expected to exist, and assuming there are no right-handed neutrinos)
Astronomy
Messier 32, a magnitude 9.0 galaxy in the constellation Andromeda which is a companion to M31.
The New General Catalogue object NGC 32, a star in the constellation Pegasus
In music
A thirty-second note or demisemiquaver is a note played for 1/32 of the duration of a whole note
The number of completed, numbered piano sonatas by Ludwig van Beethoven
"32 Footsteps", a song by They Might Be Giants
"The Chamber of 32 Doors", a s
|
https://en.wikipedia.org/wiki/34%20%28number%29
|
34 (thirty-four) is the natural number following 33 and preceding 35.
In mathematics
34 is the ninth distinct semiprime, with four divisors including 1 and itself. Specifically, 34 is the ninth distinct semiprime, it being the sixth of the form . Its neighbors 33 and 35 are also distinct semiprimes with four divisors each, where 34 is the smallest number to be surrounded by numbers with the same number of divisors it has. This is the first distinct semiprime treble cluster, the next being (85, 86, 87).
The number 34 has an aliquot sum of 20, and is the seventh member in the aliquot sequence (34, 20, 22, 14, 10, 8, 7, 1, 0) that belongs to the prime 7-aliquot tree.
Its reduced totient and Euler totient values are both 16 (or 42 = 24). The sum of all its divisors aside from one equals 53, which is the sixteenth prime number.
There is no solution to the equation φ(x) = 34, making 34 a nontotient. Nor is there a solution to the equation x − φ(x) = 34, making 34 a noncototient.
It is the third Erdős–Woods number, following 22 and 16.
It is the ninth Fibonacci number and a companion Pell number.
Since it is an odd-indexed Fibonacci number, 34 is a Markov number.
34 is also the fourth heptagonal number.
This number is also the magic constant of Queens Problem for .
There are 34 topologically distinct convex heptahedra, excluding mirror images.
34 is the magic constant of a normal magic square, and magic octagram (see accompanying images).
In science
The atomic number of selenium
Messier object M34, a magnitude 6.0 open cluster in the constellation Perseus
The New General Catalogue object NGC 34, a peculiar galaxy in the constellation Cetus
The human vertebral column is made up of up to 34 vertebrae.
Literature
In The Count of Monte Cristo, Number 34 is how Edmond Dantès is referred to during his imprisonment in the Château d'If.
Transportation
34th Street (Manhattan), a major cross-town street in New York City
34th Street station (disambiguation), multiple rail stations
In other fields
34 is also:
The traffic code of Istanbul, Turkey
"#34", a song by the Dave Matthews Band
The number of the French department Hérault
+34 is the code for international direct-dial phone calls to Spain
The number used to win the 2021 Daytona 500 with Michael McDowell and Front Row Motorsports. The number and team also won at Pocono Raceway with Chris Buescher and Talladega Superspeedway with David Ragan. In 1961, Wendell Scott won with the number to become the first African American to win in NASCAR.
See also
Rule 34
References
External links
Prime Curios! 34 from the Prime Pages
Integers
|
https://en.wikipedia.org/wiki/Value%20distribution%20theory%20of%20holomorphic%20functions
|
In mathematics, the value distribution theory of holomorphic functions is a division of mathematical analysis. The purpose of the theory is to provide quantitative measures of the number of times a function f(z) assumes a value a, as z grows in size, refining the Picard theorem on behaviour close to an essential singularity. The theory exists for analytic functions (and meromorphic functions) of one complex variable z, or of several complex variables.
In the case of one variable, the term Nevanlinna theory, after Rolf Nevanlinna, is also common. The now-classical theory received renewed interest when Paul Vojta suggested some analogies to the problem of integral solutions to Diophantine equations. These turned out to involve some close parallels and to lead to fresh points of view on the Mordell conjecture and related questions.
holomorphic functions
Meromorphic functions
|
https://en.wikipedia.org/wiki/31%20%28number%29
|
31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.
In mathematics
31 is the 11th prime number. It is a superprime and a self prime (after 3, 5, and 7), as no integer added up to its base 10 digits results in 31. It is the third Mersenne prime of the form 2n − 1, and the eighth Mersenne prime exponent, in-turn yielding the maximum positive value for a 32-bit signed binary integer in computing: 2,147,483,647. After 3, it is the second Mersenne prime not to be a double Mersenne prime, while the 31st prime number (127) is the second double Mersenne prime, following 7. On the other hand, the thirty-first triangular number is the perfect number 496, of the form 2(5 − 1)(25 − 1) by the Euclid-Euler theorem. 31 is also a primorial prime like its twin prime (29), as well as both a lucky prime and a happy number like its dual permutable prime in decimal (13).
31 is the number of regular polygons with an odd number of sides that are known to be constructible with compass and straightedge, from combinations of known Fermat primes of the form 22n + 1 (they are 3, 5, 17, 257 and 65537).
31 is the 11th and final consecutive supersingular prime. After 31, the only supersingular primes are 41, 47, 59, and 71.
31 is the first prime centered pentagonal number, the fifth centered triangular number, and a centered decagonal number.
For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals.
At 31, the Mertens function sets a new low of −4, a value which is not subceded until 110.
31 is a repdigit in base 2 (11111) and in base 5 (111).
The cube root of 31 is the value of correct to four significant figures:
The first five Euclid numbers of the form p1 × p2 × p3 × ... × pn + 1 (with pn the nth prime) are prime:
3 = 2 + 1
7 = 2 × 3 + 1
31 = 2 × 3 × 5 + 1
211 = 2 × 3 × 5 × 7 + 1 and
2311 = 2 × 3 × 5 × 7 × 11 + 1
The following term, 30031 = 59 × 509 = 2 × 3 × 5 × 7 × 11 × 13 + 1, is composite. The next prime number of this form has a largest prime p of 31: 2 × 3 × 5 × 7 × 11 × 13 × ... × 31 + 1 ≈ 8.2 × 1033.
While 13 and 31 in base-ten are the proper first duo of two-digit permutable primes and emirps with distinct digits in base ten, 11 is the only two-digit permutable prime that is its own permutable prime. Meanwhile 1310 in ternary is 1113 and 3110 in quinary is 1115, with 1310 in quaternary represented as 314 and 3110 as 1334 (their mirror permutations 3314 and 134, equivalent to 61 and 7 in decimal, respectively, are also prime). (11, 13) are the third twin prime pair, formed by the fifth and sixth prime numbers, whose indices add to 11, itself the prime index of 31.
The numbers 31, 331, 3331, , , , and are all prime. For a time it was thought that every number of the form 3w1 would be prime. However, the next nine numbers of the sequence are composite; their factorisations are:
= 17 ×
= 673 ×
= 307 ×
= 19 × 83 ×
= 523 × 3049 ×
= 607 ×
|
https://en.wikipedia.org/wiki/Degenerate%20conic
|
In geometry, a degenerate conic is a conic (a second-degree plane curve, defined by a polynomial equation of degree two) that fails to be an irreducible curve. This means that the defining equation is factorable over the complex numbers (or more generally over an algebraically closed field) as the product of two linear polynomials.
Using the alternative definition of the conic as the intersection in three-dimensional space of a plane and a double cone, a conic is degenerate if the plane goes through the vertex of the cones.
In the real plane, a degenerate conic can be two lines that may or may not be parallel, a single line (either two coinciding lines or the union of a line and the line at infinity), a single point (in fact, two complex conjugate lines), or the null set (twice the line at infinity or two parallel complex conjugate lines).
All these degenerate conics may occur in pencils of conics. That is, if two real non-degenerated conics are defined by quadratic polynomial equations and , the conics of equations form a pencil, which contains one or three degenerate conics. For any degenerate conic in the real plane, one may choose and so that the given degenerate conic belongs to the pencil they determine.
Examples
The conic section with equation is degenerate as its equation can be written as , and corresponds to two intersecting lines forming an "X". This degenerate conic occurs as the limit case in the pencil of hyperbolas of equations The limiting case is an example of a degenerate conic consisting of twice the line at infinity.
Similarly, the conic section with equation , which has only one real point, is degenerate, as is factorable as over the complex numbers. The conic consists thus of two complex conjugate lines that intersect in the unique real point, , of the conic.
The pencil of ellipses of equations degenerates, for , into two parallel lines and, for , into a double line.
The pencil of circles of equations degenerates for into two lines, the line at infinity and the line of equation .
Classification
Over the complex projective plane there are only two types of degenerate conics – two different lines, which necessarily intersect in one point, or one double line. Any degenerate conic may be transformed by a projective transformation into any other degenerate conic of the same type.
Over the real affine plane the situation is more complicated. A degenerate real conic may be:
Two intersecting lines, such as
Two parallel lines, such as
A double line (multiplicity 2), such as
Two intersecting complex conjugate lines (only one real point), such as
Two parallel complex conjugate lines (no real point), such as
A single line and the line at infinity
Twice the line at infinity (no real point in the affine plane)
For any two degenerate conics of the same class, there are affine transformations mapping the first conic to the second one.
Discriminant
Non-degenerate real conics can be classified a
|
https://en.wikipedia.org/wiki/Directrix
|
In mathematics, a directrix is a curve associated with a process generating a geometric object, such as:
Directrix (conic section)
Directrix (generatrix)
Directrix (rational normal scroll)
Other uses
Directrix is a spaceship in the Lensman series of novels by E. E. Smith.
Directrix is the name of a Dubai-based alternative rock band.
|
https://en.wikipedia.org/wiki/Unitary
|
Unitary may refer to:
Mathematics
Unitary divisor
Unitary element
Unitary group
Unitary matrix
Unitary morphism
Unitary operator
Unitary transformation
Unitary representation
Unitarity (physics)
E-unitary inverse semigroup
Politics
Unitary authority
Unitary state
See also
Unital (disambiguation)
Unitarianism, belief that God is one entity
|
https://en.wikipedia.org/wiki/Jo%C3%ABl-Fran%C3%A7ois%20Durand
|
Joël-François Durand (born 17 September 1954) is a French composer.
Biography
Born in Orléans, Durand studied mathematics, music education and piano in Paris, then composition with Brian Ferneyhough in Freiburg im Breisgau, Germany (1981–84), and at Stony Brook University, New York, with Arel and Semegen (1984–86) . Between 1979 and 1984 he attended masterclasses by György Ligeti and Luciano Berio at the Centre Acanthes in Aix-en-Provence, and with Luigi Nono at the Freiburg Musikhochschule. In 1981–82 he was awarded a scholarship from the DAAD (German Academic Exchange Service) . In 1982 he received a Darmstadt Institute Scholarship for his String Trio, and in 1983 his piano piece ...d'asiles déchirés... was awarded a prize at the Third International K.H. Stockhausen Composition Competition in Brescia, Italy. He left Europe in 1984 to pursue a PhD in Composition (awarded in 1988) at Stony Brook University (USA), where he studied composition with Bülent Arel and electronic music with Daria Semegen.
Durand was awarded scholarships from the Fulbright Foundation (1984) and from the French Ministry of Culture (1985). He received the Kranichsteiner Musikpreis from the Darmstadt Internationalen Ferienkurse in 1990. He was appointed assistant professor in composition at the University of Washington in 1991, and was invited to teach at the University of California, San Diego in the autumn of 1992 . During the 1980-90s, he was regularly invited as a lecturer at the summer courses at Darmstädter Ferienkurse. He taught at the annual mastercourse at the Fondation Royaumont in France in 1993. He is currently Professor of Composition , and since 2002 Associate Director of the University of Washington School of Music .
In parallel to his activity as composer, Durand designs and manufactures high-end tonearms for record players. He founded the company Durand Tonearms LLC in 2009.
Works
Orchestral works
Piano Concerto (1993) – 22'
Five Musical Tales (1998) – 14'
Athanor (2000–2001) – 20'
Le Tombeau de Rameau III (2014) – 16'
Tropes de : Bussy (2017–18) – 25'
Chamber music and works for ensemble
String Trio (1981) – 7'
So er (1985) for 20 instruments – 11'
Lichtung (1987) for 10 instruments – 12'
Die innere Grenze (1988) for string sextet – 25'
Un feu distinct (1991) for flute, clarinet, piano, violin and cello – 15'
B.F., ein Mittelpunkt (1992) for 8 instruments – 3'
La terre et le feu (1999) for oboe and ensemble – 18'
La mesure des choses III. La mesure de la terre et du feu (1999) for oboe and viola – 12'
Cinq Duos (1999) for violin and viola – 14'
In the mirror land (2003) for flute and oboe; also versions for flute and B clarinet and for B clarinet and oboe – 6'
Ombre/Miroir (2004) for flute and 14 instruments – 14'
String Quartet (2005) – 21'
Le Tombeau de Rameau (2008) for flute, viola, and harp – 23'
Hermetic Definition (2012–13) for 10 instruments – 25'
Cage 100 Party Piece (2013) for ensemble – 1'
Mundus Imaginalis (2015) for 14
|
https://en.wikipedia.org/wiki/Rigged%20Hilbert%20space
|
In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis. Such spaces were introduced to study spectral theory in the broad sense. They bring together the 'bound state' (eigenvector) and 'continuous spectrum', in one place.
Motivation
A function such as
is an eigenfunction of the differential operator
on the real line , but isn't square-integrable for the usual Borel measure on . To properly consider this function as an eigenfunction requires some way of stepping outside the strict confines of the Hilbert space theory. This was supplied by the apparatus of Schwartz distributions, and a generalized eigenfunction theory was developed in the years after 1950.
Functional analysis approach
The concept of rigged Hilbert space places this idea in an abstract functional-analytic framework. Formally, a rigged Hilbert space consists of a Hilbert space , together with a subspace which carries a finer topology, that is one for which the natural inclusion
is continuous. It is no loss to assume that is dense in for the Hilbert norm. We consider the inclusion of dual spaces in . The latter, dual to in its 'test function' topology, is realised as a space of distributions or generalised functions of some sort, and the linear functionals on the subspace of type
for in are faithfully represented as distributions (because we assume dense).
Now by applying the Riesz representation theorem we can identify with . Therefore, the definition of rigged Hilbert space is in terms of a sandwich:
The most significant examples are those for which is a nuclear space; this comment is an abstract expression of the idea that consists of test functions and of the corresponding distributions. Also, a simple example is given by Sobolev spaces: Here (in the simplest case of Sobolev spaces on )
where .
Formal definition (Gelfand triple)
A rigged Hilbert space is a pair with a Hilbert space, a dense subspace, such that is given a topological vector space structure for which the inclusion map is continuous.
Identifying with its dual space , the adjoint to is the map
The duality pairing between and is then compatible with the inner product on , in the sense that:
whenever and . In the case of complex Hilbert spaces, we use a Hermitian inner product; it will be complex linear in (math convention) or (physics convention), and conjugate-linear (complex anti-linear) in the other variable.
The triple is often named the "Gelfand triple" (after the mathematician Israel Gelfand).
Note that even though is isomorphic to (via Riesz representation) if it happens that is a Hilbert space in its own right, this isomorphism is not the same as the composition of the inclusion with its adjoint
References
J.-P. Antoine, Quantum Mechanics Beyond Hilbert Space (1996), appearing in Irreversibility and Causality,
|
https://en.wikipedia.org/wiki/Index%20set
|
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be indexed or labeled by means of the elements of a set , then is an index set. The indexing consists of a surjective function from onto , and the indexed collection is typically called an indexed family, often written as .
Examples
An enumeration of a set gives an index set , where is the particular enumeration of .
Any countably infinite set can be (injectively) indexed by the set of natural numbers .
For , the indicator function on is the function given by
The set of all such indicator functions, , is an uncountable set indexed by .
Other uses
In computational complexity theory and cryptography, an index set is a set for which there exists an algorithm that can sample the set efficiently; e.g., on input , can efficiently select a poly(n)-bit long element from the set.
See also
Friendly-index set
References
Mathematical notation
Basic concepts in set theory
|
https://en.wikipedia.org/wiki/Comparison%20of%20topologies
|
In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can be used for comparison of the topologies.
Definition
A topology on a set may be defined as the collection of subsets which are considered to be "open". An alternative definition is that it is the collection of subsets which are considered "closed". These two ways of defining the topology are essentially equivalent because the complement of an open set is closed and vice versa. In the following, it doesn't matter which definition is used.
Let τ1 and τ2 be two topologies on a set X such that τ1 is contained in τ2:
.
That is, every element of τ1 is also an element of τ2. Then the topology τ1 is said to be a coarser (weaker or smaller) topology than τ2, and τ2 is said to be a finer (stronger or larger) topology than τ1.
If additionally
we say τ1 is strictly coarser than τ2 and τ2 is strictly finer than τ1.
The binary relation ⊆ defines a partial ordering relation on the set of all possible topologies on X.
Examples
The finest topology on X is the discrete topology; this topology makes all subsets open. The coarsest topology on X is the trivial topology; this topology only admits the empty set
and the whole space as open sets.
In function spaces and spaces of measures there are often a number of possible topologies. See topologies on the set of operators on a Hilbert space for some intricate relationships.
All possible polar topologies on a dual pair are finer than the weak topology and coarser than the strong topology.
The complex vector space Cn may be equipped with either its usual (Euclidean) topology, or its Zariski topology. In the latter, a subset V of Cn is closed if and only if it consists of all solutions to some system of polynomial equations. Since any such V also is a closed set in the ordinary sense, but not vice versa, the Zariski topology is strictly weaker than the ordinary one.
Properties
Let τ1 and τ2 be two topologies on a set X. Then the following statements are equivalent:
τ1 ⊆ τ2
the identity map idX : (X, τ2) → (X, τ1) is a continuous map.
the identity map idX : (X, τ1) → (X, τ2) is a strongly/relatively open map.
(The identity map idX is surjective and therefore it is strongly open if and only if it is relatively open.)
Two immediate corollaries of the above equivalent statements are
A continuous map f : X → Y remains continuous if the topology on Y becomes coarser or the topology on X finer.
An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser.
One can also compare topologies using neighborhood bases. Let τ1 and τ2 be two topologies on a set X and let Bi(x) be a local base for the topology τi at x ∈ X for i = 1,2. Then τ1 ⊆ τ2 if and only if for all x ∈ X, each open set U1 in B1(x) contains some open set U2 in B2(x). Intuitively, this makes sense: a finer topology should have sm
|
https://en.wikipedia.org/wiki/Chern%E2%80%93Gauss%E2%80%93Bonnet%20theorem
|
In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler–Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its curvature form (an analytical invariant).
It is a highly non-trivial generalization of the classic Gauss–Bonnet theorem (for 2-dimensional manifolds / surfaces) to higher even-dimensional Riemannian manifolds. In 1943, Carl B. Allendoerfer and André Weil proved a special case for extrinsic manifolds. In a classic paper published in 1944, Shiing-Shen Chern proved the theorem in full generality connecting global topology with local geometry.
Riemann–Roch and Atiyah–Singer are other generalizations of the Gauss–Bonnet theorem.
Statement
One useful form of the Chern theorem is that
where denotes the Euler characteristic of . The Euler class is defined as
where we have the Pfaffian . Here is a compact orientable 2n-dimensional Riemannian manifold without boundary, and is the associated curvature form of the Levi-Civita connection. In fact, the statement holds with the curvature form of any metric connection on the tangent bundle, as well as for other vector bundles over .
Since the dimension is 2n, we have that is an -valued 2-differential form on (see special orthogonal group). So can be regarded as a skew-symmetric 2n × 2n matrix whose entries are 2-forms, so it is a matrix over the commutative ring . Hence the Pfaffian is a 2n-form. It is also an invariant polynomial.
However, Chern's theorem in general is that for any closed orientable n-dimensional ,
where the above pairing (,) denotes the cap product with the Euler class of the tangent bundle .
Proofs
In 1944, the general theorem was first proved by S. S. Chern in a classic paper published by the Princeton University math department.
In 2013, a proof of the theorem via supersymmetric Euclidean field theories was also found.
Applications
The Chern–Gauss–Bonnet theorem can be seen as a special instance in the theory of characteristic classes. The Chern integrand is the Euler class. Since it is a top-dimensional differential form, it is closed. The naturality of the Euler class means that when changing the Riemannian metric, one stays in the same cohomology class. That means that the integral of the Euler class remains constant as the metric is varied and is thus a global invariant of the smooth structure.
The theorem has also found numerous applications in physics, including:
adiabatic phase or Berry's phase,
string theory,
condensed matter physics,
topological quantum field theory,
topological phases of matter (see the 2016 Nobel Prize in physics by Duncan Haldane et al.).
Special cases
Four-dimensional manifolds
In dimension , for a compact oriente
|
https://en.wikipedia.org/wiki/Dirichlet%20series
|
In mathematics, a Dirichlet series is any series of the form
where s is complex, and is a complex sequence. It is a special case of general Dirichlet series.
Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet.
Combinatorial importance
Dirichlet series can be used as generating series for counting weighted sets of objects with respect to a weight which is combined multiplicatively when taking Cartesian products.
Suppose that A is a set with a function w: A → N assigning a weight to each of the elements of A, and suppose additionally that the fibre over any natural number under that weight is a finite set. (We call such an arrangement (A,w) a weighted set.) Suppose additionally that an is the number of elements of A with weight n. Then we define the formal Dirichlet generating series for A with respect to w as follows:
Note that if A and B are disjoint subsets of some weighted set (U, w), then the Dirichlet series for their (disjoint) union is equal to the sum of their Dirichlet series:
Moreover, if (A, u) and (B, v) are two weighted sets, and we define a weight function by
for all a in A and b in B, then we have the following decomposition for the Dirichlet series of the Cartesian product:
This follows ultimately from the simple fact that
Examples
The most famous example of a Dirichlet series is
whose analytic continuation to (apart from a simple pole at ) is the Riemann zeta function.
Provided that is real-valued at all natural numbers , the respective real and imaginary parts of the Dirichlet series have known formulas where we write :
Treating these as formal Dirichlet series for the time being in order to be able to ignore matters of convergence, note that we have:
as each natural number has a unique multiplicative decomposition into powers of primes. It is this bit of combinatorics which inspires the Euler product formula.
Another is:
where is the Möbius function. This and many of the following series may be obtained by applying Möbius inversion and Dirichlet convolution to known series. For example, given a Dirichlet character one has
where is a Dirichlet L-function.
If the arithmetic function has a Dirichlet inverse function , i.e., if there exists an inverse function such that the Dirichlet convolution of f with its inverse yields the multiplicative identity
, then the DGF of the inverse function is given by the reciprocal of F:
Other identities include
where is the totient function,
where Jk is the Jordan function, and
where σa(n) is the divisor function. By specialization to the divisor function d = σ0 we have
The logarithm of the zeta function is given by
Similarly, we have that
Here, Λ(n) is the von Man
|
https://en.wikipedia.org/wiki/Gift%20wrapping%20algorithm
|
In computational geometry, the gift wrapping algorithm is an algorithm for computing the convex hull of a given set of points.
Planar case
In the two-dimensional case the algorithm is also known as Jarvis march, after R. A. Jarvis, who published it in 1973; it has O(nh) time complexity, where n is the number of points and h is the number of points on the convex hull. Its real-life performance compared with other convex hull algorithms is favorable when n is small or h is expected to be very small with respect to n. In general cases, the algorithm is outperformed by many others ( See Convex hull algorithms).
Algorithm
For the sake of simplicity, the description below assumes that the points are in general position, i.e., no three points are collinear. The algorithm may be easily modified to deal with collinearity, including the choice whether it should report only extreme points (vertices of the convex hull) or all points that lie on the convex hull. Also, the complete implementation must deal with degenerate cases when the convex hull has only 1 or 2 vertices, as well as with the issues of limited arithmetic precision, both of computer computations and input data.
The gift wrapping algorithm begins with i=0 and a point p0 known to be on the convex hull, e.g., the leftmost point, and selects the point pi+1 such that all points are to the right of the line pi pi+1. This point may be found in O(n) time by comparing polar angles of all points with respect to point pi taken for the center of polar coordinates. Letting i=i+1, and repeating with until one reaches ph=p0 again yields the convex hull in h steps. In two dimensions, the gift wrapping algorithm is similar to the process of winding a string (or wrapping paper) around the set of points.
The approach can be extended to higher dimensions.
Pseudocode
algorithm jarvis(S) is
// S is the set of points
// P will be the set of points which form the convex hull. Final set size is i.
pointOnHull = leftmost point in S // which is guaranteed to be part of the CH(S)
i := 0
repeat
P[i] := pointOnHull
endpoint := S[0] // initial endpoint for a candidate edge on the hull
for j from 0 to |S| do
// endpoint == pointOnHull is a rare case and can happen only when j == 1 and a better endpoint has not yet been set for the loop
if (endpoint == pointOnHull) or (S[j] is on left of line from P[i] to endpoint) then
endpoint := S[j] // found greater left turn, update endpoint
i := i + 1
pointOnHull = endpoint
until endpoint = P[0] // wrapped around to first hull point
Complexity
The inner loop checks every point in the set S, and the outer loop repeats for each point on the hull. Hence the total run time is . The run time depends on the size of the output, so Jarvis's march is an output-sensitive algorithm.
However, because the running time depends linearly on the number of hull v
|
https://en.wikipedia.org/wiki/Beth%20number
|
In mathematics, particularly in set theory, the beth numbers are a certain sequence of infinite cardinal numbers (also known as transfinite numbers), conventionally written , where is the second Hebrew letter (beth). The beth numbers are related to the aleph numbers (), but unless the generalized continuum hypothesis is true, there are numbers indexed by that are not indexed by .
Definition
Beth numbers are defined by transfinite recursion:
where is an ordinal and is a limit ordinal.
The cardinal is the cardinality of any countably infinite set such as the set of natural numbers, so that .
Let be an ordinal, and be a set with cardinality . Then,
denotes the power set of (i.e., the set of all subsets of ),
the set denotes the set of all functions from to {0,1},
the cardinal is the result of cardinal exponentiation, and
is the cardinality of the power set of .
Given this definition,
are respectively the cardinalities of
so that the second beth number is equal to , the cardinality of the continuum (the cardinality of the set of the real numbers), and the third beth number is the cardinality of the power set of the continuum.
Because of Cantor's theorem, each set in the preceding sequence has cardinality strictly greater than the one preceding it. For infinite limit ordinals, λ, the corresponding beth number is defined to be the supremum of the beth numbers for all ordinals strictly smaller than λ:
One can also show that the von Neumann universes have cardinality .
Relation to the aleph numbers
Assuming the axiom of choice, infinite cardinalities are linearly ordered; no two cardinalities can fail to be comparable. Thus, since by definition no infinite cardinalities are between and , it follows that
Repeating this argument (see transfinite induction) yields
for all ordinals .
The continuum hypothesis is equivalent to
The generalized continuum hypothesis says the sequence of beth numbers thus defined is the same as the sequence of aleph numbers, i.e.,
for all ordinals .
Specific cardinals
Beth null
Since this is defined to be , or aleph null, sets with cardinality include:
the natural numbers N
the rational numbers Q
the algebraic numbers
the computable numbers and computable sets
the set of finite sets of integers
the set of finite multisets of integers
the set of finite sequences of integers
Beth one
Sets with cardinality include:
the transcendental numbers
the irrational numbers
the real numbers R
the complex numbers C
the uncomputable real numbers
Euclidean space Rn
the power set of the natural numbers (the set of all subsets of the natural numbers)
the set of sequences of integers (i.e. all functions N → Z, often denoted ZN)
the set of sequences of real numbers, RN
the set of all real analytic functions from R to R
the set of all continuous functions from R to R
the set of all functions from R to R with at most countable discontinuities
the set of finite subsets of real numbers
the set
|
https://en.wikipedia.org/wiki/Illinois%20Mathematics%20and%20Science%20Academy
|
The Illinois Mathematics and Science Academy, or IMSA, is a three-year residential public secondary education institution in Aurora, Illinois, United States, with an enrollment of approximately 650 students.
Enrollment is generally offered to incoming sophomores, although younger students who have had the equivalent of one year of algebra and a 9th-grade science equivalent are eligible to apply. All applicants undergo a competitive admissions process involving the review of grades, teacher evaluations, student essays, and SAT scores. Historically, approximately one-third of applicants in any given year are admitted. Due to its nature as a public institution, there are no charges related to tuition, room, and board; however, there is an annual student fee that may be reduced or waived based on family income. IMSA has been consistently ranked by Newsweek as one of the top ten high schools in the country for math and science, and some of its graduates have become leaders in a variety of fields. It is the top-rated public high school in Illinois on Niche.com.
History
The school's founding president was former Batavia Superintendent Stephanie Pace Marshall, who was involved with the project from the start and with the school’s first legal counsel Richard L. Horwitz, helped form IMSA's original legislation, governing by-laws and slogan. Marshall retired from the position on June 30, 2007, and was later named President Emerita by the Board of Trustees.
Although the school received a budget cut in financial year 2002, its budget has since increased with the support of House Minority Leader Tom Cross. IMSA's chairperson was Paula Olszewski-Kubilius.
Admission
Prospective students, who are usually freshmen in high school but in some cases may be eighth graders, must complete an application to be considered for admission to IMSA.
The application process consists of an official transcript of the student's last 2½ years of school, scores from the SAT I, four student essays, three teacher evaluations in science, mathematics, and English, and a list of awards and extracurricular activities.
Historic admission statistics
In order to draw greater numbers of applications and "transform teaching and learning," IMSA has an outreach program run by the Center for Teaching and Learning (formerly known as The Center for Advancement and Renewal of Learning and Teaching (The Center@IMSA), then "Professional Field Services (PFS)"). Some students who are invited to attend IMSA are admitted on the condition that they successfully complete a three-week, intensive preparation course, known as EXCEL, over the summer. IMSA has a fairly low retention rate; the average retention rate per class is 85%. The reasons for this may include the difficulty of the IMSA curriculum, home-sickness, disciplinary expulsion, student's family moving out of state, and the inability for Illinois students to matriculate to IMSA after their sophomore year.
Academics
Students at IMSA take rig
|
https://en.wikipedia.org/wiki/S%C3%A9minaire%20de%20G%C3%A9om%C3%A9trie%20Alg%C3%A9brique%20du%20Bois%20Marie
|
In mathematics, the Séminaire de Géométrie Algébrique du Bois Marie (SGA) was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris. (The name came from the small wood on the estate in Bures-sur-Yvette where the IHÉS was located from 1962.) The seminar notes were eventually published in twelve volumes, all except one in the Springer Lecture Notes in Mathematics series.
Style
The material has a reputation of being hard to read for a number of reasons. More elementary or foundational parts were relegated to the EGA series of Grothendieck and Jean Dieudonné, causing long strings of logical dependencies in the statements. The style is very abstract and makes heavy use of category theory. Moreover, an attempt was made to achieve maximally general statements, while assuming that the reader is aware of the motivations and concrete examples.
First publication
The original notes to SGA were published in fascicles by the IHÉS, most of which went through two or three revisions. These were published as the seminar proceeded, beginning in the early 60's and continuing through most of the decade. They can still be found in large math libraries, but distribution was limited. In the late 60's and early 70's, the original seminar notes were comprehensively revised and rewritten to take into account later developments. In addition, a new volume, SGA 4½, was compiled by Pierre Deligne and published in 1977; it contains simplified and new results by Deligne within the scope of SGA4 as well as some material from SGA5, which had not yet appeared at that time. The revised notes, except for SGA2, were published by Springer in its Lecture Notes in Mathematics series.
After a dispute with Springer, Grothendieck refused the permission for reprints of the series. While these later revisions were more widely distributed than the original fascicles, they are still uncommon outside of libraries.
References to SGA typically mean the later, revised editions and not the original fascicles; some of the originals were labelled by capital letters, thus for example S.G.A.D. = SGA3 and S.G.A.A. = SGA4.
Series titles
The volumes of the SGA series are the following:
SGA1 Revêtements étales et groupe fondamental, 1960–1961 (Étale coverings and the fundamental group), Lecture Notes in Mathematics 224, 1971
SGA2 Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, 1961–1962 (Local cohomology of coherent sheaves and global and local Lefschetz theorems), North Holland 1968
SGA3 Schémas en groupes, 1962–1964 (Group schemes), Lecture Notes in Mathematics 151, 152 and 153, 1970
SGA4 Théorie des topos et cohomologie étale des schémas, 1963–1964 (Topos theory and étale cohomology), Lecture Notes in Mathematics 269, 270 and 305, 1972/3
SGA4½ Cohomologie étale (Étale cohomology), Lecture Notes in Mathema
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.