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https://en.wikipedia.org/wiki/Adams%20operation
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In mathematics, an Adams operation, denoted ψk for natural numbers k, is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams. The basic idea is to implement some fundamental identities in symmetric function theory, at the level of vector bundles or other representing object in more abstract theories.
Adams operations can be defined more generally in any λ-ring.
Adams operations in K-theory
Adams operations ψk on K theory (algebraic or topological) are characterized by the following properties.
ψk are ring homomorphisms.
ψk(l)= lk if l is the class of a line bundle.
ψk are functorial.
The fundamental idea is that for a vector bundle V on a topological space X, there is an analogy between Adams operators and exterior powers, in which
ψk(V) is to Λk(V)
as
the power sum Σ αk is to the k-th elementary symmetric function σk
of the roots α of a polynomial P(t). (Cf. Newton's identities.) Here Λk denotes the k-th exterior power. From classical algebra it is known that the power sums are certain integral polynomials Qk in the σk. The idea is to apply the same polynomials to the Λk(V), taking the place of σk. This calculation can be defined in a K-group, in which vector bundles may be formally combined by addition, subtraction and multiplication (tensor product). The polynomials here are called Newton polynomials (not, however, the Newton polynomials of interpolation theory).
Justification of the expected properties comes from the line bundle case, where V is a Whitney sum of line bundles. In this special case the result of any Adams operation is naturally a vector bundle, not a linear combination of ones in K-theory. Treating the line bundle direct factors formally as roots is something rather standard in algebraic topology (cf. the Leray–Hirsch theorem). In general a mechanism for reducing to that case comes from the splitting principle for vector bundles.
Adams operations in group representation theory
The Adams operation has a simple expression in group representation theory. Let G be a group and ρ a representation of G with character χ. The representation ψk(ρ) has character
References
Algebraic topology
Symmetric functions
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https://en.wikipedia.org/wiki/Cohomology%20operation
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In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if F is a functor defining a cohomology theory, then a cohomology operation should be a natural transformation from F to itself. Throughout there have been two basic points:
the operations can be studied by combinatorial means; and
the effect of the operations is to yield an interesting bicommutant theory.
The origin of these studies was the work of Pontryagin, Postnikov, and Norman Steenrod, who first defined the Pontryagin square, Postnikov square, and Steenrod square operations for singular cohomology, in the case of mod 2 coefficients. The combinatorial aspect there arises as a formulation of the failure of a natural diagonal map, at cochain level. The general theory of the Steenrod algebra of operations has been brought into close relation with that of the symmetric group.
In the Adams spectral sequence the bicommutant aspect is implicit in the use of Ext functors, the derived functors of Hom-functors; if there is a bicommutant aspect, taken over the Steenrod algebra acting, it is only at a derived level. The convergence is to groups in stable homotopy theory, about which information is hard to come by. This connection established the deep interest of the cohomology operations for homotopy theory, and has been a research topic ever since. An extraordinary cohomology theory has its own cohomology operations, and these may exhibit a richer set on constraints.
Formal definition
A cohomology operation of type
is a natural transformation of functors
defined on CW complexes.
Relation to Eilenberg–MacLane spaces
Cohomology of CW complexes is representable by an Eilenberg–MacLane space, so by the Yoneda lemma a cohomology operation of type is given by a homotopy class of maps . Using representability once again, the cohomology operation is given by an element of .
Symbolically, letting denote the set of homotopy classes of maps from to ,
See also
Secondary cohomology operation
References
Algebraic topology
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https://en.wikipedia.org/wiki/Probabilistic%20argument
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Probabilistic argument may refer to:
Probabilistic argument, any argument involving probability theory
Probabilistic method, a method of non-constructive existence proof in mathematics
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https://en.wikipedia.org/wiki/T-square%20%28fractal%29
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In mathematics, the T-square is a two-dimensional fractal. It has a boundary of infinite length bounding a finite area. Its name comes from the drawing instrument known as a T-square.
Algorithmic description
It can be generated from using this algorithm:
Image 1:
Start with a square. (The black square in the image)
Image 2:
At each convex corner of the previous image, place another square, centered at that corner, with half the side length of the square from the previous image.
Take the union of the previous image with the collection of smaller squares placed in this way.
Images 3–6:
Repeat step 2.
The method of creation is rather similar to the ones used to create a Koch snowflake or a Sierpinski triangle, "both based on recursively drawing equilateral triangles and the Sierpinski carpet."
Properties
The T-square fractal has a fractal dimension of ln(4)/ln(2) = 2. The black surface extent is almost everywhere in the bigger square, for once a point has been darkened, it remains black for every other iteration; however some points remain white.
The fractal dimension of the boundary equals .
Using mathematical induction one can prove that for each n ≥ 2 the number of new squares that are added at stage n equals .
The T-Square and the chaos game
The T-square fractal can also be generated by an adaptation of the chaos game, in which a point jumps repeatedly half-way towards the randomly chosen vertices of a square. The T-square appears when the jumping point is unable to target the vertex directly opposite the vertex previously chosen. That is, if the current vertex is v[i] and the previous vertex was v[i-1], then v[i] ≠ v[i-1] + vinc, where vinc = 2 and modular arithmetic means that 3 + 2 = 1, 4 + 2 = 2:
If vinc is given different values, allomorphs of the T-square appear that are computationally equivalent to the T-square but very different in appearance:
T-square fractal and Sierpiński triangle
The T-square fractal can be derived from the Sierpiński triangle, and vice versa, by adjusting the angle at which sub-elements of the original fractal are added from the center outwards.
See also
List of fractals by Hausdorff dimension
The Toothpick sequence generates a similar pattern
H tree
References
Further reading
Iterated function system fractals
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https://en.wikipedia.org/wiki/Manifold%20decomposition
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In topology, a branch of mathematics, a manifold M may be decomposed or split by writing M as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form M.
Manifold decomposition works in two directions: one can start with the smaller pieces and build up a manifold, or start with a large manifold and decompose it. The latter has proven a very useful way to study manifolds: without tools like decomposition, it is sometimes very hard to understand a manifold. In particular, it has been useful in attempts to classify 3-manifolds and also in proving the higher-dimensional Poincaré conjecture.
The table below is a summary of the various manifold-decomposition techniques. The column labeled "M" indicates what kind of manifold can be decomposed; the column labeled "How it is decomposed" indicates how, starting with a manifold, one can decompose it into smaller pieces; the column labeled "The pieces" indicates what the pieces can be; and the column labeled "How they are combined" indicates how the smaller pieces are combined to make the large manifold.
See also
Surgery theory
Geometric topology
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https://en.wikipedia.org/wiki/Vaughan%20Jones
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Sir Vaughan Frederick Randal Jones (31 December 19526 September 2020) was a New Zealand mathematician known for his work on von Neumann algebras and knot polynomials. He was awarded a Fields Medal in 1990.
Early life
Jones was born in Gisborne, New Zealand, on 31 December 1952. He was brought up in Cambridge, New Zealand, where he attended St Peter's School. He subsequently transferred to Auckland Grammar School after winning the Gillies Scholarship, and graduated in 1969 from Auckland Grammar. He went on to complete his undergraduate studies at the University of Auckland, obtaining a BSc in 1972 and an MSc in 1973. For his graduate studies, he went to Switzerland, where he completed his PhD at the University of Geneva in 1979. His thesis, titled Actions of finite groups on the hyperfinite II1 factor, was written under the supervision of André Haefliger, and won him the Vacheron Constantin Prize.
Career
Jones moved to the United States in 1980. There, he taught at the University of California, Los Angeles (1980–1981), and the University of Pennsylvania (1981–1985), before being appointed as professor of mathematics at the University of California, Berkeley. His work on knot polynomials, with the discovery of what is now called the Jones polynomial, was from an unexpected direction with origins in the theory of von Neumann algebras, an area of analysis already much developed by Alain Connes. It led to the solution of a number of classical problems of knot theory, to increased interest in low-dimensional topology, and the development of quantum topology.
Jones taught at Vanderbilt University as Stevenson Distinguished Professor of mathematics from 2011 until his death. He remained Professor Emeritus at University of California, Berkeley, where he had been on the faculty from 1985 to 2011 and was a Distinguished Alumni Professor at the University of Auckland.
Jones was made an honorary vice-president for life of the International Guild of Knot Tyers in 1992. The Jones Medal, created by the Royal Society of New Zealand in 2010, is named after him.
Personal life
Jones met his wife, Martha Myers, during a ski camp for foreign students while they were studying in Switzerland. She was there as a Fulbright scholar, and subsequently became an associate professor of medicine, health and society. Together, they have three children.
Jones died on 6 September 2020 at age 67 from health complications resulting from a severe ear infection.
Jones was a certified barista.
Honours and awards
1990awarded the Fields Medal
1990elected Fellow of the Royal Society
1991awarded the Rutherford Medal by the Royal Society of New Zealand
1991awarded the degree of Doctor of Science by the University of Auckland
1992elected to the Australian Academy of Science as a Corresponding Fellow
1992awarded a Miller Professorship at the University of California Berkeley
2002appointed Distinguished Companion of the New Zealand Order of Merit (DCNZM) in the 200
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https://en.wikipedia.org/wiki/Asymmetric%20relation
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In mathematics, an asymmetric relation is a binary relation on a set where for all if is related to then is not related to
Formal definition
A binary relation on is any subset of Given write if and only if which means that is shorthand for The expression is read as " is related to by " The binary relation is called if for all if is true then is false; that is, if then
This can be written in the notation of first-order logic as
A logically equivalent definition is:
for all at least one of and is ,
which in first-order logic can be written as:
An example of an asymmetric relation is the "less than" relation between real numbers: if then necessarily is not less than The "less than or equal" relation on the other hand, is not asymmetric, because reversing for example, produces and both are true.
Asymmetry is not the same thing as "not symmetric": the less-than-or-equal relation is an example of a relation that is neither symmetric nor asymmetric. The empty relation is the only relation that is (vacuously) both symmetric and asymmetric.
Properties
A relation is asymmetric if and only if it is both antisymmetric and irreflexive.
Restrictions and converses of asymmetric relations are also asymmetric. For example, the restriction of from the reals to the integers is still asymmetric, and the inverse of is also asymmetric.
A transitive relation is asymmetric if and only if it is irreflexive: if and transitivity gives contradicting irreflexivity.
As a consequence, a relation is transitive and asymmetric if and only if it is a strict partial order.
Not all asymmetric relations are strict partial orders. An example of an asymmetric non-transitive, even antitransitive relation is the relation: if beats then does not beat and if beats and beats then does not beat
An asymmetric relation need not have the connex property. For example, the strict subset relation is asymmetric, and neither of the sets and is a strict subset of the other. A relation is connex if and only if its complement is asymmetric.
See also
Tarski's axiomatization of the reals – part of this is the requirement that over the real numbers be asymmetric.
References
Binary relations
Asymmetry
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https://en.wikipedia.org/wiki/Hyperbolic%20motion
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In geometry, hyperbolic motions are isometric automorphisms of a hyperbolic space. Under composition of mappings, the hyperbolic motions form a continuous group. This group is said to characterize the hyperbolic space. Such an approach to geometry was cultivated by Felix Klein in his Erlangen program. The idea of reducing geometry to its characteristic group was developed particularly by Mario Pieri in his reduction of the primitive notions of geometry to merely point and motion.
Hyperbolic motions are often taken from inversive geometry: these are mappings composed of reflections in a line or a circle (or in a hyperplane or a hypersphere for hyperbolic spaces of more than two dimensions). To distinguish the hyperbolic motions, a particular line or circle is taken as the absolute. The proviso is that the absolute must be an invariant set of all hyperbolic motions. The absolute divides the plane into two connected components, and hyperbolic motions must not permute these components.
One of the most prevalent contexts for inversive geometry and hyperbolic motions is in the study of mappings of the complex plane by Möbius transformations. Textbooks on complex functions often mention two common models of hyperbolic geometry: the Poincaré half-plane model where the absolute is the real line on the complex plane, and the Poincaré disk model where the absolute is the unit circle in the complex plane.
Hyperbolic motions can also be described on the hyperboloid model of hyperbolic geometry.
This article exhibits these examples of the use of hyperbolic motions: the extension of the metric to the half-plane, and in the location of a quasi-sphere of a hypercomplex number system.
Motions on the hyperbolic plane
Every motion (transformation or isometry) of the hyperbolic plane to itself can be realized as the composition of at most three reflections. In n-dimensional hyperbolic space, up to n+1 reflections might be required. (These are also true for Euclidean and spherical geometries, but the classification below is different.)
All the isometries of the hyperbolic plane can be classified into these classes:
Orientation preserving
the identity isometry — nothing moves; zero reflections; zero degrees of freedom.
inversion through a point (half turn) — two reflections through mutually perpendicular lines passing through the given point, i.e. a rotation of 180 degrees around the point; two degrees of freedom.
rotation around a normal point — two reflections through lines passing through the given point (includes inversion as a special case); points move on circles around the center; three degrees of freedom.
"rotation" around an ideal point (horolation) — two reflections through lines leading to the ideal point; points move along horocycles centered on the ideal point; two degrees of freedom.
translation along a straight line — two reflections through lines perpendicular to the given line; points off the given line move along hypercycles; three degr
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https://en.wikipedia.org/wiki/SIAM%20Journal%20on%20Computing
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The SIAM Journal on Computing is a scientific journal focusing on the mathematical and formal aspects of computer science. It is published by the Society for Industrial and Applied Mathematics (SIAM).
Although its official ISO abbreviation is SIAM J. Comput., its publisher and contributors frequently use the shorter abbreviation SICOMP.
SICOMP typically hosts the special issues of the IEEE Annual Symposium on Foundations of Computer Science (FOCS) and the Annual ACM Symposium on Theory of Computing (STOC), where about 15% of papers published in FOCS and STOC each year are invited to these special issues. For example, Volume 48 contains 11 out of 85 papers published in FOCS 2016.
References
External links
SIAM Journal on Computing
bibliographic information on DBLP
Computer science journals
Academic journals established in 1972
Computing
Bimonthly journals
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https://en.wikipedia.org/wiki/Weighted%20geometric%20mean
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In statistics, the weighted geometric mean is a generalization of the geometric mean using the weighted arithmetic mean.
Given a sample and weights , it is calculated as:
The second form above illustrates that the logarithm of the geometric mean is the weighted arithmetic mean of the logarithms of the individual values.
If all the weights are equal, the weighted geometric mean simplifies to the ordinary unweighted geometric mean.
See also
Average
Central tendency
Summary statistics
Weighted arithmetic mean
Weighted harmonic mean
External links
Non-Newtonian calculus website
Means
Mathematical analysis
Non-Newtonian calculus
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https://en.wikipedia.org/wiki/SKI%20combinator%20calculus
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The SKI combinator calculus is a combinatory logic system and a computational system. It can be thought of as a computer programming language, though it is not convenient for writing software. Instead, it is important in the mathematical theory of algorithms because it is an extremely simple Turing complete language. It can be likened to a reduced version of the untyped lambda calculus. It was introduced by Moses Schönfinkel and Haskell Curry.
All operations in lambda calculus can be encoded via abstraction elimination into the SKI calculus as binary trees whose leaves are one of the three symbols S, K, and I (called combinators).
Notation
Although the most formal representation of the objects in this system requires binary trees, for simpler typesetting they are often represented as parenthesized expressions, as a shorthand for the tree they represent. Any subtrees may be parenthesized, but often only the right-side subtrees are parenthesized, with left associativity implied for any unparenthesized applications. For example, ISK means ((IS)K). Using this notation, a tree whose left subtree is the tree KS and whose right subtree is the tree SK can be written as KS(SK). If more explicitness is desired, the implied parentheses can be included as well: ((KS)(SK)).
Informal description
Informally, and using programming language jargon, a tree (xy) can be thought of as a function x applied to an argument y. When evaluated (i.e., when the function is "applied" to the argument), the tree "returns a value", i.e., transforms into another tree. The "function", "argument" and the "value" are either combinators or binary trees. If they are binary trees, they may be thought of as functions too, if needed.
The evaluation operation is defined as follows:
(x, y, and z represent expressions made from the functions S, K, and I, and set values):
I returns its argument:
Ix = x
K, when applied to any argument x, yields a one-argument constant function Kx, which, when applied to any argument y, returns x:
Kxy = x
S is a substitution operator. It takes three arguments and then returns the first argument applied to the third, which is then applied to the result of the second argument applied to the third. More clearly:
Sxyz = xz(yz)
Example computation: SKSK evaluates to KK(SK) by the S-rule. Then if we evaluate KK(SK), we get K by the K-rule. As no further rule can be applied, the computation halts here.
For all trees x and all trees y, SKxy will always evaluate to y in two steps, Ky(xy) = y, so the ultimate result of evaluating SKxy will always equal the result of evaluating y. We say that SKx and I are "functionally equivalent" because they always yield the same result when applied to any y.
From these definitions it can be shown that SKI calculus is not the minimum system that can fully perform the computations of lambda calculus, as all occurrences of I in any expression can be replaced by (SKK) or (SKS) or (SK whatever) and the resulting expression
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https://en.wikipedia.org/wiki/Iterative%20Viterbi%20decoding
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Iterative Viterbi decoding is an algorithm that spots the subsequence S of an observation O = {o1, ..., on} having the highest average probability (i.e., probability scaled by the length of S) of being generated by a given hidden Markov model M with m states. The algorithm uses a modified Viterbi algorithm as an internal step.
The scaled probability measure was first proposed by John S. Bridle. An early algorithm to solve this problem, sliding window, was proposed by Jay G. Wilpon et al., 1989, with constant cost T = mn2/2.
A faster algorithm consists of an iteration of calls to the Viterbi algorithm, reestimating a filler score until convergence.
The algorithm
A basic (non-optimized) version, finding the sequence s with the smallest normalized distance from some subsequence of t is:
// input is placed in observation s[1..n], template t[1..m],
// and [[distance matrix]] d[1..n,1..m]
// remaining elements in matrices are solely for internal computations
(int, int, int) AverageSubmatchDistance(char s[0..(n+1)], char t[0..(m+1)], int d[1..n,0..(m+1)]) {
// score, subsequence start, subsequence end
declare int e, B, E
t'[0] := t'[m+1] := s'[0] := s'[n+1] := 'e'
e := random()
do
e' := e
for i := 1 to n do d'[i,0] := d'[i,m+1] := e
(e, B, E) := ViterbiDistance(s', t', d')
e := e/(E-B+1)
until (e == e')
return (e, B, E)
}
The ViterbiDistance() procedure returns the tuple (e, B, E), i.e., the Viterbi score "e" for the match of t and the selected entry (B) and exit (E) points from it. "B" and "E" have to be recorded using a simple modification to Viterbi.
A modification that can be applied to CYK tables, proposed by Antoine Rozenknop, consists in subtracting e from all elements of the initial matrix d.
References
Silaghi, M., "Spotting Subsequences matching a HMM using the Average Observation Probability Criteria with application to Keyword Spotting", AAAI, 2005.
Rozenknop, Antoine, and Silaghi, Marius; "Algorithme de décodage de treillis selon le critère de coût moyen pour la reconnaissance de la parole", TALN 2001.
Further reading
Error detection and correction
Markov models
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https://en.wikipedia.org/wiki/Augmented%20truncated%20cube
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In geometry, the augmented truncated cube is one of the Johnson solids (). As its name suggests, it is created by attaching a square cupola () onto one octagonal face of a truncated cube.
References
Norman W. Johnson, "Convex Solids with Regular Faces", Canadian Journal of Mathematics, 18, 1966, pages 169–200. Contains the original enumeration of the 92 solids and the conjecture that there are no others.
The first proof that there are only 92 Johnson solids.
External links
Johnson solids
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https://en.wikipedia.org/wiki/Biaugmented%20truncated%20cube
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In geometry, the biaugmented truncated cube is one of the Johnson solids (). As its name suggests, it is created by attaching two square cupolas () onto two parallel octagonal faces of a truncated cube.
External links
Johnson solids
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https://en.wikipedia.org/wiki/Augmented%20truncated%20dodecahedron
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In geometry, the augmented truncated dodecahedron is one of the Johnson solids (). As its name suggests, it is created by attaching a pentagonal cupola () onto one decagonal face of a truncated dodecahedron.
External links
Johnson solids
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https://en.wikipedia.org/wiki/Parabiaugmented%20truncated%20dodecahedron
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In geometry, the parabiaugmented truncated dodecahedron is one of the Johnson solids (). As its name suggests, it is created by attaching two pentagonal cupolas () onto two parallel decagonal faces of a truncated dodecahedron.
External links
Johnson solids
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https://en.wikipedia.org/wiki/Metabiaugmented%20truncated%20dodecahedron
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In geometry, the metabiaugmented truncated dodecahedron is one of the Johnson solids (). As its name suggests, it is created by attaching two pentagonal cupolas () onto two nonadjacent, nonparallel decagonal faces of a truncated dodecahedron.
External links
Johnson solids
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https://en.wikipedia.org/wiki/Triaugmented%20truncated%20dodecahedron
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In geometry, the triaugmented truncated dodecahedron is one of the Johnson solids (); of them, it has the greatest volume in proportion to the cube of the side length. As its name suggests, it is created by attaching three pentagonal cupolas () onto three nonadjacent decagonal faces of a truncated dodecahedron.
External links
Johnson solids
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https://en.wikipedia.org/wiki/Seifert%E2%80%93Weber%20space
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In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds.
It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert–Weber space. Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space.
With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five. Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold.
It is a (finite volume) quotient space of the (non-finite volume) order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.
The Seifert–Weber space is a rational homology sphere, and its first homology group is isomorphic to . William Thurston conjectured that the Seifert–Weber space is not a Haken manifold, that is, it does not contain any incompressible surfaces; proved the conjecture with the aid of their computer software Regina.
References
External links
Regina – Support Data: Weber-Seifert dodecahedral space
The Weber–Seifert dodecahedral space: Answering a computational challenge
3-manifolds
Riemannian geometry
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https://en.wikipedia.org/wiki/Erich%20Hecke
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Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician known for his work in number theory and the theory of modular forms.
Biography
Hecke was born in Buk, Province of Posen, German Empire (now Poznań, Poland). He obtained his doctorate in Göttingen under the supervision of David Hilbert.
Kurt Reidemeister and Heinrich Behnke were among his students.
In 1933 Hecke signed the Loyalty Oath of German Professors to Adolf Hitler and the National Socialist State, but was later known as being opposed to the Nazis.
Hecke died in Copenhagen, Denmark.
André Weil, in the foreword to his text Basic Number Theory says: "To improve upon Hecke, in a treatment along classical lines of the theory of algebraic numbers, would be a futile and impossible task", referring to Hecke's book "Lectures on the Theory of Algebraic Numbers."
Research
His early work included establishing the functional equation for the Dedekind zeta function, with a proof based on theta functions. The method extended to the L-functions associated to a class of characters now known as Hecke characters or idele class characters; such L-functions are now known as Hecke L-functions. He devoted most of his research to the theory of modular forms, creating the general theory of cusp forms (holomorphic, for GL(2)), as it is now understood in the classical setting.
Recognition
He was a Plenary Speaker of the ICM in 1936 in Oslo.
See also
List of things named after Erich Hecke
Hecke algebra (disambiguation)
Tate's thesis
References
External links
1887 births
1947 deaths
People from Poznań County
People from the Province of Posen
20th-century German mathematicians
Number theorists
Academic staff of the University of Hamburg
Academic staff of the University of Basel
Academic staff of the University of Göttingen
University of Breslau alumni
University of Göttingen alumni
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https://en.wikipedia.org/wiki/MacTutor%20History%20of%20Mathematics%20Archive
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The MacTutor History of Mathematics Archive is a website maintained by John J. O'Connor and Edmund F. Robertson and hosted by the University of St Andrews in Scotland. It contains detailed biographies on many historical and contemporary mathematicians, as well as information on famous curves and various topics in the history of mathematics.
The History of Mathematics archive was an outgrowth of Mathematical MacTutor system, a HyperCard database by the same authors, which won them the European Academic Software award in 1994. In the same year, they founded their web site. it has biographies on over 2800 mathematicians and scientists.
In 2015, O'Connor and Robertson won the Hirst Prize of the London Mathematical Society for their work. The citation for the Hirst Prize calls the archive "the most widely used and influential web-based resource in history of mathematics".
See also
Mathematics Genealogy Project
MathWorld
PlanetMath
References
External links
Mathematical MacTutor system
Works about the history of mathematics
Mathematics websites
University of St Andrews
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https://en.wikipedia.org/wiki/Dirichlet%20integral
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In mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line:
This integral is not absolutely convergent, meaning is not Lebesgue-integrable, because the Dirichlet integral is infinite in the sense of Lebesgue integration. It is, however, finite in the sense of the improper Riemann integral or the generalized Riemann or Henstock–Kurzweil integral. This can be seen by using Dirichlet's test for improper integrals.
It is a good illustration of special techniques for evaluating definite integrals. The sine integral, an antiderivative of the sinc function, is not an elementary function. However the improper definite integral can be determined in several ways: the Laplace transform, double integration, differentiating under the integral sign, contour integration, and the Dirichlet kernel.
Evaluation
Laplace transform
Let be a function defined whenever Then its Laplace transform is given by
if the integral exists.
A property of the Laplace transform useful for evaluating improper integrals is
provided exists.
In what follows, one needs the result which is the Laplace transform of the function (see the section 'Differentiating under the integral sign' for a derivation) as well as a version of Abel's theorem (a consequence of the final value theorem for the Laplace transform).
Therefore,
Double integration
Evaluating the Dirichlet integral using the Laplace transform is equivalent to calculating the same double definite integral by changing the order of integration, namely,
Differentiation under the integral sign (Feynman's trick)
First rewrite the integral as a function of the additional variable namely, the Laplace transform of So let
In order to evaluate the Dirichlet integral, we need to determine The continuity of can be justified by applying the dominated convergence theorem after integration by parts. Differentiate with respect to and apply the Leibniz rule for differentiating under the integral sign to obtain
Now, using Euler's formula one can express the sine function in terms of complex exponentials:
Therefore,
Integrating with respect to gives
where is a constant of integration to be determined. Since using the principal value. This means that for
Finally, by continuity at we have as before.
Complex contour integration
Consider
As a function of the complex variable it has a simple pole at the origin, which prevents the application of Jordan's lemma, whose other hypotheses are satisfied.
Define then a new function
The pole has been moved to the negative imaginary axis, so can be integrated along the semicircle of radius centered at extending in the positive imaginary direction, and closed along the real axis. One then takes the limit
The complex integral is zero by the residue theorem, as there are no poles inside the integr
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https://en.wikipedia.org/wiki/Projective%20hierarchy
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In the mathematical field of descriptive set theory, a subset of a Polish space is projective if it is for some positive integer . Here is
if is analytic
if the complement of , , is
if there is a Polish space and a subset such that is the projection of onto ; that is,
The choice of the Polish space in the third clause above is not very important; it could be replaced in the definition by a fixed uncountable Polish space, say Baire space or Cantor space or the real line.
Relationship to the analytical hierarchy
There is a close relationship between the relativized analytical hierarchy on subsets of Baire space (denoted by lightface letters and ) and the projective hierarchy on subsets of Baire space (denoted by boldface letters and ). Not every subset of Baire space is . It is true, however, that if a subset X of Baire space is then there is a set of natural numbers A such that X is . A similar statement holds for sets. Thus the sets classified by the projective hierarchy are exactly the sets classified by the relativized version of the analytical hierarchy. This relationship is important in effective descriptive set theory.
A similar relationship between the projective hierarchy and the relativized analytical hierarchy holds for subsets of Cantor space and, more generally, subsets of any effective Polish space.
Table
References
Descriptive set theory
Mathematical logic hierarchies
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https://en.wikipedia.org/wiki/Nagata%E2%80%93Smirnov%20metrization%20theorem
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In topology, the Nagata–Smirnov metrization theorem characterizes when a topological space is metrizable. The theorem states that a topological space is metrizable if and only if it is regular, Hausdorff and has a countably locally finite (that is, -locally finite) basis.
A topological space is called a regular space if every non-empty closed subset of and a point p not contained in admit non-overlapping open neighborhoods.
A collection in a space is countably locally finite (or -locally finite) if it is the union of a countable family of locally finite collections of subsets of
Unlike Urysohn's metrization theorem, which provides only a sufficient condition for metrizability, this theorem provides both a necessary and sufficient condition for a topological space to be metrizable. The theorem is named after Junichi Nagata and Yuriĭ Mikhaĭlovich Smirnov, whose (independent) proofs were published in 1950 and 1951, respectively.
See also
Notes
References
.
.
General topology
Theorems in topology
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https://en.wikipedia.org/wiki/Pointed%20space
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In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as that remains unchanged during subsequent discussion, and is kept track of during all operations.
Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map between a pointed space with basepoint and a pointed space with basepoint is a based map if it is continuous with respect to the topologies of and and if This is usually denoted
Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint.
The pointed set concept is less important; it is anyway the case of a pointed discrete space.
Pointed spaces are often taken as a special case of the relative topology, where the subset is a single point. Thus, much of homotopy theory is usually developed on pointed spaces, and then moved to relative topologies in algebraic topology.
Category of pointed spaces
The class of all pointed spaces forms a category Top with basepoint preserving continuous maps as morphisms. Another way to think about this category is as the comma category, ( Top) where is any one point space and Top is the category of topological spaces. (This is also called a coslice category denoted Top.) Objects in this category are continuous maps Such maps can be thought of as picking out a basepoint in Morphisms in ( Top) are morphisms in Top for which the following diagram commutes:
It is easy to see that commutativity of the diagram is equivalent to the condition that preserves basepoints.
As a pointed space, is a zero object in Top, while it is only a terminal object in Top.
There is a forgetful functor Top Top which "forgets" which point is the basepoint. This functor has a left adjoint which assigns to each topological space the disjoint union of and a one-point space whose single element is taken to be the basepoint.
Operations on pointed spaces
A subspace of a pointed space is a topological subspace which shares its basepoint with so that the inclusion map is basepoint preserving.
One can form the quotient of a pointed space under any equivalence relation. The basepoint of the quotient is the image of the basepoint in under the quotient map.
One can form the product of two pointed spaces as the topological product with serving as the basepoint.
The coproduct in the category of pointed spaces is the , which can be thought of as the 'one-point union' of spaces.
The smash product of two pointed spaces is essentially the quotient of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into a symmetric monoidal category with the pointed 0-sphere as the unit object, but this is false for general spaces: the associativ
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https://en.wikipedia.org/wiki/Basepoint
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Basepoint may refer to a point singled out in a:
Pointed set, or in a
Pointed space
See also
Origin (mathematics)
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https://en.wikipedia.org/wiki/Hyperbolic%203-manifold
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In mathematics, more precisely in topology and differential geometry, a hyperbolic 3-manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all its sectional curvatures equal to −1. It is generally required that this metric be also complete: in this case the manifold can be realised as a quotient of the 3-dimensional hyperbolic space by a discrete group of isometries (a Kleinian group).
Hyperbolic 3-manifolds of finite volume have a particular importance in 3-dimensional topology as follows from Thurston's geometrisation conjecture proved by Perelman. The study of Kleinian groups is also an important topic in geometric group theory.
Importance in topology
Hyperbolic geometry is the most rich and least understood of the eight geometries in dimension 3 (for example, for all other geometries it is not hard to give an explicit enumeration of the finite-volume manifolds with this geometry, while this is far from being the case for hyperbolic manifolds). After the proof of the Geometrisation conjecture, understanding the topological properties of hyperbolic 3-manifolds is thus a major goal of 3-dimensional topology. Recent breakthroughs of Kahn–Markovic, Wise, Agol and others have answered most long-standing open questions on the topic but there are still many less prominent ones which have not been solved.
In dimension 2 almost all closed surfaces are hyperbolic (all but the sphere, projective plane, torus and Klein bottle). In dimension 3 this is far from true: there are many ways to construct infinitely many non-hyperbolic closed manifolds. On the other hand, the heuristic statement that "a generic 3-manifold tends to be hyperbolic" is verified in many contexts. For example, any knot which is not either a satellite knot or a torus knot is hyperbolic. Moreover, almost all Dehn surgeries on a hyperbolic knot yield a hyperbolic manifold. A similar result is true of links (Thurston's hyperbolic Dehn surgery theorem), and since all 3-manifolds are obtained as surgeries on a link in the 3-sphere this gives a more precise sense to the informal statement. Another sense in which "almost all" manifolds are hyperbolic in dimension 3 is that of random models. For example, random Heegaard splittings of genus at least 2 are almost surely hyperbolic (when the complexity of the gluing map goes to infinity).
The relevance of the hyperbolic geometry of a 3-manifold to its topology also comes from the Mostow rigidity theorem, which states that the hyperbolic structure of a hyperbolic 3-manifold of finite volume is uniquely determined by its homotopy type. In particular geometric invariant such as the volume can be used to define new topological invariants.
Structure
Manifolds of finite volume
In this case one important tool to understand the geometry of a manifold is the thick-thin decomposition. It states that a hyperbolic 3-manifold of finite volume has a decomposition into two parts:
the thick part,
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https://en.wikipedia.org/wiki/Matrix%20unit
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In linear algebra, a matrix unit is a matrix with only one nonzero entry with value 1. The matrix unit with a 1 in the ith row and jth column is denoted as . For example, the 3 by 3 matrix unit with i = 1 and j = 2 is
A vector unit is a standard unit vector.
A single-entry matrix generalizes the matrix unit for matrices with only one nonzero entry of any value, not necessarily of value 1.
Properties
The set of m by n matrix units is a basis of the space of m by n matrices.
The product of two matrix units of the same square shape satisfies the relation
where is the Kronecker delta.
The group of scalar n-by-n matrices over a ring R is the centralizer of the subset of n-by-n matrix units in the set of n-by-n matrices over R.
The matrix norm (induced by the same two vector norms) of a matrix unit is equal to 1.
When multiplied by another matrix, it isolates a specific row or column in arbitrary position. For example, for any 3-by-3 matrix A:
References
Sparse matrices
1 (number)
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https://en.wikipedia.org/wiki/Abraham%20Wald
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Abraham Wald (; , ; – ) was a Jewish Hungarian mathematician who contributed to decision theory, geometry and econometrics, and founded the field of sequential analysis. One of his well-known statistical works was written during World War II on how to minimize the damage to bomber aircraft and took into account the survivorship bias in his calculations. He spent his research career at Columbia University. He was the grandson of Rabbi Moshe Shmuel Glasner.
Life and career
Wald was born on 31 October 1902 in Kolozsvár, Transylvania, in the Kingdom of Hungary. A religious Jew, he did not attend school on Saturdays, as was then required by the Hungarian school system, and so he was homeschooled by his parents until college. His parents were quite knowledgeable and competent as teachers.
In 1928, he graduated in mathematics from the King Ferdinand I University. In 1927, he had entered graduate school at the University of Vienna, from which he graduated in 1931 with a Ph.D. in mathematics. His advisor there was Karl Menger.
Despite Wald's brilliance, he could not obtain a university position because of Austrian discrimination against Jews. However, Oskar Morgenstern created a position for Wald in economics. When Nazi Germany annexed Austria in 1938, the discrimination against Jews intensified. In particular, Wald and his family were persecuted as Jews. Wald immigrated to the United States at the invitation of the Cowles Commission for Research in Economics, to work on econometrics research.
During World War II, Wald was a member of the Statistical Research Group (SRG) at Columbia University, where he applied his statistical skills to various wartime problems. They included methods of sequential analysis and sampling inspection. One of the problems that the SRG worked on was to examine the distribution of damage to aircraft returning after flying missions to provide advice on how to minimize bomber losses to enemy fire. Wald derived a useful means of estimating the damage distribution for all aircraft that flew from the data on the damage distribution of all aircraft that returned. His work is considered seminal in the discipline of operational research, which was then fledgling.
Wald and his wife died in 1950 when the Air India plane (VT-CFK, a DC-3 aircraft) in which they were travelling crashed near the Rangaswamy Pillar in the northern part of the Nilgiri Mountains, in southern India, on an extensive lecture tour at the invitation of the Indian government. He had visited the Indian Statistical Institute at Calcutta and was to attend the Indian Science Congress at Bangalore in January. Their two children were back at home in the United States.
After his death, Wald was criticized by Sir Ronald A. Fisher FRS. Fisher attacked Wald for being a mathematician without scientific experience who had written an incompetent book on statistics. Fisher particularly criticized Wald's work on the design of experiments and alleged ignorance of the basic id
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https://en.wikipedia.org/wiki/Eccentricity%20%28mathematics%29
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In mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes its shape.
One can think of the eccentricity as a measure of how much a conic section deviates from being circular. In particular:
The eccentricity of a circle is 0.
The eccentricity of an ellipse which is not a circle is between 0 and 1.
The eccentricity of a parabola is 1.
The eccentricity of a hyperbola is greater than 1.
The eccentricity of a pair of lines is
Two conic sections with the same eccentricity are similar.
Definitions
Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity, commonly denoted as .
The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented with its axis vertical, the eccentricity is
where β is the angle between the plane and the horizontal and α is the angle between the cone's slant generator and the horizontal. For the plane section is a circle, for a parabola. (The plane must not meet the vertex of the cone.)
The linear eccentricity of an ellipse or hyperbola, denoted (or sometimes or ), is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis : that is, (lacking a center, the linear eccentricity for parabolas is not defined). It is worth to note that a parabola can be treated as an ellipse or a hyperbola, but with one focal point at infinity.
Alternative names
The eccentricity is sometimes called the first eccentricity to distinguish it from the second eccentricity and third eccentricity defined for ellipses (see below). The eccentricity is also sometimes called the numerical eccentricity.
In the case of ellipses and hyperbolas the linear eccentricity is sometimes called the half-focal separation.
Notation
Three notational conventions are in common use:
for the eccentricity and for the linear eccentricity.
for the eccentricity and for the linear eccentricity.
or for the eccentricity and for the linear eccentricity (mnemonic for half-focal separation).
This article uses the first notation.
Values
Here, for the ellipse and the hyperbola, is the length of the semi-major axis and is the length of the semi-minor axis.
When the conic section is given in the general quadratic form
the following formula gives the eccentricity if the conic section is not a parabola (which has eccentricity equal to 1), not a degenerate hyperbola or degenerate ellipse, and not an imaginary ellipse:
where if the determinant of the 3×3 matrix
is negative or if that determinant is positive.
Ellipses
The eccentricity of an ellipse is strictly less than 1. When circles (which have eccentricity 0) are counted as ellipses, the eccentricity of an ellipse is greater than or
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https://en.wikipedia.org/wiki/Ellipse%20%28disambiguation%29
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In mathematics, an ellipse is a geometrical figure.
Ellipse may also refer to:
MacAdam ellipse, an area in a chromaticity diagram
Elliptic leaf shape
Superellipse, a geometric figure
As a name, it may also be:
The Ellipse, an area in Washington, D.C., United States
Ellipse Programmé, a French animation studio
Elipse, a Yugoslav rock band
Ellipse, a 2009 album by Imogen Heap
"Ellipse", a song from the album In Silence We Yearn by Oh Hiroshima
Explorer Ellipse, an American homebuilt aircraft design
La société Ellipse, a French aircraft manufacturer
Similar terms
Ellipsis, a punctuation mark
Ellipsis, a rhetorical suppression of words to give an expression more liveliness
Eclipse, an astronomical event
Elliptical (trainer), a stationary exercise machine
See also
Ellipsis (disambiguation)
Oval (disambiguation)
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https://en.wikipedia.org/wiki/Dirichlet%27s%20principle
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In mathematics, and particularly in potential theory, Dirichlet's principle is the assumption that the minimizer of a certain energy functional is a solution to Poisson's equation.
Formal statement
Dirichlet's principle states that, if the function is the solution to Poisson's equation
on a domain of with boundary condition
on the boundary ,
then u can be obtained as the minimizer of the Dirichlet energy
amongst all twice differentiable functions such that on (provided that there exists at least one function making the Dirichlet's integral finite). This concept is named after the German mathematician Peter Gustav Lejeune Dirichlet.
History
The name "Dirichlet's principle" is due to Riemann, who applied it in the study of complex analytic functions.
Riemann (and others such as Gauss and Dirichlet) knew that Dirichlet's integral is bounded below, which establishes the existence of an infimum; however, he took for granted the existence of a function that attains the minimum. Weierstrass published the first criticism of this assumption in 1870, giving an example of a functional that has a greatest lower bound which is not a minimum value. Weierstrass's example was the functional
where is continuous on , continuously differentiable on , and subject to boundary conditions , where and are constants and . Weierstrass showed that , but no admissible function can make equal 0. This example did not disprove Dirichlet's principle per se, since the example integral is different from Dirichlet's integral. But it did undermine the reasoning that Riemann had used, and spurred interest in proving Dirichlet's principle as well as broader advancements in the calculus of variations and ultimately functional analysis.
In 1900, Hilbert later justified Riemann's use of Dirichlet's principle by developing the direct method in the calculus of variations.
See also
Dirichlet problem
Hilbert's twentieth problem
Plateau's problem
Green's first identity
Notes
References
Calculus of variations
Partial differential equations
Harmonic functions
Mathematical principles
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https://en.wikipedia.org/wiki/Lickorish%E2%80%93Wallace%20theorem
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In mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted.
The theorem was proved in the early 1960s by W. B. R. Lickorish and Andrew H. Wallace, independently and by different methods. Lickorish's proof rested on the Lickorish twist theorem, which states that any orientable automorphism of a closed orientable surface is generated by Dehn twists along 3g − 1 specific simple closed curves in the surface, where g denotes the genus of the surface. Wallace's proof was more general and involved adding handles to the boundary of a higher-dimensional ball.
A corollary of the theorem is that every closed, orientable 3-manifold bounds a simply-connected compact 4-manifold.
By using his work on automorphisms of non-orientable surfaces, Lickorish also showed that every closed, non-orientable, connected 3-manifold is obtained by Dehn surgery on a link in the non-orientable 2-sphere bundle over the circle. Similar to the orientable case, the surgery can be done in a special way which allows the conclusion that every closed, non-orientable 3-manifold bounds a compact 4-manifold.
References
3-manifolds
Theorems in topology
Theorems in geometry
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https://en.wikipedia.org/wiki/Dehn%20surgery
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In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link. It is often conceptualized as two steps: drilling then filling.
Definitions
Given a 3-manifold and a link , the manifold drilled along is obtained by removing an open tubular neighborhood of from . If , the drilled manifold has torus boundary components . The manifold drilled along is also known as the link complement, since if one removed the corresponding closed tubular neighborhood from , one obtains a manifold diffeomorphic to .
Given a 3-manifold whose boundary is made of 2-tori , we may glue in one solid torus by a homeomorphism (resp. diffeomorphism) of its boundary to each of the torus boundary components of the original 3-manifold. There are many inequivalent ways of doing this, in general. This process is called Dehn filling.
Dehn surgery on a 3-manifold containing a link consists of drilling out a tubular neighbourhood of the link together with Dehn filling on all the components of the boundary corresponding to the link.
In order to describe a Dehn surgery (see ), one picks two oriented simple closed curves and on the corresponding boundary torus of the drilled 3-manifold, where is a meridian of (a curve staying in a small ball in and having linking number +1 with or, equivalently, a curve that bounds a disc that intersects once the component ) and is a longitude of (a curve travelling once along or, equivalently, a curve on such that the algebraic intersection is equal to +1).
The curves and generate the fundamental group of the torus , and they form a basis of its first homology group. This gives any simple closed curve on the torus two coordinates and , so that . These coordinates only depend on the homotopy class of .
We can specify a homeomorphism of the boundary of a solid torus to by having the meridian curve of the solid torus map to a curve homotopic to . As long as the meridian maps to the surgery slope , the resulting Dehn surgery will yield a 3-manifold that will not depend on the specific gluing (up to homeomorphism). The ratio is called the surgery coefficient of .
In the case of links in the 3-sphere or more generally an oriented integral homology sphere, there is a canonical choice of the longitudes : every longitude is chosen so that it is null-homologous in the knot complement—equivalently, if it is the boundary of a Seifert surface.
When the ratios are all integers (note that this condition does not depend on the choice of the longitudes, since it corresponds to the new meridians intersecting exactly once the ancient meridians), the surgery is called an integral surgery.
Such surgeries are closely related to handlebodies, cobordism and Morse functions.
Examples
If all surgery coefficients are infinite, then each new meridian is homotopic to the ancient meridian . Therefore the homeomor
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https://en.wikipedia.org/wiki/Adjunction%20space
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In mathematics, an adjunction space (or attaching space) is a common construction in topology where one topological space is attached or "glued" onto another. Specifically, let X and Y be topological spaces, and let A be a subspace of Y. Let f : A → X be a continuous map (called the attaching map). One forms the adjunction space X ∪f Y (sometimes also written as X +f Y) by taking the disjoint union of X and Y and identifying a with f(a) for all a in A. Formally,
where the equivalence relation ~ is generated by a ~ f(a) for all a in A, and the quotient is given the quotient topology. As a set, X ∪f Y consists of the disjoint union of X and (Y − A). The topology, however, is specified by the quotient construction.
Intuitively, one may think of Y as being glued onto X via the map f.
Examples
A common example of an adjunction space is given when Y is a closed n-ball (or cell) and A is the boundary of the ball, the (n−1)-sphere. Inductively attaching cells along their spherical boundaries to this space results in an example of a CW complex.
Adjunction spaces are also used to define connected sums of manifolds. Here, one first removes open balls from X and Y before attaching the boundaries of the removed balls along an attaching map.
If A is a space with one point then the adjunction is the wedge sum of X and Y.
If X is a space with one point then the adjunction is the quotient Y/A.
Properties
The continuous maps h : X ∪f Y → Z are in 1-1 correspondence with the pairs of continuous maps hX : X → Z and hY : Y → Z that satisfy hX(f(a))=hY(a) for all a in A.
In the case where A is a closed subspace of Y one can show that the map X → X ∪f Y is a closed embedding and (Y − A) → X ∪f Y is an open embedding.
Categorical description
The attaching construction is an example of a pushout in the category of topological spaces. That is to say, the adjunction space is universal with respect to the following commutative diagram:
Here i is the inclusion map and ϕX, ϕY are the maps obtained by composing the quotient map with the canonical injections into the disjoint union of X and Y. One can form a more general pushout by replacing i with an arbitrary continuous map g—the construction is similar. Conversely, if f is also an inclusion the attaching construction is to simply glue X and Y together along their common subspace.
See also
Quotient space
Mapping cylinder
References
Stephen Willard, General Topology, (1970) Addison-Wesley Publishing Company, Reading Massachusetts. (Provides a very brief introduction.)
Ronald Brown, "Topology and Groupoids" pdf available , (2006) available from amazon sites. Discusses the homotopy type of adjunction spaces, and uses adjunction spaces as an introduction to (finite) cell complexes.
J.H.C. Whitehead "Note on a theorem due to Borsuk" Bull AMS 54 (1948), 1125-1132 is the earliest outside reference I know of using the term "adjuction space".
Topology
Topological spaces
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https://en.wikipedia.org/wiki/Atoroidal
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In mathematics, an atoroidal 3-manifold is one that does not contain an essential torus.
There are two major variations in this terminology: an essential torus may be defined geometrically, as an embedded, non-boundary parallel, incompressible torus, or it may be defined algebraically, as a subgroup of its fundamental group that is not conjugate to a peripheral subgroup (i.e., the image of the map on fundamental group induced by an inclusion of a boundary component). The terminology is not standardized, and different authors require atoroidal 3-manifolds to satisfy certain additional restrictions. For instance:
gives a definition of atoroidality that combines both geometric and algebraic aspects, in terms of maps from a torus to the manifold and the induced maps on the fundamental group. He then notes that for irreducible boundary-incompressible 3-manifolds this gives the algebraic definition.
uses the algebraic definition without additional restrictions.
uses the geometric definition, restricted to irreducible manifolds.
requires the algebraic variant of atoroidal manifolds (which he calls simply atoroidal) to avoid being one of three kinds of fiber bundle. He makes the same restriction on geometrically atoroidal manifolds (which he calls topologically atoroidal) and in addition requires them to avoid incompressible boundary-parallel embedded Klein bottles. With these definitions, the two kinds of atoroidality are equivalent except on certain Seifert manifolds.
A 3-manifold that is not atoroidal is called toroidal.
References
3-manifolds
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https://en.wikipedia.org/wiki/Boundary%20parallel
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In mathematics, a closed n-manifold N embedded in an (n + 1)-manifold M is boundary parallel (or ∂-parallel, or peripheral) if there is an isotopy of N onto a boundary component of M.
An example
Consider the annulus . Let π denote the projection map
If a circle S is embedded into the annulus so that π restricted to S is a bijection, then S is boundary parallel. (The converse is not true.)
If, on the other hand, a circle S is embedded into the annulus so that π restricted to S is not surjective, then S is not boundary parallel. (Again, the converse is not true.)
Geometric topology
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https://en.wikipedia.org/wiki/Comenius%20University%20Faculty%20of%20Mathematics%2C%20Physics%20and%20Informatics
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The Faculty of Mathematics, Physics and Informatics (FMPH; ; ; colloquial: Matfyz) is one of 13 faculties of the Comenius University in Bratislava, the capital of Slovakia. The faculty provides higher education in mathematics, physics and informatics, as well as teacher training in subjects related to these branches of study. It was established in 1980 by separating from the Faculty of Natural Sciences under the name of Faculty of Mathematics and Physics (). Its name was changed to the contemporary name in 2000.
In 2015, Faculty of Mathematics, Physics and Informatics was ranked first in the group of natural sciences in the ranking of faculties in Slovakia by the Academic Ranking and Rating Agency (ARRA).
The campus is located in Mlynská dolina in Bratislava, along with the Faculty of Natural Sciences of the Comenius University, the Faculty of Informatics and Information Technologies and the Faculty of Electrical Engineering and Information Technology of the Slovak University of Technology.
Endowment of the faculty in 2015 was €11.7 million.
Departments
Mathematics
Department of Algebra, Geometry and Didactics of Mathematics
Department of Applied Mathematics and Statistics
Department of Mathematical Analysis and Numerical Mathematics
Physics
Department of Astronomy, Physics of the Earth and Meteorology
Department of Experimental Physics
Department of Nuclear Physics and Biophysics
Department of Theoretical Physics and Didactics of Physics
Informatics
Department of Applied Informatics
Department of Informatics
Department of Informatics Education
Study programmes
The faculty offers undergraduate and postgraduate education in the fields of mathematics, physics, informatics and teaching.
Mathematics
Undergraduate
Mathematics
Economic and Financial Mathematics
Managerial Mathematics
Insurance Mathematics
Postgraduate
Computer Graphics and Geometry
Economic and Financial Mathematics
Mathematics
Managerial Mathematics
Probability and Mathematical Statistics
Physics
Undergraduate
Physics
Renewable Energy Sources and Environmental Physics
Biomedical Physics
Postgraduate
Astronomy and Astrophysics
Biophysics and Chemical Physics
Biomedical Physics
Environmental Physics, Renewable Energy Sources, Meteorology and Climatology
Nuclear and Sub-nuclear Physics
Optics, Lasers and Optical Spectroscopy
Physics of the Earth
Plasma Physics
Solid State Physics
Theoretical Physics
Informatics
Undergraduate
Informatics
Applied Informatics
Bioinformatics
Data science
Postgraduate
Informatics
Applied Informatics
Cognitive Science
Teacher training
Teacher training forms a separate group of study. Physical education instruction is provided by the Faculty of Physical Education and Sports and English is provided by the Faculty of Arts. Special teacher training is also offered for those graduates of the faculty following a non-teaching course of study who are interested in gaining the Teacher Certif
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https://en.wikipedia.org/wiki/Department%20of%20Computer%20Science%2C%20FMPI%2C%20Comenius%20University
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The Department of Computer Science is a department of the Faculty of Mathematics, Physics and Informatics at the Comenius University in Bratislava, the capital of Slovakia. It is headed by Prof. RNDr. Branislav Rovan, Phd.
Educational and scientific achievements
The first comprehensive computer science curriculum in Czechoslovakia (now Slovakia) was introduced at the Faculty in 1973. The department, established in 1974, continues to be responsible for organizing the major part of the undergraduate and graduate computer science education to this date. The distinguishing feature of the curriculum has been a balanced coverage of the mathematical foundations, theoretical computer science, and practical computer science. The part of the curriculum covered by the department at present includes courses on computer architecture, system software, networks, databases, software design, design and analysis of algorithms, formal languages, computational complexity, discrete mathematics, cryptology, data security and others.
The department succeeded several times in project applications within the TEMPUS Programme of the EU. The projects CIPRO and „Neumann Network“ helped to build the departmental hardware infrastructure and to establish the expertise in Unix workstation technology, networking, and structured document processing. The CUSTU PARLAB parallel computing laboratory run jointly with the Department of Informatics of the Faculty of Electrical Engineering and Informatics of the Slovak Technical University also resulted from one of these projects. Furthermore, the department participated in project LEARN-ED under the COPERNICUS Programme and built a multimedia laboratory.
The department has been involved in the organization of one of the top European conferences in theoretical computer science ? MFCS ? each time it took place in Slovakia. Further conferences recently organized or co-organized by the department include SOFSEM '98 and DISC '99. Besides, the department houses the secretariat of the European Association for Theoretical Computer Science, of the Slovak Society for Computer Science and of the Association for Security of Information Technologies (ASIT).
Research topics
Research in theoretical computer science and discrete mathematics has the longest tradition in the department. Most notably, the result of Róbert Szelepcsényi on the closure of nondeterministic space under complement, independently obtained also by N. Immerman, brought the Gödel Prize of the ACM and EATCS to both of them in 1995. More recently research in parallel and distributed computing, cryptology and information security, and in software development has been initiated. The department is involved in international cooperation on the development of the structured document editor within the Euromath Project.
The department has many international contacts, succeeded in research project application (project ALTEC ? „Algorithms for Future Technologies“) with partners from E
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https://en.wikipedia.org/wiki/Pointed%20set
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In mathematics, a pointed set (also based set or rooted set) is an ordered pair where is a set and is an element of called the base point, also spelled basepoint.
Maps between pointed sets and —called based maps, pointed maps, or point-preserving maps—are functions from to that map one basepoint to another, i.e. maps such that . Based maps are usually denoted .
Pointed sets are very simple algebraic structures. In the sense of universal algebra, a pointed set is a set together with a single nullary operation which picks out the basepoint. Pointed maps are the homomorphisms of these algebraic structures.
The class of all pointed sets together with the class of all based maps forms a category. Every pointed set can be converted to an ordinary set by forgetting the basepoint (the forgetful functor is faithful), but the reverse is not true. In particular, the empty set cannot be pointed, because it has no element that can be chosen as the basepoint.
Categorical properties
The category of pointed sets and based maps is equivalent to the category of sets and partial functions. The base point serves as a "default value" for those arguments for which the partial function is not defined. One textbook notes that "This formal completion of sets and partial maps by adding 'improper', 'infinite' elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science." This category is also isomorphic to the coslice category (), where is (a functor that selects) a singleton set, and (the identity functor of) the category of sets. This coincides with the algebraic characterization, since the unique map extends the commutative triangles defining arrows of the coslice category to form the commutative squares defining homomorphisms of the algebras.
There is a faithful functor from pointed sets to usual sets, but it is not full and these categories are not equivalent.
The category of pointed sets is a pointed category. The pointed singleton sets are both initial objects and terminal objects, i.e. they are zero objects. The category of pointed sets and pointed maps has both products and coproducts, but it is not a distributive category. It is also an example of a category where is not isomorphic to .
Applications
Many algebraic structures rely on a distinguished point. For example, groups are pointed sets by choosing the identity element as the basepoint, so that group homomorphisms are point-preserving maps. This observation can be restated in category theoretic terms as the existence of a forgetful functor from groups to pointed sets.
A pointed set may be seen as a pointed space under the discrete topology or as a vector space over the field with one element.
As "rooted set" the notion naturally appears in the study of antimatroids and transportation polytopes.
See also
Notes
References
Further reading
External links
Pullbacks in Category of Sets and Partial Functions
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https://en.wikipedia.org/wiki/Projective%20differential%20geometry
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In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties of mathematical objects such as functions, diffeomorphisms, and submanifolds, that are invariant under transformations of the projective group. This is a mixture of the approaches from Riemannian geometry of studying invariances, and of the Erlangen program of characterizing geometries according to their group symmetries.
The area was much studied by mathematicians from around 1890 for a generation (by J. G. Darboux, George Henri Halphen, Ernest Julius Wilczynski, E. Bompiani, G. Fubini, Eduard Čech, amongst others), without a comprehensive theory of differential invariants emerging. Élie Cartan formulated the idea of a general projective connection, as part of his method of moving frames; abstractly speaking, this is the level of generality at which the Erlangen program can be reconciled with differential geometry, while it also develops the oldest part of the theory (for the projective line), namely the Schwarzian derivative, the simplest projective differential invariant.
Further work from the 1930s onwards was carried out by J. Kanitani, Shiing-Shen Chern, A. P. Norden, G. Bol, S. P. Finikov and G. F. Laptev. Even the basic results on osculation of curves, a manifestly projective-invariant topic, lack any comprehensive theory. The ideas of projective differential geometry recur in mathematics and its applications, but the formulations given are still rooted in the language of the early twentieth century.
See also
Affine geometry of curves
References
Ernest Julius Wilczynski Projective differential geometry of curves and ruled surfaces (Leipzig: B.G. Teubner,1906)
Further reading
Notes on Projective Differential Geometry by Michael Eastwood
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https://en.wikipedia.org/wiki/Graphical%20Models
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Graphical Models is an academic journal in computer graphics and geometry processing publisher by Elsevier. , its editor-in-chief is Bedrich Benes of the Purdue University.
History
This journal has gone through multiple names. Founded in 1972 as Computer Graphics and Image Processing by Azriel Rosenfeld, it became the first journal to focus on computer image analysis.
Its first change of name came in 1983, when it became Computer Vision, Graphics, and Image Processing. In 1991 it split into two journals, CVGIP: Graphical Models and Image Processing,
and CVGIP: Image Understanding, which later became Computer Vision and Image Understanding. Meanwhile, in 1995, the journal Graphical Models and Image Processing removed the "CVGIP" prefix from its former name, and finally took its current title, Graphical Models, in 2002.
Ranking
Although initially ranked by SCImago Journal Rank as a top-quartile journal in 1999 in its main topic areas, computer graphics and computer-aided design, and then for many years ranked as second-quartile, by 2020 it had fallen to the third quartile.
References
Geometry processing
Computer science journals
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https://en.wikipedia.org/wiki/Hodge%20index%20theorem
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In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanned by such curves (up to linear equivalence) has a one-dimensional subspace on which it is positive definite (not uniquely determined), and decomposes as a direct sum of some such one-dimensional subspace, and a complementary subspace on which it is negative definite.
In a more formal statement, specify that V is a non-singular projective surface, and let H be the divisor class on V of a hyperplane section of V in a given projective embedding. Then the intersection
where d is the degree of V (in that embedding). Let D be the vector space of rational divisor classes on V, up to algebraic equivalence. The dimension of D is finite and is usually denoted by ρ(V). The Hodge index theorem says that the subspace spanned by H in D has a complementary subspace on which the intersection pairing is negative definite. Therefore, the signature (often also called index) is (1,ρ(V)-1).
The abelian group of divisor classes up to algebraic equivalence is now called the Néron-Severi group; it is known to be a finitely-generated abelian group, and the result is about its tensor product with the rational number field. Therefore, ρ(V) is equally the rank of the Néron-Severi group (which can have a non-trivial torsion subgroup, on occasion).
This result was proved in the 1930s by W. V. D. Hodge, for varieties over the complex numbers, after it had been a conjecture for some time of the Italian school of algebraic geometry (in particular, Francesco Severi, who in this case showed that ρ < ∞). Hodge's methods were the topological ones brought in by Lefschetz. The result holds over general (algebraically closed) fields.
References
, see Ch. V.1
Algebraic surfaces
Geometry of divisors
Intersection theory
Theorems in algebraic geometry
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https://en.wikipedia.org/wiki/A%20Mathematician%27s%20Apology
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A Mathematician's Apology is a 1940 essay by British mathematician G. H. Hardy, which offers a defence of the pursuit of mathematics. Central to Hardy's "apology" – in the sense of a formal justification or defence (as in Plato's Apology of Socrates) – is an argument that mathematics has value independent of possible applications. Hardy located this value in the beauty of mathematics, and gave some examples of and criteria for mathematical beauty. The book also includes a brief autobiography, and gives the layman an insight into the mind of a working mathematician.
Background
Hardy felt the need to justify his life's work in mathematics at this time mainly for two reasons. Firstly, at age 62, Hardy felt the approach of old age (he had survived a heart attack in 1939) and the decline of his mathematical creativity and skills.
By devoting time to writing the Apology, Hardy was admitting that his own time as a creative mathematician was finished. In his foreword to the 1967 edition of the book, C. P. Snow describes the Apology as
"a passionate lament for creative powers that used to be and that will never come again".
In Hardy's words, "Exposition, criticism, appreciation, is work for second-rate minds. [...] It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done."
Secondly, at the start of World War II, Hardy, a committed pacifist, wanted to justify his belief that mathematics should be pursued for its own sake rather than for the sake of its applications. He began writing on this subject when he was invited to contribute an article to Eureka, the journal of The Archimedeans (the Cambridge University student mathematical society). One of the topics the editor suggested was "something about mathematics and the war", and the result was the article "Mathematics in war-time". Hardy later incorporated this article into A Mathematician's Apology.
He wanted to write a book in which he would explain his mathematical philosophy to the next generation of mathematicians; that would defend mathematics by elaborating on the merits of pure mathematics solely, without having to resort to the attainments of applied mathematics in order to justify the overall importance of mathematics; and that would inspire the upcoming generations of pure mathematicians. Hardy was an atheist, and makes his justification not to God but to his fellow men.
Hardy initially submitted A Mathematician's Apology to Cambridge University Press with the intention of personally paying for its printing, but the Press decided to fund publication with an initial run of four thousand copies.
Summary
One of the main themes of the book is the beauty that mathematics possesses, which Hardy compares to painting and poetry. For Hardy, the most beautiful mathematics was that which had
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https://en.wikipedia.org/wiki/Thom%20space
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In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space.
Construction of the Thom space
One way to construct this space is as follows. Let
be a rank n real vector bundle over the paracompact space B. Then for each point b in B, the fiber is an -dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let be the unit ball bundle with respect to our orthogonal structure, and let be the unit sphere bundle, then the Thom space is the quotient of topological spaces. is a pointed space with the image of in the quotient as basepoint. If B is compact, then is the one-point compactification of E.
For example, if E is the trivial bundle , then and . Writing for B with a disjoint basepoint, is the smash product of and ; that is, the n-th reduced suspension of .
The Thom isomorphism
The significance of this construction begins with the following result, which belongs to the subject of cohomology of fiber bundles. (We have stated the result in terms of coefficients to avoid complications arising from orientability; see also Orientation of a vector bundle#Thom space.)
Let be a real vector bundle of rank n. Then there is an isomorphism, now called a Thom isomorphism
for all k greater than or equal to 0, where the right hand side is reduced cohomology.
This theorem was formulated and proved by René Thom in his famous 1952 thesis.
We can interpret the theorem as a global generalization of the suspension isomorphism on local trivializations, because the Thom space of a trivial bundle on B of rank k is isomorphic to the kth suspension of , B with a disjoint point added (cf. #Construction of the Thom space.) This can be more easily seen in the formulation of the theorem that does not make reference to Thom space:
In concise terms, the last part of the theorem says that u freely generates as a right -module. The class u is usually called the Thom class of E. Since the pullback is a ring isomorphism, is given by the equation:
In particular, the Thom isomorphism sends the identity element of to u. Note: for this formula to make sense, u is treated as an element of (we drop the ring )
Significance of Thom's work
In his 1952 paper, Thom showed that the Thom class, the Stiefel–Whitney classes, and the Steenrod operations were all related. He used these ideas to prove in the 1954 paper Quelques propriétés globales des variétés differentiables that the cobordism groups could be computed as the homotopy groups of certain Thom spaces MG(n). The proof depends on and is intimately related to the transversality properties of smooth manifolds—see Thom transversality theorem. By reversing this construction, John Milnor and Sergei Novikov (among many others) w
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https://en.wikipedia.org/wiki/Newton%27s%20method%20in%20optimization
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In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function , which are solutions to the equation . As such, Newton's method can be applied to the derivative of a twice-differentiable function to find the roots of the derivative (solutions to ), also known as the critical points of . These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article. This is relevant in optimization, which aims to find (global) minima of the function .
Newton's method
The central problem of optimization is minimization of functions. Let us first consider the case of univariate functions, i.e., functions of a single real variable. We will later consider the more general and more practically useful multivariate case.
Given a twice differentiable function , we seek to solve the optimization problem
Newton's method attempts to solve this problem by constructing a sequence from an initial guess (starting point) that converges towards a minimizer of by using a sequence of second-order Taylor approximations of around the iterates. The second-order Taylor expansion of around is
The next iterate is defined so as to minimize this quadratic approximation in , and setting . If the second derivative is positive, the quadratic approximation is a convex function of , and its minimum can be found by setting the derivative to zero. Since
the minimum is achieved for
Putting everything together, Newton's method performs the iteration
Geometric interpretation
The geometric interpretation of Newton's method is that at each iteration, it amounts to the fitting of a parabola to the graph of at the trial value , having the same slope and curvature as the graph at that point, and then proceeding to the maximum or minimum of that parabola (in higher dimensions, this may also be a saddle point), see below. Note that if happens to a quadratic function, then the exact extremum is found in one step.
Higher dimensions
The above iterative scheme can be generalized to dimensions by replacing the derivative with the gradient (different authors use different notation for the gradient, including ), and the reciprocal of the second derivative with the inverse of the Hessian matrix (different authors use different notation for the Hessian, including ). One thus obtains the iterative scheme
Often Newton's method is modified to include a small step size instead of :
This is often done to ensure that the Wolfe conditions, or much simpler and efficient Armijo's condition, are satisfied at each step of the method. For step sizes other than 1, the method is often referred to as the relaxed or damped Newton's method.
Convergence
If is a strongly convex function with Lipschitz Hessian, then provided that is close enough to , the sequence generated by Newton's method will converge to the (nec
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https://en.wikipedia.org/wiki/Algebra%20bundle
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In mathematics, an algebra bundle is a fiber bundle whose fibers are algebras and local trivializations respect the algebra structure. It follows that the transition functions are algebra isomorphisms. Since algebras are also vector spaces, every algebra bundle is a vector bundle.
Examples include the tensor-algebra bundle, exterior bundle, and symmetric bundle associated to a given vector bundle, as well as the Clifford bundle associated to any Riemannian vector bundle.
See also
Lie algebra bundle
References
.
.
.
.
Vector bundles
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https://en.wikipedia.org/wiki/Moment%20problem
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In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure μ to the sequence of moments
More generally, one may consider
for an arbitrary sequence of functions Mn.
Introduction
In the classical setting, μ is a measure on the real line, and M is the sequence { xn : n = 0, 1, 2, ... }. In this form the question appears in probability theory, asking whether there is a probability measure having specified mean, variance and so on, and whether it is unique.
There are three named classical moment problems: the Hamburger moment problem in which the support of μ is allowed to be the whole real line; the Stieltjes moment problem, for [0, +∞); and the Hausdorff moment problem for a bounded interval, which without loss of generality may be taken as [0, 1].
Existence
A sequence of numbers mn is the sequence of moments of a measure μ if and only if a certain positivity condition is fulfilled; namely, the Hankel matrices Hn,
should be positive semi-definite. This is because a positive-semidefinite Hankel matrix corresponds to a linear functional such that and (non-negative for sum of squares of polynomials). Assume can be extended to . In the univariate case, a non-negative polynomial can always be written as a sum of squares. So the linear functional is positive for all the non-negative polynomials in the univariate case. By Haviland's theorem, the linear functional has a measure form, that is . A condition of similar form is necessary and sufficient for the existence of a measure supported on a given interval [a, b].
One way to prove these results is to consider the linear functional that sends a polynomial
to
If mkn are the moments of some measure μ supported on [a, b], then evidently
Vice versa, if () holds, one can apply the M. Riesz extension theorem and extend to a functional on the space of continuous functions with compact support C0([a, b]), so that
By the Riesz representation theorem, () holds iff there exists a measure μ supported on [a, b], such that
for every ƒ ∈ C0([a, b]).
Thus the existence of the measure is equivalent to (). Using a representation theorem for positive polynomials on [a, b], one can reformulate () as a condition on Hankel matrices.
See and for more details.
Uniqueness (or determinacy)
The uniqueness of μ in the Hausdorff moment problem follows from the Weierstrass approximation theorem, which states that polynomials are dense under the uniform norm in the space of continuous functions on [0, 1]. For the problem on an infinite interval, uniqueness is a more delicate question; see Carleman's condition, Krein's condition and . There are distributions, such as log-normal distributions, which have finite moments for all the positive integers but where other distributions have the same moments.
Formal solution
When the solution exists, it can be formally written using derivatives of the Dirac delta function asThe expression can be derived
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https://en.wikipedia.org/wiki/Fatness
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Fatness may refer to:
Obesity, a medical condition where excess body fat has accumulated to the extent that it may have a negative impact on health
The property of a fat object, in geometry, referring to an object in two or more dimensions whose lengths in the different dimensions are similar
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https://en.wikipedia.org/wiki/List%20of%20metropolitan%20areas%20in%20Sweden
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Sweden has three metropolitan areas consisting of the areas surrounding the three largest cities, Stockholm, Gothenburg and Malmö. The statistics have been retrieved from Statistics Sweden and the statistics released on 10 November 2014. The official land areas for each municipality have also been retrieved from Statistics Sweden, the agency that defines these areas.
Population centers
, Sweden had 2 metropolitan areas with a population of over 1,000,000 people each.
The following table shows the populations of the top ten metropolitan areas.
Metropolitan Stockholm
Metropolitan Stockholm (also known as Greater Stockholm or, in Swedish, Storstockholm), is a metropolitan area surrounding the Swedish capital of Stockholm. Since 2005, Metropolitan Stockholm is defined by official Swedish Statistics as all of Stockholm County. It is the largest of the three metropolitan areas in Sweden.
Metropolitan Stockholm is divided into 5 areas: Stockholm City Centre, Söderort, Västerort of Stockholm Municipality; and the northern suburbs and southern suburbs, which consists of several municipalities.
km2 Population per km2
Metropolitan Gothenburg
Metropolitan Gothenburg (Storgöteborg or literally Greater Gothenburg), is a metropolitan area surrounding the city of Gothenburg in Sweden. The metropolitan region is located in Västra Götaland County, except for the municipality of Kungsbacka, which is located to the south in Halland County. As of 2005, the municipalities of Alingsås and Lilla Edet were added to the region. The region is often used for statistical measures, and estimates in the 1960s predicted that the region would have about one million inhabitants in the year 2000. The region is the second largest metropolitan area in Sweden after Metropolitan Stockholm.
km2 Population per km2
Metropolitan Malmö
Greater Malmö (Stormalmö), also known as Metropolitan Malmö is the metropolitan area of Malmö in Sweden. The area is located in Southwestern Scania (Sydvästra Skåne), which is often considered synonymous with Greater Malmö, and it is part of the wider transnational Öresund Region. Besides Malmö, large towns in Greater Malmö includes Lund and Trelleborg, the former of which was the seat of the historical Catholic Archdiocese of Lund.
Since the 1970s, improvements in highways and the regional and InterRegio train networks means the commuting area has grown to include Ystad, Skurup, Sjöbo, Eslöv, Höör, Landskrona and Helsingborg, though only some of these are included in official definitions of Greater Malmö. It's not uncommon to live in Malmö and work either in Ystad or Helsingborg, or vice versa, but these towns have kept their mental allegiance with older divisions of Scania. Commuting across the Öresund has become more common, both through the Øresund Bridge and the HH Ferry route, at which car ferries departs every 12 minutes in summer (every 15 minutes in winter).
Statistics Sweden, which sets the official definitions for all metropolitan
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https://en.wikipedia.org/wiki/Mandelbrot%20Competition
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Named in honor of Benoit Mandelbrot, the Mandelbrot Competition was a mathematics competition founded by Sam Vandervelde, Richard Rusczyk and Sandor Lehoczky that operated from 1990 to 2019. It allowed high school students to compete individually and in four-person teams.
Competition
The Mandelbrot was a "correspondence competition," meaning that the competition was sent to a school's coach and students competed at their own school on a predetermined date. Individual results and team answers were then sent back to the contest coordinators. The most notable aspects of the Mandelbrot competition were the difficulty of the problems (much like the American Mathematics Competition and harder American Invitational Mathematics Examination problems) and the proof-based team round. Many past medalists at the International Mathematics Olympiad first tried their skills on the Mandelbrot Competition.
History
The Mandelbrot Competition was started by Sam Vandervelde, Richard Rusczyk, and Sandor Lehoczky while they were undergraduates in the early 1990s. Vandervelde ran the competition until its completion in 2019. Rusczyk now manages Art of Problem Solving Inc. and Lehoczky enjoys a successful career on Wall Street.
Contest format
The individual competition consisted of seven questions of varying value, worth a total of 14 points, that students had 40 minutes to answer. The team competition was a proof-based competition, where many questions were asked about a particular situation, and a team of four students was given 60 minutes to answer.
Divisions
The Mandelbrot Competition had two divisions, referred to as National and Regional. Questions at the National level were more difficult than those at the Regional level, but generally had overlap or concerned similar topics. For example, in the individual competition, the National competition would remove some of the easier Regional questions, and add some harder questions. In the team competition, the topic would be the same but the National level would be given fewer hints.
Results
Results would be published after each annual iteration of the contest, and in its final iteration, the results were published online with School leaderboards and Individual leaderboards divided by region and national. However, since mandelbrot.org is not maintained any more, it can only be visited here.
References
Mathematics competitions
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https://en.wikipedia.org/wiki/Volume%20integral
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In mathematics (particularly multivariable calculus), a volume integral (∭) refers to an integral over a 3-dimensional domain; that is, it is a special case of multiple integrals. Volume integrals are especially important in physics for many applications, for example, to calculate flux densities, or to calculate mass from a corresponding density function.
In coordinates
It can also mean a triple integral within a region of a function and is usually written as:
A volume integral in cylindrical coordinates is
and a volume integral in spherical coordinates (using the ISO convention for angles with as the azimuth and measured from the polar axis (see more on conventions)) has the form
Example
Integrating the equation over a unit cube yields the following result:
So the volume of the unit cube is 1 as expected. This is rather trivial however, and a volume integral is far more powerful. For instance if we have a scalar density function on the unit cube then the volume integral will give the total mass of the cube. For example for density function:
the total mass of the cube is:
See also
Divergence theorem
Surface integral
Volume element
External links
Multivariable calculus
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https://en.wikipedia.org/wiki/Greater%20Hartford%20Academy%20of%20Mathematics%20and%20Science
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The Academy of Aerospace and Engineering (also known as AAE, Aerospace, and Aerospace and Engineering) is a regional magnet high school located in Windsor, Connecticut. The school's half-day program operates as the Greater Hartford Academy of Mathematics And Science (also known as GHAMAS). The building houses a grade 6-12 program. It is run by the Capitol Region Education Council (CREC), one of 6 Regional Educational Service Centers (RESC) in Connecticut. Trinity College has been involved in some of the projects with GHAMAS, such as the Brain Bee, a neuroscience competition. Hartford Hospital is involved in school activities as well.
The Academy of Aerospace and Engineering was built as GHAMAS in 1999. Labs at the Academy include the Robotics, Physics, Earth Science, Biology, Cell Culture, Greenhouse & Potting, Biochemistry, Chemistry, Special Instrumentation, and Engineering Labs. There are also several smaller student laboratories which are used by students to conduct independent research through a senior design and research course called Capstone. Occasionally, speakers from industry or academia come to lecture full-day and morning half-day students (grades 9 and 10) about the field that they work in and educate them to possible careers in that field. Students partake in a variety of clubs at the high school level, including competitive FIRST Tech Challenge robotics and debate teams. Select students pursue scientific research and engineering projects throughout the year and present their work at the Connecticut Science and Engineering Fair. Each year, some students that have presented exemplary work are chosen by CSEF to compete in the International Science and Engineering Fair
AAE has historically been an exclusively half-day program operating as GHAMAS and is now solely a full-day program operating as AAE. Since the fall of 2011, the school holds 9-12 grade half-day, and 6-12 grade full-day students. Ninth and tenth-grade students take three foundation math (Algebra I, Geometry, Algebra II, Pre-calculus, or higher) and science (Physics, Earth Science, Biology, and Chemistry) courses in the morning, followed by humanities and other classes at their sending district's high school or with the full-day program. Half-day juniors and seniors take these humanities at their home schools during the morning and join the AAE juniors and seniors for up to four advanced elective courses in the afternoon, such as Molecular and Cellular Biology, Anatomy, Zoology, or Astronomy, along with Advanced Placement curricula.
AAE is a member of the NCSSSMST. This is an organization of secondary schools that promote Mathematics, Science, and Technology schools. Greater Hartford Academy of Math and Science has been involved as a NASA Explorer School. It is one of only three such schools in Connecticut. The director of both the high school and middle school academies is Adam Johnson.
History
On January 9, 2010, a bus carrying GHAMAS students to a robotics competit
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https://en.wikipedia.org/wiki/Amsterdam%2C%20Saskatchewan
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Amsterdam is a hamlet within the Rural Municipality of Buchanan No. 304, Saskatchewan, Canada. Listed as a designated place by Statistics Canada, the hamlet had a population of 25 in the Canada 2016 Census. The hamlet is located 63.9 km north of the city of Yorkton and 1.5 km west of Highway 9. The community was founded at the turn of the 20th century by Dutch immigrants, hence the name. In its prime, the community had a post office, grain elevator, garage, and a school. It, like many small towns, has been hit hard by the gradual trend toward urbanization. The hamlet now has fewer than 25 people; most are of Ukrainian descent.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Amsterdam had a population of 30 living in 13 of its 14 total private dwellings, a change of from its 2016 population of 25. With a land area of , it had a population density of in 2021.
See also
List of communities in Saskatchewan
Hamlets of Saskatchewan
Block settlements
References
Buchanan No. 304, Saskatchewan
Designated places in Saskatchewan
Organized hamlets in Saskatchewan
Division No. 9, Saskatchewan
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https://en.wikipedia.org/wiki/William%20Chauvenet
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William Chauvenet (24 May 1820 in Milford, Pennsylvania – 13 December 1870 in St. Paul, Minnesota) was a professor of mathematics, astronomy, navigation, and surveying who was instrumental in the establishment of the U.S. Naval Academy at Annapolis, Maryland, and later the second chancellor of Washington University in St. Louis.
Early life
William Chauvenet was born on a farm near Milford, Pennsylvania to Guillaume Marc Chauvenet, a former soldier of Napoleon's army reconverted in silk trade after the Emperor's fall, and Mary B. Kerr and was raised in Philadelphia. He entered Yale University at age 16, and graduated in 1840 with high honors. While at Yale, Chauvenet contributed to the school newspaper and was a pianist with the Beethoven Society. He was one of eight founding members of the Skull and Bones Society.
United States Navy
In 1841, he was appointed a professor of mathematics in the United States Navy, and for a while served on the USS Mississippi teaching math. His professorship led Chauvenet to see the necessity of a United States naval academy. While others had proposed the idea, no one had actually seen it through. In 1842, he was appointed head of the naval asylum in Philadelphia, Pennsylvania. At the Naval Asylum, prospective officers took an eight-month course before sailing. Chauvenet felt the course was lacking and drew up his own plan for a two-year course. Presenting to several secretaries of the navy, the course was finally accepted in 1845.
He was instrumental in the 1845 founding of the United States Naval Academy at Annapolis, Maryland, and taught there for years. He was president of the academic board and in 1851 was part of a board that recommended the course of study be extended to four years. Chauvenet taught in many subjects, including mathematics, surveying, astronomy, and navigation. He helped to establish an astronomical observatory at the naval academy. Chauvenet's contributions were so important that in 1890, Admiral S.R. Franklin proclaimed him "Father of the Naval Academy". A bronze plaque was installed with this inscription in 1916, at the behest of Congress.
In 1855, he declined Yale's offer of a professorship of mathematics to continue working at the Naval Academy in Annapolis.
Washington University
In 1859, Yale again came calling, offering this time the professorship of astronomy and natural philosophy. Instead, Chauvenet took a job offered by Washington University in St. Louis: professor of mathematics and astronomy. He brought with him a deep love of music and a familiarity with the classics, in addition to being an outstanding figure in the world of science, noted by many historians as one of the foremost mathematical minds in the U.S. before the Civil War. It was Chauvenet who mathematically confirmed James B. Eads' plans for the first bridge to span the Mississippi River at St. Louis. The directors of the University chose him to be chancellor after his friend and Yale classmate Joseph Hoyt die
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https://en.wikipedia.org/wiki/Hyperbolic%20link
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In mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component.
As a consequence of the work of William Thurston, it is known that every knot is precisely one of the following: hyperbolic, a torus knot, or a satellite knot. As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links.
As a consequence of Thurston's hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link enables one to obtain many more hyperbolic 3-manifolds.
Examples
Borromean rings are hyperbolic.
Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco.
41 knot (the figure-eight knot)
52 knot (the three-twist knot)
61 knot (the stevedore knot)
62 knot
63 knot
74 knot
10 161 knot (the "Perko pair" knot)
12n242 knot
See also
SnapPea
Hyperbolic volume (knot)
Further reading
Colin Adams (1994, 2004) The Knot Book, American Mathematical Society, .
William Menasco (1984) "Closed incompressible surfaces in alternating knot and link complements", Topology 23(1):37–44.
William Thurston (1978-1981) The geometry and topology of three-manifolds, Princeton lecture notes.
External links
Colin Adams, Handbook of Knot Theory
3-manifolds
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https://en.wikipedia.org/wiki/Excision%20theorem
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In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space and subspaces and such that is also a subspace of , the theorem says that under certain circumstances, we can cut out (excise) from both spaces such that the relative homologies of the pairs into are isomorphic.
This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute.
Theorem
Statement
If are as above, we say that can be excised if the inclusion map of the pair into induces an isomorphism on the relative homologies:
The theorem states that if the closure of is contained in the interior of , then can be excised.
Often, subspaces that do not satisfy this containment criterion still can be excised—it suffices to be able to find a deformation retract of the subspaces onto subspaces that do satisfy it.
Proof Sketch
The proof of the excision theorem is quite intuitive, though the details are rather involved. The idea is to subdivide the simplices in a relative cycle in to get another chain consisting of "smaller" simplices, and continuing the process until each simplex in the chain lies entirely in the interior of or the interior of . Since these form an open cover for and simplices are compact, we can eventually do this in a finite number of steps. This process leaves the original homology class of the chain unchanged (this says the subdivision operator is chain homotopic to the identity map on homology).
In the relative homology , then, this says all the terms contained entirely in the interior of can be dropped without affecting the homology class of the cycle. This allows us to show that the inclusion map is an isomorphism, as each relative cycle is equivalent to one that avoids entirely.
Applications
Eilenberg–Steenrod Axioms
The excision theorem is taken to be one of the Eilenberg–Steenrod Axioms.
Mayer-Vietoris Sequences
The Mayer–Vietoris sequence may be derived with a combination of excision theorem and the long-exact sequence.
Suspension Theorem for Homology
The excision theorem may be used to derive the suspension theorem for homology, which says for all , where is the suspension of .
Invariance of Dimension
If nonempty open sets and
are homeomorphic, then m = n. This follows from the excision theorem, the long exact sequence for the pair , and the fact that deformation retracts onto a sphere.
In particular, is not homeomorphic to if .
See also
Homotopy excision theorem
References
Bibliography
Joseph J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag,
Allen Hatcher, Algebraic Topology. Cambridge University Press, Cambridge, 2002.
Homology theory
Theorems in topology
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https://en.wikipedia.org/wiki/Relative%20homology
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In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.
Definition
Given a subspace , one may form the short exact sequence
where denotes the singular chains on the space X. The boundary map on descends to and therefore induces a boundary map on the quotient. If we denote this quotient by , we then have a complex
By definition, the th relative homology group of the pair of spaces is
One says that relative homology is given by the relative cycles, chains whose boundaries are chains on A, modulo the relative boundaries (chains that are homologous to a chain on A, i.e., chains that would be boundaries, modulo A again).
Properties
The above short exact sequences specifying the relative chain groups gives rise to a chain complex of short exact sequences. An application of the snake lemma then yields a long exact sequence
The connecting map takes a relative cycle, representing a homology class in , to its boundary (which is a cycle in A).
It follows that , where is a point in X, is the n-th reduced homology group of X. In other words, for all . When , is the free module of one rank less than . The connected component containing becomes trivial in relative homology.
The excision theorem says that removing a sufficiently nice subset leaves the relative homology groups unchanged. Using the long exact sequence of pairs and the excision theorem, one can show that is the same as the n-th reduced homology groups of the quotient space .
Relative homology readily extends to the triple for .
One can define the Euler characteristic for a pair by
The exactness of the sequence implies that the Euler characteristic is additive, i.e., if , one has
Local homology
The -th local homology group of a space at a point , denoted
is defined to be the relative homology group . Informally, this is the "local" homology of close to .
Local homology of the cone CX at the origin
One easy example of local homology is calculating the local homology of the cone (topology) of a space at the origin of the cone. Recall that the cone is defined as the quotient space
where has the subspace topology. Then, the origin is the equivalence class of points . Using the intuition that the local homology group of at captures the homology of "near" the origin, we should expect this is the homology of since has a homotopy retract to . Computing the local homology can then be done using the long exact sequence in homology
Because the cone of a space is contractible, the middle homology groups are all zero, giving the isomorphism
since is contractible to .
In algebraic geometry
Note the previous construction can be proven in algebraic geometry using the affine cone of a pr
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https://en.wikipedia.org/wiki/MSRI
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MSRI may refer to:
Malaysian Social Research Institute, Kuala Lumpur, assists refugees
Mathematical Sciences Research Institute, California, undertakes research in mathematics
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https://en.wikipedia.org/wiki/Ideal%20theory
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In mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only considers ideals in commutative rings.)
Throughout the articles, rings refer to commutative rings. See also the article ideal (ring theory) for basic operations such as sum or products of ideals.
Ideals in a finitely generated algebra over a field
Ideals in a finitely generated algebra over a field (that is, a quotient of a polynomial ring over a field) behave somehow nicer than those in a general commutative ring. First, in contrast to the general case, if is a finitely generated algebra over a field, then the radical of an ideal in is the intersection of all maximal ideals containing the ideal (because is a Jacobson ring). This may be thought of as an extension of Hilbert's Nullstellensatz, which concerns the case when is a polynomial ring.
Topology determined by an ideal
If I is an ideal in a ring A, then it determines the topology on A where a subset U of A is open if, for each x in U,
for some integer . This topology is called the I-adic topology. It is also called an a-adic topology if is generated by an element .
For example, take , the ring of integers and an ideal generated by a prime number p. For each integer , define when , prime to . Then, clearly,
where denotes an open ball of radius with center . Hence, the -adic topology on is the same as the metric space topology given by . As a metric space, can be completed. The resulting complete metric space has a structure of a ring that extended the ring structure of ; this ring is denoted as and is called the ring of p-adic integers.
Ideal class group
In a Dedekind domain A (e.g., a ring of integers in a number field or the coordinate ring of a smooth affine curve) with the field of fractions , an ideal is invertible in the sense: there exists a fractional ideal (that is, an A-submodule of ) such that , where the product on the left is a product of submodules of K. In other words, fractional ideals form a group under a product. The quotient of the group of fractional ideals by the subgroup of principal ideals is then the ideal class group of A.
In a general ring, an ideal may not be invertible (in fact, already the definition of a fractional ideal is not clear). However, over a Noetherian integral domain, it is still possible to develop some theory generalizing the situation in Dedekind domains. For example, Ch. VII of Bourbaki's Algèbre commutative gives such a theory.
The ideal class group of A, when it can be defined, is closely related to the Picard group of the spectrum of A (often the two are the same; e.g., for Dedekind domains).
In algebraic number theory, especially in class field theory, it is more convenient to use a generalization of an ideal class group called an idele class group.
Closure operations
There a
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https://en.wikipedia.org/wiki/Computational%20topology
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Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational geometry and computational complexity theory.
A primary concern of algorithmic topology, as its name suggests, is to develop efficient algorithms for solving problems that arise naturally in fields such as computational geometry, graphics, robotics, structural biology and chemistry, using methods from computable topology.
Major algorithms by subject area
Algorithmic 3-manifold theory
A large family of algorithms concerning 3-manifolds revolve around normal surface theory, which is a phrase that encompasses several techniques to turn problems in 3-manifold theory into integer linear programming problems.
Rubinstein and Thompson's 3-sphere recognition algorithm. This is an algorithm that takes as input a triangulated 3-manifold and determines whether or not the manifold is homeomorphic to the 3-sphere. It has exponential run-time in the number of tetrahedral simplexes in the initial 3-manifold, and also an exponential memory profile. Moreover, it is implemented in the software package Regina. Saul Schleimer went on to show the problem lies in the complexity class NP. Furthermore, Raphael Zentner showed that the problem lies in the complexity class coNP, provided that the generalized Riemann hypothesis holds. He uses instanton gauge theory, the geometrization theorem of 3-manifolds, and subsequent work of Greg Kuperberg on the complexity of knottedness detection.
The connect-sum decomposition of 3-manifolds is also implemented in Regina, has exponential run-time and is based on a similar algorithm to the 3-sphere recognition algorithm.
Determining that the Seifert-Weber 3-manifold contains no incompressible surface has been algorithmically implemented by Burton, Rubinstein and Tillmann and based on normal surface theory.
The Manning algorithm is an algorithm to find hyperbolic structures on 3-manifolds whose fundamental group have a solution to the word problem.
At present the JSJ decomposition has not been implemented algorithmically in computer software. Neither has the compression-body decomposition. There are some very popular and successful heuristics, such as SnapPea which has much success computing approximate hyperbolic structures on triangulated 3-manifolds. It is known that the full classification of 3-manifolds can be done algorithmically, in fact, it is known that deciding whether two closed, oriented 3-manifolds given by triangulations (simplicial complexes) are equivalent (homeomorphic) is elementary recursive. This generalizes the result on 3-sphere recognition.
Conversion algorithms
SnapPea implements an algorithm to convert a planar knot or link diagram into a cusped triangulation. This algorithm has a roughly linear run-time in the number of crossings in the diagram, and low memory profile. The algorithm is similar to the Wirthinger algorithm for constructin
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https://en.wikipedia.org/wiki/Connected%20component
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Connected component may refer to:
Connected component (graph theory), a set of vertices in a graph that are linked to each other by paths
Connected component (topology), a maximal subset of a topological space that cannot be covered by the union of two disjoint open sets
See also
Connected-component labeling, an algorithm for finding contiguous subsets of pixels in a digital image
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https://en.wikipedia.org/wiki/Supertrace
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In the theory of superalgebras, if A is a commutative superalgebra, V is a free right A-supermodule and T is an endomorphism from V to itself, then the supertrace of T, str(T) is defined by the following trace diagram:
More concretely, if we write out T in block matrix form after the decomposition into even and odd subspaces as follows,
then the supertrace
str(T) = the ordinary trace of T00 − the ordinary trace of T11.
Let us show that the supertrace does not depend on a basis.
Suppose e1, ..., ep are the even basis vectors and ep+1, ..., ep+q are the odd basis vectors. Then, the components of T, which are elements of A, are defined as
The grading of Tij is the sum of the gradings of T, ei, ej mod 2.
A change of basis to e1', ..., ep', e(p+1)', ..., e(p+q)' is given by the supermatrix
and the inverse supermatrix
where of course, AA−1 = A−1A = 1 (the identity).
We can now check explicitly that the supertrace is basis independent. In the case where T is even, we have
In the case where T is odd, we have
The ordinary trace is not basis independent, so the appropriate trace to use in the Z2-graded setting is the supertrace.
The supertrace satisfies the property
for all T1, T2 in End(V). In particular, the supertrace of a supercommutator is zero.
In fact, one can define a supertrace more generally for any associative superalgebra E over a commutative superalgebra A as a linear map tr: E -> A which vanishes on supercommutators. Such a supertrace is not uniquely defined; it can always at least be modified by multiplication by an element of A.
Physics applications
In supersymmetric quantum field theories, in which the action integral is invariant under a set of symmetry transformations (known as supersymmetry transformations) whose algebras are superalgebras, the supertrace has a variety of applications. In such a context, the supertrace of the mass matrix for the theory can be written as a sum over spins of the traces of the mass matrices for particles of different spin:
In anomaly-free theories where only renormalizable terms appear in the superpotential, the above supertrace can be shown to vanish, even when supersymmetry is spontaneously broken.
The contribution to the effective potential arising at one loop (sometimes referred to as the Coleman-Weinberg potential) can also be written in terms of a supertrace. If is the mass matrix for a given theory, the one-loop potential can be written as
where and are the respective tree-level mass matrices for the separate bosonic and fermionic degrees of freedom in the theory and is a cutoff scale.
See also
Berezinian
References
Super linear algebra
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https://en.wikipedia.org/wiki/Joachim%20Lambek
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Joachim "Jim" Lambek (5 December 1922 – 23 June 2014) was a Canadian mathematician. He was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his PhD degree in 1950 with Hans Zassenhaus as advisor.
Biography
Lambek was born in Leipzig, Germany, where he attended a Gymnasium. He came to England in 1938 as a refugee on the Kindertransport. From there he was interned as an enemy alien and deported to a prison work camp in New Brunswick, Canada. There, he began in his spare time a mathematical apprenticeship with Fritz Rothberger, also
interned, and wrote the McGill Junior Matriculation in fall of 1941. In the spring of 1942, he was released and settled in Montreal, where he entered studies at McGill University, graduating with an honours mathematics degree in 1945 and an MSc a year later. In 1950, he completed his doctorate under Hans Zassenhaus becoming McGill's first PhD in mathematics.
Lambek became assistant professor at McGill; he was made a full professor in 1963. He spent his sabbatical year 1965–66 in at the Institute for Mathematical Research at ETH Zurich, where Beno Eckmann had gathered together a group of researchers interested in algebraic topology and category theory, including Bill Lawvere. There Lambek reoriented his research into category theory.
Lambek retired in 1992 but continued his involvement at McGill's mathematics department. In 2000 a festschrift celebrating Lambek's contributions to mathematical structures in computer science was published. On the occasion of Lambek's 90th birthday, a collection Categories and Types in Logic, Language, and Physics was produced in tribute to him.
Scholarly work
Lambek's PhD thesis investigated vector fields using the biquaternion algebra over Minkowski space, as well as semigroup immersion in a group. The second component was published by the Canadian Journal of Mathematics. He later returned to biquaternions when in 1995 he contributed "If Hamilton had prevailed: Quaternions in Physics", which exhibited the Riemann–Silberstein bivector to express the free-space electromagnetic equations.
Lambek supervised 17 doctoral students, and has 75 doctoral descendants as of 2020. He has over 100 publications listed in the Mathematical Reviews, including 6 books. His earlier work was mostly in module theory, especially torsion theories, non-commutative localization, and injective modules. One of his earliest papers, , proved the Lambek–Moser theorem about integer sequences. In 1963 he published an important result, now known as Lambek's theorem, on character modules characterizing flatness of a module. His more recent work is in pregroups and formal languages; his earliest works in this field were probably and . He is noted, among other things, for the Lambek calculus, an effort to capture mathematical aspects of natural language syntax in logical form, and a work that has been very influential in computational linguistics, as well as for developing
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https://en.wikipedia.org/wiki/Calculus%20of%20structures
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The calculus of structures is a proof calculus with deep inference for studying the structural proof theory of noncommutative logic. The calculus has since been applied to study linear logic, classical logic, modal logic, and process calculi, and many benefits are claimed to follow in these investigations from the way in which deep inference is made available in the calculus.
References
Alessio Guglielmi (2004)., 'A System of Interaction and Structure'. ACM Transactions on Computational Logic.
Kai Brünnler (2004). Deep Inference and Symmetry in Classical Proofs. Logos Verlag.
External links
Calculus of structures homepage
CoS in Maude: page documenting implementations of logical systems in the calculus of structures, using the Maude system.
Logical calculi
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https://en.wikipedia.org/wiki/Deep%20inference
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Deep inference names a general idea in structural proof theory that breaks with the classical sequent calculus by generalising the notion of structure to permit inference to occur in contexts of high structural complexity. The term deep inference is generally reserved for proof calculi where the structural complexity is unbounded; in this article we will use non-shallow inference to refer to calculi that have structural complexity greater than the sequent calculus, but not unboundedly so, although this is not at present established terminology.
Deep inference is not important in logic outside of structural proof theory, since the phenomena that lead to the proposal of formal systems with deep inference are all related to the cut-elimination theorem. The first calculus of deep inference was proposed by Kurt Schütte, but the idea did not generate much interest at the time.
Nuel Belnap proposed display logic in an attempt to characterise the essence of structural proof theory. The calculus of structures was proposed in order to give a cut-free characterisation of noncommutative logic. Cirquent calculus was developed as a system of deep inference allowing to explicitly account for the possibility of subcomponent-sharing.
Notes
Further reading
Kai Brünnler, "Deep Inference and Symmetry in Classical Proofs" (Ph.D. thesis 2004), also published in book form by Logos Verlag ().
Deep Inference and the Calculus of Structures Intro and reference web page about ongoing research in deep inference.
Proof theory
Inference
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https://en.wikipedia.org/wiki/Proof%20calculus
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In mathematical logic, a proof calculus or a proof system is built to prove statements.
Overview
A proof system includes the components:
Formal language: The set L of formulas admitted by the system, for example, propositional logic or first-order logic.
Rules of inference: List of rules that can be employed to prove theorems from axioms and theorems.
Axioms: Formulas in L assumed to be valid. All theorems are derived from axioms.
A formal proof of a well-formed formula in a proof system is a set of axioms and rules of inference of proof system that infers that the well-formed formula is a theorem of proof system.
Usually a given proof calculus encompasses more than a single particular formal system, since many proof calculi are under-determined and can be used for radically different logics. For example, a paradigmatic case is the sequent calculus, which can be used to express the consequence relations of both intuitionistic logic and relevance logic. Thus, loosely speaking, a proof calculus is a template or design pattern, characterized by a certain style of formal inference, that may be specialized to produce specific formal systems, namely by specifying the actual inference rules for such a system. There is no consensus among logicians on how best to define the term.
Examples of proof calculi
The most widely known proof calculi are those classical calculi that are still in widespread use:
The class of Hilbert systems, of which the most famous example is the 1928 Hilbert–Ackermann system of first-order logic;
Gerhard Gentzen's calculus of natural deduction, which is the first formalism of structural proof theory, and which is the cornerstone of the formulae-as-types correspondence relating logic to functional programming;
Gentzen's sequent calculus, which is the most studied formalism of structural proof theory.
Many other proof calculi were, or might have been, seminal, but are not widely used today.
Aristotle's syllogistic calculus, presented in the Organon, readily admits formalisation. There is still some modern interest in syllogisms, carried out under the aegis of term logic.
Gottlob Frege's two-dimensional notation of the Begriffsschrift (1879) is usually regarded as introducing the modern concept of quantifier to logic.
C.S. Peirce's existential graph easily might have been seminal, had history worked out differently.
Modern research in logic teems with rival proof calculi:
Several systems have been proposed that replace the usual textual syntax with some graphical syntax. proof nets and cirquent calculus are among such systems.
Recently, many logicians interested in structural proof theory have proposed calculi with deep inference, for instance display logic, hypersequents, the calculus of structures, and bunched implication.
See also
Propositional proof system
Proof nets
Cirquent calculus
Calculus of structures
Formal proof
Method of analytic tableaux
Resolution (logic)
References
Proof theory
Logical calculi
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https://en.wikipedia.org/wiki/Solid%20torus
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In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product of the disk and the circle, endowed with the product topology.
A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus.
A solid torus is a torus plus the volume inside the torus. Real-world objects that approximate a solid torus include O-rings, non-inflatable lifebuoys, ring doughnuts, and bagels.
Topological properties
The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to , the ordinary torus.
Since the disk is contractible, the solid torus has the homotopy type of a circle, . Therefore the fundamental group and homology groups are isomorphic to those of the circle:
See also
Cheerios
Hyperbolic Dehn surgery
Reeb foliation
Whitehead manifold
Donut
References
3-manifolds
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https://en.wikipedia.org/wiki/Northeast%20Region%2C%20Brazil
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The Northeast Region of Brazil (; ) is one of the five official and political regions of the country according to the Brazilian Institute of Geography and Statistics. Of Brazil's twenty-six states, it comprises nine: Maranhão, Piauí, Ceará, Rio Grande do Norte, Paraíba, Pernambuco, Alagoas, Sergipe and Bahia, along with the Fernando de Noronha archipelago (formerly a separate territory, now part of Pernambuco).
Chiefly known as Nordeste ("Northeast") in Brazil, this region was the first to be colonized by the Portuguese and other European peoples, playing a crucial role in the country's history. Nordeste'''s dialects and rich culture, including its folklore, cuisines, music and literature, became the most easily distinguishable across the country. To this day, Nordeste is known for its history and culture, as well as for its natural environment and its hot weather.Nordeste stretches from the Atlantic seaboard in the northeast and southeast, northwest and west to the Amazon Basin and south through the Espinhaço highlands in southern Bahia. It encloses the São Francisco River and drainage basin, which were instrumental in the exploration, settlement and economic development of the region. The region lies entirely within the earth's tropical zone and encompasses Caatinga, Atlantic Forest and part of the Cerrado ecoregions. The climate is hot and semi-arid, varying from xeric in Caatinga, to mesic in Cerrado and hydric in the Atlantic Forest.
The Northeast Region represents 18% of Brazilian territory, has a population of 53.6 million people, 28% of the total population of the country, and contributes 13.4% (2011) of Brazil's GDP. Nearly three quarters of the population live in urban areas clustered along the Atlantic coast and about 15 million people live in the hinterland. It is an impoverished region: 43.5% of the population lives in poverty, defined as less than $2/day.
The capital of each state is also its largest city. These include Salvador, Recife, Fortaleza and São Luís, all of which are coastal cities with a population above one million.Nordeste has nine international airports, and the region has the second largest number of passengers (roughly 20%) in Brazil.
Geography
Zona da Mata ("Forest Zone")
The Zona da Mata comprises the rainforest zones of Nordeste (part of the Atlantic Forest or Mata Atlântica) in the humid eastern coast, where the region's largest capital cities are also located. The forest area was much larger before suffering from centuries of deforestation and exploration. For many years, sugar cane cultivation in this region was the mainstay of Brazil's economy, being superseded only when coffee production developed in the late 19th century. Sugar cane is cultivated on large estates whose owners maintain tremendous political influence.
Agreste
Since the escarpment does not generate any further rainfall on its slopes from the lifting of the trade winds, annual rainfall decreases steadily inland. After a relatively sho
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https://en.wikipedia.org/wiki/Club%20set
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In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal that is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name club is a contraction of "closed and unbounded".
Formal definition
Formally, if is a limit ordinal, then a set is closed in if and only if for every if then Thus, if the limit of some sequence from is less than then the limit is also in
If is a limit ordinal and then is unbounded in if for any there is some such that
If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals).
For example, the set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded.
If is an uncountable initial ordinal, then the set of all limit ordinals is closed unbounded in In fact a club set is nothing else but the range of a normal function (i.e. increasing and continuous).
More generally, if is a nonempty set and is a cardinal, then (the set of subsets of of cardinality ) is club if every union of a subset of is in and every subset of of cardinality less than is contained in some element of (see stationary set).
The closed unbounded filter
Let be a limit ordinal of uncountable cofinality For some , let be a sequence of closed unbounded subsets of Then is also closed unbounded. To see this, one can note that an intersection of closed sets is always closed, so we just need to show that this intersection is unbounded. So fix any and for each n < ω choose from each an element which is possible because each is unbounded. Since this is a collection of fewer than ordinals, all less than their least upper bound must also be less than so we can call it This process generates a countable sequence The limit of this sequence must in fact also be the limit of the sequence and since each is closed and is uncountable, this limit must be in each and therefore this limit is an element of the intersection that is above which shows that the intersection is unbounded. QED.
From this, it can be seen that if is a regular cardinal, then is a non-principal -complete proper filter on the set (that is, on the poset ).
If is a regular cardinal then club sets are also closed under diagonal intersection.
In fact, if is regular and is any filter on closed under diagonal intersection, containing all sets of the form for then must include all club sets.
See also
References
Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. .
Lévy, Azriel (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002, Dover.
Ordinal numbers
Set theory
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https://en.wikipedia.org/wiki/Mostowski%20collapse%20lemma
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In mathematical logic, the Mostowski collapse lemma, also known as the Shepherdson–Mostowski collapse, is a theorem of set theory introduced by and .
Statement
Suppose that R is a binary relation on a class X such that
R is set-like: R−1[x] = {y : y R x} is a set for every x,
R is well-founded: every nonempty subset S of X contains an R-minimal element (i.e. an element x ∈ S such that R−1[x] ∩ S is empty),
R is extensional: R−1[x] ≠ R−1[y] for every distinct elements x and y of X
The Mostowski collapse lemma states that for every such R there exists a unique transitive class (possibly proper) whose structure under the membership relation is isomorphic to (X, R), and the isomorphism is unique. The isomorphism maps each element x of X to the set of images of elements y of X such that y R x (Jech 2003:69).
Generalizations
Every well-founded set-like relation can be embedded into a well-founded set-like extensional relation. This implies the following variant of the Mostowski collapse lemma: every well-founded set-like relation is isomorphic to set-membership on a (non-unique, and not necessarily transitive) class.
A mapping F such that F(x) = {F(y) : y R x} for all x in X can be defined for any well-founded set-like relation R on X by well-founded recursion. It provides a homomorphism of R onto a (non-unique, in general) transitive class. The homomorphism F is an isomorphism if and only if R is extensional.
The well-foundedness assumption of the Mostowski lemma can be alleviated or dropped in non-well-founded set theories. In Boffa's set theory, every set-like extensional relation is isomorphic to set-membership on a (non-unique) transitive class. In set theory with Aczel's anti-foundation axiom, every set-like relation is bisimilar to set-membership on a unique transitive class, hence every bisimulation-minimal set-like relation is isomorphic to a unique transitive class.
Application
Every set model of ZF is set-like and extensional. If the model is well-founded, then by the Mostowski collapse lemma it is isomorphic to a transitive model of ZF and such a transitive model is unique.
Saying that the membership relation of some model of ZF is well-founded is stronger than saying that the axiom of regularity is true in the model. There exists a model M (assuming the consistency of ZF) whose domain has a subset A with no R-minimal element, but this set A is not a "set in the model" (A is not in the domain of the model, even though all of its members are). More precisely, for no such set A there exists x in M such that A = R−1[x]. So M satisfies the axiom of regularity (it is "internally" well-founded) but it is not well-founded and the collapse lemma does not apply to it.
References
Lemmas
Lemmas in set theory
Wellfoundedness
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https://en.wikipedia.org/wiki/Suspension%20%28topology%29
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In topology, a branch of mathematics, the suspension of a topological space X is intuitively obtained by stretching X into a cylinder and then collapsing both end faces to points. One views X as "suspended" between these end points. The suspension of X is denoted by SX or susp(X).
There is a variation of the suspension for pointed space, which is called the reduced suspension and denoted by ΣX. The "usual" suspension SX is sometimes called the unreduced suspension, unbased suspension, or free suspension of X, to distinguish it from ΣX.
Free suspension
The (free) suspension of a topological space can be defined in several ways.
1. is the quotient space . In other words, it can be constructed as follows:
Construct the cylinder .
Consider the entire set as a single point ("glue" all its points together).
Consider the entire set as a single point ("glue" all its points together).
2. Another way to write this is:
Where are two points, and for each i in {0,1}, is the projection to the point (a function that maps everything to ). That means, the suspension is the result of constructing the cylinder , and then attaching it by its faces, and , to the points along the projections .
3. One can view as two cones on X, glued together at their base.
4. can also be defined as the join where is a discrete space with two points.
Properties
In rough terms, S increases the dimension of a space by one: for example, it takes an n-sphere to an (n + 1)-sphere for n ≥ 0.
Given a continuous map there is a continuous map defined by where square brackets denote equivalence classes. This makes into a functor from the category of topological spaces to itself.
Reduced suspension
If X is a pointed space with basepoint x0, there is a variation of the suspension which is sometimes more useful. The reduced suspension or based suspension ΣX of X is the quotient space:
.
This is the equivalent to taking SX and collapsing the line (x0 × I) joining the two ends to a single point. The basepoint of the pointed space ΣX is taken to be the equivalence class of (x0, 0).
One can show that the reduced suspension of X is homeomorphic to the smash product of X with the unit circle S1.
For well-behaved spaces, such as CW complexes, the reduced suspension of X is homotopy equivalent to the unbased suspension.
Adjunction of reduced suspension and loop space functors
Σ gives rise to a functor from the category of pointed spaces to itself. An important property of this functor is that it is left adjoint to the functor taking a pointed space to its loop space . In other words, we have a natural isomorphism
where and are pointed spaces and stands for continuous maps that preserve basepoints. This adjunction can be understood geometrically, as follows: arises out of if a pointed circle is attached to every non-basepoint of , and the basepoints of all these circles are identified and glued to the basepoint of . Now, to specify a pointed map from
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https://en.wikipedia.org/wiki/Nemmers%20Prize%20in%20Mathematics
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The Frederic Esser Nemmers Prize in Mathematics is awarded biennially from Northwestern University. It was initially endowed along with a companion prize, the Erwin Plein Nemmers Prize in Economics, as part of a $14 million donation from the Nemmers brothers. They envisioned creating an award that would be as prestigious as the Nobel Prize. To this end, the majority of the income earned from the endowment is returned to the principal to increase the size of the award.
As of 2023, the award carries a $300,000 stipend and the scholar spends several weeks in residence at Northwestern University.
Recipients
Following recipients received this award:
1994 Yuri I. Manin
1996 Joseph B. Keller
1998 John H. Conway
2000 Edward Witten
2002 Yakov G. Sinai
2004 Mikhail Gromov
2006 Robert Langlands
2008 Simon Donaldson
2010 Terence Tao
2012 Ingrid Daubechies
2014 Michael J. Hopkins
2016 János Kollár
2018 Assaf Naor
2020 Nalini Anantharaman
2022 Bhargav Bhatt
See also
List of mathematics awards
References
External links
Citations page
Nemmers Prize 2012
Mathematics awards
Northwestern University
1994 establishments in Illinois
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https://en.wikipedia.org/wiki/Leroy%20P.%20Steele%20Prize
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The Leroy P. Steele Prizes are awarded every year by the American Mathematical Society, for distinguished research work and writing in the field of mathematics. Since 1993, there has been a formal division into three categories.
The prizes have been given since 1970, from a bequest of Leroy P. Steele, and were set up in honor of George David Birkhoff, William Fogg Osgood and William Caspar Graustein. The way the prizes are awarded was changed in 1976 and 1993, but the initial aim of honoring expository writing as well as research has been retained. The prizes of $5,000 are not given on a strict national basis, but relate to mathematical activity in the USA, and writing in English (originally, or in translation).
Steele Prize for Lifetime Achievement
2023 Nicholas M. Katz
2022 Richard P. Stanley
2021 Spencer Bloch
2020 Karen Uhlenbeck
2019 Jeff Cheeger
2018 Jean Bourgain
2017 James G. Arthur
2016 Barry Simon
2015 Victor Kac
2014 Phillip A. Griffiths
2013 Yakov G. Sinai
2012 Ivo M. Babuška
2011 John W. Milnor
2010 William Fulton
2009 Luis Caffarelli
2008 George Lusztig
2007 Henry P. McKean
2006 Frederick W. Gehring, Dennis P. Sullivan
2005 Israel M. Gelfand
2004 Cathleen Synge Morawetz
2003 Ronald Graham, Victor Guillemin
2002 Michael Artin, Elias Stein
2001 Harry Kesten
2000 Isadore M. Singer
1999 Richard V. Kadison
1998 Nathan Jacobson
1997 Ralph S. Phillips
1996 Goro Shimura
1995 John T. Tate
1994 Louis Nirenberg
1993 Eugene B. Dynkin
Steele Prize for Mathematical Exposition
Steele Prize for Seminal Contribution to Research
Leroy P. Steele Prizes awarded prior to 1993
1992 Jacques Dixmier for his books von Neumann Algebras (Algèbres de von Neumann ), Gauthier-Villars, Paris (1957); C*-Algebras (Les C*-Algèbres et leurs Representations ), Gauthier-Villars, Paris (1964); and Enveloping Algebras (Algèbres Enveloppantes ), Gauthier-Villars, Paris (1974).
1992 James Glimm for his paper, Solution in the large for nonlinear hyperbolic systems of conservation laws, Communications on Pure and Applied Mathematics, XVIII (1965), pp. 697–715.
1992: Peter D. Lax for his numerous and fundamental contributions to the theory and applications of linear and nonlinear partial differential equations and functional analysis, for his leadership in the development of computational and applied mathematics, and for his extraordinary impact as a teacher.
1991: Jean-François Treves for Pseudodifferential and Fourier Integral Operators, Volumes 1 and 2 (Plenum Press, 1980).
1991 Eugenio Calabi for his fundamental work on global differential geometry, especially complex differential geometry.
1991 Armand Borel for his extensive contributions in geometry and topology, the theory of Lie groups, their lattices and representations and the theory of automorphic forms, the theory of algebraic groups and their representations and extensive organizational and educational efforts to develop and disseminate modern
1990 R. D. Richtmyer for his book Difference Methods for Initial-
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https://en.wikipedia.org/wiki/Induced%20metric
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In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback. It may be determined using the following formula (using the Einstein summation convention), which is the component form of the pullback operation:
Here , describe the indices of coordinates of the submanifold while the functions encode the embedding into the higher-dimensional manifold whose tangent indices are denoted , .
Example – Curve in 3D
Let
be a map from the domain of the curve with parameter into the Euclidean manifold . Here are constants.
Then there is a metric given on as
.
and we compute
Therefore
See also
First fundamental form
References
Differential geometry
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https://en.wikipedia.org/wiki/Wheeler%E2%80%93DeWitt%20equation
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The Wheeler–DeWitt equation for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts to mathematically combine the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity.
In this approach, time plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called 'problem of time'. More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergman–Komar "group" (which is the diffeomorphism group on-shell).
Motivation and background
In canonical gravity, spacetime is foliated into spacelike submanifolds. The three-metric (i.e., metric on the hypersurface) is and given by
In that equation the Latin indices run over the values 1, 2, 3 and the Greek indices run over the values 1, 2, 3, 4. The three-metric is the field, and we denote its conjugate momenta as . The Hamiltonian is a constraint (characteristic of most relativistic systems)
where and is the Wheeler–DeWitt metric. In index-free notation, the Wheeler–DeWitt metric on the space of positive definite quadratic forms g in three dimensions is
Quantization "puts hats" on the momenta and field variables; that is, the functions of numbers in the classical case become operators that modify the state function in the quantum case. Thus we obtain the operator
Working in "position space", these operators are
One can apply the operator to a general wave functional of the metric where:
which would give a set of constraints amongst the coefficients . This means the amplitudes for gravitons at certain positions is related to the amplitudes for a different number of gravitons at different positions. Or, one could use the two-field formalism, treating as an independent field so that the wave function is .
Mathematical formalism
The Wheeler–DeWitt equation is a functional differential equation. It is ill-defined in the general case, but very important in theoretical physics, especially in quantum gravity. It is a functional differential equation on the space of three dimensional spatial metrics. The Wheeler–DeWitt equation has the form of an operator acting on a wave functional; the functional reduces to a function in cosmology. Contrary to the general case, the Wheeler–DeWitt equation is well defined in minisuperspaces like the configuration space of cosmological theories. An example of such a wave function is the Hartle–Hawking state. Bryce DeWitt first published this equation in 1967 under the name "Einstein–Schrödinger equation"; it was later renamed the "Wheeler–DeWitt equation".
Hamiltonian constraint
Simply speaking, the Wheeler–DeWitt equation says
where is the Hamiltonian constraint in quantized general relativity and stands for the wave function of the universe. Unlike ordinary quantum f
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https://en.wikipedia.org/wiki/Growth%20curve
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Growth curve can refer to:
Growth curve (statistics), an empirical model of the evolution of a quantity over time.
Growth curve (biology), a statistical growth curve used to model a biological quantity.
Curve of growth (astronomy), the relation between the equivalent width and the optical depth.
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https://en.wikipedia.org/wiki/Taut%20foliation
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In mathematics, tautness is a rigidity property of foliations. A taut foliation is a codimension 1 foliation of a closed manifold with the property that every leaf meets a transverse circle. By transverse circle, is meant a closed loop that is always transverse to the tangent field of the foliation.
If the foliated manifold has non-empty tangential boundary, then a codimension 1 foliation is taut if every leaf meets a transverse circle or a transverse arc with endpoints on the tangential boundary. Equivalently, by a result of Dennis Sullivan, a codimension 1 foliation is taut if there exists a Riemannian metric that makes each leaf a minimal surface. Furthermore, for compact manifolds the existence, for every leaf , of a transverse circle meeting , implies the existence of a single transverse circle meeting every leaf.
Taut foliations were brought to prominence by the work of William Thurston and David Gabai.
Relation to Reebless foliations
Taut foliations are closely related to the concept of Reebless foliation. A taut foliation cannot have a Reeb component, since the component would act like a "dead-end" from which a transverse curve could never escape; consequently, the boundary torus of the Reeb component has no transverse circle puncturing it. A Reebless foliation can fail to be taut but the only leaves of the foliation with no puncturing transverse circle must be compact, and in particular, homeomorphic to a torus.
Properties
The existence of a taut foliation implies various useful properties about a closed 3-manifold. For example, a closed, orientable 3-manifold, which admits a taut foliation with no sphere leaf, must be irreducible, covered by , and have negatively curved fundamental group.
Rummler–Sullivan theorem
By a theorem of Hansklaus Rummler and Dennis Sullivan, the following conditions are equivalent for transversely orientable codimension one foliations of closed, orientable, smooth manifolds M:
is taut;
there is a flow transverse to which preserves some volume form on M;
there is a Riemannian metric on M for which the leaves of are least area surfaces.
References
Foliations
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https://en.wikipedia.org/wiki/Pseudogroup
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In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie to investigate symmetries of differential equations, rather than out of abstract algebra (such as quasigroup, for example). The modern theory of pseudogroups was developed by Élie Cartan in the early 1900s.
Definition
A pseudogroup imposes several conditions on a sets of homeomorphisms (respectively, diffeomorphisms) defined on open sets U of a given Euclidean space or more generally of a fixed topological space (respectively, smooth manifold). Since two homeomorphisms and compose to a homeomorphism from U to W, one asks that the pseudogroup is closed under composition and inversion. However, unlike those for a group, the axioms defining a pseudogroup are not purely algebraic; the further requirements are related to the possibility of restricting and of patching homeomorphisms (similar to the gluing axiom for sections of a sheaf).
More precisely, a pseudogroup on a topological space is a collection of homeomorphisms between open subsets of satisfying the following properties:
The domains of the elements in cover ("cover").
The restriction of an element in to any open set contained in its domain is also in ("restriction").
The composition ○ of two elements of , when defined, is in ("composition").
The inverse of an element of is in ("inverse").
The property of lying in is local, i.e. if : → is a homeomorphism between open sets of and is covered by open sets with restricted to lying in for each , then also lies in ("local").
As a consequence the identity homeomorphism of any open subset of lies in .
Similarly, a pseudogroup on a smooth manifold is defined as a collection of diffeomorphisms between open subsets of satisfying analogous properties (where we replace homeomorphisms with diffeomorphisms).
Two points in are said to be in the same orbit if an element of sends one to the other. Orbits of a pseudogroup clearly form a partition of ; a pseudogroup is called transitive if it has only one orbit.
Examples
A widespread class of examples is given by pseudogroups preserving a given geometric structure. For instance, if (X, g) is a Riemannian manifold, one has the pseudogroup of its local isometries; if (X, ω) is a symplectic manifold, one has the pseudogroup of its local symplectomorphisms; etc. These pseudogroups should be thought as the set of the local symmetries of these structures.
Pseudogroups of symmetries and geometric structures
Manifolds with additional structures can often be defined using the pseudogroups of symmetries of a fixed local model. More precisely, given a pseudogroup , a -atlas on a topological space consists of a standard atlas on such that the changes of coordinates (i.e. the transition maps) belong to . An equivalent class of Γ-a
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https://en.wikipedia.org/wiki/Direct%20integral
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In mathematics and functional analysis a direct integral or Hilbert integral is a generalization of the concept of direct sum. The theory is most developed for direct integrals of Hilbert spaces and direct integrals of von Neumann algebras. The concept was introduced in 1949 by John von Neumann in one of the papers in the series On Rings of Operators. One of von Neumann's goals in this paper was to reduce the classification of (what are now called) von Neumann algebras on separable Hilbert spaces to the classification of so-called factors. Factors are analogous to full matrix algebras over a field, and von Neumann wanted to prove a continuous analogue of the Artin–Wedderburn theorem classifying semi-simple rings.
Results on direct integrals can be viewed as generalizations of results about finite-dimensional C*-algebras of matrices; in this case the results are easy to prove directly. The infinite-dimensional case is complicated by measure-theoretic technicalities.
Direct integral theory was also used by George Mackey in his analysis of systems of imprimitivity and his general theory of induced representations of locally compact separable groups.
Direct integrals of Hilbert spaces
The simplest example of a direct integral are the L2 spaces associated to a (σ-finite) countably additive measure μ on a measurable space X. Somewhat more generally one can consider a separable Hilbert space H and the space of square-integrable H-valued functions
Terminological note: The terminology adopted by the literature on the subject is followed here, according to which a measurable space X is referred to as a Borel space and the elements of the distinguished σ-algebra of X as Borel sets, regardless of whether or not the underlying σ-algebra comes from a topological space (in most examples it does). A Borel space is standard if and only if it is isomorphic to the underlying Borel space of a Polish space; all Polish spaces of a given cardinality are isomorphic to each other (as Borel spaces). Given a countably additive measure μ on X, a measurable set is one that differs from a Borel set by a null set. The measure μ on X is a standard measure if and only if there is a null set E such that its complement X − E is a standard Borel space. All measures considered here are σ-finite.
Definition. Let X be a Borel space equipped with a countably additive measure μ. A measurable family of Hilbert spaces on (X, μ) is a family {Hx}x∈ X, which is locally equivalent to a trivial family in the following sense: There is a countable partition
by measurable subsets of X such that
where Hn is the canonical n-dimensional Hilbert space, that is
A cross-section of {Hx}x∈ X is a family {sx}x ∈ X such that sx ∈ Hx for all x ∈ X. A cross-section is measurable if and only if its restriction to each partition element Xn is measurable. We will identify measurable cross-sections s, t that are equal almost everywhere. Given a measurable family of Hilbert spaces, the direct integra
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https://en.wikipedia.org/wiki/San%20Rafael%20Department
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San Rafael is one of the departments of Mendoza Province, Argentina. The seat of the department is in the city of San Rafael.
Statistics
Geographical location: 34° 15´ to 36° southern latitude and 70° 10´ to 66° 55´ eastern longitude.
Area: 31,235 km² (20.82% of the provincial area)
Extension: 204 km from north to south. 298 km from east to west
Altitude: 750 m above sea level.
Neighboring departments
North: San Carlos Department, Santa Rosa Department, Mendoza and La Paz Department, Mendoza
East: San Luis Province and General Alvear Department
South: La Pampa Province and Malargüe Department
West: Chile.
Administration
The department is subdivided in 18 districts: Ciudad, El Cerrito, Cuadro Nacional, Las Malvinas District, El Sosneado (added in 2005, it used to belong to Cuadro Benegas, but then it was added as a district), Las Paredes, La Llave, Cuadro Benegas, Cuadro Nacional, Cañada Seca, Goudge, Jaime Prats, Monte Comán, Rama Caída, Real del Padre, Punta del Agua, Villa Atuel and Villa 25 de Mayo.
External links
Site of San Rafael government (In Spanish)
Guide to the city of San Rafael (In Spanish)
Departments of Mendoza Province
States and territories established in 1805
Wine regions of Argentina
1805 establishments in the Spanish Empire
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https://en.wikipedia.org/wiki/Zeno%20machine
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In mathematics and computer science, Zeno machines (abbreviated ZM, and also called accelerated Turing machine, ATM) are a hypothetical computational model related to Turing machines that are capable of carrying out computations involving a countably infinite number of algorithmic steps. These machines are ruled out in most models of computation.
The idea of Zeno machines was first discussed by Hermann Weyl in 1927; the name refers to Zeno's paradoxes, attributed to the ancient Greek philosopher Zeno of Elea. Zeno machines play a crucial role in some theories. The theory of the Omega Point devised by physicist Frank J. Tipler, for instance, can only be valid if Zeno machines are possible.
Definition
A Zeno machine is a Turing machine that can take an infinite number of steps, and then continue take more steps. This can be thought of as a supertask where units of time are taken to perform the -th step; thus, the first step takes 0.5 units of time, the second takes 0.25, the third 0.125 and so on, so that after one unit of time, a countably infinite number of steps will have been performed.
Infinite time Turing machines
A more formal model of the Zeno machine is the infinite time Turing machine. Defined first in unpublished work by Jeffrey Kidder and expanded upon by Joel Hamkins and Andy Lewis, in Infinite Time Turing Machines, the infinite time Turing machine is an extension of the classical Turing machine model, to include transfinite time; that is time beyond all finite time. A classical Turing machine has a status at step (in the start state, with an empty tape, read head at cell 0) and a procedure for getting from one status to the successive status. In this way the status of a Turing machine is defined for all steps corresponding to a natural number. An maintains these properties, but also defines the status of the machine at limit ordinals, that is ordinals that are neither nor the successor of any ordinal. The status of a Turing machine consists of 3 parts:
The state
The location of the read-write head
The contents of the tape
Just as a classical Turing machine has a labeled start state, which is the state at the start of a program, an has a labeled limit state which is the state for the machine at any limit ordinal. This is the case even if the machine has no other way to access this state, for example no node transitions to it. The location of the read-write head is set to zero for at any limit step. Lastly the state of the tape is determined by the limit supremum of previous tape states. For some machine , a cell and, a limit ordinal then
That is the th cell at time is the limit supremum of that same cell as the machine approaches . This can be thought of as the limit if it converges or otherwise.
Computability
Zeno machines have been proposed as a model of computation more powerful than classical Turing machines, based on their ability to solve the halting problem for classical Turing machines. Cristian
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https://en.wikipedia.org/wiki/Scott%20core%20theorem
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In mathematics, the Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott, . The precise statement is as follows:
Given a 3-manifold (not necessarily compact) with finitely generated fundamental group, there is a compact three-dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable.
A simplified proof is given in , and a stronger uniqueness statement is proven in .
References
3-manifolds
Theorems in group theory
Theorems in topology
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https://en.wikipedia.org/wiki/BTI
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BTI or Bti may refer to:
Acronyms
Bacillus thuringiensis israelensis (Bti), a bacterium
Barisan Tani Indonesia
Baron Tornado Index of tornado probability
Before the Impact, an American TV series
Beverage Testing Institute
Bicycle Technologies International, Santa Fe, New Mexico, US
Boston Theological Institute
Boyce Thompson Institute for Plant Research, Cornell University, Ithaca, New York, US
Branch Target Identification, an AMD technology for mitigating computer security exploits;
Branch Target Injection or Spectre variant 2, a security vulnerability
Breaking the Impasse, an Israel-Palestinian group
Breed Technologies, Inc., now known as Joyson Safety Systems
Btrieve Technologies, Inc.
Because the Internet, the second studio album by American singer Childish Gambino
Codes and symbols
Barter Island LRRS Airport, Alaska, US, IATA airport code
British American Tobacco PLC, NYSE symbol
The ICAO designator of the Latvian flag carrier, AirBaltic.
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https://en.wikipedia.org/wiki/Zeitschrift%20f%C3%BCr%20Angewandte%20Mathematik%20und%20Physik
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The Zeitschrift für Angewandte Mathematik und Physik (English: Journal of Applied Mathematics and Physics) is a bimonthly peer-reviewed scientific journal published by Birkhäuser Verlag. The editor-in-chief is Kaspar Nipp (ETH Zurich). It was established in 1950 and covers the fields of theoretical and applied mechanics, applied mathematics, and related topics. According to the Journal Citation Reports, the journal has a 2017 impact factor of 1.711.
References
External links
Mathematics journals
Physics journals
Academic journals established in 1950
Springer Science+Business Media academic journals
Bimonthly journals
English-language journals
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https://en.wikipedia.org/wiki/Alexander%20McAulay
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Alexander McAulay (9 December 1863 – 6 July 1931) was the first professor of mathematics and physics at the University of Tasmania, Hobart, Tasmania. He was also a proponent of dual quaternions, which he termed "octonions" or "Clifford biquaternions".
McAulay was born on 9 December 1863 and attended Kingswood School in Bath. He proceeded to Caius College, Cambridge, there taking up a study of the quaternion algebra. In 1883 he published an article "Some general theorems in quaternion integration". McAulay took his degree in 1886, and began to reflect on the instruction of students in quaternion theory. In an article "Establishment of the fundamental properties of quaternions" he suggested improvements to the texts then in use. He also wrote a technical article on integration.
Departing for Australia, he lectured at Ormond College, University of Melbourne from 1893 to 1895. As a distant correspondent, he participated in a vigorous debate about the place of quaternions in physics education. In 1893 his book Utility of Quaternions in Physics was published. A. S. Hathaway contributed a positive review and Peter Guthrie Tait praised
it in these terms:
Here, at last, we exclaim, is a man who has caught the full spirit of the quaternion system: the real aestus, the awen of the Welsh Bards, the divinus afflatus that transports the poet beyond the limits of sublunary things! Intuitively recognizing its power, he snatches up the magnificent weapon which Hamilton tenders us all, and at once dashes off to the jungle on the quest of big game.
McAulay took up the position of Professor of Physics in Tasmania from 1896 until 1929, at which time his son Alexander Leicester McAulay took over the position for the next thirty years.
Following William Kingdon Clifford who had extended quaternions to dual quaternions, McAulay made a special study of this hypercomplex number system. In 1898 McAulay published, through Cambridge University Press, his Octonions: a Development of Clifford's Biquaternions.
McAulay died on 6 July 1931.
His brother Francis Macaulay, who stayed in England, also contributed to ring theory. The University of Tasmania has commemorated the McAulays' contributions in Winter Public Lectures.
Works
1893: Utility of Quaternions in Physics, link from Project Gutenberg.
1898: Octonions: a development of Clifford's Biquaternions, link from Internet Archive
1900: "Notes on the Electromagnetic Theory of Light", Philosophical Magazine 49(5):228–242.
References
Rev N. M. Ferres (1892), Review of "On the Mathematical Theory of Electromagnetism", in Proceedings of the Royal Society, London, v.51,p. 400
Rev N. M. Ferres (1895) Preview of Octonions, Proceedings of the Royal Society 59: 169, weblink from Archive.org.
External links
Bruce Scott (1986) McAulay, Alexander (1863 – 1931) from Australian Dictionary of Biography.
1863 births
1931 deaths
British physicists
19th-century British mathematicians
19th-century Australian mathematicians
2
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https://en.wikipedia.org/wiki/Vector%20algebra
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In mathematics, vector algebra may mean:
Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space.
The algebraic operations in vector calculus, namely the specific additional structure of vectors in 3-dimensional Euclidean space of dot product and especially cross product. In this sense, vector algebra is contrasted with geometric algebra, which provides an alternative generalization to higher dimensions.
An algebra over a field, a vector space equipped with a bilinear product
Original vector algebras of the nineteenth century like quaternions, tessarines, or coquaternions, each of which has its own product. The vector algebras biquaternions and hyperbolic quaternions enabled the revolution in physics called special relativity by providing mathematical models.
Algebra
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https://en.wikipedia.org/wiki/Hyperbolic%20coordinates
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In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane
.
Hyperbolic coordinates take values in the hyperbolic plane defined as:
.
These coordinates in HP are useful for studying logarithmic comparisons of direct proportion in Q and measuring deviations from direct proportion.
For in take
and
.
The parameter u is the hyperbolic angle to (x, y) and v is the geometric mean of x and y.
The inverse mapping is
.
The function is a continuous mapping, but not an analytic function.
Alternative quadrant metric
Since HP carries the metric space structure of the Poincaré half-plane model of hyperbolic geometry, the bijective correspondence
brings this structure to Q. It can be grasped using the notion of hyperbolic motions. Since geodesics in HP are semicircles with centers on the boundary, the geodesics in Q are obtained from the correspondence and turn out to be rays from the origin or petal-shaped curves leaving and re-entering the origin. And the hyperbolic motion of HP given by a left-right shift corresponds to a squeeze mapping applied to Q.
Since hyperbolas in Q correspond to lines parallel to the boundary of HP, they are horocycles in the metric geometry of Q.
If one only considers the Euclidean topology of the plane and the topology inherited by Q, then the lines bounding Q seem close to Q. Insight from the metric space HP shows that the open set Q has only the origin as boundary when viewed through the correspondence. Indeed, consider rays from the origin in Q, and their images, vertical rays from the boundary R of HP. Any point in HP is an infinite distance from the point p at the foot of the perpendicular to R, but a sequence of points on this perpendicular may tend in the direction of p. The corresponding sequence in Q tends along a ray toward the origin. The old Euclidean boundary of Q is no longer relevant.
Applications in physical science
Fundamental physical variables are sometimes related by equations of the form k = x y. For instance, V = I R (Ohm's law), P = V I (electrical power), P V = k T (ideal gas law), and f λ = v (relation of wavelength, frequency, and velocity in the wave medium). When the k is constant, the other variables lie on a hyperbola, which is a horocycle in the appropriate Q quadrant.
For example, in thermodynamics the isothermal process explicitly follows the hyperbolic path and work can be interpreted as a hyperbolic angle change. Similarly, a given mass M of gas with changing volume will have variable density δ = M / V, and the ideal gas law may be written P = k T δ so that an isobaric process traces a hyperbola in the quadrant of absolute temperature and gas density.
For hyperbolic coordinates in the theory of relativity see the History section.
Statistical applications
Comparative study of population density in the quadrant begins with selecting a reference nation, region, or urban area whose population and area are taken as the point (1,1
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https://en.wikipedia.org/wiki/Normal%20map
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Normal map may refer to:
Normal mapping in 3D computer graphics
Normal invariants in mathematical surgery theory
Normal matrix in linear algebra
Normal operator in functional analysis
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https://en.wikipedia.org/wiki/Community%20areas%20in%20Chicago
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The city of Chicago is divided into 77 community areas for statistical and planning purposes. Census data and other statistics are tied to the areas, which serve as the basis for a variety of urban planning initiatives on both the local and regional levels. The areas' boundaries do not generally change, allowing comparisons of statistics across time. The areas are distinct from but related to the more numerous neighborhoods of Chicago; an area often corresponds to a neighborhood or encompasses several neighborhoods, but the areas do not always correspond to popular conceptions of the neighborhoods due to a number of factors including historical evolution and choices made by the creators of the areas. , Near North Side is the most populous of the areas with over 105,000 residents, while Burnside is the least populous with just over 2,500. Other geographical divisions of Chicago exist, such as the "sides" created by the branches of the Chicago River, the wards of the Chicago City Council, and the parishes of the Roman Catholic Church.
The Social Science Research Committee at the University of Chicago defined the community areas in the 1920s based on neighborhoods or groups of related neighborhoods within the city. In this effort it was led by sociologists Robert E. Park and Ernest Burgess, who believed that physical contingencies created areas that would inevitably form a common identity. Except for the addition of two areas (O'Hare from land annexed by the city in 1956 and Edgewater's separation from Uptown in 1980) and expansions due to minor annexations, the areas' boundaries have never been revised to reflect change but instead have been kept stable. The areas have become a part of the culture of Chicago, contributing to its perception as a "city of neighborhoods" and breaking it down into smaller regions for easier analysis and local planning. Nevertheless, Park's and Burgess's ideas on the inevitability of physically related areas forming a common bond have been questioned, and the unchanging nature of the areas has at times been considered problematic with major subsequent changes in the urban landscape such as the construction of expressways.
History
During the 19th century wards were used by the Census Bureau for data at the level below cities. This was problematic as wards were political subdivisions and thus changed after each census, limiting their utility for comparisons over time. Census tracts were first used in Chicago in the 1910 Census. However, by the 1920s the Social Science Research Committee at the University of Chicago wanted divisions that were more natural and manageable than the arbitrarily-designated and numerous census tracts. The sociologist Robert E. Park led this charge, considering physical barriers such as railroads and the Chicago River to form distinctive and consistent areas within the city, which he deemed "natural" areas that would eventually merge into a distinctive identity. Ernest Burgess, a colleague of P
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https://en.wikipedia.org/wiki/Prime%20Obsession
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Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics (2003) is a historical book on mathematics by John Derbyshire, detailing the history of the Riemann hypothesis, named for Bernhard Riemann, and some of its applications.
The book was awarded the Mathematical Association of America's inaugural Euler Book Prize in 2007.
Overview
The book is written such that even-numbered chapters present historical elements related to the development of the conjecture, and odd-numbered chapters deal with the mathematical and technical aspects. Despite the title, the book provides biographical information on many iconic mathematicians including Euler, Gauss, and Lagrange.
In chapter 1, "Card Trick", Derbyshire introduces the idea of an infinite series and the ideas of convergence and divergence of these series. He imagines that there is a deck of cards stacked neatly together, and that one pulls off the top card so that it overhangs from the deck. Explaining that it can overhang only as far as the center of gravity allows, the card is pulled so that exactly half of it is overhanging. Then, without moving the top card, he slides the second card so that it is overhanging too at equilibrium. As he does this more and more, the fractional amount of overhanging cards as they accumulate becomes less and less. He explores various types of series such as the harmonic series.
In chapter 2, Bernhard Riemann is introduced and a brief historical account of Eastern Europe in the 18th Century is discussed.
In chapter 3, the Prime Number Theorem (PNT) is introduced. The function which mathematicians use to describe the number of primes in N numbers, π(N), is shown to behave in a logarithmic manner, as so:
where log is the natural logarithm.
In chapter 4, Derbyshire gives a short biographical history of Carl Friedrich Gauss and Leonard Euler, setting up their involvement in the Prime Number Theorem.
In chapter 5, the Riemann Zeta Function is introduced:
In chapter 7, the sieve of Eratosthenes is shown to be able to be simulated using the Zeta function. With this, the following statement which becomes the pillar stone of the book is asserted:
Following the derivation of this finding, the book delves into how this is manipulated to expose the PNT's nature.
Audience and reception
According to reviewer S. W. Graham, the book is written at a level that is suitable for advanced undergraduate students of mathematics. In contrast, James V. Rauff recommends it to "anyone interested in the history and mathematics of the Riemann hypothesis".
Reviewer Don Redmond writes that, while the even-numbered chapters explain the history well, the odd-numbered chapters present the mathematics too informally to be useful, failing to provide insight to readers who do not already understand the mathematics, and failing even to explain the importance of the Riemann hypothesis. Graham adds that the level of mathematics is inconsistent, with detailed explanations o
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https://en.wikipedia.org/wiki/Cayley%27s%20formula
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In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer , the number of trees on labeled vertices is .
The formula equivalently counts the number of spanning trees of a complete graph with labeled vertices .
Proof
Many proofs of Cayley's tree formula are known.
One classical proof of the formula uses Kirchhoff's matrix tree theorem, a formula for the number of spanning trees in an arbitrary graph involving the determinant of a matrix. Prüfer sequences yield a bijective proof of Cayley's formula. Another bijective proof, by André Joyal, finds a one-to-one transformation between n-node trees with two distinguished nodes and maximal directed pseudoforests.
A proof by double counting due to Jim Pitman counts in two different ways the number of different sequences of directed edges that can be added to an empty graph on n vertices to form from it a rooted tree; see .
History
The formula was first discovered by Carl Wilhelm Borchardt in 1860, and proved via a determinant. In a short 1889 note, Cayley extended the formula in several directions, by taking into account the degrees of the vertices. Although he referred to Borchardt's original paper, the name "Cayley's formula" became standard in the field.
Other properties
Cayley's formula immediately gives the number of labelled rooted forests on n vertices, namely .
Each labelled rooted forest can be turned into a labelled tree with one extra vertex, by adding a vertex with label and connecting it to all roots of the trees in the forest.
There is a close connection with rooted forests and parking functions, since the number of parking functions on n cars is also . A bijection between rooted forests and parking functions was given by M. P. Schützenberger in 1968.
Generalizations
The following generalizes Cayley's formula to labelled forests:
Let Tn,k be the number of labelled forests on n vertices with k connected components,
such that vertices 1, 2, ..., k all belong to different connected components.
Then .
References
Trees (graph theory)
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https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Hopf%20theorem
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In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincaré and Heinz Hopf.
The Poincaré–Hopf theorem is often
illustrated by the special case of the hairy ball theorem, which simply states that there is no smooth vector field on an even-dimensional n-sphere having no sources or sinks.
Formal statement
Let be a differentiable manifold, of dimension , and a vector field on . Suppose that is an isolated zero of , and fix some local coordinates near . Pick a closed ball centered at , so that is the only zero of in . Then the index of at , , can be defined as the degree of the map from the boundary of to the -sphere given by .
Theorem. Let be a compact differentiable manifold. Let be a vector field on with isolated zeroes. If has boundary, then we insist that be pointing in the outward normal direction along the boundary. Then we have the formula
where the sum of the indices is over all the isolated zeroes of and is the Euler characteristic of . A particularly useful corollary is when there is a non-vanishing vector field implying Euler characteristic 0.
The theorem was proven for two dimensions by Henri Poincaré and later generalized to higher dimensions by Heinz Hopf.
Significance
The Euler characteristic of a closed surface is a purely topological concept, whereas the index of a vector field is purely analytic. Thus, this theorem establishes a deep link between two seemingly unrelated areas of mathematics. It is perhaps as interesting that the proof of this theorem relies heavily on integration, and, in particular, Stokes' theorem, which states that the integral of the exterior derivative of a differential form is equal to the integral of that form over the boundary. In the special case of a manifold without boundary, this amounts to saying that the integral is 0. But by examining vector fields in a sufficiently small neighborhood of a source or sink, we see that sources and sinks contribute integer amounts (known as the index) to the total, and they must all sum to 0. This result may be considered one of the earliest of a whole series of theorems establishing deep relationships between geometric and analytical or physical concepts. They play an important role in the modern study of both fields.
Sketch of proof
Embed M in some high-dimensional Euclidean space. (Use the Whitney embedding theorem.)
Take a small neighborhood of M in that Euclidean space, Nε. Extend the vector field to this neighborhood so that it still has the same zeroes and the zeroes have the same indices. In addition, make sure that the extended vector field at the boundary of Nε is directed outwards.
The sum of indices of the zeroes of the old (and new) vector field is equal to the degree of the Gauss map from the boundary of Nε to the sphere. Thus, the sum of the indices
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https://en.wikipedia.org/wiki/Regions%20of%20Brazil
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Brazil is geopolitically divided into five regions (also called macroregions), by the Brazilian Institute of Geography and Statistics, which are formed by the federative units of Brazil. Although officially recognized, the division is merely academic, considering geographic, social and economic factors, among others, and has no political effects other than orientating Federal-level government programs. Under the state level, they are further divided into intermediate regions and even further into immediate regions.
The five regions
North Region
Area: 3,689,637.9 km2 (45.27%)
Population: 17,707,783 (4,6 people/km2; 6.2%; 2016)
GDP: R$ 308 billion / US$94,8 billion (2016; 4.7%) (5th)
Climate: Equatorial
States: Acre, Amapá, Amazonas, Pará, Rondônia, Roraima, Tocantins
Largest Cities: Manaus (2,094,391); Belém (1,446,042); Porto Velho (511,219); Ananindeua (510,834); Macapá (465,495); Rio Branco (377,057); Boa Vista (326,419); Santarém (294,447); Palmas (279 856).
Economy: Iron, Copper, Gold, Bauxite, Manganese, Açaí, Pineapple, Energy production, electronic manufacturing, tourism.
Transport: Mainly rivers (which are abundant in the region). Highways are scarce and present mainly in the east. Airplanes are commonly used in small remote communities and sometimes in the larger cities.
Vegetation: Almost the entire region is covered by Amazon Rainforest, except the state of Tocantins, which has savanna-like vegetation (cerrado). Most of the native vegetation still remains.
Notable characteristics: Presence of the Amazon Rainforest, which is the vegetation dominant in every state but Tocantins. Cities are spread far apart in the region, and it has the lowest population density of the country. There are very few paved highways in the region, as it is almost isolated from the rest of the country. It is also the biggest region of Brazil, being responsible for almost half of the Brazilian territorial extension. Economic growth is above national average (especially in Amazonas and in Tocantins).
Northeast Region
Area: 1,561,177 km2 (18.3%)
Population: 53,340,945 (30.55 people/km2; 29%; 2009)
GDP: R$437 billion / US$273,1 billion (2009; ~12%) (3rd)
Climate: Hot all year long. Tropical near the coast and semi-arid in the interior; semi-equatorial in the far west of the region.
States: Alagoas, Bahia, Ceará, Maranhão, Paraíba, Pernambuco, Piauí, Rio Grande do Norte, Sergipe
Largest Cities: Salvador (2,676,606); Fortaleza (2,447,409); Recife (1,536,934); São Luís (1,011,943); Maceió (932,608); Natal (789,836); Teresina (714,583); João Pessoa (595,429); Jaboatão dos Guararapes; (580,795); Feira de Santana (481,137); Aracaju (461,083); Olinda (368,666); Campina Grande (354,546).
Economy: Tourism, tropical fruits (coconut, papaya, melon, banana, mango, pineapple), cocoa, cashew nuts, soybeans, cotton, sugarcane, machinery manufacturing, textiles,wind energy production, Região Nordeste bate recorde na geração de energia eólica e solar salt extraction
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https://en.wikipedia.org/wiki/Northrop%20Tacit%20Blue
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The Northrop Tacit Blue was a technology demonstrator aircraft created to demonstrate that a low-observable stealth surveillance aircraft with a low-probability-of-intercept radar (LPIR) and other sensors could operate close to the forward line of battle with a high degree of survivability.
Development
Unveiled by the U.S. Air Force on 30 April 1996, the Tacit Blue Technology Demonstration Program was designed to prove that such an aircraft could continuously monitor the ground situation deep behind the battlefield and provide targeting information in real time to a ground command center.
In December 1976, DARPA and the U.S. Air Force initiated the Battlefield Surveillance Aircraft-Experimental (BSAX) program, which was part of a larger Air Force program called Pave Mover. The BSAX program's goal was to develop an efficient stealth reconnaissance aircraft with a low probability of intercept radar and other sensors that could operate close to the forward line of battle with a high degree of survivability.
Tacit Blue represented the "black" component in the larger "Assault Breaker" program, which intended to validate the concept of massed standoff attacks on advancing armored formations using smart munitions. The Pave Mover radar demonstrators provided the non-stealth portion of the program's targeting system, whereas Tacit Blue was intended to demonstrate a similar but stealth capability, while validating a number of innovative stealth technology advances.
The radar sensor technology developed for Tacit Blue evolved into the radar now being used by the E-8 Joint STARS aircraft.
Tacit Blue was given the designation of "YF-117D" by the Air Force, implying it was a variant of the Lockheed F-117 stealth fighter.
Design
Tacit Blue, nicknamed "the whale" (and sometimes also called an "alien school bus" for its only slightly rounded-off rectangular shape), featured a straight tapered wing with a V-tail mounted on an oversized fuselage with a curved shape. It was the first stealth aircraft to feature curved surfaces for radar cross-section reduction. Northrop would use this stealth technology on the B-2 bomber. A single flush inlet on the top of the fuselage provided air to two medium-bypass turbofan engines. Tacit Blue employed a quadruply redundant digital fly-by-wire flight control system to help stabilize the aircraft about its longitudinal and directional axes.
Operational history
The aircraft made its first successful flight on February 5, 1982, in Area 51, at Groom Lake, Nevada, flown by Northrop test pilot Richard G. Thomas. The aircraft subsequently logged 135 flights over a three-year period. The aircraft often flew three to four flights weekly and several times flew more than once a day.
Another Tacit Blue test pilot, Ken Dyson, told CNN in 2014 that Northrop had manufactured additional major components for the jet, which amounted to half of a second plane. "If we lost one, we could have a second one up and flying in short order," Dy
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https://en.wikipedia.org/wiki/Hankel%20contour
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In mathematics, a Hankel contour is a path in the complex plane which extends from
(+∞,δ), around the origin counter clockwise and back to
(+∞,−δ), where δ is an arbitrarily small positive number. The contour thus remains arbitrarily close to the real axis but without crossing the real axis except for negative values of x. The Hankel contour can also be represented by a path that has mirror images just above and below the real axis, connected to a circle of radius ε, centered at the origin, where ε is an arbitrarily small number. The two linear portions of the contour are said to be a distance of δ from the real axis. Thus, the total distance between the linear portions of the contour is 2δ. The contour is traversed in the positively-oriented sense, meaning that the circle around the origin is traversed counter-clockwise.
Use of Hankel contours is one of the methods of contour integration. This type of path for contour integrals was first used by Hermann Hankel in his investigations of the Gamma function.
The Hankel contour is used to evaluate integrals such as the Gamma function, the Riemann zeta function, and other Hankel functions (which are Bessel functions of the third kind).
Applications
The Hankel contour and the Gamma function
The Hankel contour is helpful in expressing and solving the Gamma function in the complex t-plane. The Gamma function can be defined for any complex value in the plane if we evaluate the integral along the Hankel contour. The Hankel contour is especially useful for expressing the Gamma function for any complex value because the end points of the contour vanish, and thus allows the fundamental property of the Gamma function to be satisfied, which states .
Derivation of the contour integral expression of the Gamma function
Note that the formal representation of the Gamma function is .
To satisfy the fundamental property of the Gamma function, it follows that
after multiplying both sides by z.
Thus, given that the endpoints of the Hankel contour vanish, the left- and right-hand sides reduce to
.
Using differential equations,
becomes the general solution. While A is constant with respect to t, it holds that A may fluctuate depending on the complex number z. Since A(z) is arbitrary, a complex exponential in z may be absorbed into the definition of A(z). Substituting f(t) into the original integral then gives .
By integrating along the Hankel contour, the contour integral expression of the Gamma function becomes .
References
Further reading
Schmelzer, Thomas; Trefethen, Lloyd N. (2007-01). "Computing the Gamma Function Using Contour Integrals and Rational Approximations". SIAM Journal on Numerical Analysis. 45 (2): 558–571. . .
Hugh L. Montgomery; Robert C. Vaughan (2007). Multiplicative number theory I. Classical theory. Cambridge tracts in advanced mathematics. 97. p. 515. .
External links
http://mathworld.wolfram.com/HankelContour.html
NIST Digital Library of Mathematical Functions:Gamma Fun
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https://en.wikipedia.org/wiki/Function%20of%20a%20real%20variable
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In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers , or a subset of that contains an interval of positive length. Most real functions that are considered and studied are differentiable in some interval.
The most widely considered such functions are the real functions, which are the real-valued functions of a real variable, that is, the functions of a real variable whose codomain is the set of real numbers.
Nevertheless, the codomain of a function of a real variable may be any set. However, it is often assumed to have a structure of -vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an -algebra, such as the complex numbers or the quaternions. The structure -vector space of the codomain induces a structure of -vector space on the functions. If the codomain has a structure of -algebra, the same is true for the functions.
The image of a function of a real variable is a curve in the codomain. In this context, a function that defines curve is called a parametric equation of the curve.
When the codomain of a function of a real variable is a finite-dimensional vector space, the function may be viewed as a sequence of real functions. This is often used in applications.
Real function
A real function is a function from a subset of to where denotes as usual the set of real numbers. That is, the domain of a real function is a subset , and its codomain is It is generally assumed that the domain contains an interval of positive length.
Basic examples
For many commonly used real functions, the domain is the whole set of real numbers, and the function is continuous and differentiable at every point of the domain. One says that these functions are defined, continuous and differentiable everywhere. This is the case of:
All polynomial functions, including constant functions and linear functions
Sine and cosine functions
Exponential function
Some functions are defined everywhere, but not continuous at some points. For example
The Heaviside step function is defined everywhere, but not continuous at zero.
Some functions are defined and continuous everywhere, but not everywhere differentiable. For example
The absolute value is defined and continuous everywhere, and is differentiable everywhere, except for zero.
The cubic root is defined and continuous everywhere, and is differentiable everywhere, except for zero.
Many common functions are not defined everywhere, but are continuous and differentiable everywhere where they are defined. For example:
A rational function is a quotient of two polynomial functions, and is not defined at the zeros of the denominator.
The tangent function is not defined for where is any integer.
The logarithm function is defined only for positive values of the variable.
Some functi
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https://en.wikipedia.org/wiki/Legendre%20chi%20function
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In mathematics, the Legendre chi function is a special function whose Taylor series is also a Dirichlet series, given by
As such, it resembles the Dirichlet series for the polylogarithm, and, indeed, is trivially expressible in terms of the polylogarithm as
The Legendre chi function appears as the discrete Fourier transform, with respect to the order ν, of the Hurwitz zeta function, and also of the Euler polynomials, with the explicit relationships given in those articles.
The Legendre chi function is a special case of the Lerch transcendent, and is given by
Identities
Integral relations
References
Special functions
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https://en.wikipedia.org/wiki/Robert%20Gunning
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Robert Gunning may refer to:
Sir Robert Gunning, 1st Baronet (1731–1816), British diplomat
Robert C. Gunning, professor of mathematics at Princeton University
Robert Halliday Gunning (1818–1900), Scottish physician
Robert Gunning, American businessman, creator of the Gunning fog index of readability
Robert Gunning, musician, guitarist for The Infected
Sir Robert Gunning, 3rd Baronet (1795–1862), of the Gunning baronets, MP for Northampton
Sir Robert Charles Gunning, 8th Baronet (1901–1989), of the Gunning baronets
See also
Gunning (disambiguation)
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https://en.wikipedia.org/wiki/MuMATH
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muMATH is a computer algebra system (CAS), which was developed in the late 1970s and early eighties by Albert D. Rich and David Stoutemyer of Soft Warehouse in Honolulu, Hawaii. It was implemented in the muSIMP programming language which was built on top of a LISP dialect called . Platforms supported were CP/M and TRS-DOS (since muMATH-79), Apple II (since muMATH-80) and DOS (in muMATH-83, the last version, which was published by Microsoft).
The Soft Warehouse later developed Derive, another computer algebra system. The company was purchased by Texas Instruments in 1999, and development of Derive ended in 2006.
Literature
David D. Shochat, A Symbolic Mathematics System, Creative Computing, Oct. 1982, p. 26
Gregg Williams, The muSIMP/muMATH-79 Symbolic Math system, a Review, BYTE, Nov. 1980, p. 324
Stuart Edwards, A Computer-Algebra-Based Calculating System, BYTE 12/1983, pp- 481-494 (Describes a calculator application of muSIMP / muMATH doing automatic unit conversion.)
Computer algebra systems
CP/M software
Discontinued software
Lisp (programming language) software
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