source
stringlengths 31
168
| text
stringlengths 51
3k
|
|---|---|
https://en.wikipedia.org/wiki/Combinatorial%20number%20system
|
In mathematics, and in particular in combinatorics, the combinatorial number system of degree k (for some positive integer k), also referred to as combinadics, or the Macaulay representation of an integer, is a correspondence between natural numbers (taken to include 0) N and k-combinations. The combinations are represented as strictly decreasing sequences ck > ... > c2 > c1 ≥ 0 where each ci corresponds to the index of a chosen element in a given k-combination. Distinct numbers correspond to distinct k-combinations, and produce them in lexicographic order. The numbers less than correspond to all of }. The correspondence does not depend on the size n of the set that the k-combinations are taken from, so it can be interpreted as a map from N to the k-combinations taken from N; in this view the correspondence is a bijection.
The number N corresponding to (ck, ..., c2, c1) is given by
.
The fact that a unique sequence corresponds to any non-negative number N was first observed by D. H. Lehmer. Indeed, a greedy algorithm finds the k-combination corresponding to N: take ck maximal with , then take ck−1 maximal with , and so forth. Finding the number N, using the formula above, from the k-combination (ck, ..., c2, c1) is also known as "ranking", and the opposite operation (given by the greedy algorithm) as "unranking"; the operations are known by these names in most computer algebra systems, and in computational mathematics.
The originally used term "combinatorial representation of integers" was shortened to "combinatorial number system" by Knuth,
who also gives a much older reference;
the term "combinadic" is introduced by James McCaffrey (without reference to previous terminology or work).
Unlike the factorial number system, the combinatorial number system of degree k is not a mixed radix system: the part of the number N represented by a "digit" ci is not obtained from it by simply multiplying by a place value.
The main application of the combinatorial number system is that it allows rapid computation of the k-combination that is at a given position in the lexicographic ordering, without having to explicitly list the preceding it; this allows for instance random generation of k-combinations of a given set. Enumeration of k-combinations has many applications, among which are software testing, sampling, quality control, and the analysis of lottery games.
Ordering combinations
A k-combination of a set S is a subset of S with k (distinct) elements. The main purpose of the combinatorial number system is to provide a representation, each by a single number, of all possible k-combinations of a set S of n elements. Choosing, for any n, } as such a set, it can be arranged that the representation of a given k-combination C is independent of the value of n (although n must of course be sufficiently large); in other words considering C as a subset of a larger set by increasing n will not change the number that represents C. Thus for the combinatori
|
https://en.wikipedia.org/wiki/Wolstenholme%27s%20theorem
|
In mathematics, Wolstenholme's theorem states that for a prime number , the congruence
holds, where the parentheses denote a binomial coefficient. For example, with p = 7, this says that 1716 is one more than a multiple of 343. The theorem was first proved by Joseph Wolstenholme in 1862. In 1819, Charles Babbage showed the same congruence modulo p2, which holds for . An equivalent formulation is the congruence
for , which is due to Wilhelm Ljunggren (and, in the special case , to J. W. L. Glaisher) and is inspired by Lucas' theorem.
No known composite numbers satisfy Wolstenholme's theorem and it is conjectured that there are none (see below). A prime that satisfies the congruence modulo p4 is called a Wolstenholme prime (see below).
As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) harmonic numbers:
(Congruences with fractions make sense, provided that the denominators are coprime to the modulus.)
For example, with p=7, the first of these says that the numerator of 49/20 is a multiple of 49, while the second says the numerator of 5369/3600 is a multiple of 7.
Wolstenholme primes
A prime p is called a Wolstenholme prime iff the following condition holds:
If p is a Wolstenholme prime, then Glaisher's theorem holds modulo p4. The only known Wolstenholme primes so far are 16843 and 2124679 ; any other Wolstenholme prime must be greater than 109. This result is consistent with the heuristic argument that the residue modulo p4 is a pseudo-random multiple of p3. This heuristic predicts that the number of Wolstenholme primes between K and N is roughly ln ln N − ln ln K. The Wolstenholme condition has been checked up to 109, and the heuristic says that there should be roughly one Wolstenholme prime between 109 and 1024. A similar heuristic predicts that there are no "doubly Wolstenholme" primes, for which the congruence would hold modulo p5.
A proof of the theorem
There is more than one way to prove Wolstenholme's theorem. Here is a proof that directly establishes Glaisher's version using both combinatorics and algebra.
For the moment let p be any prime, and let a and b be any non-negative integers. Then a set A with ap elements can be divided into a rings of length p, and the rings can be rotated separately. Thus, the a-fold direct sum of the cyclic group of order p acts on the set A, and by extension it acts on the set of subsets of size bp. Every orbit of this group action has pk elements, where k is the number of incomplete rings, i.e., if there are k rings that only partly intersect a subset B in the orbit. There are orbits of size 1 and there are no orbits of size p. Thus we first obtain Babbage's theorem
Examining the orbits of size p2, we also obtain
Among other consequences, this equation tells us that the case a=2 and b=1 implies the general case of the second form of Wolstenholme's theorem.
Switching from combinatorics to algebra, both sides of this co
|
https://en.wikipedia.org/wiki/Hurwitz%27s%20automorphisms%20theorem
|
In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1). A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve. The theorem is named after Adolf Hurwitz, who proved it in .
Hurwitz's bound also holds for algebraic curves over a field of characteristic 0, and over fields of positive characteristic p>0 for groups whose order is coprime to p, but can fail over fields of positive characteristic p>0 when p divides the group order. For example, the double cover of the projective line y2 = xp −x branched at all points defined over the prime field has genus g=(p−1)/2 but is acted on by the group SL2(p) of order p3−p.
Interpretation in terms of hyperbolicity
One of the fundamental themes in differential geometry is a trichotomy between the Riemannian manifolds of positive, zero, and negative curvature K. It manifests itself in many diverse situations and on several levels. In the context of compact Riemann surfaces X, via the Riemann uniformization theorem, this can be seen as a distinction between the surfaces of different topologies:
X a sphere, a compact Riemann surface of genus zero with K > 0;
X a flat torus, or an elliptic curve, a Riemann surface of genus one with K = 0;
and X a hyperbolic surface, which has genus greater than one and K < 0.
While in the first two cases the surface X admits infinitely many conformal automorphisms (in fact, the conformal automorphism group is a complex Lie group of dimension three for a sphere and of dimension one for a torus), a hyperbolic Riemann surface only admits a discrete set of automorphisms. Hurwitz's theorem claims that in fact more is true: it provides a uniform bound on the order of the automorphism group as a function of the genus and characterizes those Riemann surfaces for which the bound is sharp.
Statement and proof
Theorem: Let be a smooth connected Riemann surface of genus . Then its automorphism group has size at most .
Proof: Assume for now that is finite (this will be proved at the end).
Consider the quotient map . Since acts by holomorphic functions, the quotient is locally of the form and the quotient is a smooth Riemann surface. The quotient map is a branched cover, and we will see below that the ramification points correspond to the orbits that have a non-trivial stabiliser. Let be the genus of .
By the Riemann-Hurwitz formula, where the sum is over the ramification points for the quotient map . The ramification index at is just the order of the stabiliser group, since where the number of pre-images of (the number of points in the orbit
|
https://en.wikipedia.org/wiki/Order%20%28ring%20theory%29
|
In mathematics, an order in the sense of ring theory is a subring of a ring , such that
is a finite-dimensional algebra over the field of rational numbers
spans over , and
is a -lattice in .
The last two conditions can be stated in less formal terms: Additively, is a free abelian group generated by a basis for over .
More generally for an integral domain contained in a field , we define to be an -order in a -algebra if it is a subring of which is a full -lattice.
When is not a commutative ring, the idea of order is still important, but the phenomena are different. For example, the Hurwitz quaternions form a maximal order in the quaternions with rational co-ordinates; they are not the quaternions with integer coordinates in the most obvious sense. Maximal orders exist in general, but need not be unique: there is in general no largest order, but a number of maximal orders. An important class of examples is that of integral group rings.
Examples
Some examples of orders are:
If is the matrix ring over , then the matrix ring over is an -order in
If is an integral domain and a finite separable extension of , then the integral closure of in is an -order in .
If in is an integral element over , then the polynomial ring is an -order in the algebra
If is the group ring of a finite group , then is an -order on
A fundamental property of -orders is that every element of an -order is integral over .
If the integral closure of in is an -order then this result shows that must be the maximal -order in . However this hypothesis is not always satisfied: indeed need not even be a ring, and even if is a ring (for example, when is commutative) then need not be an -lattice.
Algebraic number theory
The leading example is the case where is a number field and is its ring of integers. In algebraic number theory there are examples for any other than the rational field of proper subrings of the ring of integers that are also orders. For example, in the field extension of Gaussian rationals over , the integral closure of is the ring of Gaussian integers and so this is the unique maximal -order: all other orders in are contained in it. For example, we can take the subring of complex numbers of the form , with and integers.
The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.
See also
Hurwitz quaternion order – An example of ring order
Notes
References
Ring theory
|
https://en.wikipedia.org/wiki/Lw%C3%B3w-Warsaw%20School
|
Lwow–Warsaw School may refer to:
Lwów–Warsaw school of logic
Lwów School of Mathematics
Warsaw School of Mathematics
Lwów–Warsaw School of History
|
https://en.wikipedia.org/wiki/Exotic%20sphere
|
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere. That is, M is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic").
The first exotic spheres were constructed by in dimension as -bundles over . He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finite abelian group if the dimension is not 4. The classification of exotic spheres by showed that the oriented exotic 7-spheres are the non-trivial elements of a cyclic group of order 28 under the operation of connected sum.
Specifically, this means that the elements of this group (n ≠ 4) are the equivalence classes of smooth structures on Sn, where two structures are considered equivalent if there is an orientation preserving diffeomorphism carrying one structure onto the other. The group operation is defined by [x] + [y] = [x + y],
where x and y are arbitrary representatives of their equivalence classes, and "x + y" denotes the smooth structure on the smooth Sn that is the connected sum of x and y. It is necessary to show that such a definition does not depend on the choices made; indeed this can be shown.
Introduction
The unit n-sphere, , is the set of all (n+1)-tuples of real numbers, such that the sum . For instance, is a circle, while is the surface of an ordinary ball of radius one in 3 dimensions. Topologists consider a space, X, to be an n-sphere if there is a homeomorphism between them, i.e. every point in X may be assigned to exactly one point in the unit n-sphere in a bicontinuous (i.e. continuous and invertible with continuous inverse) manner. For example, a point x on an n-sphere of radius r can be matched with a point on the unit n-sphere by adjusting its distance from the origin by . Similarly, an n-cube of any radius can be continuously transformed to an n-sphere.
In differential topology, the relevant notion of sameness is witnessed by a diffeomorphism, which is a homeomorphism with the additional condition that it is smooth, that is, it should have derivatives of all orders everywhere. To calculate derivatives, one needs to have local coordinate systems defined consistently in X. Mathematicians were surprised in 1956 when Milnor showed that consistent coordinate systems could be set up on the 7-sphere in two different ways that were equivalent in the continuous sense, but not in the differentiable sense. Milnor and others set about trying to discover how many such exotic spheres could exist in each dimension and to understand how they relate to each other. No exotic structures are possible on the 1-, 2-, 3-, 5-, 6-, 12-, 56- or 61-spheres. Some higher
|
https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler%20divergence
|
In mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted , is a type of statistical distance: a measure of how one probability distribution is different from a second, reference probability distribution . A simple interpretation of the KL divergence of from is the expected excess surprise from using as a model when the actual distribution is . While it is a measure of how different two distributions are, and in some sense is thus a "distance", it is not actually a metric, which is the most familiar and formal type of distance. In particular, it is not symmetric in the two distributions (in contrast to variation of information), and does not satisfy the triangle inequality. Instead, in terms of information geometry, it is a type of divergence, a generalization of squared distance, and for certain classes of distributions (notably an exponential family), it satisfies a generalized Pythagorean theorem (which applies to squared distances).
In the simple case, a relative entropy of 0 indicates that the two distributions in question have identical quantities of information. Relative entropy is a nonnegative function of two distributions or measures. It has diverse applications, both theoretical, such as characterizing the relative (Shannon) entropy in information systems, randomness in continuous time-series, and information gain when comparing statistical models of inference; and practical, such as applied statistics, fluid mechanics, neuroscience and bioinformatics.
Introduction and context
Consider two probability distributions and . Usually, represents the data, the observations, or a measured probability distribution. Distribution represents instead a theory, a model, a description or an approximation of . The Kullback–Leibler divergence is then interpreted as the average difference of the number of bits required for encoding samples of using a code optimized for rather than one optimized for . Note that the roles of and can be reversed in some situations where that is easier to compute, such as with the expectation–maximization (EM) algorithm and evidence lower bound (ELBO) computations.
Etymology
The relative entropy was introduced by Solomon Kullback and Richard Leibler in as "the mean information for discrimination between and per observation from ", where one is comparing two probability measures , and are the hypotheses that one is selecting from measure (respectively). They denoted this by , and defined the "'divergence' between and " as the symmetrized quantity , which had already been defined and used by Harold Jeffreys in 1948. In , the symmetrized form is again referred to as the "divergence", and the relative entropies in each direction are referred to as a "directed divergences" between two distributions; Kullback preferred the term discrimination information. The term "divergence" is in contrast to a distance (metric), since the symmetrized divergen
|
https://en.wikipedia.org/wiki/G%20Ring
|
G Ring may refer to:
, a planetary ring system around Saturn.
G-ring or Grothendieck ring, a type of commutative ring in algebra
|
https://en.wikipedia.org/wiki/Leibniz%27s%20rule
|
Leibniz's rule (named after Gottfried Wilhelm Leibniz) may refer to one of the following:
Product rule in differential calculus
General Leibniz rule, a generalization of the product rule
Leibniz integral rule
The alternating series test, also called Leibniz's rule
See also
Leibniz (disambiguation)
Leibniz' law (disambiguation)
List of things named after Gottfried Leibniz
|
https://en.wikipedia.org/wiki/General%20Leibniz%20rule
|
In calculus, the general Leibniz rule, named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if and are -times differentiable functions, then the product is also -times differentiable and its th derivative is given by
where is the binomial coefficient and denotes the jth derivative of f (and in particular ).
The rule can be proven by using the product rule and mathematical induction.
Second derivative
If, for example, , the rule gives an expression for the second derivative of a product of two functions:
More than two factors
The formula can be generalized to the product of m differentiable functions f1,...,fm.
where the sum extends over all m-tuples (k1,...,km) of non-negative integers with and
are the multinomial coefficients. This is akin to the multinomial formula from algebra.
Proof
The proof of the general Leibniz rule proceeds by induction. Let and be -times differentiable functions. The base case when claims that:
which is the usual product rule and is known to be true. Next, assume that the statement holds for a fixed that is, that
Then,
And so the statement holds for and the proof is complete.
Multivariable calculus
With the multi-index notation for partial derivatives of functions of several variables, the Leibniz rule states more generally:
This formula can be used to derive a formula that computes the symbol of the composition of differential operators. In fact, let P and Q be differential operators (with coefficients that are differentiable sufficiently many times) and Since R is also a differential operator, the symbol of R is given by:
A direct computation now gives:
This formula is usually known as the Leibniz formula. It is used to define the composition in the space of symbols, thereby inducing the ring structure.
See also
References
Articles containing proofs
Differentiation rules
Gottfried Wilhelm Leibniz
Mathematical identities
Theorems in analysis
Theorems in calculus
|
https://en.wikipedia.org/wiki/Discrepancy%20theory
|
In mathematics, discrepancy theory describes the deviation of a situation from the state one would like it to be in. It is also called the theory of irregularities of distribution. This refers to the theme of classical discrepancy theory, namely distributing points in some space such that they are evenly distributed with respect to some (mostly geometrically defined) subsets. The discrepancy (irregularity) measures how far a given distribution deviates from an ideal one.
Discrepancy theory can be described as the study of inevitable irregularities of distributions, in measure-theoretic and combinatorial settings. Just as Ramsey theory elucidates the impossibility of total disorder, discrepancy theory studies the deviations from total uniformity.
A significant event in the history of discrepancy theory was the 1916 paper of Weyl on the uniform distribution of sequences in the unit interval.
Theorems
Discrepancy theory is based on the following classic theorems:
The theorem of van Aardenne–Ehrenfest
Axis-parallel rectangles in the plane (Roth, Schmidt)
Discrepancy of half-planes (Alexander, Matoušek)
Arithmetic progressions (Roth, Sarkozy, Beck, Matousek & Spencer)
Beck–Fiala theorem
Six Standard Deviations Suffice (Spencer)
Major open problems
The unsolved problems relating to discrepancy theory include:
Axis-parallel rectangles in dimensions three and higher (folklore)
Komlós conjecture
Heilbronn triangle problem on the minimum area of a triangle determined by three points from an n-point set
Applications
Applications for discrepancy theory include:
Numerical integration: Monte Carlo methods in high dimensions.
Computational geometry: Divide-and-conquer algorithm.
Image processing: Halftoning
Random trial formulation: Randomized controlled trial
See also
Discrepancy of hypergraphs
References
Further reading
Diophantine approximation
Unsolved problems in mathematics
Discrepancy theory
Measure theory
|
https://en.wikipedia.org/wiki/Lyapunov%20time
|
In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. It is named after the Russian mathematician Aleksandr Lyapunov. It is defined as the inverse of a system's largest Lyapunov exponent.
Use
The Lyapunov time mirrors the limits of the predictability of the system. By convention, it is defined as the time for the distance between nearby trajectories of the system to increase by a factor of e. However, measures in terms of 2-foldings and 10-foldings are sometimes found, since they correspond to the loss of one bit of information or one digit of precision respectively.
While it is used in many applications of dynamical systems theory, it has been particularly used in celestial mechanics where it is important for the problem of the stability of the Solar System. However, empirical estimation of the Lyapunov time is often associated with computational or inherent uncertainties.
Examples
Typical values are:
See also
Belousov–Zhabotinsky reaction
Molecular chaos
Three-body problem
References
Dynamical systems
|
https://en.wikipedia.org/wiki/Type-2%20Gumbel%20distribution
|
In probability theory, the Type-2 Gumbel probability density function is
for
.
For the mean is infinite. For the variance is infinite.
The cumulative distribution function is
The moments exist for
The distribution is named after Emil Julius Gumbel (1891 – 1966).
Generating random variates
Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate
has a Type-2 Gumbel distribution with parameter and . This is obtained by applying the inverse transform sampling-method.
Related distributions
The special case b = 1 yields the Fréchet distribution.
Substituting and yields the Weibull distribution. Note, however, that a positive k (as in the Weibull distribution) would yield a negative a and hence a negative probability density, which is not allowed.
Based on The GNU Scientific Library, used under GFDL.
See also
Extreme value theory
Gumbel distribution
Continuous distributions
|
https://en.wikipedia.org/wiki/Dirichlet%20distribution
|
In probability and statistics, the Dirichlet distribution (after Peter Gustav Lejeune Dirichlet), often denoted , is a family of continuous multivariate probability distributions parameterized by a vector of positive reals. It is a multivariate generalization of the beta distribution, hence its alternative name of multivariate beta distribution (MBD). Dirichlet distributions are commonly used as prior distributions in Bayesian statistics, and in fact, the Dirichlet distribution is the conjugate prior of the categorical distribution and multinomial distribution.
The infinite-dimensional generalization of the Dirichlet distribution is the Dirichlet process.
Definitions
Probability density function
The Dirichlet distribution of order K ≥ 2 with parameters α1, ..., αK > 0 has a probability density function with respect to Lebesgue measure on the Euclidean space RK-1 given by
where belong to the standard simplex, or in other words:
The normalizing constant is the multivariate beta function, which can be expressed in terms of the gamma function:
Support
The support of the Dirichlet distribution is the set of K-dimensional vectors whose entries are real numbers in the interval [0,1] such that , i.e. the sum of the coordinates is equal to 1. These can be viewed as the probabilities of a K-way categorical event. Another way to express this is that the domain of the Dirichlet distribution is itself a set of probability distributions, specifically the set of K-dimensional discrete distributions. The technical term for the set of points in the support of a K-dimensional Dirichlet distribution is the open standard (K − 1)-simplex, which is a generalization of a triangle, embedded in the next-higher dimension. For example, with K = 3, the support is an equilateral triangle embedded in a downward-angle fashion in three-dimensional space, with vertices at (1,0,0), (0,1,0) and (0,0,1), i.e. touching each of the coordinate axes at a point 1 unit away from the origin.
Special cases
A common special case is the symmetric Dirichlet distribution, where all of the elements making up the parameter vector have the same value. The symmetric case might be useful, for example, when a Dirichlet prior over components is called for, but there is no prior knowledge favoring one component over another. Since all elements of the parameter vector have the same value, the symmetric Dirichlet distribution can be parametrized by a single scalar value α, called the concentration parameter. In terms of α, the density function has the form
When α=1, the symmetric Dirichlet distribution is equivalent to a uniform distribution over the open standard (K − 1)-simplex, i.e. it is uniform over all points in its support. This particular distribution is known as the flat Dirichlet distribution. Values of the concentration parameter above 1 prefer variates that are dense, evenly distributed distributions, i.e. all the values within a single sample are similar to each other.
|
https://en.wikipedia.org/wiki/Landau%20distribution
|
In probability theory, the Landau distribution is a probability distribution named after Lev Landau.
Because of the distribution's "fat" tail, the moments of the distribution, like mean or variance, are undefined. The distribution is a particular case of stable distribution.
Definition
The probability density function, as written originally by Landau, is defined by the complex integral:
where a is an arbitrary positive real number, meaning that the integration path can be any parallel to the imaginary axis, intersecting the real positive semi-axis, and refers to the natural logarithm.
In other words it is the Laplace transform of the function .
The following real integral is equivalent to the above:
The full family of Landau distributions is obtained by extending the original distribution to a location-scale family of stable distributions with parameters and , with characteristic function:
where and , which yields a density function:
Taking and we get the original form of above.
Properties
Translation: If then .
Scaling: If then .
Sum: If and then .
These properties can all be derived from the characteristic function.
Together they imply that the Landau distributions are closed under affine transformations.
Approximations
In the "standard" case and , the pdf can be approximated using Lindhard theory which says:
where is Euler's constant.
A similar approximation of for and is:
Related distributions
The Landau distribution is a stable distribution with stability parameter and skewness parameter both equal to 1.
References
Continuous distributions
Probability distributions with non-finite variance
Power laws
Stable distributions
Lev Landau
|
https://en.wikipedia.org/wiki/Stable%20distribution
|
In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. A random variable is said to be stable if its distribution is stable. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution, after Paul Lévy, the first mathematician to have studied it.
Of the four parameters defining the family, most attention has been focused on the stability parameter, (see panel). Stable distributions have , with the upper bound corresponding to the normal distribution, and to the Cauchy distribution. The distributions have undefined variance for , and undefined mean for . The importance of stable probability distributions is that they are "attractors" for properly normed sums of independent and identically distributed (iid) random variables. The normal distribution defines a family of stable distributions. By the classical central limit theorem the properly normed sum of a set of random variables, each with finite variance, will tend toward a normal distribution as the number of variables increases. Without the finite variance assumption, the limit may be a stable distribution that is not normal. Mandelbrot referred to such distributions as "stable Paretian distributions", after Vilfredo Pareto. In particular, he referred to those maximally skewed in the positive direction with as "Pareto–Lévy distributions", which he regarded as better descriptions of stock and commodity prices than normal distributions.
Definition
A non-degenerate distribution is a stable distribution if it satisfies the following property:
Since the normal distribution, the Cauchy distribution, and the Lévy distribution all have the above property, it follows that they are special cases of stable distributions.
Such distributions form a four-parameter family of continuous probability distributions parametrized by location and scale parameters μ and c, respectively, and two shape parameters and , roughly corresponding to measures of asymmetry and concentration, respectively (see the figures).
The characteristic function of any probability distribution is the Fourier transform of its probability density function . The density function is therefore the inverse Fourier transform of the characteristic function:
Although the probability density function for a general stable distribution cannot be written analytically, the general characteristic function can be expressed analytically. A random variable X is called stable if its characteristic function can be written as
where is just the sign of and
μ ∈ R is a shift parameter, , called the skewness parameter, is a measure of asymmetry. Notice that in this context the usual skewness is not well defined, as for the distribution does not admit 2nd or higher moments, and the usual skewness definition is the 3rd central moment.
The reason this gives a s
|
https://en.wikipedia.org/wiki/VSEPR%20theory
|
Valence shell electron pair repulsion (VSEPR) theory ( , ), is a model used in chemistry to predict the geometry of individual molecules from the number of electron pairs surrounding their central atoms. It is also named the Gillespie-Nyholm theory after its two main developers, Ronald Gillespie and Ronald Nyholm.
The premise of VSEPR is that the valence electron pairs surrounding an atom tend to repel each other. The greater the repulsion, the higher in energy (less stable) the molecule is. Therefore, the VSEPR-predicted molecular geometry of a molecule is the one that has as little of this repulsion as possible. Gillespie has emphasized that the electron-electron repulsion due to the Pauli exclusion principle is more important in determining molecular geometry than the electrostatic repulsion.
The insights of VSEPR theory are derived from topological analysis of the electron density of molecules. Such quantum chemical topology (QCT) methods include the electron localization function (ELF) and the quantum theory of atoms in molecules (AIM or QTAIM).
History
The idea of a correlation between molecular geometry and number of valence electron pairs (both shared and unshared pairs) was originally proposed in 1939 by Ryutaro Tsuchida in Japan, and was independently presented in a Bakerian Lecture in 1940 by Nevil Sidgwick and Herbert Powell of the University of Oxford. In 1957, Ronald Gillespie and Ronald Sydney Nyholm of University College London refined this concept into a more detailed theory, capable of choosing between various alternative geometries.
Overview
VSEPR theory is used to predict the arrangement of electron pairs around central atoms in molecules, especially simple and symmetric molecules. A central atom is defined in this theory as an atom which is bonded to two or more other atoms, while a terminal atom is bonded to only one other atom. For example in the molecule methyl isocyanate (H3C-N=C=O), the two carbons and one nitrogen are central atoms, and the three hydrogens and one oxygen are terminal atoms. The geometry of the central atoms and their non-bonding electron pairs in turn determine the geometry of the larger whole molecule.
The number of electron pairs in the valence shell of a central atom is determined after drawing the Lewis structure of the molecule, and expanding it to show all bonding groups and lone pairs of electrons. In VSEPR theory, a double bond or triple bond is treated as a single bonding group. The sum of the number of atoms bonded to a central atom and the number of lone pairs formed by its nonbonding valence electrons is known as the central atom's steric number.
The electron pairs (or groups if multiple bonds are present) are assumed to lie on the surface of a sphere centered on the central atom and tend to occupy positions that minimize their mutual repulsions by maximizing the distance between them. The number of electron pairs (or groups), therefore, determines the overall geometry that they will
|
https://en.wikipedia.org/wiki/Journal%20of%20Recreational%20Mathematics
|
The Journal of Recreational Mathematics was an American journal dedicated to recreational mathematics, started in 1968. It had generally been published quarterly by the Baywood Publishing Company, until it ceased publication with the last issue (volume 38, number 2) published in 2014. The initial publisher (of volumes 1–5) was Greenwood Periodicals.
Harry L. Nelson was primary editor for five years (volumes 9 through 13, excepting volume 13, number 4, when the initial editor returned as lead) and Joseph Madachy, the initial lead editor and editor of a predecessor called Recreational Mathematics Magazine which ran during the years 1961 to 1964, was the editor for many years. Charles Ashbacher and Colin Singleton took over as editors when Madachy retired (volume 30 number 1). The final editors were Ashbacher and Lamarr Widmer. The journal has from its inception also listed associate editors, one of whom was Leo Moser.
The journal contains:
Original articles
Book reviews
Alphametics And Solutions To Alphametics
Problems And Conjectures
Solutions To Problems And Conjectures
Proposer's And Solver's List For Problems And Conjectures
Indexing
The journal is indexed in:
Academic Search Premier
Book Review Index
International Bibliography of Periodical Literature
International Bibliography of Book Reviews
Readers' Guide to Periodical Literature
The Gale Group
References
Recreational mathematics
Mathematics journals
Academic journals established in 1968
Publications disestablished in 2014
Quarterly journals
Defunct journals of the United States
|
https://en.wikipedia.org/wiki/Terence%20Tao
|
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes topics in harmonic analysis, partial differential equations, algebraic combinatorics, arithmetic combinatorics, geometric combinatorics, probability theory, compressed sensing and analytic number theory.
Tao was born to ethnic Chinese immigrant parents and raised in Adelaide. Tao won the Fields Medal in 2006 and won the Royal Medal and Breakthrough Prize in Mathematics in 2014. He is also a 2006 MacArthur Fellow. Tao has been the author or co-author of over three hundred research papers. He is widely regarded as one of the greatest living mathematicians and has been referred to as the "Mozart of mathematics."
Life and career
Family
Tao's parents are first-generation immigrants from Hong Kong to Australia. Tao's father, Billy Tao, was a Chinese paediatrician who was born in Shanghai and earned his medical degree (MBBS) from the University of Hong Kong in 1969. Tao's mother, Grace Leong, was born in Hong Kong; she received a first-class honours degree in mathematics and physics at the University of Hong Kong. She was a secondary school teacher of mathematics and physics in Hong Kong. Billy and Grace met as students at the University of Hong Kong. They then emigrated from Hong Kong to Australia in 1972.
Tao also has two brothers, Trevor and Nigel, who are living in Australia. Both formerly represented the states at the International Mathematical Olympiad. Furthermore, Trevor has been representing Australia internationally in chess and holds the title of Chess International Master. Tao speaks Cantonese but cannot write Chinese. Tao is married to Laura Tao, an electrical engineer at NASA's Jet Propulsion Laboratory. They live in Los Angeles, California, and have two children: Riley and daughter Madeleine.
Childhood
A child prodigy, Tao exhibited extraordinary mathematical abilities from an early age, attending university-level mathematics courses at the age of 9. He is one of only three children in the history of the Johns Hopkins' Study of Exceptional Talent program to have achieved a score of 700 or greater on the SAT math section while just eight years old; Tao scored a 760. Julian Stanley, Director of the Study of Mathematically Precocious Youth, stated that Tao had the greatest mathematical reasoning ability he had found in years of intensive searching.
Tao was the youngest participant to date in the International Mathematical Olympiad, first competing at the age of ten; in 1986, 1987, and 1988, he won a bronze, silver, and gold medal, respectively. Tao remains the youngest winner of each of the three medals in the Olympiad's history, having won the gold medal at the age of 13 in 1988.
Career
At age 14, Tao attended the Research Science Institute, a summer program for secondary students. In 1991, he received h
|
https://en.wikipedia.org/wiki/Inner%20model
|
In set theory, a branch of mathematical logic, an inner model for a theory T is a substructure of a model M of a set theory that is both a model for T and contains all the ordinals of M.
Definition
Let be the language of set theory. Let S be a particular set theory, for example the ZFC axioms and let (possibly the same as S) also be a theory in .
If is a model for , and is an -structure such that
is a substructure of , i.e. the interpretation of in is
is a model of
the domain of is a transitive class of
contains all -ordinals
then we say that is an inner model of (in ). Usually will equal (or subsume) , so that is a model for 'inside' the model of .
If only conditions 1 and 2 hold, N is called a standard model of T (in M), a standard submodel of T (if S = T and) N is a set in M. A model N of T in M is called transitive when it is standard and condition 3 holds. If the axiom of foundation is not assumed (that is, is not in S) all three of these concepts are given the additional condition that N be well-founded. Hence inner models are transitive, transitive models are standard, and standard models are well-founded.
The assumption that there exists a standard submodel of ZFC (in a given universe) is stronger than the assumption that there exists a model. In fact, if there is a standard submodel, then there is a smallest standard submodel
called the minimal model contained in all standard submodels. The minimal submodel contains no standard submodel (as it is minimal) but (assuming the consistency of ZFC) it contains
some model of ZFC by the Gödel completeness theorem. This model is necessarily not well-founded otherwise its Mostowski collapse would be a standard submodel. (It is not well-founded as a relation in the universe, though it
satisfies the axiom of foundation so is "internally" well-founded. Being well-founded is not an absolute property.)
In particular in the minimal submodel there is a model of ZFC but there is no standard submodel of ZFC.
Use
Usually when one talks about inner models of a theory, the theory one is discussing is ZFC or some extension of ZFC (like ZFC + a measurable cardinal). When no theory is mentioned, it is usually assumed that the model under discussion is an inner model of ZFC. However, it is not uncommon to talk about inner models of subtheories of ZFC (like ZF or KP) as well.
Related ideas
It was proved by Kurt Gödel that any model of ZF has a least inner model of ZF (which is also an inner model of ZFC + GCH), called the constructible universe, or L.
There is a branch of set theory called inner model theory that studies ways of constructing least inner models of theories extending ZF. Inner model theory has led to the discovery of the exact consistency strength of many important set theoretical properties.
See also
Countable transitive models and generic filters
References
Inner model theory
|
https://en.wikipedia.org/wiki/Linearization
|
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. This method is used in fields such as engineering, physics, economics, and ecology.
Linearization of a function
Linearizations of a function are lines—usually lines that can be used for purposes of calculation. Linearization is an effective method for approximating the output of a function at any based on the value and slope of the function at , given that is differentiable on (or ) and that is close to . In short, linearization approximates the output of a function near .
For example, . However, what would be a good approximation of ?
For any given function , can be approximated if it is near a known differentiable point. The most basic requisite is that , where is the linearization of at . The point-slope form of an equation forms an equation of a line, given a point and slope . The general form of this equation is: .
Using the point , becomes . Because differentiable functions are locally linear, the best slope to substitute in would be the slope of the line tangent to at .
While the concept of local linearity applies the most to points arbitrarily close to , those relatively close work relatively well for linear approximations. The slope should be, most accurately, the slope of the tangent line at .
Visually, the accompanying diagram shows the tangent line of at . At , where is any small positive or negative value, is very nearly the value of the tangent line at the point .
The final equation for the linearization of a function at is:
For , . The derivative of is , and the slope of at is .
Example
To find , we can use the fact that . The linearization of at is , because the function defines the slope of the function at . Substituting in , the linearization at 4 is . In this case , so is approximately . The true value is close to 2.00024998, so the linearization approximation has a relative error of less than 1 millionth of a percent.
Linearization of a multivariable function
The equation for the linearization of a function at a point is:
The general equation for the linearization of a multivariable function at a point is:
where is the vector of variables, is the gradient, and is the linearization point of interest
.
Uses of linearization
Linearization makes it possible to use tools for studying linear systems to analyze the behavior of a nonlinear function near a given point. The linearization of a function is the first order term of its Taylor expansion around the point of interest. For a system defined by the equation
,
the linearized system can be written as
where is the point of interest and
|
https://en.wikipedia.org/wiki/Arbitrarily%20large
|
In mathematics, the phrases arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear of the fact that an object is large, small and long with little limitation or restraint, respectively. The use of "arbitrarily" often occurs in the context of real numbers (and its subsets thereof), though its meaning can differ from that of "sufficiently" and "infinitely".
Examples
The statement
" is non-negative for arbitrarily large ."
is a shorthand for:
"For every real number , is non-negative for some value of greater than ."
In the common parlance, the term "arbitrarily long" is often used in the context of sequence of numbers. For example, to say that there are "arbitrarily long arithmetic progressions of prime numbers" does not mean that there exists any infinitely long arithmetic progression of prime numbers (there is not), nor that there exists any particular arithmetic progression of prime numbers that is in some sense "arbitrarily long". Rather, the phrase is used to refer to the fact that no matter how large a number is, there exists some arithmetic progression of prime numbers of length at least .
Similar to arbitrarily large, one can also define the phrase " holds for arbitrarily small real numbers", as follows:
In other words:
However small a number, there will be a number smaller than it such that holds.
Arbitrarily large vs. sufficiently large vs. infinitely large
While similar, "arbitrarily large" is not equivalent to "sufficiently large". For instance, while it is true that prime numbers can be arbitrarily large (since there are infinitely many of them due to Euclid's theorem), it is not true that all sufficiently large numbers are prime.
As another example, the statement " is non-negative for arbitrarily large ." could be rewritten as:
However, using "sufficiently large", the same phrase becomes:
Furthermore, "arbitrarily large" also does not mean "infinitely large". For example, although prime numbers can be arbitrarily large, an infinitely large prime number does not exist—since all prime numbers (as well as all other integers) are finite.
In some cases, phrases such as "the proposition is true for arbitrarily large " are used primarily for emphasis, as in " is true for all , no matter how large is." In these cases, the phrase "arbitrarily large" does not have the meaning indicated above (i.e., "however large a number, there will be some larger number for which still holds."). Instead, the usage in this case is in fact logically synonymous with "all".
See also
Sufficiently large
Mathematical jargon
References
Mathematical terminology
|
https://en.wikipedia.org/wiki/Logarithmic%20distribution
|
In probability and statistics, the logarithmic distribution (also known as the logarithmic series distribution or the log-series distribution) is a discrete probability distribution derived from the Maclaurin series expansion
From this we obtain the identity
This leads directly to the probability mass function of a Log(p)-distributed random variable:
for k ≥ 1, and where 0 < p < 1. Because of the identity above, the distribution is properly normalized.
The cumulative distribution function is
where B is the incomplete beta function.
A Poisson compounded with Log(p)-distributed random variables has a negative binomial distribution. In other words, if N is a random variable with a Poisson distribution, and Xi, i = 1, 2, 3, ... is an infinite sequence of independent identically distributed random variables each having a Log(p) distribution, then
has a negative binomial distribution. In this way, the negative binomial distribution is seen to be a compound Poisson distribution.
R. A. Fisher described the logarithmic distribution in a paper that used it to model relative species abundance.
See also
Poisson distribution (also derived from a Maclaurin series)
References
Further reading
Discrete distributions
Logarithms
|
https://en.wikipedia.org/wiki/Yule%E2%80%93Simon%20distribution
|
In probability and statistics, the Yule–Simon distribution is a discrete probability distribution named after Udny Yule and Herbert A. Simon. Simon originally called it the Yule distribution.
The probability mass function (pmf) of the Yule–Simon (ρ) distribution is
for integer and real , where is the beta function. Equivalently the pmf can be written in terms of the rising factorial as
where is the gamma function. Thus, if is an integer,
The parameter can be estimated using a fixed point algorithm.
The probability mass function f has the property that for sufficiently large k we have
This means that the tail of the Yule–Simon distribution is a realization of Zipf's law: can be used to model, for example, the relative frequency of the th most frequent word in a large collection of text, which according to Zipf's law is inversely proportional to a (typically small) power of .
Occurrence
The Yule–Simon distribution arose originally as the limiting distribution of a particular model studied by Udny Yule in 1925 to analyze the growth in the number of species per genus in some higher taxa of biotic organisms. The Yule model makes use of two related Yule processes, where a Yule process is defined as a continuous time birth process which starts with one or more individuals. Yule proved that when time goes to infinity, the limit distribution of the number of species in a genus selected uniformly at random has a specific form and exhibits a power-law behavior in its tail. Thirty years later, the Nobel laureate Herbert A. Simon proposed a time-discrete preferential attachment model to describe the appearance of new words in a large piece of a text. Interestingly enough, the limit distribution of the number of occurrences of each word, when the number of words diverges, coincides with that of the number of species belonging to the randomly chosen genus in the Yule model, for a specific choice of the parameters. This fact explains the designation Yule–Simon distribution that is commonly assigned to that limit distribution. In the context of random graphs, the Barabási–Albert model also exhibits an asymptotic degree distribution that equals the Yule–Simon distribution in correspondence of a specific choice of the parameters and still presents power-law characteristics for more general choices of the parameters. The same happens also for other preferential attachment random graph models.
The preferential attachment process can also be studied as an urn process in which balls are added to a growing number of urns, each ball being allocated to an urn with probability linear in the number (of balls) the urn already contains.
The distribution also arises as a compound distribution, in which the parameter of a geometric distribution is treated as a function of random variable having an exponential distribution. Specifically, assume that follows an exponential distribution with scale or rate :
with density
Then a Yule–Simon distributed vari
|
https://en.wikipedia.org/wiki/JTS
|
JTS may refer to:
Alfa Romeo JTS engine, an automobile engine
Java Topology Suite (JTS Topology Suite), a software library
Janesville Transit System, Wisconsin, US
Jakarta Taipei School, Indonesia
Java transaction service, a software library
Jewish Theological Seminary of America, New York City
Jimmy Two-Shoes, a Canadian animated series
Journal of Transatlantic Studies
Journal of Traumatic Stress, US
JT Storage, a d=former US hard drive manufacturer
Jabhat Tahrir Souriya (Syrian Liberation Front), an Islamist group in the Syrian Civil War
|
https://en.wikipedia.org/wiki/Schur%20multiplier
|
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group of a group G. It was introduced by in his work on projective representations.
Examples and properties
The Schur multiplier of a finite group G is a finite abelian group whose exponent divides the order of G. If a Sylow p-subgroup of G is cyclic for some p, then the order of is not divisible by p. In particular, if all Sylow p-subgroups of G are cyclic, then is trivial.
For instance, the Schur multiplier of the nonabelian group of order 6 is the trivial group since every Sylow subgroup is cyclic. The Schur multiplier of the elementary abelian group of order 16 is an elementary abelian group of order 64, showing that the multiplier can be strictly larger than the group itself. The Schur multiplier of the quaternion group is trivial, but the Schur multiplier of dihedral 2-groups has order 2.
The Schur multipliers of the finite simple groups are given at the list of finite simple groups. The covering groups of the alternating and symmetric groups are of considerable recent interest.
Relation to projective representations
Schur's original motivation for studying the multiplier was to classify projective representations of a group, and the modern formulation of his definition is the second cohomology group . A projective representation is much like a group representation except that instead of a homomorphism into the general linear group , one takes a homomorphism into the projective general linear group . In other words, a projective representation is a representation modulo the center.
showed that every finite group G has associated to it at least one finite group C, called a Schur cover, with the property that every projective representation of G can be lifted to an ordinary representation of C. The Schur cover is also known as a covering group or Darstellungsgruppe. The Schur covers of the finite simple groups are known, and each is an example of a quasisimple group. The Schur cover of a perfect group is uniquely determined up to isomorphism, but the Schur cover of a general finite group is only determined up to isoclinism.
Relation to central extensions
The study of such covering groups led naturally to the study of central and stem extensions.
A central extension of a group G is an extension
where is a subgroup of the center of C.
A stem extension of a group G is an extension
where is a subgroup of the intersection of the center of C and the derived subgroup of C; this is more restrictive than central.
If the group G is finite and one considers only stem extensions, then there is a largest size for such a group C, and for every C of that size the subgroup K is isomorphic to the Schur multiplier of G. If the finite group G is moreover perfect, then C is unique up to isomorphism and is itself perfect. Such C are often called universal perfect central extensions of G, or covering group (as it is a discrete analog of
|
https://en.wikipedia.org/wiki/Axiom%20of%20real%20determinacy
|
In mathematics, the axiom of real determinacy (abbreviated as ADR) is an axiom in set theory. It states the following:
The axiom of real determinacy is a stronger version of the axiom of determinacy (AD), which makes the same statement about games where both players choose integers; ADR is inconsistent with the axiom of choice. It also implies the existence of inner models with certain large cardinals.
ADR is equivalent to AD plus the axiom of uniformization.
See also
AD+
Axiom of projective determinacy
Topological game
Axioms of set theory
Determinacy
|
https://en.wikipedia.org/wiki/Dedekind%20sum
|
In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function D of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They have subsequently been much studied in number theory, and have occurred in some problems of topology. Dedekind sums have a large number of functional equations; this article lists only a small fraction of these.
Dedekind sums were introduced by Richard Dedekind in a commentary on fragment XXVIII of Bernhard Riemann's collected papers.
Definition
Define the sawtooth function as
We then let
be defined by
the terms on the right being the Dedekind sums. For the case a = 1, one often writes
s(b, c) = D(1, b; c).
Simple formulae
Note that D is symmetric in a and b, and hence
and that, by the oddness of (( )),
D(−a, b; c) = −D(a, b; c),
D(a, b; −c) = D(a, b; c).
By the periodicity of D in its first two arguments, the third argument being the length of the period for both,
D(a, b; c) = D(a+kc, b+lc; c), for all integers k,l.
If d is a positive integer, then
D(ad, bd; cd) = dD(a, b; c),
D(ad, bd; c) = D(a, b; c), if (d, c) = 1,
D(ad, b; cd) = D(a, b; c), if (d, b) = 1.
There is a proof for the last equality making use of
Furthermore, az = 1 (mod c) implies D(a, b; c) = D(1, bz; c).
Alternative forms
If b and c are coprime, we may write s(b, c) as
where the sum extends over the c-th roots of unity other than 1, i.e. over all such that and .
If b, c > 0 are coprime, then
Reciprocity law
If b and c are coprime positive integers then
Rewriting this as
it follows that the number 6c s(b,c) is an integer.
If k = (3, c) then
and
A relation that is prominent in the theory of the Dedekind eta function is the following. Let q = 3, 5, 7 or 13 and let n = 24/(q − 1). Then given integers a, b, c, d with ad − bc = 1 (thus belonging to the modular group), with c chosen so that c = kq for some integer k > 0, define
Then nδ is an even integer.
Rademacher's generalization of the reciprocity law
Hans Rademacher found the following generalization of the reciprocity law for Dedekind sums: If a, b, and c are pairwise coprime positive integers, then
Hence, the above triple sum vanishes if and only if (a, b, c) is a Markov triple, i.e. a solution of the Markov equation
References
Further reading
Tom M. Apostol, Modular functions and Dirichlet Series in Number Theory (1990), Springer-Verlag, New York. (See chapter 3.)
Matthias Beck and Sinai Robins, Dedekind sums: a discrete geometric viewpoint, (2005 or earlier)
Hans Rademacher and Emil Grosswald, Dedekind Sums, Carus Math. Monographs, 1972. .
Number theory
Modular forms
|
https://en.wikipedia.org/wiki/Fifth%20government%20of%20Jordi%20Pujol
|
The colors indicate the political party affiliation of each member:
So the statistics of the Government composition are:
Cabinets of Catalonia
|
https://en.wikipedia.org/wiki/Gyroelongated%20square%20pyramid
|
In geometry, the gyroelongated square pyramid is one of the Johnson solids (). As its name suggests, it can be constructed by taking a square pyramid and "gyroelongating" it, which in this case involves joining a square antiprism to its base.
Applications
The Gyroelongated square pyramid represents the capped square antiprismatic molecular geometry:
Dual polyhedron
The dual of the gyroelongated square pyramid has 9 faces: 4 kites, 1 square and 4 pentagonal.
See also
Gyroelongated square bipyramid
External links
Johnson solids
Pyramids and bipyramids
|
https://en.wikipedia.org/wiki/Elongated%20pentagonal%20pyramid
|
In geometry, the elongated pentagonal pyramid is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a pentagonal pyramid () by attaching a pentagonal prism to its base.
Formulae
The following formulae for the height (), surface area () and volume () can be used if all faces are regular, with edge length :
Dual polyhedron
The dual of the elongated pentagonal pyramid has 11 faces: 5 triangular, 1 pentagonal and 5 trapezoidal. It is topologically identical to the Johnson solid.
See also
Elongated pentagonal bipyramid
References
External links
Johnson solids
Self-dual polyhedra
Pyramids and bipyramids
|
https://en.wikipedia.org/wiki/Gyroelongated%20pentagonal%20pyramid
|
In geometry, the gyroelongated pentagonal pyramid is one of the Johnson solids (). As its name suggests, it is formed by taking a pentagonal pyramid and "gyroelongating" it, which in this case involves joining a pentagonal antiprism to its base.
It can also be seen as a diminished icosahedron, an icosahedron with the top (a pentagonal pyramid, ) chopped off by a plane. Other Johnson solids can be formed by cutting off multiple pentagonal pyramids from an icosahedron: the pentagonal antiprism and metabidiminished icosahedron (two pyramids removed), and the tridiminished icosahedron (three pyramids removed).
Dual polyhedron
The dual of the gyroelongated pentagonal pyramid has 11 faces: 5 kites, 1 regular pentagonal and 5 irregular pentagons.
External links
Johnson solids
Pyramids and bipyramids
|
https://en.wikipedia.org/wiki/Tridiminished%20icosahedron
|
In geometry, the tridiminished icosahedron is one of the Johnson solids (). The name refers to one way of constructing it, by removing three pentagonal pyramids () from a regular icosahedron, which replaces three sets of five triangular faces from the icosahedron with three mutually adjacent pentagonal faces.
Related polytopes
The tridiminished icosahedron is the vertex figure of the snub 24-cell, a uniform 4-polytope (4-dimensional polytope).
See also
Diminished icosahedron (J11)
Metabidiminished icosahedron (J62)
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Metabidiminished%20icosahedron
|
In geometry, the metabidiminished icosahedron is one of the Johnson solids (). The name refers to one way of constructing it, by removing two pentagonal pyramids () from a regular icosahedron, replacing two sets of five triangular faces of the icosahedron with two adjacent pentagonal faces. If two pentagonal pyramids are removed to form nonadjacent pentagonal faces, the result is instead the pentagonal antiprism.
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Kripke%E2%80%93Platek%20set%20theory
|
The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek.
The theory can be thought of as roughly the predicative part of ZFC and is considerably weaker than it.
Axioms
In its formulation, a Δ0 formula is one all of whose quantifiers are bounded. This means any quantification is the form or (See the Lévy hierarchy.)
Axiom of extensionality: Two sets are the same if and only if they have the same elements.
Axiom of induction: φ(a) being a formula, if for all sets x the assumption that φ(y) holds for all elements y of x entails that φ(x) holds, then φ(x) holds for all sets x.
Axiom of empty set: There exists a set with no members, called the empty set and denoted {}.
Axiom of pairing: If x, y are sets, then so is {x, y}, a set containing x and y as its only elements.
Axiom of union: For any set x, there is a set y such that the elements of y are precisely the elements of the elements of x.
Axiom of Δ0-separation: Given any set and any Δ0 formula φ(x), there is a subset of the original set containing precisely those elements x for which φ(x) holds. (This is an axiom schema.)
Axiom of Δ0-collection: Given any Δ0 formula φ(x, y), if for every set x there exists a set y such that φ(x, y) holds, then for all sets X there exists a set Y such that for every x in X there is a y in Y such that φ(x, y) holds.
Some but not all authors include an
Axiom of infinity
KP with infinity is denoted by KPω. These axioms lead to close connections between KP, generalized recursion theory, and the theory of admissible ordinals.
KP can be studied as a constructive set theory by dropping the law of excluded middle, without changing any axioms.
Empty set
If any set is postulated to exist, such as in the axiom of infinity, then the axiom of empty set is redundant because it is equal to the subset . Furthermore, the existence of a member in the universe of discourse, i.e., ∃x(x=x), is implied in certain formulations of first-order logic, in which case the axiom of empty set follows from the axiom of Δ0-separation, and is thus redundant.
Comparison with Zermelo-Fraenkel set theory
As noted, the above are weaker than ZFC as they exclude the power set axiom, choice, and sometimes infinity. Also the axioms of separation and collection here are weaker than the corresponding axioms in ZFC because the formulas φ used in these are limited to bounded quantifiers only.
The axiom of induction in the context of KP is stronger than the usual axiom of regularity, which amounts to applying induction to the complement of a set (the class of all sets not in the given set).
Related definitions
A set is called admissible if it is transitive and is a model of Kripke–Platek set theory.
An ordinal number is called an admissible ordinal if L is an admissible set.
L is called an amenable set if it is a standard model of KP set theory without the axiom of Δ0-collection.
Theorems
Admissible sets
The ordinal
|
https://en.wikipedia.org/wiki/Pericyclic%20reaction
|
In organic chemistry, a pericyclic reaction is the type of organic reaction wherein the transition state of the molecule has a cyclic geometry, the reaction progresses in a concerted fashion, and the bond orbitals involved in the reaction overlap in a continuous cycle at the transition state. Pericyclic reactions stand in contrast to linear reactions, encompassing most organic transformations and proceeding through an acyclic transition state, on the one hand and coarctate reactions, which proceed through a doubly cyclic, concerted transition state on the other hand. Pericyclic reactions are usually rearrangement or addition reactions. The major classes of pericyclic reactions are given in the table below (the three most important classes are shown in bold). Ene reactions and cheletropic reactions are often classed as group transfer reactions and cycloadditions/cycloeliminations, respectively, while dyotropic reactions and group transfer reactions (if ene reactions are excluded) are rarely encountered.
In general, these are considered to be equilibrium processes, although it is possible to push the reaction in one direction by designing a reaction by which the product is at a significantly lower energy level; this is due to a unimolecular interpretation of Le Chatelier's principle. There is thus a set of "retro" pericyclic reactions.
Mechanism of pericyclic reaction
By definition, pericyclic reactions proceed through a concerted mechanism involving a single, cyclic transition state. Because of this, prior to a systematic understanding of pericyclic processes through the principle of orbital symmetry conservation, they were facetiously referred to as 'no-mechanism reactions'. However, reactions for which pericyclic mechanisms can be drawn often have related stepwise mechanisms proceeding through radical or dipolar intermediates that are also viable. Some classes of pericyclic reactions, such as the [2+2] ketene cycloaddition reactions, can be 'controversial' because their mechanism is sometimes not definitively known to be concerted (or may depend on the reactive system). Moreover, pericyclic reactions also often have metal-catalyzed analogs, although usually these are also not technically pericyclic, since they proceed via metal-stabilized intermediates, and therefore are not concerted.
Despite these caveats, the theoretical understanding of pericyclic reactions is probably among the most sophisticated and well-developed in all of organic chemistry. The understanding of how orbitals interact in the course of a pericyclic process has led to the Woodward–Hoffmann rules, a simple set of criteria to predict whether a pericyclic mechanism for a reaction is likely or favorable. For instance, these rules predict that the [4+2] cycloaddition of butadiene and ethylene under thermal conditions is likely a pericyclic process, while the [2+2] cycloaddition of two ethylene molecules is not. These are consistent with experimental data, supporting
|
https://en.wikipedia.org/wiki/Statistics%20South%20Africa
|
Statistics South Africa (frequently shortened to Stats SA) is the national statistical service of South Africa with the goal of producing timely, accurate and official statistics, in order to advance economic growth, development and democracy. To this end, Statistics South Africa produces official demographic, economic and social censuses and surveys. To date Statistics South Africa has produced three censuses, in 1996, 2001 , 2011and 2022. Stats SA was previously known as the "Central Statistical Service", shortly after the end of apartheid and also it absorbed the statistical services of the former Transkei, Bophuthatswana, Venda and Ciskei.
Surveys conducted
1999 Survey of Activities of Young People, or the SAYP.
South African National Census of 2001
2007 Community Survey
South African National Census of 2011
South African National Census of 2022
References
External links
Statistics Act, no. 6 of 1999 from polity.org.za
Government departments of South Africa
Scientific organisations based in South Africa
South Africa
|
https://en.wikipedia.org/wiki/Markov%20decision%20process
|
In mathematics, a Markov decision process (MDP) is a discrete-time stochastic control process. It provides a mathematical framework for modeling decision making in situations where outcomes are partly random and partly under the control of a decision maker. MDPs are useful for studying optimization problems solved via dynamic programming. MDPs were known at least as early as the 1950s; a core body of research on Markov decision processes resulted from Ronald Howard's 1960 book, Dynamic Programming and Markov Processes. They are used in many disciplines, including robotics, automatic control, economics and manufacturing. The name of MDPs comes from the Russian mathematician Andrey Markov as they are an extension of Markov chains.
At each time step, the process is in some state , and the decision maker may choose any action that is available in state . The process responds at the next time step by randomly moving into a new state , and giving the decision maker a corresponding reward .
The probability that the process moves into its new state is influenced by the chosen action. Specifically, it is given by the state transition function . Thus, the next state depends on the current state and the decision maker's action . But given and , it is conditionally independent of all previous states and actions; in other words, the state transitions of an MDP satisfy the Markov property.
Markov decision processes are an extension of Markov chains; the difference is the addition of actions (allowing choice) and rewards (giving motivation). Conversely, if only one action exists for each state (e.g. "wait") and all rewards are the same (e.g. "zero"), a Markov decision process reduces to a Markov chain.
Definition
A Markov decision process is a 4-tuple , where:
is a set of states called the state space,
is a set of actions called the action space (alternatively, is the set of actions available from state ),
is the probability that action in state at time will lead to state at time ,
is the immediate reward (or expected immediate reward) received after transitioning from state to state , due to action
The state and action spaces may be finite or infinite, for example the set of real numbers. Some processes with countably infinite state and action spaces can be reduced to ones with finite state and action spaces.
A policy function is a (potentially probabilistic) mapping from state space () to action space ().
Optimization objective
The goal in a Markov decision process is to find a good "policy" for the decision maker: a function that specifies the action that the decision maker will choose when in state . Once a Markov decision process is combined with a policy in this way, this fixes the action for each state and the resulting combination behaves like a Markov chain (since the action chosen in state is completely determined by and reduces to , a Markov transition matrix).
The objective is to choose a policy that will maximiz
|
https://en.wikipedia.org/wiki/Optimization%20problem
|
In mathematics, engineering, computer science and economics, an optimization problem is the problem of finding the best solution from all feasible solutions.
Optimization problems can be divided into two categories, depending on whether the variables are continuous or discrete:
An optimization problem with discrete variables is known as a discrete optimization, in which an object such as an integer, permutation or graph must be found from a countable set.
A problem with continuous variables is known as a continuous optimization, in which an optimal value from a continuous function must be found. They can include constrained problems and multimodal problems.
Continuous optimization problem
The standard form of a continuous optimization problem is
where
is the objective function to be minimized over the -variable vector ,
are called inequality constraints
are called equality constraints, and
and .
If , the problem is an unconstrained optimization problem. By convention, the standard form defines a minimization problem. A maximization problem can be treated by negating the objective function.
Combinatorial optimization problem
Formally, a combinatorial optimization problem is a quadruple , where
is a set of instances;
given an instance , is the set of feasible solutions;
given an instance and a feasible solution of , denotes the measure of , which is usually a positive real.
is the goal function, and is either or .
The goal is then to find for some instance an optimal solution, that is, a feasible solution with
For each combinatorial optimization problem, there is a corresponding decision problem that asks whether there is a feasible solution for some particular measure . For example, if there is a graph which contains vertices and , an optimization problem might be "find a path from to that uses the fewest edges". This problem might have an answer of, say, 4. A corresponding decision problem would be "is there a path from to that uses 10 or fewer edges?" This problem can be answered with a simple 'yes' or 'no'.
In the field of approximation algorithms, algorithms are designed to find near-optimal solutions to hard problems. The usual decision version is then an inadequate definition of the problem since it only specifies acceptable solutions. Even though we could introduce suitable decision problems, the problem is more naturally characterized as an optimization problem.
See also
− the optimum need not be found, just a "good enough" solution.
References
External links
Computational problems
|
https://en.wikipedia.org/wiki/Invariant%20%28mathematics%29
|
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.
Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects.
Examples
A simple example of invariance is expressed in our ability to count. For a finite set of objects of any kind, there is a number to which we always arrive, regardless of the order in which we count the objects in the set. The quantity—a cardinal number—is associated with the set, and is invariant under the process of counting.
An identity is an equation that remains true for all values of its variables. There are also inequalities that remain true when the values of their variables change.
The distance between two points on a number line is not changed by adding the same quantity to both numbers. On the other hand, multiplication does not have this same property, as distance is not invariant under multiplication.
Angles and ratios of distances are invariant under scalings, rotations, translations and reflections. These transformations produce similar shapes, which is the basis of trigonometry. In contrast, angles and ratios are not invariant under non-uniform scaling (such as stretching). The sum of a triangle's interior angles (180°) is invariant under all the above operations. As another example, all circles are similar: they can be transformed into each other and the ratio of the circumference to the diameter is invariant (denoted by the Greek letter π (pi)).
Some more complicated examples:
The real part and the absolute value of a complex number are invariant under complex conjugation.
The degree of a polynomial is invariant under a linear change of variables.
The dimension and homology groups of a topological object are invariant under homeomorphism.
The number of fixed points of a dynamical system is invariant under many mathematical operations.
Euclidean distance is invariant under orthogonal transformations.
Euclidean area is invariant under linear maps which have determinant ±1 (see ).
Some invariants of projective transformations include collinearity of three or more points, concurrency
|
https://en.wikipedia.org/wiki/Pisano%20period
|
In number theory, the nth Pisano period, written as (n), is the period with which the sequence of Fibonacci numbers taken modulo n repeats. Pisano periods are named after Leonardo Pisano, better known as Fibonacci. The existence of periodic functions in Fibonacci numbers was noted by Joseph Louis Lagrange in 1774.
Definition
The Fibonacci numbers are the numbers in the integer sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, ...
defined by the recurrence relation
For any integer n, the sequence of Fibonacci numbers Fi taken modulo n is periodic.
The Pisano period, denoted (n), is the length of the period of this sequence. For example, the sequence of Fibonacci numbers modulo 3 begins:
0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, ...
This sequence has period 8, so (3) = 8.
Properties
With the exception of (2) = 3, the Pisano period (n) is always even.
A simple proof of this can be given by observing that (n) is equal to the order of the Fibonacci matrix
in the general linear group GL2(ℤn) of invertible 2 by 2 matrices in the finite ring ℤn of integers modulo n. Since Q has determinant −1, the determinant of Q(n) is (−1)(n), and since this must equal 1 in ℤn, either n ≤ 2 or (n) is even.
If m and n are coprime, then (mn) is the least common multiple of (m) and (n), by the Chinese remainder theorem. For example, (3) = 8 and (4) = 6 imply (12) = 24. Thus the study of Pisano periods may be reduced to that of Pisano periods of prime powers q = pk, for k ≥ 1.
If p is prime, (pk) divides pk–1 (p). It is unknown if
for every prime p and integer k > 1. Any prime p providing a counterexample would necessarily be a Wall–Sun–Sun prime, and conversely every Wall–Sun–Sun prime p gives a counterexample (set k = 2).
So the study of Pisano periods may be further reduced to that of Pisano periods of primes. In this regard, two primes are anomalous. The prime 2 has an odd Pisano period, and the prime 5 has period that is relatively much larger than the Pisano period of any other prime. The periods of powers of these primes are as follows:
If n = 2k, then (n) = 3·2k–1 = = .
if n = 5k, then (n) = 20·5k–1 = = 4n.
From these it follows that if n = 2·5k then (n) = 6n.
The remaining primes all lie in the residue classes or . If p is a prime different from 2 and 5, then the modulo p analogue of Binet's formula implies that (p) is the multiplicative order of a root of modulo p. If , these roots belong to (by quadratic reciprocity). Thus their order, (p) is a divisor of p − 1. For example, (11) = 11 − 1 = 10 and (29) = (29 − 1)/2 = 14.
If the roots modulo p of do not belong to (by quadratic reciprocity again), and belong to the finite field
As the Frobenius automorphism exchanges these roots, it follows that, denoting them by r and s, we have r p = s, and thus r p+1 = –1. That is r 2(p+1) = 1, and the Pisano period, which is the order
|
https://en.wikipedia.org/wiki/Sylvester%27s%20law%20of%20inertia
|
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if is the symmetric matrix that defines the quadratic form, and is any invertible matrix such that is diagonal, then the number of negative elements in the diagonal of is always the same, for all such ; and the same goes for the number of positive elements.
This property is named after James Joseph Sylvester who published its proof in 1852.
Statement
Let be a symmetric square matrix of order with real entries. Any non-singular matrix of the same size is said to transform into another symmetric matrix , also of order , where is the transpose of . It is also said that matrices and are congruent. If is the coefficient matrix of some quadratic form of , then is the matrix for the same form after the change of basis defined by .
A symmetric matrix can always be transformed in this way into a diagonal matrix
which has only entries , , along the diagonal. Sylvester's law of inertia states that the number of diagonal entries of each kind is an invariant of , i.e. it does not depend on the matrix used.
The number of s, denoted , is called the positive index of inertia of , and the number of s, denoted , is called the negative index of inertia. The number of s, denoted , is the dimension of the null space of , known as the nullity of . These numbers satisfy an obvious relation
The difference, , is usually called the signature of . (However, some authors use that term for the triple
consisting of the nullity and the positive and negative indices of inertia of ; for a non-degenerate form of a given dimension these are equivalent data, but in general the triple yields more data.)
If the matrix has the property that every principal upper left minor is non-zero then the negative index of inertia is equal to the number of sign changes in the sequence
Statement in terms of eigenvalues
The law can also be stated as follows: two symmetric square matrices of the same size have the same number of positive, negative and zero eigenvalues if and only if they are congruent (, for some non-singular ).
The positive and negative indices of a symmetric matrix are also the number of positive and negative eigenvalues of . Any symmetric real matrix has an eigendecomposition of the form where is a diagonal matrix containing the eigenvalues of , and is an orthonormal square matrix containing the eigenvectors. The matrix can be written where is diagonal with entries , and is diagonal with . The matrix transforms to .
Law of inertia for quadratic forms
In the context of quadratic forms, a real quadratic form in variables (or on an -dimensional real vector space) can by a suitable change of basis (by non-singular linear transformation from to ) be brought to the diagonal form
with each . Sylvester's law of inertia states that the number of c
|
https://en.wikipedia.org/wiki/Square%20pyramid
|
In geometry, a square pyramid is a pyramid with a square base, having a total of five faces. If the pyramid's apex lies on a line erected perpendicularly from the center of the square, it is a right square pyramid with four isosceles triangles; otherwise, it is an oblique square pyramid. When all of the pyramid's edges are equal in length, its triangles are all equilateral, and it is called an equilateral square pyramid.
Square pyramids have arisen throughout the history of architecture, with an example being the Egyptian pyramid. They also crop up in chemistry in square pyramidal molecular structures. Square pyramids are often used in the construction of other polyhedra.
Properties
Right and equilateral square pyramid
A square pyramid has eight edges, five faces that include four triangles, and one square as the base. It also has five vertices, one of which is an apex, a vertex where all lateral edges meet. A lateral edge is a segment line between the apex and another vertex at the square base. When the apex is perpendicularly above the center of the base, all the lateral edges have the same length, and the faces other than the base are congruent isosceles triangles, a right square pyramid. A square pyramid where the apex is not perpendicularly above the center of the base, and the faces are not an isosceles triangle is an oblique square pyramid.
When all edges have the same length, the four isosceles triangles become equilateral, and therefore all the faces are regular polygons. This pyramid is known as an equilateral square pyramid. Its dihedral angle between two adjacent triangular faces and between the base and triangular face is
respectively. It has three-dimensional symmetry group of cyclic group pyramidal symmetry, a symmetry of order eight; this means that it is symmetrical as one rotates it for every quarter-turn of a full angle around the axis of symmetry, two vertical planes passing through the diagonals of the square base, and two other vertical planes passing through the midpoint of the opposite edges of the square base. A convex polyhedra that has regular polygons as its faces is called Johnson solid, and because an equilateral square pyramid has these properties, it is a Johnson solid that numbered as among them.
Surface area and volume
The slant height of a right square pyramid can be considered as the height of an isosceles triangle, which can be obtained by applying the Pythagorean theorem, that is:
Here, is the length of the base of an isosceles triangle, and is the length of the side of an isosceles triangle, a lateral edge of a right square pyramid. The height of a right square pyramid can be considered and obtained similarly, and substitute the formula of the slant height gives
A surface area is the total area of the faces of a polyhedron. Hence, the surface area of a square pyramid can be expressed as the sum of four times the area of a triangle and the area of a square. The area of a triangle is the half
|
https://en.wikipedia.org/wiki/Pentagonal%20pyramid
|
In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point (the apex). Like any pyramid, it is self-dual.
The regular pentagonal pyramid has a base that is a regular pentagon and lateral faces that are equilateral triangles. It is one of the Johnson solids ().
It can be seen as the "lid" of an icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid,
More generally an order-2 vertex-uniform pentagonal pyramid can be defined with a regular pentagonal base and 5 isosceles triangle sides of any height.
Cartesian coordinates
The pentagonal pyramid can be seen as the "lid" of a regular icosahedron; the rest of the icosahedron forms a gyroelongated pentagonal pyramid, J11. From the Cartesian coordinates of the icosahedron, Cartesian coordinates for a pentagonal pyramid with edge length 2 may be inferred as
where (sometimes written as φ) is the golden ratio.
The height H, from the midpoint of the pentagonal face to the apex, of a pentagonal pyramid with edge length a may therefore be computed as:
Its surface area A can be computed as the area of the pentagonal base plus five times the area of one triangle:
Its volume can be calculated as:
Related polyhedra
The pentagrammic star pyramid has the same vertex arrangement, but connected onto a pentagram base:
Example
References
External links
Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra ( VRML model)
Pyramids and bipyramids
Self-dual polyhedra
Prismatoid polyhedra
Johnson solids
|
https://en.wikipedia.org/wiki/Triangular%20cupola
|
In geometry, the triangular cupola is one of the Johnson solids (). It can be seen as half a cuboctahedron.
Formulae
The following formulae for the volume (), the surface area () and the height () can be used if all faces are regular, with edge length a:
Dual polyhedron
The dual of the triangular cupola has 6 triangular and 3 kite faces:
Related polyhedra and honeycombs
The triangular cupola can be augmented by 3 square pyramids, leaving adjacent coplanar faces. This isn't a Johnson solid because of its coplanar faces. Merging those coplanar triangles into larger ones, topologically this is another triangular cupola with isosceles trapezoidal side faces. If all the triangles are retained and the base hexagon is replaced by 6 triangles, it generates a coplanar deltahedron with 22 faces.
The triangular cupola can form a tessellation of space with square pyramids and/or octahedra, the same way octahedra and cuboctahedra can fill space.
The family of cupolae with regular polygons exists up to n=5 (pentagons), and higher if isosceles triangles are used in the cupolae.
References
External links
Prismatoid polyhedra
Johnson solids
|
https://en.wikipedia.org/wiki/Square%20cupola
|
In geometry, the square cupola, sometimes called lesser dome, is one of the Johnson solids (). It can be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagon.
Formulae
The following formulae for the circumradius, surface area, volume, and height can be used if all faces are regular, with edge length a:
Related polyhedra and honeycombs
Other convex cupolae
Dual polyhedron
The dual of the square cupola has 8 triangular and 4 kite faces:
Crossed square cupola
The crossed square cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex square cupola. It can be obtained as a slice of the nonconvex great rhombicuboctahedron or quasirhombicuboctahedron, analogously to how the square cupola may be obtained as a slice of the rhombicuboctahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is an octagram.
It may be seen as a cupola with a retrograde square base, so that the squares and triangles connect across the bases in the opposite way to the square cupola, hence intersecting each other.
Honeycombs
The square cupola is a component of several nonuniform space-filling lattices:
with tetrahedra;
with cubes and cuboctahedra; and
with tetrahedra, square pyramids and various combinations of cubes, elongated square pyramids and elongated square bipyramids.
References
External links
Prismatoid polyhedra
Johnson solids
|
https://en.wikipedia.org/wiki/Pentagonal%20rotunda
|
In geometry, the pentagonal rotunda is one of the Johnson solids (). It can be seen as half of an icosidodecahedron, or as half of a pentagonal orthobirotunda. It has a total of 17 faces.
Formulae
The following formulae for volume, surface area, circumradius, and height are valid if all faces are regular, with edge length a:
Dual polyhedron
The dual of the pentagonal rotunda has 20 faces: 10 triangular, 5 rhombic, and 5 kites.
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Elongated%20square%20cupola
|
In geometry, the elongated square cupola is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a square cupola () by attaching an octagonal prism to its base. The solid can be seen as a rhombicuboctahedron with its "lid" (another square cupola) removed.
Formulae
The following formulae for volume, surface area and circumradius can be used if all faces are regular, with edge length a:
Dual polyhedron
The dual of the elongated square cupola has 20 faces: 8 isosceles triangles, 4 kites, 8 quadrilaterals.
Related polyhedra and honeycombs
The elongated square cupola forms space-filling honeycombs with tetrahedra and cubes; with cubes and cuboctahedra; and with tetrahedra, elongated square pyramids, and elongated square bipyramids. (The latter two units can be decomposed into cubes and square pyramids.)
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Elongated%20square%20gyrobicupola
|
In geometry, the elongated square gyrobicupola or pseudo-rhombicuboctahedron is one of the Johnson solids (). It is not usually considered to be an Archimedean solid, even though its faces consist of regular polygons that meet in the same pattern at each of its vertices, because unlike the 13 Archimedean solids, it lacks a set of global symmetries that map every vertex to every other vertex (though Grünbaum has suggested it should be added to the traditional list of Archimedean solids as a 14th example). It strongly resembles, but should not be mistaken for, the rhombicuboctahedron, which is an Archimedean solid. It is also a canonical polyhedron.
This shape may have been discovered by Johannes Kepler in his enumeration of the Archimedean solids, but its first clear appearance in print appears to be the work of Duncan Sommerville in 1905. It was independently rediscovered by J. C. P. Miller by 1930 (by mistake while attempting to construct a model of the rhombicuboctahedron) and again by V. G. Ashkinuse in 1957.
Construction and relation to the rhombicuboctahedron
As the name suggests, it can be constructed by elongating a square gyrobicupola (J29) and inserting an octagonal prism between its two halves.
The solid can also be seen as the result of twisting one of the square cupolae (J4) on a rhombicuboctahedron (one of the Archimedean solids; a.k.a. the elongated square orthobicupola) by 45 degrees. It is therefore a gyrate rhombicuboctahedron. Its similarity to the rhombicuboctahedron gives it the alternative name pseudo-rhombicuboctahedron. It has occasionally been referred to as "the fourteenth Archimedean solid".
This property does not carry over to its pentagonal-faced counterpart, the gyrate rhombicosidodecahedron.
Symmetry and classification
The pseudo-rhombicuboctahedron possesses D4d symmetry. It is locally vertex-regular – the arrangement of the four faces incident on any vertex is the same for all vertices; this is unique among the Johnson solids. However, the manner in which it is "twisted" gives it a distinct "equator" and two distinct "poles", which in turn divide its vertices into 8 "polar" vertices (4 per pole) and 16 "equatorial" vertices. It is therefore not vertex-transitive, and consequently not usually considered to be one of the Archimedean solids.
With faces colored by its D4d symmetry, it can look like this:
There are 8 (green) squares around its equator, 4 (red) triangles and 4 (yellow) squares above and below, and one (blue) square on each pole.
Related polyhedra and honeycombs
The elongated square gyrobicupola can form a space-filling honeycomb with the regular tetrahedron, cube, and cuboctahedron. It can also form another honeycomb with the tetrahedron, square pyramid and various combinations of cubes, elongated square pyramids, and elongated square bipyramids.
The pseudo great rhombicuboctahedron is a nonconvex analog of the pseudo-rhombicuboctahedron, constructed in a similar way from the nonconvex great
|
https://en.wikipedia.org/wiki/Elongated%20pentagonal%20rotunda
|
In geometry, the elongated pentagonal rotunda is one of the Johnson solids (J21). As the name suggests, it can be constructed by elongating a pentagonal rotunda (J6) by attaching a decagonal prism to its base. It can also be seen as an elongated pentagonal orthobirotunda (J42) with one pentagonal rotunda removed.
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:
Dual polyhedron
The dual of the elongated pentagonal rotunda has 30 faces: 10 isosceles triangles, 10 rhombi, and 10 quadrilaterals.
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Gyroelongated%20square%20cupola
|
In geometry, the gyroelongated square cupola is one of the Johnson solids (J23). As the name suggests, it can be constructed by gyroelongating a square cupola (J4) by attaching an octagonal antiprism to its base. It can also be seen as a gyroelongated square bicupola (J45) with one square bicupola removed.
Area and Volume
The surface area is,
The volume is the sum of the volume of a square cupola and the volume of an octagonal prism,
Dual polyhedron
The dual of the gyroelongated square cupola has 20 faces: 8 kites, 4 rhombi, and 8 pentagons.
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Gyroelongated%20pentagonal%20rotunda
|
In geometry, the gyroelongated pentagonal rotunda is one of the Johnson solids (J25). As the name suggests, it can be constructed by gyroelongating a pentagonal rotunda (J6) by attaching a decagonal antiprism to its base. It can also be seen as a gyroelongated pentagonal birotunda (J48) with one pentagonal rotunda removed.
Area and Volume
With edge length a, the surface area is
and the volume is
Dual polyhedron
The dual of the gyroelongated pentagonal rotunda has 30 faces: 10 pentagons, 10 rhombi, and 10 quadrilaterals.
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Square%20orthobicupola
|
In geometry, the square orthobicupola is one of the Johnson solids (). As the name suggests, it can be constructed by joining two square cupolae () along their octagonal bases, matching like faces. A 45-degree rotation of one cupola before the joining yields a square gyrobicupola ().
The square orthobicupola is the second in an infinite set of orthobicupolae.
The square orthobicupola can be elongated by the insertion of an octagonal prism between its two cupolae to yield a rhombicuboctahedron, or collapsed by the removal of an irregular hexagonal prism to yield an elongated square dipyramid (), which itself is merely an elongated octahedron.
It can be constructed from the disphenocingulum () by replacing the band of up-and-down triangles by a band of rectangles, while fixing two opposite sphenos.
Related polyhedra and honeycombs
The square orthobicupola forms space-filling honeycombs with tetrahedra; with cubes and cuboctahedra; with tetrahedra and cubes; with square pyramids, tetrahedra and various combinations of cubes, elongated square pyramids and/or elongated square bipyramids.
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Square%20gyrobicupola
|
In geometry, the square gyrobicupola is one of the Johnson solids (). Like the square orthobicupola (), it can be obtained by joining two square cupolae () along their bases. The difference is that in this solid, the two halves are rotated 45 degrees with respect to one another.
The square gyrobicupola is the second in an infinite set of gyrobicupolae.
Related to the square gyrobicupola is the elongated square gyrobicupola. This polyhedron is created when an octagonal prism is inserted between the two halves of the square gyrobicupola. It is argued whether or not the elongated square gyrobicupola is an Archimedean solid because, although it meets every other standard necessary to be an Archimedean solid, it is not highly symmetric.
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:
Related polyhedra and honeycombs
The square gyrobicupola forms space-filling honeycombs with tetrahedra, cubes and cuboctahedra; and with tetrahedra, square pyramids, and elongated square bipyramids. (The latter unit can be decomposed into elongated square pyramids, cubes, and/or square pyramids).
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Pentagonal%20orthobirotunda
|
In geometry, the pentagonal orthobirotunda is one of the Johnson solids (). It can be constructed by joining two pentagonal rotundae () along their decagonal faces, matching like faces.
Related polyhedra
The pentagonal orthobirotunda is also related to an Archimedean solid, the icosidodecahedron, which can also be called a pentagonal gyrobirotunda, similarly created by two pentagonal rotunda but with a 36-degree rotation.
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Augmented%20tridiminished%20icosahedron
|
In geometry, the augmented tridiminished icosahedron is one of the
Johnson solids (). It can be obtained by joining a tetrahedron to another Johnson solid, the tridiminished icosahedron ().
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Elongated%20pentagonal%20gyrobirotunda
|
In geometry, the elongated pentagonal gyrobirotunda is one of the Johnson solids (). As the name suggests, it can be constructed by elongating a "pentagonal gyrobirotunda," or icosidodecahedron (one of the Archimedean solids), by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae () through 36 degrees before inserting the prism yields an elongated pentagonal orthobirotunda ().
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Elongated%20pentagonal%20orthobirotunda
|
In geometry, the elongated pentagonal orthobirotunda is one of the Johnson solids (). Its Conway polyhedron notation is at5jP5. As the name suggests, it can be constructed by elongating a pentagonal orthobirotunda () by inserting a decagonal prism between its congruent halves. Rotating one of the pentagonal rotundae () through 36 degrees before inserting the prism yields the elongated pentagonal gyrobirotunda ().
Formulae
The following formulae for volume and surface area can be used if all faces are regular, with edge length a:
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Gyroelongated%20pentagonal%20birotunda
|
In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda (either or the icosidodecahedron) by inserting a decagonal antiprism between its two halves.
The gyroelongated pentagonal birotunda is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form. In the illustration to the right, each pentagonal face on the bottom half of the figure is connected by a path of two triangular faces to a pentagonal face above it and to the left. In the figure of opposite chirality (the mirror image of the illustrated figure), each bottom pentagon would be connected to a pentagonal face above it and to the right. The two chiral forms of are not considered different Johnson solids.
Area and Volume
With edge length a, the surface area is
and the volume is
See also
Birotunda
External links
Johnson solids
Chiral polyhedra
|
https://en.wikipedia.org/wiki/Gyroelongated%20square%20bicupola
|
In geometry, the gyroelongated square bicupola is one of the Johnson solids (). As the name suggests, it can be constructed by gyroelongating a square bicupola ( or ) by inserting an octagonal antiprism between its congruent halves.
The gyroelongated square bicupola is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form. In the illustration to the right, each square face on the left half of the figure is connected by a path of two triangular faces to a square face below it and to the left. In the figure of opposite chirality (the mirror image of the illustrated figure), each square on the left would be connected to a square face above it and to the right. The two chiral forms of are not considered different Johnson solids.
Area and Volume
With edge length a, the surface area is
and the volume is
References
External links
Johnson solids
Chiral polyhedra
|
https://en.wikipedia.org/wiki/Twelfth%20root%20of%20two
|
The twelfth root of two or (or equivalently ) is an algebraic irrational number, approximately equal to 1.0594631. It is most important in Western music theory, where it represents the frequency ratio (musical interval) of a semitone () in twelve-tone equal temperament. This number was proposed for the first time in relationship to musical tuning in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals (frequency ratios) as consisting of different numbers of a single interval, the equal tempered semitone (for example, a minor third is 3 semitones, a major third is 4 semitones, and perfect fifth is 7 semitones). A semitone itself is divided into 100 cents (1 cent = ).
Numerical value
The twelfth root of two to 20 significant figures is . Fraction approximations in increasing order of accuracy include , , , , and .
, its numerical value has been computed to at least twenty billion decimal digits.
The equal-tempered chromatic scale
A musical interval is a ratio of frequencies and the equal-tempered chromatic scale divides the octave (which has a ratio of 2:1) into twelve equal parts. Each note has a frequency that is 2 times that of the one below it.
Applying this value successively to the tones of a chromatic scale, starting from A above middle C (known as A4) with a frequency of 440 Hz, produces the following sequence of pitches:
The final A (A5: 880 Hz) is exactly twice the frequency of the lower A (A4: 440 Hz), that is, one octave higher.
Other tuning scales
Other tuning scales use slightly different interval ratios:
The just or Pythagorean perfect fifth is 3/2, and the difference between the equal tempered perfect fifth and the just is a grad, the twelfth root of the Pythagorean comma ().
The equal tempered Bohlen–Pierce scale uses the interval of the thirteenth root of three ().
Stockhausen's Studie II (1954) makes use of the twenty-fifth root of five (), a compound major third divided into 5×5 parts.
The delta scale is based on ≈.
The gamma scale is based on ≈.
The beta scale is based on ≈.
The alpha scale is based on ≈.
Pitch adjustment
Since the frequency ratio of a semitone is close to 106% (), increasing or decreasing the playback speed of a recording by 6% will shift the pitch up or down by about one semitone, or "half-step". Upscale reel-to-reel magnetic tape recorders typically have pitch adjustments of up to ±6%, generally used to match the playback or recording pitch to other music sources having slightly different tunings (or possibly recorded on equipment that was not running at quite the right speed). Modern recording studios utilize digital pitch shifting to achieve similar results, ranging from cents up to several half-steps (note that reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not).
History
Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1
|
https://en.wikipedia.org/wiki/Toe%20%28automotive%29
|
In automotive engineering, toe, also known as tracking, is the symmetric angle that each wheel makes with the longitudinal axis of the vehicle, as a function of static geometry, and kinematic and compliant effects. This can be contrasted with steer, which is the antisymmetric angle, i.e. both wheels point to the left or right, in parallel (roughly). Negative toe, or toe out, is the front of the wheel pointing away from the centreline of the vehicle. Positive toe, or toe in, is the front of the wheel pointing towards the centreline of the vehicle. Historically, and still commonly in the United States, toe was specified as the linear difference (either inches or millimeters) of the distance between the two front-facing and rear-facing tire centerlines at the outer diameter and axle-height; since the toe angle in that case depends on the tire diameter, the linear dimension toe specification for a particular vehicle is for specified tires.
Description
In a rear-wheel drive vehicle, increased front toe-in provides greater straight-line stability at the cost of some sluggishness of turning response. Performance vehicles may run zero front toe or even some toe-out for a better response to steering inputs. The wear on the tires is marginally increased as the tires are under slight side slip conditions when the steering is set straight ahead. On front-wheel drive vehicles, the situation is more complex. Rear toe-in provides better stability during cornering.
Toe is usually adjustable in production automobiles, even though caster angle and camber angle are often not adjustable. Maintenance of front-end alignment, which used to involve all three adjustments, currently involves only setting the toe; in most cases, even for a car in which caster or camber are adjustable, only the toe will need adjustment. Toe may only be adjustable on the front wheels.
One related concept is that the proper toe for straight-line travel of a vehicle will not be correct while turning, since the inside wheel must travel around a smaller radius than the outside wheel; to compensate for this, the steering linkage typically conforms more or less to Ackermann steering geometry, modified to suit the characteristics of the individual vehicle.
Road–rail vehicles
The front rail wheels of road–rail vehicles are often set to toe-in by a distance of 6 mm over 1 metre. Unlike other forms of rolling stock, road-rail vehicles do not always have a common axle between the rail wheels and the toe-in angle prevents the vehicle from hunting when on-rail.
Interaction with camber
When a wheel is set up to have some camber angle, the interaction between the tire and road surface causes the wheel to tend to want to roll in a curve, as if it were part of a conical surface (camber thrust). This tendency to turn increases the rolling resistance as well as increasing tire wear. A small degree of toe (toe-out for negative camber, toe-in for positive camber) will cancel this turning tendency, reducing
|
https://en.wikipedia.org/wiki/Pentagonal%20cupola
|
In geometry, the pentagonal cupola is one of the Johnson solids (). It can be obtained as a slice of the rhombicosidodecahedron. The pentagonal cupola consists of 5 equilateral triangles, 5 squares, 1 pentagon, and 1 decagon.
Formulae
The following formulae for volume, surface area and circumradius can be used if all faces are regular, with edge length a:
The height of the pentagonal cupola is
.
Related polyhedra
Dual polyhedron
The dual of the pentagonal cupola has 10 triangular faces and 5 kite faces:
Other convex cupolae
Crossed pentagrammic cupola
In geometry, the crossed pentagrammic cupola is one of the nonconvex Johnson solid isomorphs, being topologically identical to the convex pentagonal cupola. It can be obtained as a slice of the nonconvex great rhombicosidodecahedron or quasirhombicosidodecahedron, analogously to how the pentagonal cupola may be obtained as a slice of the rhombicosidodecahedron. As in all cupolae, the base polygon has twice as many edges and vertices as the top; in this case the base polygon is a decagram.
It may be seen as a cupola with a retrograde pentagrammic base, so that the squares and triangles connect across the bases in the opposite way to the pentagrammic cuploid, hence intersecting each other more deeply.
References
External links
Prismatoid polyhedra
Johnson solids
|
https://en.wikipedia.org/wiki/Diminished%20rhombicosidodecahedron
|
In geometry, the diminished rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with one pentagonal cupola removed.
Related Johnson solids are:
: parabidiminished rhombicosidodecahedron with two opposing cupolae removed, and
: metabidiminished rhombicosidodecahedron with two non-opposing cupolae removed, and
: tridiminished rhombicosidodecahedron with three cupola removed.
External links
Editable printable net of a diminished rhombicosidodecahedron with interactive 3D view
Johnson solids
|
https://en.wikipedia.org/wiki/Gyrate%20rhombicosidodecahedron
|
In geometry, the gyrate rhombicosidodecahedron is one of the Johnson solids (). It is also a canonical polyhedron.
Related polyhedron
It can be constructed as a rhombicosidodecahedron with one pentagonal cupola rotated through 36 degrees. They have the same faces around each vertex, but vertex configurations along the rotation become a different order, .
Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are:
The parabigyrate rhombicosidodecahedron () where two opposing cupolae are rotated;
The metabigyrate rhombicosidodecahedron () where two non-opposing cupolae are rotated;
And the trigyrate rhombicosidodecahedron () where three cupolae are rotated.
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Dedekind%20zeta%20function
|
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the field of rational numbers Q). It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζK(s) = 0 and 0 < Re(s) < 1, then Re(s) = 1/2.
The Dedekind zeta function is named for Richard Dedekind who introduced it in his supplement to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie.
Definition and basic properties
Let K be an algebraic number field. Its Dedekind zeta function is first defined for complex numbers s with real part Re(s) > 1 by the Dirichlet series
where I ranges through the non-zero ideals of the ring of integers OK of K and NK/Q(I) denotes the absolute norm of I (which is equal to both the index [OK : I] of I in OK or equivalently the cardinality of quotient ring OK / I). This sum converges absolutely for all complex numbers s with real part Re(s) > 1. In the case K = Q, this definition reduces to that of the Riemann zeta function.
Euler product
The Dedekind zeta function of has an Euler product which is a product over all the non-zero prime ideals of
This is the expression in analytic terms of the uniqueness of prime factorization of ideals in . For is non-zero.
Analytic continuation and functional equation
Erich Hecke first proved that ζK(s) has an analytic continuation to the complex plane as a meromorphic function, having a simple pole only at s = 1. The residue at that pole is given by the analytic class number formula and is made up of important arithmetic data involving invariants of the unit group and class group of K.
The Dedekind zeta function satisfies a functional equation relating its values at s and 1 − s. Specifically, let ΔK denote the discriminant of K, let r1 (resp. r2) denote the number of real places (resp. complex places) of K, and let
and
where Γ(s) is the gamma function. Then, the functions
satisfy the functional equation
Special values
Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode (at least conjecturally) important arithmetic data of the field K. For example, the analytic class number formula relates the residue at s = 1 to the class number h(K) of K, the regulator R(K) of K, the number w(K) of roots of unity in K, the absolute discriminant of K, and the number of real and complex places of K. Another example is at s = 0 where it has a zero whose order r is equal to the rank of the unit group of OK and the leading term is given by
It follows from the functional equation that .
Combining the functional equation and the fact that Γ(s) is infinite at all integers less than or equal
|
https://en.wikipedia.org/wiki/Weil%20pairing
|
In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual. It was introduced by André Weil (1940) for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function.
Formulation
Choose an elliptic curve E defined over a field K, and an integer n > 0 (we require n to be coprime to char(K) if char(K) > 0) such that K contains a primitive nth root of unity. Then the n-torsion on is known to be a Cartesian product of two cyclic groups of order n. The Weil pairing produces an n-th root of unity
by means of Kummer theory, for any two points , where and .
A down-to-earth construction of the Weil pairing is as follows. Choose a function F in the function field of E over the algebraic closure of K with divisor
So F has a simple zero at each point P + kQ, and a simple pole at each point kQ if these points are all distinct. Then F is well-defined up to multiplication by a constant. If G is the translation of F by Q, then by construction G has the same divisor, so the function G/F is constant.
Therefore if we define
we shall have an n-th root of unity (as translating n times must give 1) other than 1. With this definition it can be shown that w is alternating and bilinear, giving rise to a non-degenerate pairing on the n-torsion.
The Weil pairing does not extend to a pairing on all the torsion points (the direct limit of n-torsion points) because the pairings for different n are not the same. However
they do fit together to give a pairing Tℓ(E) × Tℓ(E) → Tℓ(μ) on the Tate module Tℓ(E) of the elliptic curve E (the inverse limit of the ℓn-torsion points) to the Tate module Tℓ(μ) of the multiplicative group (the inverse limit of ℓn roots of unity).
Generalisation to abelian varieties
For abelian varieties over an algebraically closed field K, the Weil pairing is a nondegenerate pairing
for all n prime to the characteristic of K. Here denotes the dual abelian variety of A. This is the so-called Weil pairing for higher dimensions. If A is equipped with a polarisation
,
then composition gives a (possibly degenerate) pairing
If C is a projective, nonsingular curve of genus ≥ 0 over k, and J its Jacobian, then the theta-divisor of J induces a principal polarisation of J, which in this particular case happens to be an isomorphism (see autoduality of Jacobians). Hence, composing the Weil pairing for J with the polarisation gives a nondegenerate pairing
for all n prime to the characteristic of k.
As in the case of elliptic curves, explicit formulae for this pairing can be given in terms of divisors of C.
Applications
The pairing is used in number theory and algebraic geometry, a
|
https://en.wikipedia.org/wiki/Parabidiminished%20rhombicosidodecahedron
|
In geometry, the parabidiminished rhombicosidodecahedron is one of the Johnson solids (). It is also a canonical polyhedron.
It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae removed. Related Johnson solids are the diminished rhombicosidodecahedron () where one cupola is removed, the metabidiminished rhombicosidodecahedron () where two non-opposing cupolae are removed, and the tridiminished rhombicosidodecahedron () where three cupolae are removed.
Example
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Metabidiminished%20rhombicosidodecahedron
|
In geometry, the metabidiminished rhombicosidodecahedron is one of the Johnson solids ().
It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae () removed.
Related Johnson solids are:
The diminished rhombicosidodecahedron () where one cupola is removed,
The parabidiminished rhombicosidodecahedron () where two opposing cupolae are removed,
The gyrate bidiminished rhombicosidodecahedron () where two non-opposing cupolae are removed and a third is rotated 36 degrees,
And the tridiminished rhombicosidodecahedron () where three cupolae are removed.
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Tridiminished%20rhombicosidodecahedron
|
In geometry, the tridiminished rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with three pentagonal cupolae removed.
Related Johnson solids are:
: diminished rhombicosidodecahedron with one cupola removed,
: parabidiminished rhombicosidodecahedron with two opposing cupolae removed, and
: metabidiminished rhombicosidodecahedron with two non-opposing cupolae removed.
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Trigyrate%20rhombicosidodecahedron
|
In geometry, the trigyrate rhombicosidodecahedron is one of the Johnson solids (). It contains 20 triangles, 30 squares and 12 pentagons. It is also a canonical polyhedron.
It can be constructed as a rhombicosidodecahedron with three pentagonal cupolae rotated through 36 degrees. Related Johnson solids are:
The gyrate rhombicosidodecahedron () where one cupola is rotated;
The parabigyrate rhombicosidodecahedron () where two opposing cupolae are rotated;
And the metabigyrate rhombicosidodecahedron () where two non-opposing cupolae are rotated.
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
|
https://en.wikipedia.org/wiki/Snub%20disphenoid
|
In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some vertices have four faces and others have five. It is a dodecahedron, one of the eight deltahedra (convex polyhedra with equilateral triangle faces), and is the 84th Johnson solid (non-uniform convex polyhedra with regular faces). It can be thought of as a square antiprism where both squares are replaced with two equilateral triangles.
The snub disphenoid is also the vertex figure of the isogonal 13-5 step prism, a polychoron constructed from a 13-13 duoprism by selecting a vertex on a tridecagon, then selecting the 5th vertex on the next tridecagon, doing so until reaching the original tridecagon. It cannot be made uniform, however, because the snub disphenoid has no circumscribed sphere.
History and naming
This shape was called a Siamese dodecahedron in the paper by Hans Freudenthal and B. L. van der Waerden (1947) which first described the set of eight convex deltahedra. The dodecadeltahedron name was given to the same shape by , referring to the fact that it is a 12-sided deltahedron. There are other simplicial dodecahedra, such as the hexagonal bipyramid, but this is the only one that can be realized with equilateral faces. Bernal was interested in the shapes of holes left in irregular close-packed arrangements of spheres, so he used a restrictive definition of deltahedra, in which a deltahedron is a convex polyhedron with triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no room to pack another sphere inside the cage created by this system of spheres. This restrictive definition disallows the triangular bipyramid (as forming two tetrahedral holes rather than a single hole), pentagonal bipyramid (because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and icosahedron (because it has interior room for another sphere). Bernal writes that the snub disphenoid is "a very common coordination for the calcium ion in crystallography". In coordination geometry, it is usually known as the trigonal dodecahedron or simply as the dodecahedron.
The snub disphenoid name comes from Norman Johnson's 1966 classification of the Johnson solids, convex polyhedra all of whose faces are regular. It exists first in a series of polyhedra with axial symmetry, so also can be given the name digonal gyrobianticupola.
Properties
The snub disphenoid is 4-connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the pentagonal bipy
|
https://en.wikipedia.org/wiki/Education%20in%20Taiwan
|
The educational system in Taiwan is the responsibility of the Ministry of Education. The system produces pupils with some of the highest test scores in the world, especially in mathematics and science. Former president Ma Ying-jeou announced in January 2011 that the government would begin the phased implementation of a twelve-year compulsory education program by 2014.
In 2015, Taiwanese students achieved one of the world's best results in mathematics, science and literacy, as tested by the Programme for International Student Assessment (PISA), a worldwide evaluation of 15-year-old school pupils' scholastic performance. Taiwan is one of the top-performing OECD countries in reading literacy, mathematics and sciences with the average student scoring 523.7, compared with the OECD average of 493, placing it seventh in the world and has one of the world's most highly educated labor forces among OECD countries. Although current law mandates only nine years of schooling, 95 percent junior high school students go on to a senior vocational high school, trade school, junior college, or university.
In Taiwan, adhering to the Confucian paradigm for education where parents believe that receiving a good education is a very high priority for Taiwanese families and an important goal in their children's life. Many parents in Taiwan believe that effort and persistence matters more than innate ability if their children want to receive better grades in school. These beliefs are shared by the teachers and guidance counselors and the schools as they regularly keep the parents abreast on their child's overall academic performance in the school. Many parents have high expectations for their children, emphasize academic achievement and actively intervene in their children's academic progress by making sure that their children receive top grades and would go on to great sacrifices including borrowing money to put their child through university.
Due to its role in promoting Taiwan's economic development, high test results, and high university entrance rate, Taiwan's education system has been praised. 45 percent of Taiwanese aged 25 to 64 hold a bachelor's degree or higher. Furthermore, the education system has been criticized for its overemphasis on rote memorization and excessive academic pressure it places on students. Students in Taiwan are faced with immense pressure to succeed academically from their parents, teachers, peers, and society in order to secure prestigious white collar job positions while eschewing vocational education, critical thinking, and creativity. With a narrow bandwidth of prestigious job positions and a far greater number of university graduates seeking them, many have been employed in lesser positions with salaries far below their expectations. Taiwan's universities have also been criticized for not keeping up with the technological trends and employment demands in its fast moving job market referring to a skills mismatch cited by a number of
|
https://en.wikipedia.org/wiki/Snub%20square%20antiprism
|
In geometry, the snub square antiprism is one of the Johnson solids ().
It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids, although it is a relative of the icosahedron that has fourfold symmetry instead of threefold.
Construction
The snub square antiprism is constructed as its name suggests, a square antiprism which is snubbed, and represented as ss{2,8}, with s{2,8} as a square antiprism. It can be constructed in Conway polyhedron notation as sY4 (snub square pyramid).
It can also be constructed as a square gyrobianticupolae, connecting two anticupolae with gyrated orientations.
Cartesian coordinates
Let k ≈ 0.82354 be the positive root of the cubic polynomial
Furthermore, let h ≈ 1.35374 be defined by
Then, Cartesian coordinates of a snub square antiprism with edge length 2 are given by the union of the orbits of the points
under the action of the group generated by a rotation around the z-axis by 90° and by a rotation by 180° around a straight line perpendicular to the z-axis and making an angle of 22.5° with the x-axis.
We may then calculate the surface area of a snub square antiprism of edge length a as
and its volume as
where ξ ≈ 3.60122 is the greatest real root of the polynomial
Snub antiprisms
Similarly constructed, the ss{2,6} is a snub triangular antiprism (a lower symmetry octahedron), and result as a regular icosahedron. A snub pentagonal antiprism, ss{2,10}, or higher n-antiprisms can be similar constructed, but not as a convex polyhedron with equilateral triangles. The preceding Johnson solid, the snub disphenoid also fits constructionally as ss{2,4}, but one has to retain two degenerate digonal faces (drawn in red) in the digonal antiprism.
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Arnoldi%20iteration
|
In numerical linear algebra, the Arnoldi iteration is an eigenvalue algorithm and an important example of an iterative method. Arnoldi finds an approximation to the eigenvalues and eigenvectors of general (possibly non-Hermitian) matrices by constructing an orthonormal basis of the Krylov subspace, which makes it particularly useful when dealing with large sparse matrices.
The Arnoldi method belongs to a class of linear algebra algorithms that give a partial result after a small number of iterations, in contrast to so-called direct methods which must complete to give any useful results (see for example, Householder transformation). The partial result in this case being the first few vectors of the basis the algorithm is building.
When applied to Hermitian matrices it reduces to the Lanczos algorithm. The Arnoldi iteration was invented by W. E. Arnoldi in 1951.
Krylov subspaces and the power iteration
An intuitive method for finding the largest (in absolute value) eigenvalue of a given m × m matrix is the power iteration: starting with an arbitrary initial vector b, calculate normalizing the result after every application of the matrix A.
This sequence converges to the eigenvector corresponding to the eigenvalue with the largest absolute value, . However, much potentially useful computation is wasted by using only the final result, . This suggests that instead, we form the so-called Krylov matrix:
The columns of this matrix are not in general orthogonal, but we can extract an orthogonal basis, via a method such as Gram–Schmidt orthogonalization. The resulting set of vectors is thus an orthogonal basis of the Krylov subspace, . We may expect the vectors of this basis to span good approximations of the eigenvectors corresponding to the largest eigenvalues, for the same reason that approximates the dominant eigenvector.
The Arnoldi iteration
The Arnoldi iteration uses the modified Gram–Schmidt process to produce a sequence of orthonormal vectors, q1, q2, q3, ..., called the Arnoldi vectors, such that for every n, the vectors q1, ..., qn span the Krylov subspace . Explicitly, the algorithm is as follows:
Start with an arbitrary vector q1 with norm 1.
Repeat for k = 2, 3, ...
qk := A qk−1
for j from 1 to k − 1
hj,k−1 := qj* qk
qk := qk − hj,k−1 qj
hk,k−1 :=
qk := qk / hk,k−1
The j-loop projects out the component of in the directions of . This ensures the orthogonality of all the generated vectors.
The algorithm breaks down when qk is the zero vector. This happens when the minimal polynomial of A is of degree k. In most applications of the Arnoldi iteration, including the eigenvalue algorithm below and GMRES, the algorithm has converged at this point.
Every step of the k-loop takes one matrix-vector product and approximately 4mk floating point operations.
In the programming language Python with support of the NumPy library:
import numpy as np
def arnoldi_iteration(A, b, n: int):
"""Compute a basis o
|
https://en.wikipedia.org/wiki/Hoeffding%27s%20inequality
|
In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Wassily Hoeffding in 1963.
Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's inequality. It is similar to the Chernoff bound, but tends to be less sharp, in particular when the variance of the random variables is small. It is similar to, but incomparable with, one of Bernstein's inequalities.
Statement
Let be independent random variables such that almost surely. Consider the sum of these random variables,
Then Hoeffding's theorem states that, for all ,
Here is the expected value of .
Note that the inequalities also hold when the have been obtained using sampling without replacement; in this case the random variables are not independent anymore. A proof of this statement can be found in Hoeffding's paper. For slightly better bounds in the case of sampling without replacement, see for instance the paper by .
Example
Suppose and for all i. This can occur when Xi are independent Bernoulli random variables, though they need not be identically distributed. Then we get the inequality
for all . This is a version of the additive Chernoff bound which is more general, since it allows for random variables that take values between zero and one, but also weaker, since the Chernoff bound gives a better tail bound when the random variables have small variance.
General case of bounded from above random variables
Hoeffding's inequality can be extended to the case of bounded from above random variables.
Let be independent random variables such that and almost surely.
Denote by
Hoeffding's inequality for bounded from aboved random variables states that for all ,
In particular, if for all ,
then for all ,
General case of sub-Gaussian random variables
The proof of Hoeffding's inequality can be generalized to any sub-Gaussian distribution. Recall that a random variable is called sub-Gaussian, if
for some c>0. For any bounded variable , for for some sufficiently large . Then for all so taking yields
for . So every bounded variable is sub-Gaussian.
For a random variable , the following norm is finite if and only if is sub-Gaussian:
Then let be zero-mean independent sub-Gaussian random variables, the general version of the Hoeffding's inequality states that:
where c > 0 is an absolute constant.
Proof
The proof of Hoeffding's inequality follows similarly to concentration inequalities like Chernoff bounds. The main difference is the use of Hoeffding's Lemma:
Suppose is a real random variable such that almost surely. Then
Using this lemma, we can prove Hoeffding's inequality. As in the theorem statement, suppose are independent random variables such that almost surely for all i, and let .
Then for , Markov's inequality and the independence
|
https://en.wikipedia.org/wiki/167%20%28number%29
|
167 (one hundred [and] sixty-seven) is the natural number following 166 and preceding 168.
In mathematics
167 is an emirp, an isolated prime, a Chen prime, a Gaussian prime, a safe prime, and an Eisenstein prime with no imaginary part and a real part of the form .
167 is the smallest number which requires six terms when expressed using the greedy algorithm as a sum of squares, 167 = 144 + 16 + 4 + 1 + 1 + 1,
although by Lagrange's four-square theorem its non-greedy expression as a sum of squares can be shorter, e.g. 167 = 121 + 36 + 9 + 1.
167 is a full reptend prime in base 10, since the decimal expansion of 1/167 repeats the following 166 digits: 0.00598802395209580838323353293413173652694610778443113772455089820359281437125748502994 0119760479041916167664670658682634730538922155688622754491017964071856287425149700...
167 is a highly cototient number, as it is the smallest number k with exactly 15 solutions to the equation x - φ(x) = k. It is also a strictly non-palindromic number.
167 is the smallest multi-digit prime such that the product of digits is equal to the number of digits times the sum of the digits, i. e., 1×6×7 = 3×(1+6+7)
167 is the smallest positive integer d such that the imaginary quadratic field Q() has class number = 11.
In astronomy
167 Urda is a main belt asteroid
167P/CINEOS is a periodic comet in the Solar System
IC 167 is interacting galaxies
In the military
Marine Light Attack Helicopter Squadron 167 is a United States Marine Corps helicopter squadron
Martin Model 167 was a U.S.-designed light bomber during World War II
was a U.S. Navy Diver-class rescue and salvage ship during World War II
was a U.S. Navy during World War II
was a U.S. Navy during World War II
was a U.S. Navy during World War I
was a U.S. Navy during World War II
was a transport ship during World War II
was a U.S. Navy during World War II
In sports
Martina Navratilova has 167 tennis titles, an all-time record for men or women
In transportation
London Buses route 167
167th Street is an elevated local station in the Bronx on the IRT Jerome Avenue Line, , of the New York City Subway.
167th Street is an underground local station in the Bronx on the IND Concourse Line, , of the New York City Subway.
List of highways numbered 167
In other fields
167 is also:
The year AD 167 or 167 BC
The Universal Disk Format (or ECMA-167) format of a file system for optical media storage
C167 family is a 16-bit microcontroller architecture from Infineon
Pips are dots on the face of a die, denoting its value. The pip count at the start of a backgammon game is 167
See also
M167 (disambiguation)
List of highways numbered 167
United States Supreme Court cases, Volume 167
United Nations Security Council Resolution 167
External links
Prime curiosities: 167
References
Integers
|
https://en.wikipedia.org/wiki/Noam%20Elkies
|
Noam David Elkies (born August 25, 1966) is a professor of mathematics at Harvard University. At the age of 26, he became the youngest professor to receive tenure at Harvard. He is also a pianist, chess national master and a chess composer.
Early life
Elkies was born to an engineer father and a piano teacher mother. He attended Stuyvesant High School in New York City for three years before graduating in 1982 at age 15. A child prodigy, in 1981, at age 14, Elkies was awarded a gold medal at the 22nd International Mathematical Olympiad, receiving a perfect score of 42, one of the youngest to ever do so. He went on to Columbia University, where he won the Putnam competition at the age of sixteen years and four months, making him one of the youngest Putnam Fellows in history. Elkies was a Putnam Fellow twice more during his undergraduate years. He graduated valedictorian of his class in 1985. He then earned his PhD in 1987 under the supervision of Benedict Gross and Barry Mazur at Harvard University.
From 1987 to 1990, Elkies was a junior fellow of the Harvard Society of Fellows.
Work in mathematics
In 1987, Elkies proved that an elliptic curve over the rational numbers is supersingular at infinitely many primes. In 1988, he found a counterexample to Euler's sum of powers conjecture for fourth powers. His work on these and other problems won him recognition and a position as an associate professor at Harvard in 1990. In 1993, Elkies was made a full, tenured professor at the age of 26. This made him the youngest full professor in the history of Harvard. Along with A. O. L. Atkin he extended Schoof's algorithm to create the Schoof–Elkies–Atkin algorithm.
Elkies also studies the connections between music and mathematics; he is on the advisory board of the Journal of Mathematics and Music. He has discovered many new patterns in Conway's Game of Life and has studied the mathematics of still life patterns in that cellular automaton rule. Elkies is an associate of Harvard's Lowell House.
Elkies is one of the principal investigators of the Simons Collaboration on Arithmetic Geometry, Number Theory, and Computation, a large multi-university collaboration involving Boston University, Brown, Dartmouth, Harvard, and MIT.
Elkies is the discoverer (or joint-discoverer) of many current and past record-holding elliptic curves, including the curve with the highest-known lower bound (≥28) on its rank, and the curve with the highest-known exact rank (=20).
Music
Elkies is a bass-baritone and plays the piano for the Harvard Glee Club. Jameson N. Marvin, former director of the Glee Club, compared him to "a Bach or a Mozart," citing "[h]is gifted musicality, superior musicianship and sight-reading ability."
Chess
Elkies is a composer and solver of chess problems (winning the 1996 World Chess Solving Championship). One of his problems is used by the chess trainer Mark Dvoretsky in his book "Dvoretsky's Endgame Manual". Elkies holds the title of National Master fro
|
https://en.wikipedia.org/wiki/Spherical%20cap
|
In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center of the sphere (forming a great circle), so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere.
Volume and surface area
The volume of the spherical cap and the area of the curved surface may be calculated using combinations of
The radius of the sphere
The radius of the base of the cap
The height of the cap
The polar angle between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the disk forming the base of the cap
If denotes the latitude in geographic coordinates, then , and .
The relationship between and is relevant as long as . For example, the red section of the illustration is also a spherical cap for which .
The formulas using and can be rewritten to use the radius of the base of the cap instead of , using the Pythagorean theorem:
so that
Substituting this into the formulas gives:
Deriving the surface area intuitively from the spherical sector volume
Note that aside from the calculus based argument below, the area of the spherical cap may be derived from the volume of the spherical sector, by an intuitive argument, as
The intuitive argument is based upon summing the total sector volume from that of infinitesimal triangular pyramids. Utilizing the pyramid (or cone) volume formula of , where is the infinitesimal area of each pyramidal base (located on the surface of the sphere) and is the height of each pyramid from its base to its apex (at the center of the sphere). Since each , in the limit, is constant and equivalent to the radius of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and:
Deriving the volume and surface area using calculus
The volume and area formulas may be derived by examining the rotation of the function
for , using the formulas the surface of the rotation for the area and the solid of the revolution for the volume.
The area is
The derivative of is
and hence
The formula for the area is therefore
The volume is
Applications
Volumes of union and intersection of two intersecting spheres
The volume of the union of two intersecting spheres
of radii and is
where
is the sum of the volumes of the two isolated spheres, and
the sum of the volumes of the two spherical caps forming their intersection. If is the
distance between the two sphere centers, elimination of the variables and leads
to
Volume of a spherical cap with a curved base
The volume of a spherical cap with a curved base can be calculated by considering two spheres with radii and , separated by some distance , and for which their surfaces intersect at . That is, the curvature of the base comes from sphere 2. The volume is thus the difference between sphere 2
|
https://en.wikipedia.org/wiki/Generalized%20eigenvector
|
In linear algebra, a generalized eigenvector of an matrix is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.
Let be an -dimensional vector space and let be the matrix representation of a linear map from to with respect to some ordered basis.
There may not always exist a full set of linearly independent eigenvectors of that form a complete basis for . That is, the matrix may not be diagonalizable. This happens when the algebraic multiplicity of at least one eigenvalue is greater than its geometric multiplicity (the nullity of the matrix , or the dimension of its nullspace). In this case, is called a defective eigenvalue and is called a defective matrix.
A generalized eigenvector corresponding to , together with the matrix generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace of .
Using generalized eigenvectors, a set of linearly independent eigenvectors of can be extended, if necessary, to a complete basis for . This basis can be used to determine an "almost diagonal matrix" in Jordan normal form, similar to , which is useful in computing certain matrix functions of . The matrix is also useful in solving the system of linear differential equations where need not be diagonalizable.
The dimension of the generalized eigenspace corresponding to a given eigenvalue is the algebraic multiplicity of .
Overview and definition
There are several equivalent ways to define an ordinary eigenvector. For our purposes, an eigenvector associated with an eigenvalue of an × matrix is a nonzero vector for which , where is the × identity matrix and is the zero vector of length . That is, is in the kernel of the transformation . If has linearly independent eigenvectors, then is similar to a diagonal matrix . That is, there exists an invertible matrix such that is diagonalizable through the similarity transformation . The matrix is called a spectral matrix for . The matrix is called a modal matrix for . Diagonalizable matrices are of particular interest since matrix functions of them can be computed easily.
On the other hand, if does not have linearly independent eigenvectors associated with it, then is not diagonalizable.
Definition: A vector is a generalized eigenvector of rank m of the matrix and corresponding to the eigenvalue if
but
Clearly, a generalized eigenvector of rank 1 is an ordinary eigenvector. Every × matrix has linearly independent generalized eigenvectors associated with it and can be shown to be similar to an "almost diagonal" matrix in Jordan normal form. That is, there exists an invertible matrix such that . The matrix in this case is called a generalized modal matrix for . If is an eigenvalue of algebraic multiplicity , then will have linearly independent generalized eigenvectors corresponding to . These results, in turn, provide a straightforwa
|
https://en.wikipedia.org/wiki/Bachelor%20of%20Mathematics
|
A Bachelor of Mathematics (abbreviated B.Math or BMath) is an undergraduate academic degree awarded for successfully completing a program of study in mathematics or related disciplines, such as applied mathematics, actuarial science, computational science, data analytics, financial mathematics, mathematical physics, pure mathematics, operations research or statistics. The Bachelor of Mathematics caters to high-achieving students seeking to develop a comprehensive specialised knowledge in a field of mathematics or a high level of sophistication in the applications of mathematics.
In practice, this is essentially equivalent to a Bachelor of Science or Bachelor of Arts degree with a speciality in mathematics. Relatively few institutions award Bachelor of Mathematics degrees, and the distinction between those that do and those that award B.Sc or B.A. degrees for mathematics is usually bureaucratic, rather than curriculum related.
List of institutions awarding Bachelor of Mathematics degrees
Australia
Flinders University, Adelaide, South Australia
Queensland University of Technology, Brisbane, Queensland
The Australian National University, Canberra, Australian Capital Territory (a Bachelor of Mathematical Sciences BMASC)
University of Adelaide, Adelaide, South Australia (a Bachelor of Mathematical Sciences BMathSc or Bachelor of Mathematical and Computer Sciences BMath&CompSc)
University of Newcastle, Newcastle, New South Wales
University of Western Sydney - Penrith, Parramatta, Cambelltown campuses in NSW.
Macquarie University, North Ryde, NSW.
University of Queensland, Brisbane, Queensland
University of South Australia, Adelaide, South Australia (a Bachelor of Mathematical Sciences BMathSc)
University of Wollongong, Wollongong, New South Wales
Bangladesh
University of Dhaka, Dhaka, Bangladesh
Jagannath University, Dhaka, Bangladesh
University of Chittagong, Chittagong, Bangladesh
Noakhali University of Science and Technology, Noakhali, Bangladesh
Canada
Carleton University, Ottawa, Ontario, Canada
University of Waterloo, Waterloo, Ontario, Canada (BMath or BCS - Computer Science)
University of Windsor, Windsor, Ontario, Canada
India
Indian Statistical Institute, Bangalore, India
Netherlands
Vrije Universiteit, Amsterdam, Netherlands (degree programme website)
Russia
Tomsk State University, Tomsk, Russia
Voronezh State University, Voronezh, Russia
Novosibirsk State University, Novosibirsk, Russia
United States
Black Hills State University
Philippines
Polytechnic University of the Philippines - Taguig Campus
University of the Philippines Los Baños
Ateneo de Manila University
Ghana
Kwame Nkrumah University of Science and Technology
University of Mines and Technology
South Africa
University of Johannesburg
University of Witwatersrand
University of Cape Town
Duration
A BMath program generally lasts three years with a fourth "honours" year in Australia and University of Waterloo (Canada). The BMath program at Carle
|
https://en.wikipedia.org/wiki/Poul%20Heegaard
|
Poul Heegaard (; November 2, 1871, Copenhagen - February 7, 1948, Oslo) was a Danish mathematician active in the field of topology. His 1898 thesis introduced a concept now called the Heegaard splitting of a 3-manifold. Heegaard's ideas allowed him to make a careful critique of work of Henri Poincaré. Poincaré had overlooked the possibility of the appearance of torsion in the homology groups of a space.
He later co-authored, with Max Dehn, a foundational article on combinatorial topology, in the form of an encyclopedia entry.
Heegaard studied mathematics at the University of Copenhagen, from 1889 to 1893 and following years of travelling, and teaching mathematics, he was appointed professor at University of Copenhagen in 1910. An English translation of his 1898 thesis, which laid a rigorous topological foundation for modern knot theory, may be found at https://www.maths.ed.ac.uk/~v1ranick/papers/heegaardenglish.pdf. The section on "a visually transparent representation of the complex points of an algebraic surface" is especially important.
Following a dispute with the faculty over, among other things, the hiring of Harald Bohr as professor at the University (which Heegaard opposed); Heegaard accepted a professorship at Oslo in Norway, where he worked till his retirement in 1941.
Notes
External links
"Heegaard home page"
1871 births
1948 deaths
Danish mathematicians
Topologists
Presidents of the Norwegian Mathematical Society
|
https://en.wikipedia.org/wiki/Complex%20multiplication%20of%20abelian%20varieties
|
In mathematics, an abelian variety A defined over a field K is said to have CM-type if it has a large enough commutative subring in its endomorphism ring End(A). The terminology here is from complex multiplication theory, which was developed for elliptic curves in the nineteenth century. One of the major achievements in algebraic number theory and algebraic geometry of the twentieth century was to find the correct formulations of the corresponding theory for abelian varieties of dimension d > 1. The problem is at a deeper level of abstraction, because it is much harder to manipulate analytic functions of several complex variables.
The formal definition is that
the tensor product of End(A) with the rational number field Q, should contain a commutative subring of dimension 2d over Q. When d = 1 this can only be a quadratic field, and one recovers the cases where End(A) is an order in an imaginary quadratic field. For d > 1 there are comparable cases for CM-fields, the complex quadratic extensions of totally real fields. There are other cases that reflect that A may not be a simple abelian variety (it might be a cartesian product of elliptic curves, for example). Another name for abelian varieties of CM-type is abelian varieties with sufficiently many complex multiplications.
It is known that if K is the complex numbers, then any such A has a field of definition which is in fact a number field. The possible types of endomorphism ring have been classified, as rings with involution (the Rosati involution), leading to a classification of CM-type abelian varieties. To construct such varieties in the same style as for elliptic curves, starting with a lattice Λ in Cd, one must take into account the Riemann relations of abelian variety theory.
The CM-type is a description of the action of a (maximal) commutative subring L of EndQ(A) on the holomorphic tangent space of A at the identity element. Spectral theory of a simple kind applies, to show that L acts via a basis of eigenvectors; in other words L has an action that is via diagonal matrices on the holomorphic vector fields on A. In the simple case, where L is itself a number field rather than a product of some number of fields, the CM-type is then a list of complex embeddings of L. There are 2d of those, occurring in complex conjugate pairs; the CM-type is a choice of one out of each pair. It is known that all such possible CM-types can be realised.
Basic results of Goro Shimura and Yutaka Taniyama compute the Hasse–Weil L-function of A, in terms of the CM-type and a Hecke L-function with Hecke character, having infinity-type derived from it. These generalise the results of Max Deuring for the elliptic curve case.
References
Abelian varieties
Arithmetic geometry
|
https://en.wikipedia.org/wiki/512%20%28number%29
|
512 (five hundred [and] twelve) is the natural number following 511 and preceding 513.
In mathematics
512 is a power of two: 29 (2 to the 9th power) and the cube of 8: 83.
It is the eleventh Leyland number.
It is also the third Dudeney number.
It is a self number in base 12.
It is a harshad number in decimal.
It is the cube of the sum of its digits in base 10.
It is the number of directed graphs on 3 labeled nodes.
In computing
512 bytes is a common disk sector size, and exactly a half of kibibyte.
Internet Relay Chat restricts the size of a message to 510 bytes, which fits to 512-bytes buffers when coupled with the message-separating CRLF sequence.
512 = 2·256 is the highest number of glyphs that the VGA character generator can use simultaneously.
In music
Selena Quintanilla released a song titled El Chico del Apartamento 512 (the title referring to area code 512, which serves Austin, Texas), in 1995.
Lamb of God recorded a song titled "512" for their 2015 album VII: Sturm und Drang.
Mora and Jhay Cortez recorded a song titled "512" (The number 512 in this song refers to the Percocet 512 pill, a white, round pill whose active substances are acetaminophen and oxycodone hydrochloride) in February of 2021.
References
Integers
|
https://en.wikipedia.org/wiki/Abelian%20integral
|
In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form
where is an arbitrary rational function of the two variables and , which are related by the equation
where is an irreducible polynomial in ,
whose coefficients , are rational functions of . The value of an abelian integral depends not only on the integration limits, but also on the path along which the integral is taken; it is thus a multivalued function of .
Abelian integrals are natural generalizations of elliptic integrals, which arise when
where is a polynomial of degree 3 or 4. Another special case of an abelian integral is a hyperelliptic integral, where , in the formula above, is a polynomial of degree greater than 4.
History
The theory of abelian integrals originated with a paper by Abel published in 1841. This paper was written during his stay in Paris in 1826 and presented to Augustin-Louis Cauchy in October of the same year. This theory, later fully developed by others, was one of the crowning achievements of nineteenth century mathematics and has had a major impact on the development of modern mathematics. In more abstract and geometric language, it is contained in the concept of abelian variety, or more precisely in the way an algebraic curve can be mapped into abelian varieties. Abelian integrals were later connected to the prominent mathematician David Hilbert's 16th Problem, and they continue to be considered one of the foremost challenges in contemporary mathematics.
Modern view
In the theory of Riemann surfaces, an abelian integral is a function related to the indefinite integral of a differential of the first kind. Suppose we are given a Riemann surface and on it a differential 1-form that is everywhere holomorphic on , and fix a point on , from which to integrate. We can regard
as a multi-valued function , or (better) an honest function of the chosen path drawn on from to . Since will in general be multiply connected, one should specify , but the value will in fact only depend on the homology class of .
In the case of a compact Riemann surface of genus 1, i.e. an elliptic curve, such functions are the elliptic integrals. Logically speaking, therefore, an abelian integral should be a function such as .
Such functions were first introduced to study hyperelliptic integrals, i.e., for the case where is a hyperelliptic curve. This is a natural step in the theory of integration to the case of integrals involving algebraic functions , where is a polynomial of degree . The first major insights of the theory were given by Abel; it was later formulated in terms of the Jacobian variety . Choice of gives rise to a standard holomorphic function
of complex manifolds. It has the defining property that the holomorphic 1-forms on , of which there are g independent ones if g is the genus of S, pull back to a basis for the differentials of the first kind on S.
Notes
Re
|
https://en.wikipedia.org/wiki/Differential%20of%20the%20first%20kind
|
In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic varieties), for everywhere-regular differential 1-forms. Given a complex manifold M, a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere holomorphic; on an algebraic variety V that is non-singular it would be a global section of the coherent sheaf Ω1 of Kähler differentials. In either case the definition has its origins in the theory of abelian integrals.
The dimension of the space of differentials of the first kind, by means of this identification, is the Hodge number
h1,0.
The differentials of the first kind, when integrated along paths, give rise to integrals that generalise the elliptic integrals to all curves over the complex numbers. They include for example the hyperelliptic integrals of type
where Q is a square-free polynomial of any given degree > 4. The allowable power k has to be determined by analysis of the possible pole at the point at infinity on the corresponding hyperelliptic curve. When this is done, one finds that the condition is
k ≤ g − 1,
or in other words, k at most 1 for degree of Q 5 or 6, at most 2 for degree 7 or 8, and so on (as g = [(1+ deg Q)/2]).
Quite generally, as this example illustrates, for a compact Riemann surface or algebraic curve, the Hodge number is the genus g. For the case of algebraic surfaces, this is the quantity known classically as the irregularity q. It is also, in general, the dimension of the Albanese variety, which takes the place of the Jacobian variety.
Differentials of the second and third kind
The traditional terminology also included differentials of the second kind and of the third kind. The idea behind this has been supported by modern theories of algebraic differential forms, both from the side of more Hodge theory, and through the use of morphisms to commutative algebraic groups.
The Weierstrass zeta function was called an integral of the second kind in elliptic function theory; it is a logarithmic derivative of a theta function, and therefore has simple poles, with integer residues. The decomposition of a (meromorphic) elliptic function into pieces of 'three kinds' parallels the representation as (i) a constant, plus (ii) a linear combination of translates of the Weierstrass zeta function, plus (iii) a function with arbitrary poles but no residues at them.
The same type of decomposition exists in general, mutatis mutandis, though the terminology is not completely consistent. In the algebraic group (generalized Jacobian) theory the three kinds are abelian varieties, algebraic tori, and affine spaces, and the decomposition is in terms of a composition series.
On the other hand, a meromorphic abelian differential of the second kind has traditionally been one with residues at all poles being zero. One of the third kind is one where all poles
|
https://en.wikipedia.org/wiki/Point%20group
|
In geometry, a point group is a mathematical group of symmetry operations (isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and every point group in dimension d is then a subgroup of the orthogonal group O(d). Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules.
Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to y = Mx. Each element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper rotation (determinant of M = −1).
The geometric symmetries of crystals are described by space groups, which allow translations and contain point groups as subgroups. Discrete point groups in more than one dimension come in infinite families, but from the crystallographic restriction theorem and one of Bieberbach's theorems, each number of dimensions has only a finite number of point groups that are symmetric over some lattice or grid with that number of dimensions. These are the crystallographic point groups.
Chiral and achiral point groups, reflection groups
Point groups can be classified into chiral (or purely rotational) groups and achiral groups.
The chiral groups are subgroups of the special orthogonal group SO(d): they contain only orientation-preserving orthogonal transformations, i.e., those of determinant +1. The achiral groups contain also transformations of determinant −1. In an achiral group, the orientation-preserving transformations form a (chiral) subgroup of index 2.
Finite Coxeter groups or reflection groups are those point groups that are generated purely by a set of reflectional mirrors passing through the same point. A rank n Coxeter group has n mirrors and is represented by a Coxeter-Dynkin diagram. Coxeter notation offers a bracketed notation equivalent to the Coxeter diagram, with markup symbols for rotational and other subsymmetry point groups. Reflection groups are necessarily achiral (except for the trivial group containing only the identity element).
List of point groups
One dimension
There are only two one-dimensional point groups, the identity group and the reflection group.
Two dimensions
Point groups in two dimensions, sometimes called rosette groups.
They come in two infinite families:
Cyclic groups Cn of n-fold rotation groups
Dihedral groups Dn of n-fold rotation and reflection groups
Applying the crystallographic restriction theorem restricts n to values 1, 2, 3, 4, and 6 for both families, yielding 10 groups.
The subset of pure reflectional point groups, defined by 1 or 2 mirrors, can also be given by their Coxeter group and related polygons. These include 5 crystallographic groups. The symmetry of the reflectional groups can be doubled by an isomorphism, mapping both mirrors onto each other by a bisecting mirror, doubling the symmetry order.
Thr
|
https://en.wikipedia.org/wiki/Pyotr%20Novikov
|
Pyotr Sergeyevich Novikov (; 28 August 1901, Moscow – 9 January 1975, Moscow) was a Soviet mathematician.
Novikov is known for his work on combinatorial problems in group theory: the word problem for groups, and the Burnside problem. For proving the undecidability of the word problem in groups he was awarded the Lenin Prize in 1957.
In 1953 he became a corresponding member of the Academy of Sciences of the Soviet Union and in 1960 he was elected a full member.
He was married to the mathematician Lyudmila Keldysh (1904–1976). The mathematician Sergei Novikov is his son. Sergei Adian and Albert Muchnik were among his students.
Awards and honors
Lenin Prize (1957)
Two Orders of Lenin (1961, 1971)
Order of the Red Banner of Labour
State Prize of the Russian Federation (1999, posthumous)
See also
Novikov–Boone theorem
References
External links
1901 births
1975 deaths
20th-century Russian mathematicians
Mathematicians from Moscow
Academic staff of the D. Mendeleev University of Chemical Technology of Russia
Academic staff of Moscow State Pedagogical University
Full Members of the USSR Academy of Sciences
Moscow State University alumni
Recipients of the Lenin Prize
Recipients of the Order of Lenin
Recipients of the Order of the Red Banner of Labour
State Prize of the Russian Federation laureates
Group theorists
Russian mathematicians
Soviet mathematicians
Burials at Novodevichy Cemetery
Russian scientists
|
https://en.wikipedia.org/wiki/Bruhat%20decomposition
|
In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) G = BWB of certain algebraic groups G into cells can be regarded as a general expression of the principle of Gauss–Jordan elimination, which generically writes a matrix as a product of an upper triangular and lower triangular matrices—but with exceptional cases. It is related to the Schubert cell decomposition of flag varieties: see Weyl group for this.
More generally, any group with a (B, N) pair has a Bruhat decomposition.
Definitions
G is a connected, reductive algebraic group over an algebraically closed field.
B is a Borel subgroup of G
W is a Weyl group of G corresponding to a maximal torus of B.
The Bruhat decomposition of G is the decomposition
of G as a disjoint union of double cosets of B parameterized by the elements of the Weyl group W. (Note that although W is not in general a subgroup of G, the coset wB is still well defined because the maximal torus is contained in B.)
Examples
Let G be the general linear group GLn of invertible matrices with entries in some algebraically closed field, which is a reductive group. Then the Weyl group W is isomorphic to the symmetric group Sn on n letters, with permutation matrices as representatives. In this case, we can take B to be the subgroup of upper triangular invertible matrices, so Bruhat decomposition says that one can write any invertible matrix A as a product U1PU2 where U1 and U2 are upper triangular, and P is a permutation matrix. Writing this as P = U1−1AU2−1, this says that any invertible matrix can be transformed into a permutation matrix via a series of row and column operations, where we are only allowed to add row i (resp. column i) to row j (resp. column j) if i > j (resp. i < j). The row operations correspond to U1−1, and the column operations correspond to U2−1.
The special linear group SLn of invertible matrices with determinant 1 is a semisimple group, and hence reductive. In this case, W is still isomorphic to the symmetric group Sn. However, the determinant of a permutation matrix is the sign of the permutation, so to represent an odd permutation in SLn, we can take one of the nonzero elements to be −1 instead of 1. Here B is the subgroup of upper triangular matrices with determinant 1, so the interpretation of Bruhat decomposition in this case is similar to the case of GLn.
Geometry
The cells in the Bruhat decomposition correspond to the Schubert cell decomposition of flag varieties. The dimension of the cells corresponds to the length of the word w in the Weyl group. Poincaré duality constrains the topology of the cell decomposition, and thus the algebra of the Weyl group; for instance, the top dimensional cell is unique (it represents the fundamental class), and corresponds to the longest element of a Coxeter group.
Computations
The number of cells in a given dimension of the Bruhat decomposition are the coefficients of the q-polyn
|
https://en.wikipedia.org/wiki/Cartan%20decomposition
|
In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.
Cartan involutions on Lie algebras
Let be a real semisimple Lie algebra and let be its Killing form. An involution on is a Lie algebra automorphism of whose square is equal to the identity. Such an involution is called a Cartan involution on if is a positive definite bilinear form.
Two involutions and are considered equivalent if they differ only by an inner automorphism.
Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equivalent.
Examples
A Cartan involution on is defined by , where denotes the transpose matrix of .
The identity map on is an involution. It is the unique Cartan involution of if and only if the Killing form of is negative definite or, equivalently, if and only if is the Lie algebra of a compact semisimple Lie group.
Let be the complexification of a real semisimple Lie algebra , then complex conjugation on is an involution on . This is the Cartan involution on if and only if is the Lie algebra of a compact Lie group.
The following maps are involutions of the Lie algebra of the special unitary group SU(n):
The identity involution , which is the unique Cartan involution in this case.
Complex conjugation, expressible as on .
If is odd, . The involutions (1), (2) and (3) are equivalent, but not equivalent to the identity involution since .
If is even, there is also .
Cartan pairs
Let be an involution on a Lie algebra . Since , the linear map has the two eigenvalues . If and denote the eigenspaces corresponding to +1 and -1, respectively, then . Since is a Lie algebra automorphism, the Lie bracket of two of its eigenspaces is contained in the eigenspace corresponding to the product of their eigenvalues. It follows that
, , and .
Thus is a Lie subalgebra, while any subalgebra of is commutative.
Conversely, a decomposition with these extra properties determines an involution on that is on and on .
Such a pair is also called a Cartan pair of ,
and is called a symmetric pair. This notion of a Cartan pair here is not to be confused with the distinct notion involving the relative Lie algebra cohomology .
The decomposition associated to a Cartan involution is called a Cartan decomposition of . The special feature of a Cartan decomposition is that the Killing form is negative definite on and positive definite on . Furthermore, and are orthogonal complements of each other with respect to the Killing form on .
Cartan decomposition on the Lie group level
Let be a non-compact semisimple Lie group and its Lie algebra. Let be a Cartan involution on and let be the resulting Cartan p
|
https://en.wikipedia.org/wiki/Iwasawa%20decomposition
|
In mathematics, the Iwasawa decomposition (aka KAN from its expression) of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix (QR decomposition, a consequence of Gram–Schmidt orthogonalization). It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.
Definition
G is a connected semisimple real Lie group.
is the Lie algebra of G
is the complexification of .
θ is a Cartan involution of
is the corresponding Cartan decomposition
is a maximal abelian subalgebra of
Σ is the set of restricted roots of , corresponding to eigenvalues of acting on .
Σ+ is a choice of positive roots of Σ
is a nilpotent Lie algebra given as the sum of the root spaces of Σ+
K, A, N, are the Lie subgroups of G generated by and .
Then the Iwasawa decomposition of is
and the Iwasawa decomposition of G is
meaning there is an analytic diffeomorphism (but not a group homomorphism) from the manifold to the Lie group , sending .
The dimension of A (or equivalently of ) is equal to the real rank of G.
Iwasawa decompositions also hold for some disconnected semisimple groups G, where K becomes a (disconnected) maximal compact subgroup provided the center of G is finite.
The restricted root space decomposition is
where is the centralizer of in and is the root space. The number
is called the multiplicity of .
Examples
If G=SLn(R), then we can take K to be the orthogonal matrices, A to be the positive diagonal matrices with determinant 1, and N to be the unipotent group consisting of upper triangular matrices with 1s on the diagonal.
For the case of n=2, the Iwasawa decomposition of G=SL(2,R) is in terms of
For the symplectic group G=Sp(2n, R ), a possible Iwasawa decomposition is in terms of
Non-Archimedean Iwasawa decomposition
There is an analog to the above Iwasawa decomposition for a non-Archimedean field : In this case, the group can be written as a product of the subgroup of upper-triangular matrices and the (maximal compact) subgroup , where is the ring of integers of .
See also
Lie group decompositions
Root system of a semi-simple Lie algebra
References
Lie groups
|
https://en.wikipedia.org/wiki/Hyperbolic%20angle
|
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation.
The hyperbola xy = 1 is rectangular with a semi-major axis of , analogous to the magnitude of a circular angle corresponding to the area of a circular sector in a circle with radius .
Hyperbolic angle is used as the independent variable for the hyperbolic functions sinh, cosh, and tanh, because these functions may be premised on hyperbolic analogies to the corresponding circular trigonometric functions by regarding a hyperbolic angle as defining a hyperbolic triangle.
The parameter thus becomes one of the most useful in the calculus of real variables.
Definition
Consider the rectangular hyperbola , and (by convention) pay particular attention to the branch .
First define:
The hyperbolic angle in standard position is the angle at between the ray to and the ray to , where .
The magnitude of this angle is the area of the corresponding hyperbolic sector, which turns out to be .
Note that, because of the role played by the natural logarithm:
Unlike the circular angle, the hyperbolic angle is unbounded (because is unbounded); this is related to the fact that the harmonic series is unbounded.
The formula for the magnitude of the angle suggests that, for , the hyperbolic angle should be negative. This reflects the fact that, as defined, the angle is directed.
Finally, extend the definition of hyperbolic angle to that subtended by any interval on the hyperbola. Suppose are positive real numbers such that and , so that and are points on the hyperbola and determine an interval on it. Then the squeeze mapping maps the angle to the standard position angle . By the result of Gregoire de Saint-Vincent, the hyperbolic sectors determined by these angles have the same area, which is taken to be the magnitude of the angle. This magnitude is .
Comparison with circular angle
A unit circle has a circular sector with an area half of the circular angle in radians. Analogously, a unit hyperbola has a hyperbolic sector with an area half of the hyperbolic angle.
There is also a projective resolution between circular and hyperbolic cases: both curves are conic sections, and hence are treated as projective ranges in projective geometry. Given an origin point on one of these ranges, other points correspond to angles. The idea of addition of angles, basic to science, corresponds to addition of points on one of these ranges as follows:
Circular angles can be characterised geometrically by the property that if two chords P0P1 and P0P2 subtend angles L1 and L2 at the centre of a circle, their sum is the angle subtended by a chord P0Q, where P0Q is required to be parallel to P1P2.
The same construct
|
https://en.wikipedia.org/wiki/Squeeze%20mapping
|
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping.
For a fixed positive real number , the mapping
is the squeeze mapping with parameter . Since
is a hyperbola, if and , then and the points of the image of the squeeze mapping are on the same hyperbola as is. For this reason it is natural to think of the squeeze mapping as a hyperbolic rotation, as did Émile Borel in 1914, by analogy with circular rotations, which preserve circles.
Logarithm and hyperbolic angle
The squeeze mapping sets the stage for development of the concept of logarithms. The problem of finding the area bounded by a hyperbola (such as is one of quadrature. The solution, found by Grégoire de Saint-Vincent and Alphonse Antonio de Sarasa in 1647, required the natural logarithm function, a new concept. Some insight into logarithms comes through hyperbolic sectors that are permuted by squeeze mappings while preserving their area. The area of a hyperbolic sector is taken as a measure of a hyperbolic angle associated with the sector. The hyperbolic angle concept is quite independent of the ordinary circular angle, but shares a property of invariance with it: whereas circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping. Both circular and hyperbolic angle generate invariant measures but with respect to different transformation groups. The hyperbolic functions, which take hyperbolic angle as argument, perform the role that circular functions play with the circular angle argument.
Group theory
In 1688, long before abstract group theory, the squeeze mapping was described by Euclid Speidell in the terms of the day: "From a Square and an infinite company of Oblongs on a Superficies, each Equal to that square, how a curve is begotten which shall have the same properties or affections of any Hyperbola inscribed within a Right Angled Cone."
If and are positive real numbers, the composition of their squeeze mappings is the squeeze mapping of their product. Therefore, the collection of squeeze mappings forms a one-parameter group isomorphic to the multiplicative group of positive real numbers. An additive view of this group arises from consideration of hyperbolic sectors and their hyperbolic angles.
From the point of view of the classical groups, the group of squeeze mappings is , the identity component of the indefinite orthogonal group of 2×2 real matrices preserving the quadratic form . This is equivalent to preserving the form via the change of basis
and corresponds geometrically to preserving hyperbolae. The perspective of the group of squeeze mappings as hyperbolic rotation is analogous to interpreting the group (the connected component of the definite orthogonal group) preserving quadratic form as being circular rotations.
Note that the "" notation corresponds to the fact that
|
https://en.wikipedia.org/wiki/Member
|
Member may refer to:
Military jury, referred to as "Members" in military jargon
Element (mathematics), an object that belongs to a mathematical set
In object-oriented programming, a member of a class
Field (computer science), entries in a database
Member variable, a variable that is associated with a specific object
Limb (anatomy), an appendage of the human or animal body
Euphemism for penis
Structural component of a truss, connected by nodes
User (computing), a person making use of a computing service, especially on the Internet
Member (geology), a component of a geological formation
Member of parliament
The Members, a British punk rock band
Meronymy, a semantic relationship in linguistics
Church membership, belonging to a local Christian congregation, a Christian denomination and the universal Church
Member, a participant in a club or learned society
See also
|
https://en.wikipedia.org/wiki/Racks%20and%20quandles
|
In mathematics, racks and quandles are sets with binary operations satisfying axioms analogous to the Reidemeister moves used to manipulate knot diagrams.
While mainly used to obtain invariants of knots, they can be viewed as algebraic constructions in their own right. In particular, the definition of a quandle axiomatizes the properties of conjugation in a group.
History
In 1943, Mituhisa Takasaki (高崎光久) introduced an algebraic structure which he called a Kei (圭), which would later come to be known as an involutive quandle. His motivation was to find a nonassociative algebraic structure to capture the notion of a reflection in the context of finite geometry. The idea was rediscovered and generalized in (unpublished) 1959 correspondence between John Conway and Gavin Wraith, who at the time were undergraduate students at the University of Cambridge. It is here that the modern definitions of quandles and of racks first appear. Wraith had become interested in these structures (which he initially dubbed sequentials) while at school. Conway renamed them wracks, partly as a pun on his colleague's name, and partly because they arise as the remnants (or 'wrack and ruin') of a group when one discards the multiplicative structure and considers only the conjugation structure. The spelling 'rack' has now become prevalent.
These constructs surfaced again in the 1980s: in a 1982 paper by David Joyce (where the term quandle, an arbitrary nonsense word, was coined), in a 1982 paper by Sergei Matveev (under the name distributive groupoids) and in a 1986 conference paper by Egbert Brieskorn (where they were called automorphic sets). A detailed overview of racks and their applications in knot theory may be found in the paper by Colin Rourke and Roger Fenn.
Racks
A rack may be defined as a set with a binary operation such that for every the self-distributive law holds:
and for every there exists a unique such that
This definition, while terse and commonly used, is suboptimal for certain purposes because it contains an existential quantifier which is not really necessary. To avoid this, we may write the unique such that as We then have
and thus
and
Using this idea, a rack may be equivalently defined as a set with two binary operations and such that for all
(left self-distributive law)
(right self-distributive law)
It is convenient to say that the element is acting from the left in the expression and acting from the right in the expression The third and fourth rack axioms then say that these left and right actions are inverses of each other. Using this, we can eliminate either one of these actions from the definition of rack. If we eliminate the right action and keep the left one, we obtain the terse definition given initially.
Many different conventions are used in the literature on racks and quandles. For example, many authors prefer to work with just the right action. Furthermore, the use of the symbols and is by no means
|
https://en.wikipedia.org/wiki/Siegel%20upper%20half-space
|
In mathematics, the Siegel upper half-space of degree g (or genus g) (also called the Siegel upper half-plane) is the set of g × g symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by . It is the symmetric space associated to the symplectic group .
The Siegel upper half-space has properties as a complex manifold that generalize the properties of the upper half-plane, which is the Siegel upper half-space in the special case g = 1. The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group . Just as the two-dimensional hyperbolic metric is the unique (up to scaling) metric on the upper half-plane whose isometry group is the complex automorphism group = , the Siegel upper half-space has only one metric up to scaling whose isometry group is . Writing a generic matrix Z in the Siegel upper half-space in terms of its real and imaginary parts as Z = X + iY, all metrics with isometry group are proportional to
The Siegel upper half-plane can be identified with the set of tame almost complex structures compatible with a symplectic structure , on the underlying dimensional real vector space , that is, the set of such that and for all vectors .
See also
Moduli of abelian varieties
Paramodular group, a generalization of the Siegel modular group
Siegel domain, a generalization of the Siegel upper half space
Siegel modular form, a type of automorphic form defined on the Siegel upper half-space
Siegel modular variety, a moduli space constructed as a quotient of the Siegel upper half-space
References
.
Complex analysis
Automorphic forms
Differential geometry
1939 introductions
|
https://en.wikipedia.org/wiki/Hebesphenomegacorona
|
In geometry, the hebesphenomegacorona is one of the Johnson solids (). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids. It has 21 faces, 18 triangles and 3 squares, 33 edges, and 14 vertices.
Johnson uses the prefix hebespheno- to refer to a blunt wedge-like complex formed by three adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. Likewise, the suffix -megacorona refers to a crownlike complex of 12 triangles. Joining both complexes together results in the hebesphenomegacorona.
The icosahedron can be obtained from the hebesphenomegacorona by merging the middle of the three squares into an edge, turning the neighboring two squares into triangles.
Cartesian coordinates
Let a ≈ 0.21684 be the second smallest positive root of the polynomial
Then, Cartesian coordinates of a hebesphenomegacorona with edge length 2 are given by the union of the orbits of the points
under the action of the group generated by reflections about the xz-plane and the yz-plane.
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Sphenomegacorona
|
In geometry, the sphenomegacorona is one of the Johnson solids (). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.
Johnson uses the prefix spheno- to refer to a wedge-like complex formed by two adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. Likewise, the suffix -megacorona refers to a crownlike complex of 12 triangles, contrasted with the smaller triangular complex that makes the sphenocorona. Joining both complexes together results in the sphenomegacorona.
Cartesian coordinates
Let k ≈ 0.59463 be the smallest positive root of the polynomial
Then, Cartesian coordinates of a sphenomegacorona with edge length 2 are given by the union of the orbits of the points
under the action of the group generated by reflections about the xz-plane and the yz-plane.
We may then calculate the surface area of a sphenomegacorona of edge length a as
and its volume as
where the decimal expansion of ξ is given by .
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Sphenocorona
|
In geometry, the sphenocorona is one of the Johnson solids (). It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the Platonic and Archimedean solids.
Johnson uses the prefix spheno- to refer to a wedge-like complex formed by two adjacent lunes, a lune being a square with equilateral triangles attached on opposite sides. Likewise, the suffix -corona refers to a crownlike complex of 8 equilateral triangles. Joining both complexes together results in the sphenocorona.
Cartesian coordinates
Let k ≈ 0.85273 be the smallest positive root of the quartic polynomial
Then, Cartesian coordinates of a sphenocorona with edge length 2 are given by the union of the orbits of the points
under the action of the group generated by reflections about the xz-plane and the yz-plane.
One may then calculate the surface area of a sphenocorona of edge length a as
and its volume as
Variations
The sphenocorona is also the vertex figure of the isogonal n-gonal double antiprismoid where n is an odd number greater than one, including the grand antiprism with pairs of trapezoid rather than square faces.
See also
Augmented sphenocorona
References
External links
Johnson solids
|
https://en.wikipedia.org/wiki/Overdetermined
|
Overdetermined may refer to:
Overdetermined systems in various branches of mathematics
Overdetermination in various fields of psychology or analytical thought
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.