In potential theory and optimization, polarization constants (also known as Chebyshev constants) are solutions to a max-min problem for potentials. Originally, these problems were introduced by a Japanese mathematician Makoto Ohtsuka. [1] Recently these problems got some attention as they can help to generate random points on smooth manifolds (in particular, unit sphere) with prescribed probability density function. The problem of finding the polarization constant is connected to the problem of energy minimization and, in particular to the Thomson problem. [2] [3]
From the practical point of view, these problems can be used to answer the following question: if denotes the amount of a substance received at due to an injector of the substance located at , what is the smallest number of like injectors and their optimal locations on so that a prescribed minimal amount of the substance reaches every point of ? For example, one can relate this question to treating tumors with radioactive seeds.
More precisely, for a compact set and kernel , the discrete polarization problem is the following: determine -point configurations on so that the minimum of for is as large as possible.
The Chebyshev nomenclature for this max-min problem emanates from the case when is the logarithmic kernel, for when is a subset of the complex plane, the problem is equivalent to finding the constrained -th degree Chebyshev polynomial for ; that is, the monic polynomial in the complex variable with all its zeros on having minimal uniform norm on .
If is the unit circle in the plane and , (i.e., kernel of a Riesz potential), then equally spaced points on the circle solve the point polarization problem. [4] [5]
In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions:
In numerical analysis, polynomial interpolation is the interpolation of a given data set by the polynomial of lowest possible degree that passes through the points of the dataset.
Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.
Marcel Riesz was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford algebras. He spent most of his career in Lund (Sweden).
In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure μ to the sequences of moments
In mathematics, the Lebesgue constants give an idea of how good the interpolant of a function is in comparison with the best polynomial approximation of the function. The Lebesgue constant for polynomials of degree at most n and for the set of n + 1 nodes T is generally denoted by Λn(T ). These constants are named after Henri Lebesgue.
In mathematics, Gegenbauer polynomials or ultraspherical polynomialsC(α)
n(x) are orthogonal polynomials on the interval [−1,1] with respect to the weight function (1 − x2)α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials. They are named after Leopold Gegenbauer.
The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of n electrons constrained to the surface of a unit sphere that repel each other with a force given by Coulomb's law. The physicist J. J. Thomson posed the problem in 1904 after proposing an atomic model, later called the plum pudding model, based on his knowledge of the existence of negatively charged electrons within neutrally-charged atoms.
Clenshaw–Curtis quadrature and Fejér quadrature are methods for numerical integration, or "quadrature", that are based on an expansion of the integrand in terms of Chebyshev polynomials. Equivalently, they employ a change of variables and use a discrete cosine transform (DCT) approximation for the cosine series. Besides having fast-converging accuracy comparable to Gaussian quadrature rules, Clenshaw–Curtis quadrature naturally leads to nested quadrature rules, which is important for both adaptive quadrature and multidimensional quadrature (cubature).
In computer science, a kernelization is a technique for designing efficient algorithms that achieve their efficiency by a preprocessing stage in which inputs to the algorithm are replaced by a smaller input, called a "kernel". The result of solving the problem on the kernel should either be the same as on the original input, or it should be easy to transform the output on the kernel to the desired output for the original problem.
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
In the mathematical theory of harmonic analysis, the Riesz transforms are a family of generalizations of the Hilbert transform to Euclidean spaces of dimension d > 1. They are a type of singular integral operator, meaning that they are given by a convolution of one function with another function having a singularity at the origin. Specifically, the Riesz transforms of a complex-valued function ƒ on Rd are defined by
In the mathematical fields of graph theory and combinatorics, a matching polynomial is a generating function of the numbers of matchings of various sizes in a graph. It is one of several graph polynomials studied in algebraic graph theory.
In mathematics, discrete Chebyshev polynomials, or Gram polynomials, are a type of discrete orthogonal polynomials used in approximation theory, introduced by Pafnuty Chebyshev (1864) and rediscovered by Gram (1883). They were later found to be applicable to various algebraic properties of spin angular momentum.
In the mathematical field of graph theory, the intersection number of a graph is the smallest number of elements in a representation of as an intersection graph of finite sets. In such a representation, each vertex is represented as a set, and two vertices are connected by an edge whenever their sets have a common element. Equivalently, the intersection number is the smallest number of cliques needed to cover all of the edges of .
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
In physics, the poppy-seed bagel theorem concerns interacting particles confined to a bounded surface when the particles repel each other pairwise with a magnitude that is proportional to the inverse distance between them raised to some positive power . In particular, this includes the Coulomb law observed in Electrostatics and Riesz potentials extensively studied in Potential theory. For such particles, a stable equilibrium state, which depends on the parameter , is attained when the associated potential energy of the system is minimal. For large numbers of points, these equilibrium configurations provide a discretization of which may or may not be nearly uniform with respect to the surface area of . The poppy-seed bagel theorem asserts that for a large class of sets , the uniformity property holds when the parameter is larger than or equal to the dimension of the set . For example, when the points are confined to the 2-dimensional surface of a torus embedded in 3 dimensions, one can create a large number of points that are nearly uniformly spread on the surface by imposing a repulsion proportional to the inverse square distance between the points, or any stronger repulsion. From a culinary perspective, to create the nearly perfect poppy-seed bagel where bites of equal size anywhere on the bagel would contain essentially the same number of poppy seeds, impose at least an inverse square distance repelling force on the seeds.
In mathematics, Zolotarev polynomials are polynomials used in approximation theory. They are sometimes used as an alternative to the Chebyshev polynomials where accuracy of approximation near the origin is of less importance. Zolotarev polynomials differ from the Chebyshev polynomials in that two of the coefficients are fixed in advance rather than allowed to take on any value. The Chebyshev polynomials of the first kind are a special case of Zolotarev polynomials. These polynomials were introduced by Russian mathematician Yegor Ivanovich Zolotarev in 1868.
Balanced number partitioning is a variant of multiway number partitioning in which there are constraints on the number of items allocated to each set. The input to the problem is a set of n items of different sizes, and two integers m, k. The output is a partition of the items into m subsets, such that the number of items in each subset is at most k. Subject to this, it is required that the sums of sizes in the m subsets are as similar as possible.
