Largest integral simplices with one interior integral point: Solution of Hensley's conjecture and related results

For each dimension d, d-dimensional integral simplices with exactly one interior integral point have bounded volume. This was first shown by Hensley. Explicit volume bounds were determined by Hensley, Lagarias and Ziegler, Pikhurko, and Averkov. In this paper we determine the exact upper volume bound for such simplices and characterize the volume-maximizing simplices. We also determine the sharp upper bound on the coefficient of asymmetry of an integral polytope with a single interior integral point. This result confirms a conjecture of Hensley from 1983. Moreover, for an integral simplex with precisely one interior integral point, we give bounds on the volumes of its faces, the barycentric coordinates of the interior integral point and its number of integral points. Furthermore, we prove a bound on the lattice diameter of integral polytopes with a fixed number of interior integral points. The presented results have applications in toric geometry and in integer optimization.     

积分凸性及其应用

该文在Banach空间中通过向量值函数的Bochner积分引进集合与泛函的积分凸性以及集合的积分端点等概念. 文章主要证明有限维凸集、开凸集和闭凸集均是积分凸集，下半连续凸泛函与开凸集上的上半连续凸泛函均是积分凸的, 非空紧集具有积分端点, 对紧凸集来说其积分端点集与端点集一致, 最后给出积分凸性在最优化理论方面的两个应用.     

Discrete Choquet integral and some of its symmetric extensions

Classical extensions of the Choquet integral (defined on [0,1]) to [−1,1] are the asymmetric and the symmetric Choquet integral, the second one being called also the Šipoš integral. Recently, the balancing Choquet integral was introduced as another kind of a symmetric extension of the discrete Choquet integral. We introduce and discuss a new type of such extension, the fusion Choquet integral, and discuss its properties and relationship to the balancing and the symmetric Choquet integral. The symmetric maximum introduced by Grabisch is shown to be a special case of the fusion and the balancing Choquet integral. Several extensions of OWA operators are also discussed.     

A Note on a Composition of Two Random Integral Mappings β and Some Examples

Abstract

The method of random integral representation, that is, the method of representing a given probability measure as the probability distribution of some random integral, was quite successful in the past few decades. In this note, we show that a composition of two random integral mappings ^β is again a random integral mapping. We illustrate our results with some examples.     

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H ¹(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition $\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}$ is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f?+?S f?=?g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown ${\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}$ . But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form ${\bf g}+\tilde S{\bf g}={\bf f}$ , where $\tilde S$ is a contractive operator, and we can solve it by the successive approximation.     

The nonparametric integral of the Calculus of Variations as a Weierstrass integral: Existence and representation

The definition and properties of an abstract and very general nonparametric integral of the Calculus of Variations is presented. In harmony with the Lewy-McShane approach, the nonparametric integral ∝ f, for set functions ? taking their values in a Banach space E, is defined in terms of its associated parametric integral. For the latter use is made of the abstract parametric integral proposed by Cesari in Rⁿ and then extended to Banach spaces by Breckenridge, Warner, and the authors. A condition (c) is shown to be relevant for the existence of the integral, and is preserved by the nonlinear operation f. Also, for f nonnegative, a Tonelli-type theorem is proved in the sense that the so defined Weierstrass integral ∝ f is always larger than or equal to the corresponding Lebesgue integral, and equality holds if and only if absolute continuity conditions hold. In the proof a suitable martingale is associated and a convergence theorem for martingales is applied. Applications to the calculus of variations will follow.     

The Itô integral for Brownian motion in vector lattices: Part 2

The Itô integral for Brownian motion in a vector lattice, as constructed in Part 1 of this paper, is extended to accommodate a larger class of integrands. This extension provides an analogue of the indefinite Itô integral in the classical setting which yields a local martingale. The assumption is that there exists a conditional expectation operator on the vector lattice and the construction does not depend on a probability measure space. The classical case of the extended Itô integral is a special case of the constructed integral in the vector lattice.     

The widest continuous integral

An approach to a definition of an integral, which differs from definitions of Lebesgue and Henstock-Kurzweil integrals, is considered. We use trigonometrical polynomials instead of simple functions. Let V be the space of all complex trigonometrical polynomials without the free term. The definition of a continuous integral on the space V is introduced. All continuous integrals are described in terms of norms on V. The existence of the widest continuous integral is proved, the explicit form of its norm is obtained and it is proved that this norm is equivalent to the Alexiewicz norm. It is shown that the widest continuous integral is wider than the Lebesgue integral. An analog of the fundamental theorem of calculus for the widest continuous integral is given.     

Hyperbolic beta integrals

Hyperbolic beta integrals are analogues of Euler's beta integral in which the role of Euler's gamma function is taken over by Ruijsenaars’ hyperbolic gamma function. They may be viewed as -bibasic analogues of the beta integral in which the two bases q and q? are interrelated by modular inversion, and they entail q-analogues of the beta integral for |q|=1. The integrals under consideration are the hyperbolic analogues of the Ramanujan integral, the Askey-Wilson integral and the Nassrallah-Rahman integral. We show that the hyperbolic Nassrallah-Rahman integral is a formal limit case of Spiridonov's elliptic Nassrallah-Rahman integral.     

The exterior dirichlet problem for the Laplace equation

Using a standard application of Green's theorem, the exterior Dirichlet problem for the Laplace equation in three dimensions is reduced to a pair of integral equations. One integral equation is of the second kind and the other is of the first kind. It is known that the integral equation of the second kind is not uniquely solvable, however, it has been demonstrated that the pair of integral equations has a unique solution. The present approach is based on the observation that the known function appearing in the integral equation of the second kind lies in a certain Banach space E which is a proper subspace of the Banach space of continuous complex-valued functions equipped with the maximum norm. Furthermore, it can be shown that the related integral operator when restricted to E has spectral radius less than unity. Consequently, a particular solution to the integral equation of the second kind can be obtained by the method of successive approximations and the unique solution to the problem is then obtained by using the integral equation of the first kind. Comparisons are made between the present algorithm and other known constructive methods. Finally, an example is supplied to illustrate the method of this paper.     

G可积函数的Lebesgue可测性

Botsko在连续和可导的知识基础上推广了Riemann积分,得到了一种新的积分,称为G积分．G积分既不同于Riemann积分也不同于Lebesgue积分．本文通过对G积分的研究,得到了G可积函数一定Lebesgue可测,从而有界G可积函数一定Lebesgue可积;同时我们还证明了这两个积分值相等．     

Generalizing Hopf and Lax-Oleînik formulae via conjugate integral

Using the second Fenchel conjugate transform the conjugate integral sums and the conjugate integral are introduced. An estimate of speed of convergence of the sums to the integral is obtained. In the case of a convex integrant the conjugate integral reduces to the Riemannian one. It is proved that the Fenchel conjugate transform of the conjugate integral with variable upper limit provides a formula for the viscosity solution to a Hamilton-Jacobi equation in which the Hamiltonian depends both on time and the gradient of the unknown function. In the autonomous case the obtained formula coincides with Hopf's one. Two examples are considered in which an application of the conjugate integral allows to find viscosity solutions explicitly. It is shown how the extension of the Lax-Oleînik formula to the nonautonomous case may be obtained using the generalized Hopf formula.This paper was prepared while the author was a Lise Meitner fellow at the Institut für Mathematik, Karl-Franzens-Universität Graz, Austria     

Singular integral operators in a weighted L2-space

Cauchy singular integral operators are characterized as operators in a weighted L²-space. The integral operator arises from a singular integral equation with variable coefficients. An appropriate weight function associated with the singular integral operator is constructed, and the set of polynomials orthogonal with respect to this weight function is defined. The action of the operator on polynomial sets is studied, and the definition of the operator is extended to a weighted L²-space. In this space, the operator is shown to be bounded, and, in some cases, isometric. Formulas are developed for the composition of the singular integral operator and its one sided inverse.     

On isomorphisms of integral table algebras

For integral table algebras with integral table basisT, we can consider integralR-algebraRT over a subringR of the ring of the algebraic integers. It is proved that anR-algebra isomorphism between two integral table algebras must be an integral table algebra isomorphism if it is compatible with the so-called normalizings of the integral table algebras     

Prüfer's ascent result via Inc

Let R be an integral domain whose integral closure is a Pr¨fer domain. It is proved that R ? T has the incomparability property for each integral domain T which contains R and is algebraic over R. As a corollary, one has a new proof of Pr¨fer's ascent result, which states that if R is as above and T is the integral closure of R in some field containing R, then T is a Pr¨fer domain.     

Maximal integral point sets in affine planes over finite fields

Motivated by integral point sets in the Euclidean plane, we consider integral point sets in affine planes over finite fields. An integral point set is a set of points in the affine plane over a finite field F_q, where the formally defined squared Euclidean distance of every pair of points is a square in F_q. It turns out that integral point sets over F_q can also be characterized as affine point sets determining certain prescribed directions, which gives a relation to the work of Blokhuis. Furthermore, in one important sub-case, integral point sets can be restated as cliques in Paley graphs of square order.In this article we give new results on the automorphisms of integral point sets and classify maximal integral point sets over F_q for q≤47. Furthermore, we give two series of maximal integral point sets and prove their maximality.     

On the solvability of a complete two-dimensional hypersingular integral equation

We study the solvability of a complete two-dimensional linear hypersingular integral equation that contains a hypersingular integral operator in which the integral is understood in the sense of Hadamard finite value as well as an integral operator in which the integral is understood in the sense of principal value, an integral operator with a weakly singular kernel, and an integral-free term. We consider smooth solutions in the class of functions that have Hölder continuous derivatives outside a neighborhood of the boundary. We prove the Fredholm alternative and estimate the norm of the solution in a special metric.     

On approximation and numerical solution of Fredholm integral equations of second kind using quasi-interpolation

In this article a method is presented, which can be used for the numerical treatment of integral equations. Considered is the Fredholm integral equation of second kind with continuous kernel, since this type of integral equation appears in many applications, for example when treating potential problems with integral equation methods.The method is based on the approximation of the integral operator by quasi-interpolating the density function using Gaussian kernels. We show that the approximation of the integral equation, gained with this method, for an appropriate choice of a certain parameter leads to the same numerical results as Nyström’s method with the trapezoidal rule. For this, a convergence analysis is carried out.     

A resource for introducing students to the integral concept

Many students do not have a deep understanding of the integral concept. This article defines what a deep understanding of the integral is in respect to integration involving one independent variable; briefly discusses factors which may inhibit such an understanding; and then describes the design of a mathematical resource for introducing students to the integral concept. The resource addresses a number of challenges when introducing the integral: (1) choosing an appropriate, intuitive context which gives meaning to the symbols in the integral expression; (2) aiding the transfer of the integral expression to different contexts via using the Riemann sum in an informal way so that students can see and interpret the rectangles which are inherent in this sum; and (3) the gradual formalizing of the Riemann sum and its linkage with the Fundamental Theorem of Calculus. The resource has been used over a number of years at this university amongst first-year undergraduate science and engineering students. Anecdotal evidence would suggest that the resource is beneficial.     

Boundary properties for several singular integral operators in real Clifford analysis

On the basis of the Cauchy integral formulas for regular and biregular functions, we define some Cauchy-type singular integral operators. Then we discuss the Hlder continuous property of some singular integral operators with one integral variable. Then we divide a singular integral operator with two variables into three parts and prove its Hlder continuous property on the boundary.