The Isoperimetric Problem in Spherical Cylinders

The classical isoperimetric problem for volumes is solved in × ⁿ (1). Minimizers are shown to be invariant under the group O(n) acting standardly on ⁿ , via a symmetrization argument, and are then classified. Solutions are found among two (one-parameter) families: balls and sections of the form [a, b] × ⁿ . It is shown that the minimizers may be of both types. For n= 2, it is shown that the transition between the two families occurs exactly once. Some results for general n are also presented.     

Isoperimetric Polygons of Maximum Width

The value is shown to be an upper bound on the width of any n-sided polygon with unit perimeter. This bound is reached when n is not a power of 2, and the corresponding optimal solutions are the regular polygons when n is odd and clipped regular Reuleaux polygons when n is even but not a power of 2. Using a global optimization algorithm, we show that the optimal width for the quadrilateral is with a precision of 10⁻⁴. We propose two mathematical programs to determine the maximum width when n=2 ^s with s≥3 and provide approximate, but near-optimal, solutions obtained by various heuristics and local optimization for n=8, 16, and 32. Work of the first author was supported by NSERC grant 239436-01, AFOSR FA9550-07-1-0302, and ExxonMobil. Work of the second author was supported by NSERC grant 239436-01.     

The Number of Generalized Balanced Lines

Let S be a set of r red points and b=r+2δ blue points in general position in the plane, with δ≥0. A line ℓ determined by them is balanced if in each open half-plane bounded by ℓ the difference between the number of blue points and red points is δ. We show that every set S as above has at least r balanced lines. The proof is a refinement of the ideas and techniques of Pach and Pinchasi (Discrete Comput. Geom. 25:611–628, 2001), where the result for δ=0 was proven, and introduces a new technique: sliding rotations.     

On the Number of Cylinders Touching a Ball

In this note we prove an upper bound of seven for the maximum number of unit cylinders touching a unit ball in a packing. This improves a previous bound of eight by Heppers and Szab. The value conjectured by Kuperberg in 1990 is six.     

A rapid Generalized Method of Bisection for solving Systems of Non-linear Equations

Summary A rapid Generalized Method of Bisection for solving Systems of Non-linear Equations is presented in this paper, based on the non-zero value of the topological degree. Further, while the method does not compute the topological degree, it takes care of keeping its non-zero value during the bisections and thus results in a fast bisection algorithm.     

The Generalized Prime Number Theorem for Automorphic L-Functions

Let π and π' be automorphic irreducible cuspidal representations of GLm (QA) and GLm', (QA), respectively, and L(s,π×(～π)') be the Rankin-Selberg L-function attached to π and π'. Without assuming the Generalized Ramanujan Conjecture (GRC), the author gives the generalized prime number theorem for L(s, π×(～π)') when π(=)π'. The result generalizes the corresponding result of Liu and Ye in 2007.     

The Generalized Prime Number Theorem for Automorphic L-Functions

Let π and π＇ be automorphic irreducible cuspidal representations of GLm（QA） and GLm′ （QA）, respectively, and L（s, π×π′） be the Rankin-Selberg L-function attached to π and π＇. Without assuming the Generalized Ramanujan Conjecture （GRC）, the author gives the generalized prime number theorem for L（s, π × π′） when π =π＇. The result generalizes the corresponding result of Liu and Ye in 2007.     

The Gaussian Isoperimetric Inequality and Transportation

Any probability measure on ^d which satisfies the Gaussian or exponential isoperimetric inequality fulfils a transportation inequality for a suitable cost function. Suppose that W (x) dx satisfies the Gaussian isoperimetric inequality: then a probability density function f with respect to W (x) dx has finite entropy, provided that t . This strengthens the quadratic logarithmic Sobolev inequality of Gross (Amr. J. Math 97 (1975) 1061). Let (dx) = e ^–(x) dx be a probability measure on ^d, where is uniformly convex. Talagrand's technique extends to monotone rearrangements in several dimensions (Talagrand, Geometric Funct. Anal. 6 (1996) 587), yielding a direct proof that satisfies a quadratic transportation inequality. The class of probability measures that satisfy a quadratic transportation inequality is stable under multiplication by logarithmically bounded Lipschitz densities.     

大围长图的广义无圈染色

图的顶点染色称为是r-无圈的,如果它是正常染色,使得每一个圈C上顶点的颜色数至少为min{|C|,r}.图G的r-无圈染色数是图G的r-无圈染色中所用的最少的颜色数.我们证明了对于任意的r≥4,最大度为△、围长至少为2(r-1)△的图G的r-无圈染色数至多为6(r-1)△.     

广义Stirling数偶的理论及应用

The object of this expository paper is to sum up several results concerning generalized Stirling number (GSN) pairs investigated earlier by the author. Also expounded in some detail are two kinds of extended GSN pairs with applications (illustrative examples).     

The Rotation Number for the Generalized Kronig–Penney Hamiltonians

We discuss the one-dimensional Schr?dinger operator with generalized point interaction on a lattice. We give a characterization of the band edges of its spectrum by the rotation number. Submitted: December 18, 2006. Accepted: February 23, 2007.     

一类广义Catalan数及其应用

引进了一类广义Catalan数,并赋予这类广义Catalan数组合意义,用这类广义Catalan数得到一类不定方程的解数.     

Recognition Algorithms for Orders of Small Width and Graphs of Small Dilworth Number

Partially ordered sets of small width and graphs of small Dilworth number have many interesting properties and have been well studied. Here we show that recognition of such orders and graphs can be done more efficiently than by using the well-known algorithms based on bipartite matching and matrix multiplication. In particular, we show that deciding deciding if an order has width k can be done in O(kn ²) time and whether a graph has Dilworth number k can be done in O(k ² n ²) time.For very small k we have even better results. We show that orders of width at most 3 can be recognized in O(n) time and of width at most 4 in O(nlog n).     

基于LR模糊数的广义模糊DEA模型及其求解

在广义DEA模型基础上,建立基于LR模糊数的广义模糊DEA模型.通过引入LR模糊数的加权平均值,计算了待评价决策单元能体现决策者偏好的广义模糊效率和平均广义模糊效率,对待评价决策单元进行有效性排序.最后通过实例分析,表明了该模型的实用性.     

Crystalline and Isoperimetric Square Configurations

We present crystallization results for planar atomic interactions governed by two- and three-body terms with the resulting periodicity being that of the square lattice. The emergence of a (square) Wulff shape for ground states is established by showing the optimality of ground-state configurations in terms of a discrete isoperimetric inequality. Furthermore, an n^3/4 law for the deviation from the asymptotic Wulff shape is established with an explicit constant for the leading term. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Schur-Convex Functions and Isoperimetric Inequalities

In this paper, we establish some analytic inequalities for Schur-convex functions that are made of solutions of a second order nonlinear differential equation. We apply these analytic inequalities to obtain some geometric inequalities.

     

Rienmannian Submersions and Isoperimetric Inequalities

Isoperimetric constants of the total spaces of Riemannian submersions are estimated in terms of those of the basis and fibers.