On the minimum of a positive polynomial over the standard simplex

We present a new positive lower bound for the minimum value taken by a polynomial P$P$ with integer coefficients in k$k$ variables over the standard simplex of R^k${\mathbb{R}}^k$, assuming that P$P$ is positive on the simplex. This bound depends only on the number of variables k$k$, the degree d$d$ and the bitsize τ$\tau$ of the coefficients of P$P$ and improves all the previous bounds for arbitrary polynomials which are positive over the simplex.     

碳载钴镍合金纳米电极的制备表征与电催化性能

采用电化学方法在玻碳(GC)表面电沉积CoNi合金纳米粒子,成功制得碳载CoNi合金纳米电极(CoNi/GC)。SEM结果显示,CoNi粒子呈十八面体结构,粒径约100 nm,分布较均匀。选区电子衍射(SAED)结果显示,CoNi合金纳米粒子为单晶结构。XPS结果显示,金属态的Co(0)和Ni(0)占主导地位。性能测试结果表明:CoNi/GC不但对亚硝酸钠具有较好的催化性能,相对于本体Co和本体Ni,CoNi/GC的起始还原电位(E_i)正移约90 mV,还原峰电流(j_p)增大6~14倍。而且对氧还原亦有较好的电催化活性,CoNi/GC的峰电流密度(j_p)和动力电流密度(j_k)分别是GC电极的1.7和 5.2倍。     

确定凸多面体全部顶点和非多余约束的一种算法

本文提出一种有效算法,不重复地循着由一个线性不等式系统所决定的凸多面体棱线仅一次访遍全部顶点,同时确定此系统中全部非多会约束。     

A displacement‐based finite element formulation for general polyhedra using harmonic shape functions

A displacement‐based continuous‐Galerkin finite element formulation for general polyhedra is presented for applications in nonlinear solid mechanics. The polyhedra can have an arbitrary number of vertices or faces. The faces of the polyhedra can have an arbitrary number of edges and can be nonplanar. The polyhedra can be nonconvex with only the mild restriction of star convexity with respect to the vertex‐averaged centroid. Conforming shape functions are constructed using harmonic functions defined on the undeformed configuration, thus requiring the use of a total‐Lagrangian finite element formulation in large deformation applications. For nonlinear applications with computationally intensive constitutive models, it is important to minimize the number of element quadrature points. For this reason, an integration scheme is adopted in which the number of quadrature points is equal to the number of vertices. As a first step toward verifying the element behavior in the general nonlinear setting, several linear verification examples are presented using both random Voronoi meshes and distorted hexahedral meshes. For the hexahedral meshes, results for the polyhedral formulation are compared to those of the standard trilinear hexahedral formulation. The element behavior in the nearly incompressible regime is also examined. Copyright © 2013 John Wiley & Sons, Ltd.     

Extrusion and boundary evaluation for multidimensional polyhedra

Extrusion is a basic operation allowing the generation of higher intrinsic dimension polyhedra. The paper gives closed formulas both to generate a (d+1)-dimensional polyhedron obtained by affine extrusion of a (d)-dimensional polyhedron, and to generate by rotational extrusion. Algorithms for the boundary evaluation when a decompositive representation is given are also discussed.

The representation used in the paper, based on simplicial complexes, is general and simple, and allows us to represent nonconvex, unconnected, unoriented, nonmanifold and unbounded linear polyhedra. A simplicial complex triangulating the extruded polyhedron is generated by independently extruding the simplices of the input object. The approach is very efficient because no a posteriori triangulation of the extruded polyhedron is required; furthermore, both the underlying complex and the adjacencies between cells are calculated by using closed formulas.     

Integration of polynomials over n-dimensional polyhedra

Explicit formulae and algorithms for computing integrals of polynomials over n-dimensional polyhedra are given. Two different approaches are discussed: one uses a decompositive representation, while the other one uses a boundary representation of the polyhedron. The algorithms are followed by a discussion of the complexity. In the appendix the pseudo-code and some examples of calculation are given.     

A method for detecting structure in polyhedra

A construction is given for associating a finite state automaton with a 3-dimensional polyhedron. It has the property that if a polyhedral segment is encoded as a string and used as input to the automaton, the automaton state which is reached contains information about all congruences of the segment with parts of the polyhedral surface. An application is retrieval from a polyhedral database, given as key a partial 3D view.     

On the equilibrium of funicular polyhedral frames and convex polyhedral force diagrams

This paper presents a three-dimensional extension of graphic statics using polyhedral form and force diagrams for the design of compression-only and tension-only spatial structures with externally applied loads. It explains the concept of 3D structural reciprocity based on Rankine’s original proposition for the equilibrium of spatial frames. It provides a definition for polyhedral reciprocal form and force diagrams that allows including external forces and discusses their geometrical and topological characteristics. This paper furthermore provides a geometrical procedure for constructing a pair of reciprocal polyhedral diagrams from a given polyhedron representing either the form or force diagram of a structural system. Using this method, this paper furthermore suggests a design strategy for finding complex funicular spatial forms in pure compression (or tension), based on the construction of force diagrams through the aggregation of convex polyhedral cells. Finally, it discusses the effect of changes in the geometry of the force diagram on the geometry of the form diagram and the distribution of forces in it.     

A High-Efficiency CoSe Electrocatalyst with Hierarchical Porous Polyhedron Nanoarchitecture for Accelerating Polysulfides Conversion in Li–S Batteries

Lithium–sulfur (Li–S) batteries are recognized as promising candidates for next-generation electrochemical energy storage systems owing to their high energy density and cost-effective raw materials. However, the sluggish multielectron sulfur redox reactions are the root cause of most of the issues for Li–S batteries. Herein, a high-efficiency CoSe electrocatalyst with hierarchical porous nanopolyhedron architecture (CS@HPP) derived from a metal–organic framework is presented as the sulfur host for Li–S batteries. The CS@HPP with high crystal quality and abundant reaction active sites can catalytically accelerate capture/diffusion of polysulfides and precipitation/decomposition of Li₂S. Thus, the CS@HPP sulfur cathode exhibits an excellent capacity of 1634.9 mAh g⁻¹, high rate performance, and a long cycle life with a low capacity decay of 0.04% per cycle over 1200 cycles. CoSe nanopolyhedrons are further fabricated on a carbon cloth framework (CC@CS@HPP) to unfold the electrocatalytic activity by its high electrical conductivity and large surface area. A freestanding CC@CS@HPP sulfur cathode with sulfur loading of 8.1 mg cm⁻² delivers a high areal capacity of 8.1 mAh cm⁻² under a lean electrolyte. This work will enlighten the rational design of structure–catalysis engineering of transition-metal-based nanomaterials for diverse applications.     

Positive Quadratic Differential Forms: Topological Equivalence Through Newton Polyhedra

The purpose of this article is to establish conditions under which a positive quadratic differential form is topologically equivalent to its principal part defined by Newton polyhedra. The problem is to study the simultaneous behavior of two foliations in the plane having a common point as a singularity. 2000 Mathematics Subject Classification. Primary: 37C15; Secondary: 34C20, 34A34. The first author was partially supported by FAPESP Grant No. 03/03107-9 and by CNPq Grant No. 306992/2003-5, Brazil. The second author was supported by CNPq-Brazil Grant No. 154150/01-1 and PROCAD-Brazil No. 092/01. The third author was supported by PRONEX No. 76.97.1080/00 and Fapesp-Brazil under Grant No. 97/10735-3.