Chalk it up to experience: using chalkboard paint to create mathematical manipulatives

In this article, we give two examples of creating portable chalkboards using chalkboard paint for students to use during cooperative learning. This provides a creative method for professors to facilitate active learning in the undergraduate mathematics classroom.     

Light Dialectica revisited

We upgrade the light Dialectica interpretation (Hernest, 2005) [6] by adding two more light universal quantifiers, which are both semi-computational and semi-uniform and complement each other. An illustrative example is presented for the new light quantifiers and a new application is given for the older uniform quantifier. The realizability of new light negative formulations for the Axiom of Choice and for the Independence of Premises is explored in the new setting.     

Cutting up graphs

LetΓ be infinite connected graph with more than one end. It is shown that there is a subsetd ⊂V Γ which has the following properties. (i) Bothd andd*=VΓ\d are infinite. (ii) there are only finitely many edges joiningd andd*. (iii) For eachgε AutΓ at least one ofd⊂dg, d*⊂dg, d⊂d* g, d*⊂d* g holds. Any group acting on Γ has a decomposition as a free product with amalgamation or as an HNN-group.     

净化绿色宣传

在重视生态环境的市场中,绿色可信度加上环保行动就如同上个世纪的财务指标,是影响顾客忠诚度和利润幅度的关键因素。     

光的波粒二象性探索

光具有波粒二象性,但很难从光的某一特定现象中同时看到这两种性质.如果能用粒子性解释波动性,光的两种性质就能得到完美的统一.分别对龚祖同光子模型、前进电磁场光子模型进行数学建模和MATLAB仿真,并与已有的实验结果比较.发现两个模型都存在不足,进而提出改进的光子模型,从粒子的角度,对展现光的波动性质的干涉、衍射以及偏振现象进行解释.     

Tying up baric algebras

Given two baric algebras (A ₁, ?? ₁) and (A ₂, ?? ₂) we describe a way to define a new baric algebra structure over the vector space A ₁ ?? A ₂, which we shall denote (A ₁ ? A ₂, ?? ₁ ? ?? ₂). We present some easy properties of this construction and we show that in the commutative and unital case it preserves indecomposability. Algebras of the form A ₁ ? A ₂ in the associative, coutable-dimensional, zero-characteristic case are classified.     

Spacetime and the Geometry behind it

This is the Leonardo da Vinci Lecture given in Milan in March 2006. It is a survey on the concept of space-time over the last 3000 years: it starts with Euclidean geometry, discusses the contributions of Gauss and Riemannian geometry, presents the dynamic concept of space-time in Einstein’s general relativity, describes the importance of symmetries, and ends with Calabi-Yau manifolds and their importance in today’s string theories in the attempt for a unified theory of physics. Leonardo da Vinci Lecture held on March 28, 2006     

Speeding up continuous GRASP

Continuous GRASP (C-GRASP) is a stochastic local search metaheuristic for finding cost-efficient solutions to continuous global optimization problems subject to box constraints (Hirsch et al., 2007). Like a greedy randomized adaptive search procedure (GRASP), a C-GRASP is a multi-start procedure where a starting solution for local improvement is constructed in a greedy randomized fashion. In this paper, we describe several improvements that speed up the original C-GRASP and make it more robust. We compare the new C-GRASP with the original version as well as with other algorithms from the recent literature on a set of benchmark multimodal test functions whose global minima are known. Hart’s sequential stopping rule (1998) is implemented and C-GRASP is shown to converge on all test problems.     

Similarity stabilizes blow up

The blow-up of solutions to the PDE ψ(x)u_t=[∇·A(x)∇+b(x)]u^m is studied via energy methods. The key step is a similarity transformation of the original unstable equation to a nonlocal stable one.     

Diagrams up to cohomology

We compute (under suitable assumptions) how many ways there are to take a diagram in the homotopy category of spaces and perturb it to get another diagram which looks the same up to cohomology. Sometimes there are no perturbations. This can shed light on the question of whether the -completion of the classifying space of a particular connected compact Lie group is determined up to homotopy by cohomological data.

     

Light reflection is nonlinear optimization

In this paper, we show that the near field reflector problem is a nonlinear optimization problem. From the corresponding functional and constraint function, we derive the Monge–Ampère type equation for such a problem.     

车灯光源的优化设计

通过建立直角坐标系,利用线、面方程得出以下结论:1.根据由光源发射的辐射能均匀分布在以点光源为中心的球形波阵面上,把所求的总功率之比转化为以线光源一点F′为球心,|F′B|为半径(B为灯口边缘最上的顶点)的球面被灯口所截得的两个球冠面积之比,经计算得直射光总功率反射光总功率≈0.690683.2.直射光的亮区是由圆滑动而成.3.光源放在车灯焦点处效果最佳.     

轻矩阵及其推广

在R.E.Hartwig工作的基础上进一步研究了轻矩阵,得到了轻矩阵的一些特性,且把轻矩阵推广到了广义轻矩阵,得到了广义轻矩阵的一个充要条件;部分回答了文[1]中R.E.Harwtig提出的一个公开问题：当A与B都为n除非负矩阵时,刻划方程AX=XB的非负解。     

Light leaves and Lusztig's conjecture

We define a map F with domain a certain subset of the set of light leaves (certain morphisms between Soergel bimodules introduced by the author in an earlier paper) and range the set of prime numbers. Using results of Soergel we prove the following property of F  : if the image p=F(l)$p=F(l)$ of some light leaf l under F is bigger than the Coxeter number of the corresponding Weyl group, then there is a counterexample to Lusztig's conjecture in characteristic p. We also introduce the “double leaves basis” which is an improvement of the light leaves basis that has already found interesting applications. In particular it forms a cellular basis of Soergel bimodules that allows us to produce an algorithm to find “the bad primes” for Lusztig's conjecture.