The Topology of Algebra: Combinatorics of Squaring

We study the graph each of whose edges connects an element of a given ring with the square of itself. For a finite commutative group (e.g., for the multiplicative group of coprime residue classes modulo a positive integer), we describe this graph explicitly: each of its connected components is an oriented attracting cycle equipped with identical -vertex rooted trees of special form whose roots reside on the cycle. We also compute the graphs of permutation groups on not too many elements and of the subgroups of even permutations; the connected components of these graphs are also uniformly equipped cycles.     

Higher-Order Convex Approximations of Young Measures in Optimal Control

The general theory of approximation of (possibly generalized) Young measures is presented, and concrete cases are investigated. An adjoint-operator approach, combined with quasi-interpolation of test integrands, is systematically used. Applicability is demonstrated on an optimal control problem for an elliptic system, together with one-dimensional illustrative calculations of various options.     

Weighted Young Inequalities for Convolutions

Sufficient conditions for some weighted Young inequalities type are obtained.AMS Subject Classification (1991): 42A85, 42B20, 47G10     

Projections on Convex Sets in the Relaxed Limit

The paper establishes the continuity of the best approximation, or the projection, of a function in L _p for p[1,), on a closed convex set in the space, when the set varies and converges to a limit set in the Young-measure relaxation of the space. To this end a strong-type convergence and a convexity structure are identified on the space of Young measures. The appropriate convergence of sets with respect to which the continuity holds is the Mosco convergence of sets associated with the strong-type convergence of functions.     

Ribbon tile invariants

Let be a finite set of tiles, and a set of regions tileable by . We introduce a tile counting group as a group of all linear relations for the number of times each tile can occur in a tiling of a region . We compute the tile counting group for a large set of ribbon tiles, also known as rim hooks, in a context of representation theory of the symmetric group.

The tile counting group is presented by its set of generators, which consists of certain new tile invariants. In a special case these invariants generalize the Conway-Lagarias invariant for tromino tilings and a height invariant which is related to computation of characters of the symmetric group.

The heart of the proof is the known bijection between rim hook tableaux and certain standard skew Young tableaux. We also discuss signed tilings by the ribbon tiles and apply our results to the tileability problem.

     

The Order of Best Approximation in Some Classes of Functions

Starting from the equivalence between the Ditzian–Totik modulus and , where , in this article large classes of functions are introduced for which the modulus can be easily calculated. As a consequence, very good estimates for the bestapproximation are obtained. The attempts to estimate or calculate themodulus can be a very intricateproblem.     

金属丝的杨氏模量测定实验的改进

对金属丝杨氏模量测定仪重新设计.将金属丝水平放置,在中点处悬挂重物,通过测量金属丝伸长后的转角,进而测量金属丝的杨氏模量.     

基于AVR单片机的万能材料试验机控制系统

采用AVR单片机AT90S8515作为控制核心,通过步进电机带动丝杠运动,利用光栅编码器、高精度力传感器、2个引伸计分别测量了位移、力及试件横、纵变形,在老式电子万能材料实验机上建立了新型计算机控制系统,可方便测定弹性模量和泊松比.     

测定圆柱体杨氏模量的原理分析

分析了测量圆柱体弯曲形变杨氏模量的实验原理,得出了理论公式,将测量拉压形变、弯曲形变的杨氏模量以及扭转形变的剪切模量3个实验中样品形状统一成圆柱体.     

利用光纤传感器测量金属丝的杨氏模量

将反射式光纤位移传感器应用于杨氏模量测定仪,可以精确地测定金属丝杨氏模量。