An exposition of the topological half of the Grothendieck–Hirzebruch–Riemann–Roch theorem in the fancy language of spectra

We give an informal exposition of pushforwards and orientations in generalized cohomology theories in the language of spectra. The whole note can be seen as an attempt at convincing the reader that Todd classes in Grothendieck–Hirzebruch–Riemann–Roch type formulas are not Devil’s appearances but rather that things just go in the most natural possible way.

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Normal approximation and fourth moment theorems for monochromatic triangles

Given a graph sequence denote by T₃(G_n) the number of monochromatic triangles in a uniformly random coloring of the vertices of G_n with colors. In this paper we prove a central limit theorem (CLT) for T₃(G_n) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T₃(G_n), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7].     

B(X)上保持约当积的固定点的满射

设B(X)是维数大于等于3的复Banach空间X上有界线性算子全体构成的代数.设A∈B(X),若Ax=x,则称x∈X是算子A的固定点.Fix(A)表示A的所有固定点的集合.本文刻画了B(X)上保持算子的Jordan积的固定点的满射.     

对称向量拟均衡问题有效解的存在性

利用标量化方法建立对称向量拟均衡问题有效解的存在性定理。作为标量化方法的应用，利用这一方法得到向量变分不等式和拟向量变分不等式有效解的存在性定理。     

基于摩擦磨损的KDP晶体固结磨料抛光垫优化

本文通过固结磨料球与KDP晶体对磨的单因素试验探究固结磨料球中反应物种类、磨粒浓度、反应物浓度、基体硬度对摩擦系数、磨痕截面积和磨痕处粗糙度的影响,试验结果表明:KHCO₃固结磨料球对磨后磨痕对称性好,磨痕处的粗糙度值低;磨痕截面积随磨粒和反应物浓度的增加而增大,随基体硬度的增大而降低;磨痕处粗糙度随磨粒和反应物浓度的增加先降低后上升,随基体硬度的增大先上升后降低;摩擦系数受磨粒和反应物浓度影响不明显,随基体硬度的增大而降低。选择KHCO₃作为反应物,Ⅰ基体,磨粒浓度为基体质量的100%,反应物浓度为15%制备固结磨料球与KDP晶体对磨后的磨痕轮廓对称度好且磨痕处粗糙度值低,以该组分制备固结磨料垫干式抛光KDP晶体,可实现晶体表面粗糙度Sa值为18.50 nm,材料去除率为130 nm/min的高效精密加工。     

A new idea to construct the fractal interpolation function

The aim of this paper is to present a new idea to construct the nonlinear fractal interpolation function, in which we exploit the Matkowski and the Rakotch fixed point theorems. Our technique is different from the methods presented in the previous literatures.     

Fixed Points,Continuous Selections,and Existence of Solution of Optimization-Related Problems

We propose a definition of a generalized type of Knaster–Kuratowski–Mazurkiewicz (KKM) mappings, called a weak T-KKM mapping, and a corresponding weak KKM property. A new extension of the Fan–Glicksberg fixed-point theorem is established. Sufficient conditions for the existence of a continuous selection, a fixed point of a composition, and a coincidence point are also provided. Then, we use the obtained results to study the existence of solutions to various optimization-related problems. Discussions and detailed examples are included as well to compare our results with existing ones and to explain their advantages in many situations.