Unimodality of the Freely Selfdecomposable Probability Laws

We show that any freely selfdecomposable probability law is unimodal. This is the free probabilistic analog of Yamazato’s result in (Ann. Probab. 6:523–531, 1978).     

Weyl Laws for Partially Open Quantum Maps

We study a toy model for “partially open” wave-mechanical system, like for instance a dielectric micro-cavity, in the semiclassical limit where ray dynamics is applicable. Our model is a quantized map on the 2-dimensional torus, with an additional damping at each time step, resulting in a subunitary propagator, or “damped quantum map”. We obtain analogues of Weyl’s laws for such maps in the semiclassical limit, and draw some more precise estimates when the classical dynamics is chaotic. Submitted: October 16, 2008. Accepted: April 3, 2009.     

History and Perspectives of Quantum Groups

The advent of Quantum Groups in the course of the working out the quantum analogue of the Inverse Scattering Method from the soliton theory gives an instructive example of interinfluence of different domains of mathematics. Here I give a rather personal account of this development. Leonardo da Vinci Lecture held on November 7, 2005 Received: June 2006     

Quantum set theory: Transfer Principle and De Morgan's Laws

In quantum logic, introduced by Birkhoff and von Neumann, De Morgan's Laws play an important role in the projection-valued truth value assignment of observational propositions in quantum mechanics. Takeuti's quantum set theory extends this assignment to all the set-theoretical statements on the universe of quantum sets. However, Takeuti's quantum set theory has a problem in that De Morgan's Laws do not hold between universal and existential bounded quantifiers. Here, we solve this problem by introducing a new truth value assignment for bounded quantifiers that satisfies De Morgan's Laws. To justify the new assignment, we prove the Transfer Principle, showing that this assignment of a truth value to every bounded ZFC theorem has a lower bound determined by the commutator, a projection-valued degree of commutativity, of constants in the formula. We study the most general class of truth value assignments and obtain necessary and sufficient conditions for them to satisfy the Transfer Principle, to satisfy De Morgan's Laws, and to satisfy both. For the class of assignments with polynomially definable logical operations, we determine exactly 36 assignments that satisfy the Transfer Principle and exactly 6 assignments that satisfy both the Transfer Principle and De Morgan's Laws.     

Quantum Measurements and Kolmogorov's Theory of Probability

A connection between the probability space measurability requirement and the complementarity principle in quantum mechanics is established. It is shown that measurability of the probability space implies that the results of the quantum measurement depend not only on properties of the quantum object under consideration but also on classical characteristics of the measuring instruments used. It is also shown that if the measurability requirement is taken into account, then the hypothesis that the objective reality exists does not lead to the Bell inequality.     

Phase Retrieval for Probability Distributions on Quantum Groups and Braided Groups

For nilpotent quantum groups [as introduced by Franz et al. ⁽⁷⁾], we show that (in sharp contrast to the classical case) the symmetrization of a probability distribution and the first moments of together determine uniquely the original distribution .     

A New Technique for Preserving Conservation Laws

Foundations of Computational Mathematics - This paper introduces a new symbolic-numeric strategy for finding semidiscretizations of a given PDE that preserve multiple local conservation laws. We...     

Generating M-Indeterminate Probability Densities by Way of Quantum Mechanics

Probability densities that are not uniquely determined by their moments are said to be “moment-indeterminate,” or “M-indeterminate.” Determining whether or not a density is M-indeterminate, or how to generate an M-indeterminate density, is a challenging problem with a long history. Quantum mechanics is inherently probabilistic, yet the way in which probability densities are obtained is dramatically different in comparison with standard probability theory, involving complex wave functions and operators, among other aspects. Nevertheless, the end results are standard probabilistic quantities, such as expectation values, moments and probability density functions. We show that the quantum mechanics procedure to obtain densities leads to a simple method to generate an infinite number of M-indeterminate densities. Different self-adjoint operators can lead to new classes of M-indeterminate densities. Depending on the operator, the method can produce densities that are of the Stieltjes class or new formulations that are not of the Stieltjes class. As such, the method complements and extends existing approaches and opens up new avenues for further development. The method applies to continuous and discrete probability densities. A number of examples are given.

     

基于Hadamard算子的二维离散量子行走的概率测度估计

二维离散时间量子行走是直线上的量子行走的推广.通过演化算子的作用,行走者能够按照一定规律进行移动.在本文中,我们将Hadamard算子作为控制行走者方向的硬币算子,通过与控制行走者位置的条件转移算子结合,构成完整的演化算子.通过傅里叶变换,将行走者所处的时域空间转换成频域空间后,用傅里叶积分的平稳相位法得到了行走者在t步后处于位置(x,y)的振幅以及此时的概率估计.     

Preface: Old and New Perspectives on the Tutte Polynomial

This paper introduces a special issue on the Tutte polynomial derived from the Second Workshop on Tutte Polynomials and Applications, 2005, held at the Centre de Recerca Matemàtica, Bellaterra, Catalonia. We discuss the prehistory of Tutte polynomials and two current areas of research, to what extent a graph is determined by its chromatic or Tutte polynomial and generic versions of Tutte polynomials. Received February 28, 2007     

双曲守恒律的几种新数值方法的比较研究

本文就一维线性双曲方程的光滑和间断两种初值问题的求解,对双曲守恒律的三种新数值方法,即,WENO方法、间断Galerkin方法和全局复合方法,进行了数值比较实验,在精度、计算速度等方面的比较上,对这三个方法有了一个较详细的了解,得到了一些有用的结论。     

New Perspectives on the Gender Stereotyping of Mathematics

Historically, mathematics has been stereotyped as a male domain, and there is considerable evidence to support this belief. In the last 30 years, mathematics education researchers have uncovered a range of factors contributing to the documented achievement and participation differences that favored males and sought to redress them. Mathematics as a male domain, one of the subscales of the Fennema-Sherman Mathematics Attitude Scales (1976), has been used widely to assess the extent to which mathematics is stereotyped as a masculine sphere. It has been argued that some of the items comprising the subscale are anachronistic and that the subscale scores can no longer be interpreted reliably. In this article we outline the development of two new instruments-the mathematics as a gendered domain instrument and the who and mathematics instrument-that have been designed to overcome the limitations of the original Fennema-Sherman mathematics as a male domain subscale. We also present findings from the administration of the two instruments in Australia, where they were developed, and in the United States, the site of the trials of the original Fennema-Sherman scales. The results indicate that females feel more strongly than males about some aspects of gender stereotyping in mathematics although, in general, most students feel that mathematics is gender neutral.     

New Perspectives on the Gender Stereotyping of Mathematics

Historically, mathematics has been stereotyped as a male domain, and there is considerable evidence to support this belief. In the last 30 years, mathematics education researchers have uncovered a range of factors contributing to the documented achievement and participation differences that favored males and sought to redress them. Mathematics as a male domain, one of the subscales of the Fennema-Sherman Mathematics Attitude Scales (1976), has been used widely to assess the extent to which mathematics is stereotyped as a masculine sphere. It has been argued that some of the items comprising the subscale are anachronistic and that the subscale scores can no longer be interpreted reliably. In this article we outline the development of two new instruments-the mathematics as a gendered domain instrument and the who and mathematics instrument-that have been designed to overcome the limitations of the original Fennema-Sherman mathematics as a male domain subscale. We also present findings from the administration of the two instruments in Australia, where they were developed, and in the United States, the site of the trials of the original Fennema-Sherman scales. The results indicate that females feel more strongly than males about some aspects of gender stereotyping in mathematics although, in general, most students feel that mathematics is gender neutral.     

一种新的求解双曲型守恒律的三阶中心差分格式

本文提出了一种求解双曲型守恒律新的三阶中心差分格式，主要是引入了一种推广的三阶重构，并证明了这种重构在网格边界无振荡．所提的格式保持了中心差分格式简单的优点，不需用Riemann解算器，避免了进行特征解耦．数值试验结果表明本文格式是高精度、高分辨率的。     

Changing Perspectives on Quality Education: The New Learners

Abstract

The multicultural wars remain a hot topic in the press—but on campus, the war is over: multiculturalism won. The question is no longer whether students should learn about diverse cultures, but how. The task today is that of designing and implementing programs that are educationally valuable.     

A New Method for Approximations in Probability and Operator Theories

The aim of this work is to introduce a new approach in normal approximations, approximations of semigroups of operators, and approximations by accompanying laws. We describe the method and provide several examples showing how the method works. Loosely speaking, the approach is based on multiplicative representations of differences to estimate. The method considerably reduces the technical part of proofs compared to the traditional approaches. In probability theory, it nicely applies to the nonidentically distributed case; another advantage of the method is its extendability to the case of dependent random variables.