On the Plancherel measure for linear Lie groups of rank one

In this paper we find a very explicit, simple form for the Plancherel measure for rank one, linear simple groups, including the normalizing constant.     

Stein's method and Plancherel measure of the symmetric group

We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group.

     

On some properties of the “apparent length” distribution and a compound distribution

This paper extends and supplements some of the properties of the apparent length distribution introduced by Baker (J. Appl. Statist. 27(1) (2000) 2–21). Recurrence relations for moments and formulae for moments derived via difference and differential operators will be given. We also establish maximum likelihood estimator and Bayesian estimators of a parameter of this distribution. Compound distributions based on the apparent length distribution are also discussed.     

Constructing a line segment whose length is equal to the measure of a given angle

The Möbius strip, torus, and Klein bottle are used to graphically and analytically illustrate the differences between orientable and non-orientable surfaces. An exercise/laboratory project using the non-orientable Boy surface is included.     

On the degree of weak convergence of a sequence of finite measures to the unit measure under convexity

This is a study of the degree of weak convergence under convexity of a sequence of finite measures μ_j on ^k, k 1, to the unit measure δ_x0. LetQ denote a convex and compact subset of ^k, let ƒ ε C^m(Q), m 0, satisfy a convexity condition and let μ be a finite measure on Q. Using standard moment methods, upper bounds and best upper bounds are obtained for ¦∝_Qƒdμ − ƒ(x₀)¦. They sometimes lead to sharp inequalities which are attained for particular μ and ƒ. These estimates are better than the corresponding ones found in the literature.     

On maximum length convolutional codes under a trellis complexity constraint

We look at convolutional codes with maximum possible code length for prescribed redundancy, conditioned on constraints on the free distance and on the bit-oriented trellis state complexity. Rate (n−1)/n codes have been studied to some extent in the literature, but more general rates have not been studied much. In this work we consider convolutional codes of rate (n−r)/n, r⩾1. Explicit construction techniques for free distance d_free=3 and 4 codes are described. For codes with r=2, an efficient exhaustive search algorithm is outlined. For the more general case with r⩾2, a heuristic approach is suggested. Several new codes were found for r=1 and in particular for r=2 and 3. Compared to previously known codes of similar free distance and complexity constraints, the new codes have either strictly higher rate, or the same rate but smaller low distance multiplicities.     

Young measure solutions of a class of forward-backward diffusion equations

This paper is devoted to Young measure solutions of a class of forward-backward diffusion equations. Inspired by the idea from a recent work of Demoulini, we first discuss the regular case by introducing the Young measure solutions and prove the existence for such solutions, and then approximate the extreme case by the approach of regularization and establish the existence of Young measure solutions in the class of functions with bounded variation.     

On the second solution to a critical growth Robin problem

We investigate the existence of the second mountain-pass solution to a Robin problem, where the equation is at critical growth and depends on a positive parameter λ. More precisely, we determine existence and nonexistence regions for this type of solutions, depending both on λ and on the parameter in the boundary conditions.