New Congruences Modulo 2, 4, and 8 for the Number of Tagged Parts Over the Partitions with Designated Summands

Recently, Lin introduced two new partition functions PD$_t(n)$ and PDO$_t(n)$, which count the total number of tagged parts over all partitions of $n$ with designated summands and the total number of tagged parts over all partitions of $n$ with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for PD$_t(n)$ and PDO$_t(n)$, and conjectured some congruences modulo 8. Very recently, Adansie, Chern, and Xia found two new infinite families of congruences modulo 9 for PD$_t(n)$. In this paper, we prove the congruences modulo 8 conjectured by Lin and also find many new congruences and infinite families of congruences modulo some small powers of 2.Comment: 19 page

The LHC Grid Challenge

Proc. on Lin

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On the mixing properties of piecewise expanding maps under composition with permutations, II: Maps of non-constant orientation

For an integer $m \geq 2$, let $\mathcal{P}_m$ be the partition of the unit interval $I$ into $m$ equal subintervals, and let $\mathcal{F}_m$ be the class of piecewise linear maps on $I$ with constant slope $\pm m$ on each element of $\mathcal{P}_m$. We investigate the effect on mixing properties when $f \in \mathcal{F}_m$ is composed with the interval exchange map given by a permutation $\sigma \in S_N$ interchanging the $N$ subintervals of $\mathcal{P}_N$. This extends the work in a previous paper [N.P. Byott, M. Holland and Y. Zhang, DCDS, {\bf 33}, (2013) 3365--3390], where we considered only the "stretch-and-fold" map $f_{sf}(x)=mx \bmod 1$.Comment: 27 pages 6 figure

A Tribute to Linda Buettner

On April 26, 2012, the gerontological practice, education, research, and care communities lost one of their foremost members with the passing of Dr. Linda (Lin) Lee Buettner, 56. Lin and her partner, Sue Fitzsimmons, were frequent and valued contributors, editorial board members, and reviewers for both the Journal of Gerontological Nursing and Research in Gerontological Nursing, for many years

The minimality of the map x/|x| for weighted energy

In this paper, we investigate the minimality of the map $\frac{x}{\|x\|}$ from the euclidean unit ball $\mathbf{B}^n$ to its boundary $\mathbb{S}^{n-1}$ for weighted energy functionals of the type $E\_{p,f}= \int\_{\mathbf{B}^n}f(r)\|\nabla u\|^p dx$, where $f$ is a non-negative function. We prove that in each of the two following cases: i) $p=1$ and $f$ is non-decreasing, i)) $p$ is an integer, $p \leq n-1$ and $f= r^{\alpha}$ with $\alpha \geq 0$, the map $\frac{x}{\|x\|}$ minimizes $E\_{p,f}$ among the maps in $W^{1,p}(\mathbf{B}^n, \mathbb{S}^{n-1})$ which coincide with $\frac{x}{\|x\|}$ on $\partial \mathbf{B}^n$. We also study the case where $f(r)= r^{\alpha}$ with $-n+2 < \alpha < 0$ and prove that $\frac{x}{\|x\|}$ does not minimize $E\_{p,f}$ for $\alpha$ close to $-n+2$ and when $n \geq 6$, for $\alpha$ close to $4-n$