Spinal compression fractures: no additional pain relief with use of back braces

A critical appraisal and clinical application of Li M, Law SW, Cheng J, Kee HM, Wong MS. A comparison study on the efficacy of SpinoMed® and soft lumbar orthosis for osteoporotic vertebral fracture. J. Prosthet. Orthot. Int. 2015;39(4):270-276. doi: 10.1177/030936461452820

On Searching a Table Consistent with Division Poset

Suppose $P_n=\{1,2,...,n\}$ is a partially ordered set with the partial order defined by divisibility, that is, for any two distinct elements $i,j\in P_n$ satisfying $i$ divides $j$, $i<_{P_n} j$. A table $A_n=\{a_i|i=1,2,...,n\}$ of distinct real numbers is said to be \emph{consistent} with $P_n$, provided for any two distinct elements $i,j\in \{1,2,...,n\}$ satisfying $i$ divides $j$, $a_i< a_j$. Given an real number $x$, we want to determine whether $x\in A_n$, by comparing $x$ with as few entries of $A_n$ as possible. In this paper we investigate the complexity $\tau(n)$, measured in the number of comparisons, of the above search problem. We present a $\frac{55n}{72}+O(\ln^2 n)$ search algorithm for $A_n$ and prove a lower bound $({3/4}+{17/2160})n+O(1)$ on $\tau(n)$ by using an adversary argument.Comment: 16 pages, no figure; same results, representation improved, add reference

Algebra diagrams: a HANDi introduction

A diagrammatic notation for algebra is presented – Hierarchical Al- gebra Network Diagrams, HANDi. The notation uses a 2D network notation with systematically designed icons to explicitly and coherently encode the fun- damental concepts of algebra. The structure of the diagrams is described and the rules for making derivations are presented. The key design features of HANDi are discussed and compared with the conventional formula notation in order demonstrate that the new notation is a more logical codification of intro- ductory algebra

COUPLED-CLUSTER CALCULATIONS FOR LOW-LYING ELECTRONIC STATES OF HEAVY-METAL CONTAINING MOLECULES

Coupled-cluster calculations of low-lying electronic states for heavy-metal containing diatomic molecules (e.g., PtH, ThO$^+$, ThN, BaO$^+$, CsF$^+$) are reported. Recently-developed relativistic quantum-chemical techniques have been used, including an atomic mean-field approach for efficent construction of spin-orbit integrals [1], a perturbative approach for treating spin-orbit coupling within exact-two-component equation-of-motion coupled-cluster methods [2], and a new implementation of two-component coupled-cluster methods for non-perturbative treatments of spin-orbit coupling [3]. Bond lengths, vibrational frequencies, and dipole moments of these molecules containing heavy metals are compared with experimental data to assess the accuracy and usefulness of the computational methods. \begin{thebibliography}{comment} \bibitem{Ref1} J. Liu and L. Cheng, J. Chem. Phys. submitted. \bibitem{Ref2} L. Cheng, F. Wang, J. F. Stanton, and J. Gauss, J. Chem. Phys. \textbf{148}, 044108 (2018). \bibitem{Ref3} J. Liu, Y. Shen, A. Asthana, and L. Cheng, J. Chem. Phys. \textbf{148}, 034106 (2018). \end{thebibliography

Weighted Shift Matrices: Unitary Equivalence, Reducibility and Numerical Ranges

An $n$-by-$n$ ($n\ge 3$) weighted shift matrix $A$ is one of the form [{array}{cccc}0 & a_1 & & & 0 & \ddots & & & \ddots & a_{n-1} a_n & & & 0{array}], where the $a_j$'s, called the weights of $A$, are complex numbers. Assume that all $a_j$'s are nonzero and $B$ is an $n$-by-$n$ weighted shift matrix with weights $b_1,..., b_n$. We show that $B$ is unitarily equivalent to $A$ if and only if $b_1... b_n=a_1...a_n$ and, for some fixed $k$, $1\le k \le n$, $|b_j| = |a_{k+j}|$ ($a_{n+j}\equiv a_j$) for all $j$. Next, we show that $A$ is reducible if and only if $A$ has periodic weights, that is, for some fixed $k$, $1\le k \le \lfloor n/2\rfloor$, $n$ is divisible by $k$, and $|a_j|=|a_{k+j}|$ for all $1\le j\le n-k$. Finally, we prove that $A$ and $B$ have the same numerical range if and only if $a_1...a_n=b_1...b_n$ and $S_r(|a_1|^2,..., |a_n|^2)=S_r(|b_1|^2,..., |b_n|^2)$ for all $1\le r\le \lfloor n/2\rfloor$, where $S_r$'s are the circularly symmetric functions.Comment: 27 page