# University of Wisconsin–Milwaukee

## Mathematical Sciences

The news feeds below are not published by the Mathematical Sciences Department at UW-Milwaukee, but we hope you find them informative.

Point process with last-arrival-time dependent intensity and 1-dimensional incompressible fluid system with evaporation. (arXiv:1409.5117v1 [math.AP])

Authors: Tetsuya Hattori

We consider an infinite system of quasilinear first-order partial differential equations, generalized to contain spacial integration, which describes an incompressible fluid mixture of infinite components in a line segment whose motion is driven by unbounded and space-time dependent evaporation rates. We prove unique existence of the solution to the initial-boundary value problem, with conservation-of-fluid condition at the boundary. The proof uses a map on the space of collection of characteristics, and a representation based on a non-Markovian point process with last-arrival-time dependent intensity.

A new Gershgorin-type result for the localisation of the spectrum of matrices. (arXiv:1409.5133v1 [math.SP])

We present a Gershgorin's type result on the localisation of the spectrum of a matrix. Our method is elementary and relies upon the method of Schur complements, furthermore it outperforms the one based on the Cassini ovals of Ostrovski and Brauer. Furthermore, it yields estimates that hold without major differences in the cases of both scalar and operator matrices. Several refinements of known results are obtained.

Stability Analysis for Magnetic Resonance Elastography. (arXiv:1409.5138v1 [math.AP])

Authors: Habib Ammari, Alden Waters, Hai Zhang

We consider the inverse problem of finding unknown elastic parameters from internal measurements of displacement fields for tissues. The measurements are made on the entirety of a smooth domain. Since tissues can be modeled as quasi-incompressible fluids, we examine the Stokes system and consider only the recovery of shear modulus distributions. Our main result is to establish Lipschitz stable estimates on the shear modulus distributions from internal measurements of displacement fields. These estimates imply convergence of a numerical scheme known as the Landweber iteration scheme for reconstructing the shear modulus distributions.

The ADMM penalized decoder for LDPC codes. (arXiv:1409.5140v1 [cs.IT])

Authors: Xishuo Liu, Stark C. Draper

Linear programming (LP) decoding for low-density parity-check (LDPC) codes proposed by Feldman et al. is shown to have theoretical guarantees in several regimes and empirically is not observed to suffer from an error floor. However at low signal-to-noise ratios (SNRs), LP decoding is observed to have worse error performance than belief propagation (BP) decoding. In this paper, we seek to improve LP decoding at low SNRs while still achieving good high SNR performance. We first present a new decoding framework obtained by trying to solve a non-convex optimization problem using the alternating direction method of multipliers (ADMM). This non-convex problem is constructed by adding a penalty term to the LP decoding objective. The goal of the penalty term is to make "pseudocodewords", which are the non-integer vertices of the LP relaxation to which the LP decoder fails, more costly. We name this decoder class the "ADMM penalized decoder". In our simulation results, the ADMM penalized decoder with $\ell_1$ and $\ell_2$ penalties outperforms both BP and LP decoding at all SNRs. For high SNR regimes where it is infeasible to simulate, we use an instanton analysis and show that the ADMM penalized decoder has better high SNR performance than BP decoding. We also develop a reweighted LP decoder using linear approximations to the objective with an $\ell_1$ penalty. We show that this decoder has an improved theoretical recovery threshold compared to LP decoding. In addition, we show that the empirical gain of the reweighted LP decoder is significant at low SNRs.

ADMM LP decoding of non-binary LDPC codes in $\mathbb{F}_{2^m}$. (arXiv:1409.5141v1 [cs.IT])

Authors: Xishuo Liu, Stark C. Draper

In this paper, we develop efficient decoders for non-binary low-density parity-check (LDPC) codes using the alternating direction method of multipliers (ADMM). We apply ADMM to two decoding problems. The first problem is the linear programming (LP) decoding problem. In order to develop an efficient algorithm, we focus on non-binary codes in fields of characteristic two. This allows us to use operations in $\mathbb{F}_2$ to relax further the set of constraints into a form that has a factor graph representation. Applying ADMM to the LP decoding problem yields two types of non-trivial sub-routines. The first type of sub-routine requires solving an unconstrained quadratic program. We solve this problem efficiently by leveraging new properties on constraints of the LP decoding problem. The second type of sub-routine requires Euclidean projection onto polytopes that are studied in the literature, and which can be solved efficiently using off-the-shelf techniques. These techniques scales linearly with the dimension of the vector to project. ADMM LP decoding scales linearly with block length, linearly with check degree, and quadratically with field size. The second problem we consider is a penalized LP decoding problem. This problem is obtained by incorporating a penalty term into the LP decoding objective. The ADMM algorithm for this problem requires Euclidean projection onto a polytope formed from embeddings of the non-binary single parity-check code, which can be solved by applying ADMM a second time. Empirically, this decoder significantly outperforms LP decoding at low signal-to-noise ratios.

The spectral excess theorem for distance-regular graphs having distance-$d$ graph with fewer distinct eigenvalues. (arXiv:1409.5146v1 [math.CO])

Authors: M.A. Fiol

Let $\Gamma$ be a distance-regular graph with diameter $d$ and Kneser graph $K=\Gamma_d$, the distance-$d$ graph of $\Gamma$. We say that $\Gamma$ is partially antipodal when $K$ has fewer distinct eigenvalues than $\Gamma$. In particular, this is the case of antipodal distance-regular graphs ($K$ with only two distinct eigenvalues), and the so-called half-antipodal distance-regular graphs ($K$ with only one negative eigenvalue). We provide a characterization of partially antipodal distance-regular graphs (among regular graphs with $d$ distinct eigenvalues) in terms of the spectrum and the mean number of vertices at maximal distance $d$ from every vertex. This can be seen as a general version of the so-called spectral excess theorem, which allows us to characterize those distance-regular graphs which are half-antipodal, antipodal, bipartite, or with Kneser graph being strongly regular.

A note on the Thue chromatic number of lexicographic products of graphs. (arXiv:1409.5154v1 [math.CO])

A sequence is called non-repetitive if no of its subsequences forms a repetition (a sequence $r_1,r_2,\dots,r_{2n}$ such that $r_i=r_{n+i}$ for all $1\leq i \leq n$). Let $G$ be a graph whose vertices are coloured. A colouring $\varphi$ of the graph $G$ is non-repetitive if the sequence of colours on every path in $G$ is non-repetitive. The Thue chromatic number, denoted by $\pi (G)$, is the minimum number of colours of a non-repetitive colouring of $G$. In this short note we present a general upper bound for the Thue chromatic number for the lexicographic product $G\circ H$ of graphs $G$ and $H$ with respect to some properties of the factors. This upper bound is then used to derive the exact values for $\pi(G\circ H)$ when $G$ is a complete multipartite graph and $H$ is an arbitrary graph.

A note on minimal graphs over certain unbounded domains of Hadamard manifolds. (arXiv:1409.5155v1 [math.DG])

Authors: Miriam Telichevesky

Given an unbounded domain $\Omega$ of a Hadamard manifold $M$, it makes sense to consider the problem of finding minimal graphs with prescribed continuous data on its cone-topology-boundary, i.e., on its ordinary boundary together with its asymptotic boundary. In this article it is proved that under the hypothesis that the sectional curvature of $M$ is $\le -1$ this Dirichlet problem is solvable if $\Omega$ satisfies certain convexity condition at infinity and if $\partial \Omega$ is mean convex. We also prove that mean convexity of $\partial \Omega$ is a necessary condition, extending to unbounded domains some results that are valid on bounded ones.

## Mathematics News -- ScienceDaily

Video games could dramatically streamline educational research
Scientists have figured out a dramatically easier and more cost-effective way to do research on science curriculum in the classroom -- and it could include playing video games. Called 'computational modeling,' it involves a computer 'learning' student behavior and then 'thinking' as students would. It could revolutionize the way educational research is done.
Math model designed to replace invasive kidney biopsy for lupus patients
Mathematics might be able to reduce the need for invasive biopsies in patients suffering kidney damage related to the autoimmune disease lupus. The model could also be used to monitor the effectiveness of experimental treatments for inflammation and fibrosis, researchers say.
Ebola outbreak 'out of all proportion' and severity cannot be predicted, expert says
A mathematical model that replicates Ebola outbreaks can no longer be used to ascertain the eventual scale of the current epidemic, finds new research.
Number-crunching could lead to unethical choices, says new study
Calculating the pros and cons of a potential decision is a way of decision-making. But repeated engagement with numbers-focused calculations, especially those involving money, can have unintended negative consequences.
New math and quantum mechanics: Fluid mechanics suggests alternative to quantum orthodoxy
The central mystery of quantum mechanics is that small chunks of matter sometimes seem to behave like particles, sometimes like waves. For most of the past century, the prevailing explanation of this conundrum has been what's called the "Copenhagen interpretation" -- which holds that, in some sense, a single particle really is a wave, smeared out across the universe, that collapses into a determinate location only when observed. But some founders of quantum physics -- notably Louis de Broglie -- championed an alternative interpretation, known as "pilot-wave theory," which posits that quantum particles are borne along on some type of wave. According to pilot-wave theory, the particles have definite trajectories, but because of the pilot wave's influence, they still exhibit wavelike statistics. Now a professor of applied mathematics believes that pilot-wave theory deserves a second look.
Mechanical ventilation a key indicator for pre-term children's maths problems
Both the length of time spent in hospital after birth and the use of mechanical ventilation are key indicators of reduced mathematical ability in preterm children, researchers report. Impairments in mathematic abilities are common in very preterm children. Earlier studies of children who are born very preterm (before 32 weeks of gestational age) have shown that they have a 39.4% chance of having general mathematic impairment, compared to 14.9% of those born at term (39 to 41 weeks).
Spot light on tailor-made multicyclic type of polymers
Scientists have synthesized multicyclic type of polymers for the first time offering insights for tailoring polymer properties as well as the mathematics of complex geometries.
Flu outbreak provides rare study material
Five years ago this month, one of the first U.S. outbreaks of the H1N1 virus swept through the Washington State University campus, striking some 2,000 people. A university math and biology professor has used a trove of data gathered at the time to gain insight into how only a few infected people could launch the virus's rapid spread across the university community.