# 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.

Tests for Time Series of Counts Based on the Probability Generating Function. (arXiv:1410.6172v1 [math.ST])

We propose testing procedures for the hypothesis that a given set of discrete observations may be formulated as a particular time series of counts with a specific conditional law. The new test statistics incorporate the empirical probability generating function computed from the observations. Special emphasis is given to the popular models of integer autoregression and Poisson autoregression. The asymptotic properties of the proposed test statistics are studied under the null hypothesis as well as under alternatives. A Monte Carlo power study on bootstrap versions of the new methods is included as well as real-data examples.

Invariant torsion and G_2-metrics. (arXiv:1410.6173v1 [math.DG])

Authors: Diego Conti, Thomas Bruun Madsen

We introduce and study a notion of invariant intrinsic torsion geometry which appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S^3. This space is foliated by six-dimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is invariant under SO(3). The last condition is sufficient to imply local homogeneity of such geometries, and this allows us to give a classification. We close the circle by showing that the Bryant-Salamon metric is the unique complete metric with holonomy G_2 that arises from SO(3)-structures with invariant intrinsic torsion.

ADE Double Scaled Little String Theories, Mock Modular Forms and Umbral Moonshine. (arXiv:1410.6174v1 [hep-th])

We consider double scaled little string theory on $K3$. These theories are labelled by a positive integer $k \ge 2$ and an $ADE$ root lattice with Coxeter number $k$. We count BPS fundamental string states in the holographic dual of this theory using the superconformal field theory $K3 \times \left( \frac{SL(2,\mathbb{R})_k}{U(1)} \times \frac{SU(2)_k}{U(1)} \right) \big/ \mathbb{Z}_k$. We show that the BPS fundamental string states that are counted by the second helicity supertrace of this theory give rise to weight two mixed mock modular forms. We compute the helicity supertraces using two separate techniques: a path integral analysis that leads to a modular invariant but non-holomorphic answer, and a Hamiltonian analysis of the contribution from discrete states which leads to a holomorphic but not modular invariant answer. From a mathematical point of view the Hamiltonian analysis leads to a mixed mock modular form while the path integral gives the completion of this mixed mock modular form. We also compare these weight two mixed mock modular forms to those that appear in instances of Umbral Moonshine labelled by Niemeier root lattices $X$ that are powers of $ADE$ root lattices and find that they are equal up to a constant factor that we determine. In the course of the analysis we encounter an interesting generalization of Appell-Lerch sums and generalizations of the Riemann relations of Jacobi theta functions that they obey.

On Noncommutative Finite Factorization Domains. (arXiv:1410.6178v1 [math.RA])

A domain $R$ is said to have the finite factorization property if every nonzero non-unit element of $R$ has at least one and at most finitely many distinct factorizations up to multiplication of irreducible factors by units. Let $k$ be an algebraically closed field and let $A$ be a $k$-algebra. We show that if $A$ has an associated graded ring that is a domain with the property that the dimension of each homogeneous component is finite then $A$ is a finite factorization domain. As a corollary, we show that many classes of algebras have the finite factorization property, including Weyl algebras, enveloping algebras of finite-dimensional Lie algebras, quantum affine spaces and shift algebras. This provides a termination criterion for factorization procedures over these algebras. In addition, we give explicit upper bounds on the number of distinct factorizations of an element in terms of data from the filtration.

Evaluating Prime Power Gauss and Jacobi Sums. (arXiv:1410.6179v1 [math.NT])

We show that for any mod $p^m$ characters, $\chi_1, \dots, \chi_k,$ the Jacobi sum, $$\sum_{x_1=1}^{p^m}\dots \sum_{\substack{x_k=1\\x_1+\dots+x_k=B}}^{p^m}\chi_1(x_1)\dots \chi_k(x_k),$$ has a simple evaluation when $m$ is sufficiently large (for $m\geq 2$ if $p\nmid B$). As part of the proof we give a simple evaluation of the mod $p^m$ Gauss sums when $m\geq 2$.

Instantaneous exponential lower bound for solutions to the Boltzmann equation with Maxwellian diffusion boundary conditions. (arXiv:1410.6182v1 [math-ph])

Authors: Marc Briant

We prove the immediate appearance of an exponential lower bound, uniform in time and space, for continuous mild solutions to the full Boltzmann equation in a $C^2$ convex bounded domain with the physical Maxwellian diffusion boundary conditions, under the sole assumption of regularity of the solution. We investigate a wide range of collision kernels, with and without Grad's angular cutoff assumption. In particular, the lower bound is proven to be Maxwellian in the case of cutoff collision kernels. Moreover, these results are entirely constructive if the initial distribution contains no vacuum, with explicit constants depending only on the \textit{a priori} bounds on the solution.

Maps of the solar photospheric magnetic flux are fundamental drivers for simulations of the corona and solar wind which makes photospheric simulations important predictors of solar events on Earth. However, observations of the solar photosphere are only made intermittently over small regions of the solar surface. The Air Force Data Assimilative Photospheric Flux Transport (ADAPT) model uses localized ensemble Kalman filtering techniques to adjust a set of photospheric simulations to agree with the available observations. At the same time this information is propagated to areas of the simulation that have not been observed. ADAPT implements a local ensemble transform Kalman filter (LETKF) to accomplish data assimilation, allowing the covariance structure of the flux transport model to influence assimilation of photosphere observations while eliminating spurious correlations between ensemble members arising from a limited ensemble size. We give a detailed account of the ADAPT model and the implementation of the LETKF. Advantages of the LETKF scheme over previously implemented assimilation methods are highlighted.

Fej\'er means of Vilenkin-Fourier series. (arXiv:1410.6186v1 [math.CA])

TThe main aim of this paper is to prove that there exist a martingale $f\in H_{1/2}$ such that Fej\'er means of Vilenkin-Fourier series of the martingale $f$ is not uniformly bounded in the space $L_{1/2}.$

## Mathematics News -- ScienceDaily

An effective, cost-saving way to detect natural gas pipeline leaks
Major leaks from oil and gas pipelines have led to home evacuations, explosions, millions of dollars in lawsuit payouts and valuable natural resources escaping into the air, ground and water. Scientists say they have now developed a new software-based method that finds leaks even when they're small, which could help prevent serious incidents -- and save money for customers and industry.
Technology helps even the odds for blind students
Technology to help a blind student see math clearly and pursue a degree has been uncovered by researchers. Despite losing her vision three years ago due to complications from the flu, one study entered university last fall with the specific goal of pursuing a dual degree in mathematics and business. Technology is helping her make this a reality.
New theorem determines age distribution of populations from fruit flies to humans
The initial motivation of a new study was to estimate the age structure of a fruit fly population, the result a fundamental theorem that can help determine the age distribution of essentially any group. This emerging theorem on stationary populations shows that you can determine the age distribution of a population by looking at how long they still have to live.