# 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.

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Bethe Ansatz for Yangian Invariants: Towards Super Yang-Mills Scattering Amplitudes. (arXiv:1312.1693v1 [math-ph])

We propose that Baxter's Z-invariant six-vertex model at the rational gl(2) point on a planar but in general not rectangular lattice provides a way to study Yangian invariants. These are identified with eigenfunctions of certain monodromies of an auxiliary inhomogeneous spin chain. As a consequence they are special solutions to the eigenvalue problem of the associated transfer matrix. Excitingly, this allows to construct them using Bethe ansatz techniques. Conceptually, our construction generalizes to general (super) Lie algebras and general representations. Here we present the explicit form of sample invariants for totally symmetric, finite-dimensional representations of gl(n) in terms of oscillator algebras. In particular, we discuss invariants of three- and four-site monodromies that can be understood respectively as intertwiners of the bootstrap and Yang-Baxter equation. We state a set of functional relations significant for these representations of the Yangian and discuss their solutions in terms of Bethe roots. They arrange themselves into exact strings in the complex plane. In addition, it is shown that the sample invariants can be expressed analogously to Grassmannian integrals. This aspect is closely related to a recent on-shell formulation of scattering amplitudes in planar N=4 super Yang-Mills theory.

Remote Filters and Discretely Generated Spaces. (arXiv:1312.1701v1 [math.GN])

Authors: Rodrigo Hernández-Gutiérrez

Alas, Junqueira and Wilson asked whether there is a discretely generated locally compact space whose one point compactification is not discretely generated and gave a consistent example using CH. Their construction uses a remote filter in $\omega\times{}^{\omega}2$ with a base of order type $\omega_1$ when ordered modulo compact subsets. In this paper we study the existence and preservation (under forcing extension) of similar types of filters, mainly using small uncountable cardinals. With these results we show that the CH example can be constructed in more general situations.

Generalized Taylor-Duffy Method for Efficient Evaluation of Galerkin Integrals in Boundary-Element Method Computations. (arXiv:1312.1703v1 [physics.comp-ph])

We present a generic technique, automated by computer-algebra systems and available as open-source software \cite{scuff-em}, for efficient numerical evaluation of a large family of singular and nonsingular 4-dimensional integrals over triangle-product domains, such as those arising in the boundary-element method (BEM) of computational electromagnetism. To date, practical implementation of BEM solvers has often required the aggregation of multiple disparate integral-evaluation schemes to treat all of the distinct types of integrals needed for a given BEM formulation; in contrast, our technique allows many different types of integrals to be handled by the \emph{same} algorithm and the same code implementation. Our method is a significant generalization of the Taylor--Duffy approach \cite{Taylor2003,Duffy1982}, which was originally presented for just a single type of integrand; in addition to generalizing this technique to a broad class of integrands, we also achieve a significant improvement in its efficiency by showing how the \emph{dimension} of the final numerical integral may often be reduced by one. In particular, if $n$ is the number of common vertices between the two triangles, in many cases we can reduce the dimension of the integral from $4-n$ to $3-n$, obtaining a closed-form analytical result for $n=3$ (the common-triangle case).

A Whitney map onto the Long Arc. (arXiv:1312.1704v1 [math.GN])

Authors: Rodrigo Hernández-Gutiérrez

In a recent paper, Garc\'{\i}a-Velazquez has extended the notion of Whitney map to include maps with non-metrizable codomain and left open the question of whether there is a continuum that admits such a Whitney map. In this paper, we consider two examples of hereditarily indecomposable, chainable continua of weight $\omega_1$ constructed by Michel Smith; we show that one of them admits a Whitney function onto the long arc and the other admits no Whitney maps at all.

Wijsman hyperspaces of non-separable metric spaces. (arXiv:1312.1705v1 [math.GN])

Given a metric space $\langle X,\rho \rangle$, consider its hyperspace of closed sets $CL(X)$ with the Wijsman topology $\tau_{W(\rho)}$. It is known that $\langle{CL(X),\tau_{W(\rho)}}\rangle$ is metrizable if and only if $X$ is separable and it is an open question by Di Maio and Meccariello whether this is equivalent to $\langle{CL(X),\tau_{W(\rho)}}\rangle$ being normal. In this paper we prove that if the weight of $X$ is a regular uncountable cardinal and $X$ is locally separable, then $\langle{CL(X),\tau_{W(\rho)}}\rangle$ is not normal. We also solve some questions by Cao, Junnilla and Moors regarding isolated points in Wijsman hyperspaces.

Swapping Variables for High-Dimensional Sparse Regression from Correlated Measurements. (arXiv:1312.1706v1 [math.ST])

Authors: Divyanshu Vats, Richard G. Baraniuk

We consider the high-dimensional sparse linear regression problem of accurately estimating a sparse vector using a small number of linear measurements that are contaminated by noise. It is well known that standard computationally tractable sparse regression algorithms, such as the Lasso, OMP, and their various extensions, perform poorly when the measurement matrix contains highly correlated columns. We develop a simple greedy algorithm, called SWAP, that iteratively swaps variables until a desired loss function cannot be decreased any further. SWAP is surprisingly effective in handling measurement matrices with high correlations. In particular, we prove that (i) SWAP outputs the true support, the location of the non-zero entries in the sparse vector, when initialized with the true support, and (ii) SWAP outputs the true support under a relatively mild condition on the measurement matrix when initialized with a support other than the true support. These theoretical results motivate the use of SWAP as a wrapper around various sparse regression algorithms for improved performance. We empirically show the advantages of using SWAP in sparse regression problems by comparing SWAP to several state-of-the-art sparse regression algorithms.

Focus-focus singularities in classical mechanics. (arXiv:1312.1708v1 [math.DS])

Authors: Gleb Smirnov

In this paper the local singularities of integrable Hamiltonian systems with two degrees of freedom are studied. The topological obstruction to the existence of focus-focus singularity with given complexity was found. It has been showed that only simple focus-focus singularities can appear in a typical mechanical system. The model examples of mechanical systems with complex focus-focus singularity are given.

Heaps and unpointed stable homotopy theory. (arXiv:1312.1709v1 [math.AT])

Authors: Lukáš Vokřínek

In this paper, we show how certain stability phenomena'' in unpointed model categories provide the sets of homotopy classes with the structure of abelian heaps, i.e. abelian groups without a choice of a zero. In contrast with the classical situation of stable (pointed) model categories, these sets can be empty.

## ScienceDaily: Mathematics News

The oracle of the T cell
A new online platform predicts how the human immune system reacts to foreign substances.
In the case of wholesale food distributors, it's all about location
In all but the shortest supply chains, food travels through wholesale distribution centers on its way from farm to consumer, and the location of these distributors can have a big impact on the efficiency of a food system. Now, a new mathematical model can help business owners and policy makers determine the optimal locations for such distributors.
Can iPads help students learn science? Yes, study shows
A new study shows that students grasp the unimaginable emptiness of space more effectively when they use iPads to explore 3-D simulations of the universe, compared to traditional classroom instruction.
Study highlights massive benefits of HIV treatment in South Africa
Antiretroviral therapy (ART) for the treatment of HIV infection has saved 2.8 million years of life in South Africa since 2004 and is projected to save an additional 15.1 million years of life by 2030, according to a new study published. The analysis suggests these dramatic benefits could be even greater if more aggressive HIV testing and treatment strategies are implemented.
Art could help create a better 'STEM' student
Scientists have focused on how to incorporate creativity into STEM education with the implication that doing so will increase the quality of STEM graduates. STEM studies are about problem solving, and creative endeavors are exercises in problem solving, experts say.
Forget the needle; consider the haystack
Computer scientists have developed a method to uncover hidden patterns in huge data collections. Using a mathematical method that calculates the likelihood of a pattern repeating throughout a subset of data, the researchers have been able to cut dramatically the time needed to find patterns in large collections of information such as social networks.