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math updates on arXiv.org

+ Givental J-functions, Quantum integrable systems, AGT relation with surface operator. (arXiv:1408.4132v1 [hep-th])

Authors: Satoshi Nawata

We study 4d $\mathcal{N}=2$ gauge theories with a co-dimension-two full surface operator, which exhibit a fascinating interplay of supersymmetric gauge theories, equivariant Gromov-Witten theory and geometric representation theory. For pure Yang-Mills and $\mathcal{N}=2^*$ theory, we describe a full surface operator as the 4d gauge theory coupled to a 2d $\mathcal{N}=(2,2)$ gauge theory. By supersymmetric localizations, we present the exact partition functions of both 4d and 2d theories which satisfy integrable equations. In addition, the form of the structure constants with a semi-degenerate field in SL(N,R) WZNW model is predicted from one-loop determinants of 4d gauge theories with a full surface operator via the AGT relation.

+ Efficient geodesics and an effective algorithm for distance in the complex of curves. (arXiv:1408.4133v1 [math.GT])

Authors: Joan Birman, Dan Margalit, William Menasco

We give an algorithm for determining the distance between two vertices of the complex of curves. While there already exist such algorithms, for example by Leasure, Shackleton, and Webb, our approach is new, simple, and more effective for small distances. Our method gives a new preferred finite set of geodesics between any two vertices of the complex, called efficient geodesics, which are different from the tight geodesics introduced by Masur and Minsky.

+ MICC: A tool for computing short distances in the curve complex. (arXiv:1408.4134v1 [math.GT])

Authors: Paul Glenn, William W. Menasco, Kayla Morrell, Matthew Morse

The complex of curves $\mathcal{C}(S_g)$ of a closed orientable surface of genus $g \geq 2$ is the simplicial complex having its vertices, $\mathcal{C}^0(S_g)$, are isotopy classes of essential curves in $S_g$. Two vertices co-bound an edge of the $1$-skeleton, $\mathcal{C}^1(S_g)$, if there are disjoint representatives in $S_g$. A metric is obtained on $\mathcal{C}^0(S_g)$ by assigning unit length to each edge of $\mathcal{C}^1(S_g)$. Thus, the distance between two vertices, $d(v,w)$, corresponds to the length of a geodesic---a shortest edge-path between $v$ and $w$ in $\mathcal{C}^1 (S_g)$. Recently, Birman, Margalit and the second author introduced the concept of {\em initially efficient geodesics} in $\mathcal{C}^1(S_g)$ and used them to give a new algorithm for computing the distance between vertices. In this note we introduce the software package MICC ({\em Metric in the Curve Complex}), a partial implementation of the initially efficient geodesic algorithm. We discuss the mathematics underlying MICC and give applications. In particular, we give examples of distance four vertex pairs, for $g=2$ and 3. Previously, there was only one known example, in genus $2$, due to John Hempel.

+ Recent developments of analysis for hydrodynamic flow of nematic liquid crystals. (arXiv:1408.4138v1 [math.AP])

Authors: Fanghua Lin, Changyou Wang

The study of hydrodynamics of liquid crystal leads to many fasci- nating mathematical problems, which has prompted various interesting works recently. This article reviews the static Oseen-Frank theory and surveys some recent progress on the existence, regularity, uniqueness, and large time asymp- totic of the hydrodynamic flow of nematic liquid crystals. We will also propose a few interesting questions for future investigations.

+ Global existence of weak solutions of the nematic liquid crystal flow in dimensions three. (arXiv:1408.4146v1 [math.AP])

Authors: Fanghua Lin, Changyou Wang

For any bounded smooth domain $\Omega\subset\mathbb R^3$, we establish the global existence of a weak solution $u:\Omega\times (0,+\infty)\to\mathbb R^3\times\mathbb S^2$ of the initial-boundary value (or the Cauchy) problem of the simplified Ericksen-Leslie system (1.1) modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary (or Cauchy) data $(u_0. d_0)\in {\bf H}\times H^1(\Omega,\mathbb S^2$), with $d_0(\Omega)\subset\mathbb S^2_+$

(the upper hemisphere). Furthermore, ($u,d$) satisfies the global energy inequality (1.4).

+ Analysis of graded-index optical fibers by the spectral parameter power series method. (arXiv:1408.4147v1 [physics.optics])

Authors: Raul Castillo Perez, Vladislav V. Kravchenko, Sergii M. Torba

Spectral parameter power series (SPPS) method is a recently introduced technique for solving linear differential equations and related spectral problems. In the present work we develop an approach based on the SPPS for analysis of graded-index optical fibers. The characteristic equation of the eigenvalue problem for calculation of guided modes is obtained in an analytical form in terms of SPPS. Truncation of the series and consideration in this way of the approximate characteristic equation gives us a simple and efficient numerical method for solving the problem. Comparison with the results obtained by other available techniques reveals clear advantages of the SPPS approach, in particular, with regards to accuracy. Based on the solution of the eigenvalue problem, parameters describing the dispersion are analyzed as well.

+ Lattices with many Borcherds products. (arXiv:1408.4148v1 [math.NT])

Authors: Jan Hendrik Bruinier, Stephan Ehlen, Eberhard Freitag

We prove that there are only finitely many isometry classes of even lattices $L$ of signature $(2,n)$ for which the space of cusp forms of weight $1+n/2$ for the Weil representation of the discriminant group of $L$ is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of $L$ can be realized as the divisor of a Borcherds product. We obtain similar classification results in greater generality for finite quadratic modules.

+ Global finite energy weak solutions to the compressible nematic liquid crystal flow in dimension three. (arXiv:1408.4149v1 [math.AP])

Authors: Junyu Lin, Baishun Lai, Changyou Wang

In this paper, we consider the initial and boundary value problem of a simplified compressible nematic liquid crystal flow in $\Omega\subset\mathbb R^3$. We establish the existence of global weak solutions, provided the initial orientational director field $d_0$ lies in the hemisphere $\mathbb S^2_+$.

Mathematics News -- ScienceDaily

+ How worms crawl: mathematical model challenges traditional view
A new mathematical model for earthworms and insect larvae challenges the traditional view of how these soft bodied animals get around. Researchers say that there is a far greater role for the body's mechanical properties and the local nerves which react to the surface that the animal is traveling across.
+ How children's brains memorize math facts
As children learn basic arithmetic, they gradually switch from solving problems by counting on their fingers to pulling facts from memory. The shift comes more easily for some kids than for others, but no one knows why. Now, new brain-imaging research gives the first evidence drawn from a longitudinal study to explain how the brain reorganizes itself as children learn math facts.
+ Powerful math creates 3-D shapes from simple sketches
A new graphics system that can easily produce complex 3-D shapes from simple professional sketches will be unveiled by computer scientists. The technology has the potential to dramatically simplify how designers and artists develop new product ideas. Converting an idea into a 3-D model using current commercial tools can be a complicated and painstaking process.
+ Quantum simulators explained
Everything you ever wanted to know about quantum simulators summed up in a new review. A quantum simulator is a device that actively uses quantum effects to answer questions on model systems. This review outlines various approaches used in quantum simulators.
+ 'I can't figure out how to do this!': Active-learning techniques effective for large scale classes?
In the past 10 years an active-learning course, called Active Physics, has gradually displaced lecture-based introductory course in physics at an American university. But are active-learning techniques effective when they are scaled up to large classes? A comprehensive three-year evaluation suggests that Active Physics consistently produces more proficient students with better attitudes toward learning than the lecture courses it is replacing.
+ Preterm children do not have an increased risk for dyscalculia, new research suggests
Preterm children do not suffer from dyscalculia more often than healthy full-term children, experts say, contrary to previous studies. Unlike most other studies, the researchers took the children’s IQ into consideration.
+ A mathematical theory proposed by Alan Turing in 1952 can explain the formation of fingers
Researchers have shown that BMP and WNT proteins are the so-called 'Turing molecules' for creating embryonic fingers. Findings explain why polydactyly -- the development of extra fingers or toes -- is relatively common in humans, affecting up to one in 500 births, and confirms a fundamental theory first proposed by the founding father of computer science, Alan Turing, back in 1952.
+ Numerical learning disability: Dyscalculia linked to difficulties in reading and spelling
Between three and six percent of schoolchildren suffer from an arithmetic-related learning disability. Researchers now show that these children are also more likely to exhibit deficits in reading and spelling than had been previously suspected.