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math updates on arXiv.org

+ Empirically Estimable Classification Bounds Based on a New Divergence Measure. (arXiv:1412.6534v1 [cs.IT])

Authors: Visar Berisha, Alan Wisler, Alfred O. Hero, Andreas Spanias

Information divergence functions play a critical role in statistics and information theory. In this paper we introduce a divergence function between distributions and describe a number of properties that make it appealing for classification applications. Based on an extension of a multivariate two-sample test, we identify a nonparametric estimator of the divergence that does not impose strong assumptions on the data distribution. Furthermore, we show that this measure bounds the minimum binary classification error for the case when the training and test data are drawn from the same distribution and for the case where there exists some mismatch between training and test distributions. We confirm the theoretical results by designing feature selection algorithms using the criteria from these bounds and evaluating the algorithms on a series of pathological speech classification tasks.

+ Exact integration scheme for six-node wedge element mass matrix. (arXiv:1412.6538v1 [math.NA])

Authors: Eli Hanukah

Currently, mass matrices are computed by means of sufficiently accurate numerical integration schemes. Two-point and nine-point (Gauss) quadrature remain frequently used. We derive an exact, easy to implement integration rule for six-node wedge element mass matrices based on seven points only. Both consistent and lumped mass matrices have been considered. New metric (jacobian determinant) interpolation accompanied by analytical integration permits computing effort reduction next to accuracy increase of integration rule. In addition, one and four point mass matrices integration schemes have been proposed. Accuracy superiority over equivalent schemes is established.

+ On the openness and discreteness of the mappings satisfying one inequality with respect to $p$-modulus. (arXiv:1412.6539v1 [math.CV])

Authors: Evgeny Sevost'yanov

A paper is devoted to study of topological properties of some class of space mappings. It is showed that, sense preserving mappings $f:D\rightarrow \overline{{\Bbb R}^n}$ of a domain $D\subset{\Bbb R}^n,$ $n\geqslant 2,$ satisfying some modulus inequality with respect to $p$-modulus of families of curves, are open and discrete at some restrictions on a function $Q,$ which determinate inequality mentioned above.

+ Iterated fractional Tikhonov regularization. (arXiv:1412.6540v1 [math.NA])

Authors: Davide Bianchi, Alessandro Buccini, Marco Donatelli, Stefano Serra-Capizzano

Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order. This paper provides saturation and converse results on their convergence rates. Using the same iterative refinement strategy of iterated Tikhonov regularization, new iterated fractional Tikhonov regularization methods are introduced. We show that these iterated methods are of optimal order and overcome the previous saturation results. Furthermore, nonstationary iterated fractional Tikhonov regularization methods are investigated, establishing their convergence rate under general conditions on the iteration parameters. Numerical results confirm the effectiveness of the proposed regularization iterations.

+ Game-Theoretic Analysis of the Hegselmann-Krause Model for Opinion Dynamics in Finite Dimensions. (arXiv:1412.6546v1 [cs.GT])

Authors: Seyed Rasoul Etesami, Tamer Basar

We consider the Hegselmann-Krause model for opinion dynamics and study the evolution of the system under various settings. We first analyze the termination time of the synchronous Hegselmann-Krause dynamics in arbitrary finite dimensions and show that the termination time in general only depends on the number of agents involved in the dynamics. To the best of our knowledge, that is the sharpest bound for the termination time of such dynamics that removes dependency of the termination time from the dimension of the ambient space. This answers an open question in [1] on how to obtain a tighter upper bound for the termination time. Furthermore, we study the asynchronous Hegselmann-Krause model from a novel game-theoretic approach and show that the evolution of an asynchronous Hegselmann-Krause model is equivalent to a sequence of best response updates in a well-designed potential game. We then provide a polynomial upper bound for the expected time and expected number of switching topologies until the dynamic reaches an arbitrarily small neighborhood of its equilibrium points, provided that the agents update uniformly at random. This is a step toward analysis of heterogeneous Hegselmann-Krause dynamics. Finally, we consider the heterogeneous Hegselmann-Krause dynamics and provide a necessary condition for the finite termination time of such dynamics. In particular, we sketch some future directions toward more detailed analysis of the heterogeneous Hegselmann-Krause model.

+ Stochastic partial differential equations: a rough path view. (arXiv:1412.6557v1 [math.PR])

Authors: Joscha Diehl, Peter K. Friz, Wilhelm Stannat

We discuss regular and weak solutions to rough partial differential equations (RPDEs), thereby providing a (rough path-)wise view on important classes of SPDEs. In contrast to many previous works on RPDEs, our definition gives honest meaning to RPDEs as integral equation, based on which we are able to obtain existence, uniqueness and stability results. The case of weak "rough" forward equations, may be seen as robustification of the (measure-valued) Zakai equation in the rough path sense. Feynman-Kac representation for RPDEs, in formal analogy to similar classical results in SPDE theory, play an important role.

+ Algebraic weak factorisation systems I: accessible AWFS. (arXiv:1412.6559v1 [math.CT])

Authors: John Bourke, Richard Garner

Algebraic weak factorisation systems (AWFS) refine weak factorisation systems by requiring that the assignations sending a map to its first and second factors should underlie an interacting comonad--monad pair on the arrow category. We provide a comprehensive treatment of the basic theory of AWFS---drawing on work of previous authors---and complete the theory with two main new results. The first provides a characterisation of AWFS and their morphisms in terms of their double categories of left or right maps. The second concerns a notion of cofibrant generation of an AWFS by a small double category; it states that, over a locally presentable base, any small double category cofibrantly generates an AWFS, and that the AWFS so arising are precisely those with accessible monad and comonad. Besides the general theory, numerous applications of AWFS are developed, emphasising particularly those aspects which go beyond the non-algebraic situation.

+ Algebraic weak factorisation systems II: categories of weak maps. (arXiv:1412.6560v1 [math.CT])

Authors: John Bourke, Richard Garner

We investigate the categories of weak maps associated to an algebraic weak factorisation system (AWFS) in the sense of Grandis-Tholen. For any AWFS on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the AWFS is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of "homotopy category", that freely adjoins a section for every "acyclic fibration" (=right map) of the AWFS; and using this characterisation, we give an alternate description of categories of weak maps in terms of spans with left leg an acyclic fibration. We moreover show that the 2-functor sending each AWFS on a suitable category to its cofibrant replacement comonad has a fully faithful right adjoint: so exhibiting the theory of comonads, and dually of monads, as incorporated into the theory of AWFS. We also describe various applications of the general theory: to the generalised sketches of Kinoshita-Power-Takeyama, to the two-dimensional monad theory of Blackwell-Kelly-Power, and to the theory of dg-categories.

Mathematics News -- ScienceDaily

+ Decision 'cascades' in social networks
People in social networks are often influenced by each other's decisions, resulting in a run of behaviors in which their choices become highly correlated, causing a cascade of decisions.
+ 'Microlesions' in epilepsy discovered by novel technique
Using an innovative technique combining genetic analysis and mathematical modeling with some basic sleuthing, researchers have identified previously undescribed microlesions in brain tissue from epileptic patients. The millimeter-sized abnormalities may explain why areas of the brain that appear normal can produce severe seizures in many children and adults with epilepsy.
+ Students attending summer learning programs returned to school in the fall with an advantage in math
Students attending voluntary, school district-led summer learning programs entered school in the fall with stronger mathematics skills than their peers who did not attend the programs, according to a new study.
+ Mathematicians prove the Umbral Moonshine Conjecture
Monstrous moonshine, a quirky pattern of the monster group in theoretical math, has a shadow -- umbral moonshine. Mathematicians have now proved this insight, known as the Umbral Moonshine Conjecture, offering a formula with potential applications for everything from number theory to geometry to quantum physics.
+ Basic rules for construction with a type of origami
Origami is capable of turning a simple sheet of paper into a pretty paper crane, but the principles behind it can be applied to making a microfluidic device or for storing a satellite's solar panel in a rocket's cargo bay. Researchers are turning kirigami, a related art form that allows the paper to be cut, into a technique that can be applied equally to structures on those vastly divergent length scales.
+ Physicists explain puzzling particle collisions
An anomaly spotted at the Large Hadron Collider has prompted scientists to reconsider a mathematical description of the underlying physics. By considering two forces that are distinct in everyday life but unified under extreme conditions, they have simplified one description of the interactions of elementary particles. Their new version makes specific predictions about events that future experiments should observe and could help to reveal 'new physics,' particles or processes that have yet to be discovered.
+ Theory details how 'hot' monomers affect thin-film formation
Researchers have devised a mathematical model to predict how 'hot' monomers on cold substrates affect the growth of thin films being developed for next-generation electronics.
+ Shifting boundaries and changing surfaces: Energies at work in closed flexible loop spanned by soap film
New research examines the energies at work in a closed flexible loop spanned by a soap film. While the underlying experiments are simple enough to be replicated in a kitchen sink, the research generates potentially important questions and changes how we think about different disciplines from material science to vertebrate morphogenesis.