### A study on gradient blow up for viscosity solutions of fully nonlinear, uniformly elliptic equations

Bernhard Kawohl & Nickolai Kutev
We investigate sharp conditions for boundary and interior gradient estimates of continuous viscosity solutions to fully nonlinear, uniformly elliptic equations under Dirichlet boundary conditions. When these conditions are violated, there can be blow up of the gradient in the interior or on the boundary of the domain. In particular we derive results on local and global Lipschitz continuity of continuous viscosity solutions. Lipschitz regularity near the boundary allows us to predict when the Dirichlet condition...

### Realizing Spaces as Classifying Spaces

Gregory Lupton & Samuel Bruce Smith
Which spaces occur as a classifying space for fibrations with a given fibre? We address this question in the context of rational homotopy theory. We construct an infinite family of finite complexes realized (up to rational homotopy) as classifying spaces. We also give several non-realization results, including the following: the rational homotopy types of $\mathbb{C}P^2$ and $S^4$ are not realized as the classifying space of any simply connected, rational space with finite-dimensional homotopy groups.

### A categorical model for the virtual braid group

Louis H. Kauffman & Sofia Lambropoulou

### Holomorphic automorphic forms and cohomology

Roelof W. Bruggeman, Yŏng-Ju Ch'oe & Nikolaos Diamantis
We investigate the correspondence between holomorphic automorphic forms on the upper half-plane with complex weight and parabolic cocycles. For integral weights at least 2 this correspondence is given by the Eichler integral. We use Knopp’s generalization of this integral to real weights, and apply it to complex weights that are not an integer at least 2. We show that for these weights the generalized Eichler integral gives an injection into the first cohomology group with...

### A construction of hyperbolic coxeter groups

Damian Osajda
We give a simple construction of Gromov hyperbolic Coxeter groups of arbitrarily large virtual cohomological dimension. Our construction provides new examples of such groups. Using this one can construct e.g. new groups having some interesting asphericity properties.

### Some Combinatorial Identities Related to Commuting Varieties and Hilbert Schemes

Gwyn Bellamy & Victor Ginzburg
In this article we explore some of the combinatorial consequences of recent results relating the isospectral commuting variety and the Hilbert scheme of points in the plane.

### Quantities that frequency-dependent selection maximizes

Carlo Matessi & Kristian Schneider
We consider a model of frequency-dependent selection, to which we refer as the Wildcard Model, that accommodates as particular cases a number of diverse models of biologically specific situations. Two very different particular models (Lessard, 1984; Bürger, 2005; Schneider, 2006), subsumed by the Wildcard Model, have been shown in the past to have a Lyapunov functions (LF) under appropriate genetic assumptions. We show that the Wildcard Model: (i) in continuous time is a generalized gradient...

Sophie Morier-Genoud & Valentin Ovsienko
We study the notion of $\Gamma$-graded commutative algebra for an arbitrary abelian group $\Gamma$. The main examples are the Clifford algebras already treated in [2]. We prove that the Clifford algebras are the only simple finite-dimensional associative graded commutative algebras over $\mathbb{R}$ or $\mathbb{C}$. Our approach also leads to non-associative graded commutative algebras extending the Clifford algebras.

### Categoric Aspects of Authentication

Jeroen Schillewaert & Koen Thas

### Strongly Consistent Density Estimation of Regression Redidual

László Györfi & Harro Walk
Consider the regression problem with a response variable $Y$ and with a $d$-dimensional feature vector $X$. For the regression function $m(x) = \mathbb{E}\{Y|X = x\}$, this paper investigates methods for estimating the density of the residual $Y -m(X)$ from independent and identically distributed data. For heteroscedastic regression, we prove the strong universal (density-free) $L_1$-consistency of a recursive and a nonrecursive kernel density estimate based on a regression estimate.

### Numerical Invariants and Moduli Spaces for Line Arrangements

Alexandru Dimca, Denis Ibadula & Daniela Anca Măcinic
Using several numerical invariants, we study a partition of the space of line arrangements in the complex projective plane, given by the intersection lattice types. We offer also a new characterization of the free plane curves using the Castelnuovo-Mumford regularity of the associated Milnor/Jacobian algebra.

### Cryptanalysis of Public-key Cryptosystems Based on Algebraic Geometry Codes

Irene Márquez-Corbella, Edgar Martínez-Moro & Ruud Pellikaan
This paper addresses the question of retrieving the triple $(\mathcal{X},\mathcal{P},\mathcal{E})$ from the algebraic geometry code $\mathcal{C}_L(\mathcal{X},\mathcal{P},\mathcal{E})$, where $\mathcal{X}$ is an algebraic curve over the finite field $\mathbb{F}_q, \mathcal{P}$ is an $n$-tuple of $\mathbb{F}_q$-rational points on $\mathcal{X}$ and $E$ is a divisor on $\mathcal{X}$. If deg($E$) $\geq 2g + 1$ where $g$ is the genus of $\mathcal{X}$, then there is an embedding of $\mathcal{X}$ onto $\mathcal{Y}$ in the projective space of the linear series of the...

### Extremal configurations of polygonal linkages

Giorgi Khimshiashvili, Gaiane Panina, Dirk Siersma & Alena Zhukova

### The Index of Singular Zeros of Harmonic Mappings of Anti-Analytic Degree One

Robet Luce & Olivier Sète
We study harmonic mappings of the form $f(z) = h(z) - \overline{z}$, where $h$ is an analytic function. In particular we are interested in the index (a generalized multiplicity) of the zeros of such functions. Outside the critical set of $f$, where the Jacobian of $f$ is non-vanishing, it is known that this index has similar properties as the classical multiplicity of zeros of analytic functions. Little is known about the index of zeros on...

### Positivity of the T-system cluster algebra

Philippe Di Francesco & Rinat Kedem
We give the path model solution for the cluster algebra variables of the $A_r$ T-system with generic boundary conditions. The solutions are partition functions of (strongly) non-intersecting paths on weighted graphs. The graphs are the same as those constructed for the $Q$-system in our earlier work, and depend on the seed or initial data in terms of which the solutions are given. The weights are “time-dependent” where “time” is the extra parameter which distinguishes the...

### Definable orthogonality classes in accessible categories are small

Joan Bagaria, Carles Casacuberta, Adrian R. D. Mathias & Jiří Rosický
We lower substantially the strength of the assumptions needed for the validity of certain results in category theory and homotopy theory which were known to follow from Vopenka's principle. We prove that the necessary large-cardinal hypotheses depend on the complexity of the formulas defining the given classes, in the sense of the Lévy hierarchy. For example, the statement that, for a class $\mathcal{S}$ of morphisms in an accessible category $\mathcal{C}$, the orthogonal class of objects...

### Composition of Irreducible Morphisms in Quasi-Tubes

Claudia Chaio & Piotr Malicki
We study the composition of irreducible morphisms between indecomposable modules lying in quasi-tubes of the Auslander-Reiten quivers of artin algebras $A$ in relation with the powers of the radical of their module category mod $A$.

### A 3-local identification of the alternating group of degree 8, the McLaughlin simple group and their automorphism groups

Christopher Parker & Peter Rowley
In this article we give 3-local characterizations of the alternating and symmetric groups of degree 8 and use these characterizations to recognize the sporadic simple group discovered by McLaughlin from its 3-local subgroups.

### Virtual Polytopes

Gaiane Panina & Ileana Streinu
Originating in diverse branches of mathematics, from polytope algebra and toric varieties to the theory of stressed graphs, virtual polytopes represent a natural algebraic generalization of convex polytopes. Introduced as the Grothendick group associated to the semigroup of convex polytopes, they admit a variety of geometrizations. A selection of applications demonstrates their versatility.

### Regularity and energy conservation for the compressible Euler equations

Eduard Feireisl, Piotr Gwiazda, Agnieszka Swierczewska-Gwiazda & Emil Wiedemann
We give sufficient conditions on the regularity of solutions to the inhomogeneous incompressible Euler and the compressible isentropic Euler systems in order for the energy to be conserved. Our strategy relies on commutator estimates similar to those employed by P. Constantin et al. for the homogeneous incompressible Euler equations.

### Crystal energy functions via the charge in types A and C

Cristian Lenart & Annelore Schilling
The Ram-Yip formula for Macdonald polynomials (at $t=0$) provides a statistic which we call charge. In types $A$ and $C$ it can be defined on tensor products of Kashiwara-Nakashima single column crystals. In this paper we prove that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of...

### Supertropical linear algebra

Zur Izhakian, Manfred Knebusch & Louis Rowen
The objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of "ghost surpasses." Special attention is paid to the various notions of "base," which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas the number of elements in a d-base may vary according to the d-base, it is shown that when an s-base exists,...

### Multiple Bernoulli series and volumes of moduli spaces of flat bundles over surfaces

Maria Welleda Baldoni, Arzu Boysal & Michèle Vergne
Using Szenes formula for multiple Bernoulli series, we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also multiple zeta values.

### Positive Margins and Primary Decomposition

Thomas Kahle, Johannes Rauh & Seth Sullivant
We study random walks on contingency tables with fixed marginals, corresponding to a (log-linear) hierarchical model. If the set of allowed moves is not a Markov basis, then there exist tables with the same marginals that are not connected. We study linear conditions on the values of the marginals that ensure that all tables in a given fiber are connected. We show that many graphical models have the positive margins property, which says that all...

### Geometric flows and 3-manifolds

Gerhard Huisken
The current article arose from a lecture1 given by the author in October 2005 on the work of R. Hamilton and G. Perelman on Ricci-flow and explains central analytical ingredients in geometric parabolic evolution equations that allow the application of these flows to geometric problems including the Uniformisation Theorem and the proof of the Poincare conjecture. Parabolic geometric evolution equations of second order are non-linear extensions of the ordinary heat equation to a geometric setting,...

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