3,527 Works

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,...

Time and band limiting for matrix valued functions, an example

F. A. Grünbaum, I. Pacharoni & Ignacio Nahuel Zurrián
The main purpose of this paper is to extend to a situation involving matrix valued orthogonal polynomials and spherical functions, a result that traces its origin and its importance to work of Claude Shannon in laying the mathematical foundations of information theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak. To our knowledge, this is the first example showing in a non-commutative setup that a bispectral property implies...

Quotients of Index Two and General Quotients in a Space of Orderings

Pawel Gladki & Murray Marshall
In this paper we investigate quotient structures and quotient spaces of a space of orderings arising from subgroups of index two. We provide necessary and sufficient conditions for a quotient structure to be a quotient space that, among other things, depend on the stability index of the given space. The case of the space of orderings of the field $\mathbb{Q}(x)$ is particularly important, since then the theory developed simplifies significantly. A part of the theory...

Higher Finiteness Properties of Reductive Arithmetic Groups in Positive Characteristic: the Rank Theorem

Karl-Uwe Bux, Ralf Köhl & Stefan Witzel
We show that the finiteness length of an $S$-arithmetic subgroup $\Gamma$ in a noncommutative isotropic absolutely almost simple group $\mathcal{G}$ over a global function field is one less than the sum of the local ranks of $\mathcal{G}$ taken over the places in $S$. This determines the finiteness properties for arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups. Our main tool is Behr-Harder reduction theory which we recast...

Monoid valuations and value ordered supervaluations

Zur Izhakian, Manfred Knebusch & Louis Rowen
We complement two papers on supertropical valuation theory ([IKR1], [IKR2]) by providing natural examples of m-valuations (= monoid valuations), after that of supervaluations and transmissions between them. The supervaluations discussed have values in totally ordered supertropical semirings, and the transmissions discussed respect the orderings. Basics of a theory of such semirings and transmissions are developed as far as needed.

A note on delta hedging in markets with jumps

Aleksandar Mijatović & Mikhail A. Urusov
Modelling stock prices via jump processes is common in financial markets. In practice, to hedge a contingent claim one typically uses the so-called delta-hedging strategy. This strategy stems from the Black–Merton–Scholes model where it perfectly replicates contingent claims. From the theoretical viewpoint, there is no reason for this to hold in models with jumps. However in practice the delta-hedging strategy is widely used and its potential shortcoming in models with jumps is disregarded since such...

Ghost Algebras of Double Burnside Algebras via Schur Functors

Robert Boltje & Susanne Danz
For a finite group $G$, we introduce a multiplication on the $\mathbb{Q}$-vector space with basis $\mathscr{S}_{G\times G}$, the set of subgroups of ${G \times G}$. The resulting $\mathbb{Q}$-algebra $\tilde{A}$ can be considered as a ghost algebra for the double Burnside ring $B(G,G)$ in the sense that the mark homomorphism from $B(G,G)$ to $\tilde{A}$ is a ring homomorphism. Our approach interprets $\mathbb{Q}B(G,G)$ as an algebra $eAe$, where $A$ is a twisted monoid algebra and $e$ is...

On the Directionally Newton-non-degenerate Singularities of Complex Hypersurfaces

Dmitry Kerner
We introduce a minimal generalization of Newton-non-degenerate singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called directionally Newton-non-degenerate if the local embedded topological singularity type can be restored from a collection of Newton diagrams. A singularity that is not directionally Newton-non-degenerate is called essentially Newton-degenerate. For plane curves we give an explicit and simple characterization of directionally Newton-non-degenerate singularities, for hypersurfaces we give some examples. Then we treat the question: is Newton-non-degenerate or...

Stein's method for dependent random variables occuring in statistical mechanics

Peter Eichelsbacher & Matthias Löwe
We obtain rates of convergence in limit theorems of partial sums $S_n$ for certain sequences of dependent, identically distributed random variables, which arise naturally in statistical mechanics, in particular, in the context of the Curie-Weiss models. Under appropriate assumptions there exists a real number $\alpha$, a positive number $\mu$, and a positive integer $k$ such that $(S_n-n\alpha)/n^{1-1/2k}$ converges weakly to a random variable with density proportional to $exp(-\mu|x|^{2k}/(2k)!)$. We develop Stein's method for exchangeable pairs...

A Generalization of the Discrete Version of Minkowski’s Fundamental Theorem

Bernardo González & Matthias Henze
One of the most fruitful results from Minkowski’s geometric viewpoint on number theory is his so called 1st Fundamental Theorem. It provides an optimal upper bound for the volume of an o-symmetric convex body whose only interior lattice point is the origin. Minkowski also obtained a discrete analog by proving optimal upper bounds on the number of lattice points in the boundary of such convex bodies. Whereas the volume inequality has been generalized to any...

Nonlinear matroid optimization and experimental design

Jon Lee, Shmuel Onn, Robert Weismantel, Yael Berstein, Hugo Maruri-Aguilar, Eva Riccomagno & Henry P. Wynn
We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail.

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