3,527 Works

Optimal bounds for the colored Tverberg Problem

Pavle V. M. Blagojevic, Benjamin Matschke & Günter M. Ziegler
We prove a "Tverberg type" multiple intersection theorem. It strengthens the prime case of the original Tverberg theorem from 1966, as well as the topological Tverberg theorem of Bárány et al. (1980), by adding color constraints. It also provides an improved bound for the (topological) colored Tverberg problem of Bárány & Larman (1992) that is tight in the prime case and asymptotically optimal in the general case. The proof is based on relative equivariant obstruction...

Formal punctured ribbons and two-dimensional local fields

Herbert Kurke, Denis Osipov & Alexander Zheglov
We investigate formal ribbons on curves. Roughly speaking, formal ribbon is a family of locally linearly compact vector spaces on a curve. We establish a one-to-one correspondence between formal ribbons on curves plus some geometric data and some subspaces of two-dimensional local field.

A real algebra perspective on multivariate tight wavelet frames

Maria Charina, Mihai Putinar, Claus Scheiderer & Joachim Stöckler
Recent results from real algebraic geometry and the theory of polynomial optimization are related in a new framework to the existence question of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) from are interpreted in terms of hermitian sums of squares of certain nongenative trigonometric polynomials and in terms of semi-definite programming. The latter together with the results in answer...

Module Categories for Group Algebras over Commutative Rings

David J. Benson, Srikanth Iyengar, Henning Krause & Greg Stevenson
We develop a suitable version of the stable module category of a finite group $G$ over an arbitrary commutative ring $k$. The purpose of the construction is to produce a compactly generated triangulated category whose compact objects are the finitely presented $kG$-modules. The main idea is to form a localisation of the usual version of the stable module category with respect to the filtered colimits of weakly injective modules. There is also an analogous version...

Self-dual polygons and self-dual curves

Dmitrij B. Fuks & Serge Tabachnikov

A graphical interface for the Gromov-Witten theory of curves

Renzo Cavalieri, Paul Johnson, Hannah Markwig & Dhruv Ranganathan
We explore the explicit relationship between the descendant Gromov–Witten theory of target curves, operators on Fock spaces, and tropical curve counting. We prove a classical/tropical correspondence theorem for descendant invariants and give an algorithm that establishes a tropical Gromov–Witten/Hurwitz equivalence. Tropical curve counting is related to an algebra of operators on the Fock space by means of bosonification. In this manner, tropical geometry provides a convenient “graphical user interface” for Okounkov and Pandharipande’s celebrated GW/H...

Prediction and Quantification of Individual Athletic Performance

Duncan A. J. Blythe & Franz J. Király
We present a novel, quantitative view on the human athletic performance of individuals. We obtain a predictor for athletic running performances, a parsimonious model, and a training state summary consisting of three numbers, by application of modern validation techniques and recent advances in machine learning to the thepowerof10 database of British athletes’ performances (164,746 individuals, 1,417,432 performances). Our predictor achieves a low average prediction error (out-of-sample), e.g., 3.6 min on elite Marathon performances, and a...

On Local Combinatorial Formulas for Chern Classes of Triangulated Circle Bundle

Nikolai Mnev & Georgy Sharygin
Principal circle bundle over a PL polyhedron can be triangulated and thus obtains combinatorics. The triangulation is assembled from triangulated circle bundles over simplices. To every triangulated circle bundle over a simplex we associate a necklace (in combinatorial sense). We express rational local formulas for all powers of first Chern class in the terms of mathematical expectations of parities of the associated necklaces. This rational parity is a combinatorial isomorphism invariant of triangulated circle bundle...

An introduction to heavy-tailed and sibexponential distributions

Sergey Foss, Dmitrij Koršunov & Stan Zachary
This text studies heavy-tailed distributions in probability theory, and especially convolutions of such distributions. The mail goal is to provide a complete and comprehensive introduction to the theory of long-tailed and subexponential distributions which includes many novel elements and, in particular, is based on the regular use of the principle of a single big jump.

Nonlinear Multi-Parameter Eigenvalue Problems for Systems of Nonlinear Ordinary Differential Equations Arising in Electromagnetics

Lutz Angermann, Yury V. Shestopalov, Yury G. Smirnov & Vasyl V. Yatsyk
We investigate a generalization of one-parameter eigenvalue problems arising in the theory of nonlinear waveguides to a more general nonlinear multiparameter eigenvalue problem for a nonlinear operator. Using an integral equation approach, we derive functional dispersion equations whose roots yield the desired eigenvalues. The existence and distribution of roots are verified.

Dominance and Transmissions in Supertropical Valuation Theory

Zur Izhakian, Manfred Knebusch & Louis Rowen
This paper is a sequel of [IKR1], where we defined supervaluations on a commutative ring $R$ and studied a dominance relation $\Phi >= v$ between supervaluations $\varphi$ and $\upsilon$ on $R$, aiming at an enrichment of the algebraic tool box for use in tropical geometry. A supervaluation $\varphi : R \rightarrow U$ is a multiplicative map from $R$ to a supertropical semiring $U$, cf. [IR1], [IR2], [IKR1], with further properties, which mean that $\varphi$ is...

Supertropical Matrix Algebra III : Powers of Matrices and Generalized Eigenspaces

Zur Izhakian & Louis Rowen
We investigate powers of supertropical matrices, with special attention to the role of the coefficients of the supertropical characteristic polynomial (especially the supertropical trace) in controlling the rank of a power of a matrix. This leads to a Jordan-type decomposition of supertropical matrices, together with a generalized eigenspace decomposition of a power of an arbitrary supertropical matrix.

Infeasibility certificates for linear matrix inequalities

Igor Klep & Markus Schweighofer
Farkas' lemma is a fundamental result from linear programming providing linear certi cates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry. More precisely, we show that a linear matrix inequality $L(x) \succeq 0$ is infeasible if and only if $-1$ lies in the quadratic...

Enhanced Spatial Skin-Effect for Free Vibrations of a Thick Cascade Junction with \"Super Heavy\" Concentrated Masses

Grigorij A. Čečkin & Taras A. Mel'nyk
The asymptotic behavior (as $\varepsilon \to 0$) of eigenvalues and eigenfunctions of a boundary-value problem for the Laplace operator in a thick cascade junction with concentrated masses is studied. This cascade junction consists of the junction’s body and a great number $5N= \mathcal{O}(\varepsilon^{-1})$ of $\varepsilon$-alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order $\mathcal{O}(\varepsilon)$. The mass density...

Upper tails for intersection local times of random walks in supercritical dimensions

Xia Chen & Peter Mörters
We determine the precise asymptotics of the logarithmic upper tail probability of the total intersection local time of $p$ independent random walks in $\mathbb{Z}^d$ under the assumption $p(d-2)>d$. Our approach allows a direct treatment of the infinite time horizon.

A note on k[z]-Automorphisms in Two Variables

Eric Edo, Arno Van Den Essen & Stefan Maubach
We prove that for a polynomial $f \in k[x, y, z]$ equivalent are: (1)$f$ is a $k[z]$-coordinate of $k[z][x,y]$, and (2) $k[x, y, z]/(f)\cong k^[2]$ and $f(x,y,a)$ is a coordinate in $k[x,y]$ for some $a \in k$. This solves a special case of the Abhyankar-Sathaye conjecture. As a consequence we see that a coordinate $f \in k[x,y,z]$ which is also a $k(z)$-coordinate, is a $k[z]$-coordinate. We discuss a method for constructing automorphisms of $k[x, y,...

On Concentrators and Related Approximation Constants

A. V. Bondarenko, A. Prymak & D. Radchenko
Pippenger ([Pip77]) showed the existence of (6m, 4m, 3m, 6)-concentrator for each positive integer m using a probabilistic method. We generalize his approach and prove existence of (6m, 4m, 3m, 5.05)-concentrator (which is no longer regular, but has fewer edges). We apply this result to improve the constant of approximation of almost additive set functions by additive set functions from 44.5 (established by Kalton and Roberts in [KR83]) to 39. We show a more direct...

The algebraic combinatorial approach for low-rank matrix completion

Franz J. Király, Louis Theran, Tomioka Ryota & Takeaki Uno
We propose an algebraic combinatorial framework for the problem of completing partially observed low-rank matrices. We show that the intrinsic properties of the problem, including which entries can be reconstructed, and the degrees of freedom in the reconstruction, do not depend on the values of the observed entries, but only on their position. We associate combinatorial and algebraic objects, differentials and matroids, which are descriptors of the particular reconstruction task, to the set of observed...

On conjugacy of MASAs and the outer automorphism group of the cuntz algebra

Roberto Conti, Jeong Hee Hong & Wojciech Szymanski
We investigate the structure of the outer automorphism group of the Cuntz algebra and the closely related problem of conjugacy of MASAa in $\mathcal{O}_n$. In particular, we exhibit an uncountable family of MASAs, conjugate to the standard MASA $\mathcal{D}_n$ via Bogolubov automorphisms, that are not inner conjugate to $\mathcal{D}_n$.

Right Unimodal and Bimodal Singularities inPositive Characteristic

Hong Duc Nguyen
The problem of classification of real and complex singularities was initiated by Arnol'd in the sixties who classified simple, unimodal and bimodal w.r.t. right equivalence. The classification of right simple singularities in positive characteristic was achieved by Greuel and the author in 2014. In the present paper we classify right unimodal and bimodal singularities in positive characteristic by giving explicit normal forms. Moreover we completely determine all possible adjacency diagrams of simple,unimodal and bimodal singularities....

Discrete non-commutative integrability: the proof of a conjecture by M. Kontsevich

Philippe Di Francesco & Rinat Kedem
We prove a conjecture of Kontsevich regarding the solutions of rank two recursion relations for non-commutative variables which, in the commutative case, reduce to rank two cluster algebras of affine type. The conjecture states that solutions are positive Laurent polynomials in the initial cluster variables. We prove this by use of a non-commutative version of the path models which we used for the commutative case.

Self-adjoint differential-algebraic equations

Peter Kunkel, Volker Mehrmann & Lena Scholz
Motivated from linear-quadratic optimal control problems for differential-algebraic equations (DAEs), we study the functional analytic properties of the operator associated with the necessary optimality boundary value problem and show that it is associated with a self-conjugate operator and a self-adjoint pair of matrix functions. We then study general self-adjoint pairs of matrix valued functions and derive condensed forms under orthogonal congruence transformations that preserve the self-adjointness. We analyze the relationship between self-adjoint DAEs and Hamiltonian...

Legendrian Knots in Lens Spaces

Sinem Onaran
In this note, we first classify all topological torus knots lying on the Heegaard torus in Lens spaces, and then we classify Legendrian representatives of torus knots. We show that all Legendrian torus knots in universally tight contact structures on Lens spaces are determined up to contactomorphism by their knot type, rational Thurston-Bennequin invariant and rational rotation number.

Low rank differential equations for hamiltonian matrix nearness problems

Nicola Guglielmi, Daniel Kreßner & Christian Lubich
For a Hamiltonian matrix with purely imaginary eigenvalues, we aim to determine the nearest Hamiltonian matrix such that so me or all eigenvalues leave the imaginary axis. Conversely, for a Hamiltonian matrix with all eigenvalues lying off the imaginary axis, we look for a nearest Hamiltonian matrix that has a pair of imaginary eigenvalues. The Hamiltonian matrices can be allowed to be complex or restricted to be real. Such Hamiltonian matrix nearness problems are motivated...

Alexander r-Tuples and Bier Complexes

Dusko Jojic, Ilya Nekrasov, Gaiane Panina & Rade Zivaljevic
We introduce and study Alexander $r$-Tuples $\mathcal{K} = \langle K_i \rangle ^r_{i=1}$ of simplicial complexes, as a common generalization of pairs of Alexander dual complexes (Alexander 2-tuples) and r-unavoidable complexes of [BFZ-1]. In the same vein, the Bier complexes, defined as the deleted joins $\mathcal{K}^*_\Delta$ of Alexander $r$-tuples, include both standard Bier spheres and optimal multiple chessboard complexes (Section 2.2) as interesting, special cases. Our main results are Theorem 4.3 saying that (1) the $r$-fold...

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