122 research outputs found
Tropical Carathéodory with Matroids
Bárány’s colorful generalization of Carathéodory’s Theorem combines geometrical and combinatorial constraints. Kalai–Meshulam (2005) and Holmsen (2016) generalized Bárány’s theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert–Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth
Solvability of coupled systems of generalized Hammerstein-type integral equations in the real line
In this work, we consider a generalized coupled system of integral equations of
Hammerstein-type with, eventually, discontinuous nonlinearities. The main existence tool is
Schauder’s fixed point theorem in the space of bounded and continuous functions with bounded and
continuous derivatives on R, combined with the equiconvergence at ±∞ to recover the compactness
of the correspondent operators. To the best of our knowledge, it is the first time where coupled
Hammerstein-type integral equations in real line are considered with nonlinearities depending on
several derivatives of both variables and, moreover, the derivatives can be of different order on each
variable and each equation. On the other hand, we emphasize that the kernel functions can change
sign and their derivatives in order to the first variable may be discontinuous. The last section contains
an application to a model to study the deflection of a coupled system of infinite beams
Discrete Geometry and Convexity in Honour of Imre Bárány
This special volume is contributed by the speakers of the Discrete Geometry and
Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference
is to celebrate the 70th birthday and the scientific achievements of professor
Imre Bárány, a pioneering researcher of discrete and convex geometry, topological
methods, and combinatorics. The extended abstracts presented here are written by
prominent mathematicians whose work has special connections to that of professor
Bárány. Topics that are covered include: discrete and combinatorial geometry,
convex geometry and general convexity, topological and combinatorial methods.
The research papers are presented here in two sections. After this preface and a
short overview of Imre Bárány’s works, the main part consists of 20 short but very
high level surveys and/or original results (at least an extended abstract of them)
by the invited speakers. Then in the second part there are 13 short summaries of
further contributed talks.
We would like to dedicate this volume to Imre, our great teacher, inspiring
colleague, and warm-hearted friend
Parameterised integer programming, integer cones, and related problems
Gennady ShmoninPaderborn, Univ., Diss., 200
Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics
We establish a generic upper bound ExpTime for reasoning with global
assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier
results of this kind, our bound does not require a tractable set of tableau
rules for the instance logics, so that the result applies to wider classes of
logics. Examples are Presburger modal logic, which extends graded modal logic
with linear inequalities over numbers of successors, and probabilistic modal
logic with polynomial inequalities over probabilities. We establish the
theoretical upper bound using a type elimination algorithm. We also provide a
global caching algorithm that potentially avoids building the entire
exponential-sized space of candidate states, and thus offers a basis for
practical reasoning. This algorithm still involves frequent fixpoint
computations; we show how these can be handled efficiently in a concrete
algorithm modelled on Liu and Smolka's linear-time fixpoint algorithm. Finally,
we show that the upper complexity bound is preserved under adding nominals to
the logic, i.e. in coalgebraic hybrid logic.Comment: Extended version of conference paper in FCT 201
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