174 research outputs found
Necessary and Sufficient Conditions on Partial Orders for Modeling Concurrent Computations
Partial orders are used extensively for modeling and analyzing concurrent
computations. In this paper, we define two properties of partially ordered
sets: width-extensibility and interleaving-consistency, and show that a partial
order can be a valid state based model: (1) of some synchronous concurrent
computation iff it is width-extensible, and (2) of some asynchronous concurrent
computation iff it is width-extensible and interleaving-consistent. We also
show a duality between the event based and state based models of concurrent
computations, and give algorithms to convert models between the two domains.
When applied to the problem of checkpointing, our theory leads to a better
understanding of some existing results and algorithms in the field. It also
leads to efficient detection algorithms for predicates whose evaluation
requires knowledge of states from all the processes in the system
Lower Bounds for Real Solutions to Sparse Polynomial Systems
We show how to construct sparse polynomial systems that have non-trivial
lower bounds on their numbers of real solutions. These are unmixed systems
associated to certain polytopes. For the order polytope of a poset P this lower
bound is the sign-imbalance of P and it holds if all maximal chains of P have
length of the same parity. This theory also gives lower bounds in the real
Schubert calculus through sagbi degeneration of the Grassmannian to a toric
variety, and thus recovers a result of Eremenko and Gabrielov.Comment: 31 pages. Minor revision
A parallel Buchberger algorithm for multigraded ideals
We demonstrate a method to parallelize the computation of a Gr\"obner basis
for a homogenous ideal in a multigraded polynomial ring. Our method uses
anti-chains in the lattice to separate mutually independent
S-polynomials for reduction.Comment: 8 pages, 6 figure
Rowmotion and generalized toggle groups
We generalize the notion of the toggle group, as defined in [P. Cameron-D.
Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], from
the set of order ideals of a poset to any family of subsets of a finite set. We
prove structure theorems for certain finite generalized toggle groups, similar
to the theorem of Cameron and Fon-der-Flaass in the case of order ideals. We
apply these theorems and find other results on generalized toggle groups in the
following settings: chains, antichains, and interval-closed sets of a poset;
independent sets, vertex covers, acyclic subgraphs, and spanning subgraphs of a
graph; matroids and convex geometries. We generalize rowmotion, an action
studied on order ideals in [P. Cameron-D. Fon-der-Flaass '95] and [J.
Striker-N. Williams '12], to a map we call cover-closure on closed sets of a
closure operator. We show that cover-closure is bijective if and only if the
set of closed sets is isomorphic to the set of order ideals of a poset, which
implies rowmotion is the only bijective cover-closure map.Comment: 26 pages, 5 figures, final journal versio
Existence thresholds and Ramsey properties of random posets
Let denote the power set of , ordered by inclusion, and
let denote the random poset obtained from by
retaining each element from independently at random with
probability and discarding it otherwise.
Given any fixed poset we determine the threshold for the property that
contains as an induced subposet. We also asymptotically
determine the number of copies of a fixed poset in .
Finally, we obtain a number of results on the Ramsey properties of the random
poset .Comment: 33 pages, 2 figures. Author accepted manuscript, to appear in Random
Structures and Algorithm
Estructura Combinatoria de Politopos asociados a Medidas Difusas
Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 23-11-2020This PhD thesis is devoted to the study of geometric and combinatorial aspects of polytopes associated to fuzzy measures. Fuzzy measures are an essential tool, since they generalize the concept of probability. This greater generality allows applications to be developed in various elds, from the Decision Theory to the Game Theory. The set formed by all fuzzy measures on a referential set is a polytope. In the same way, many of the most relevant subfamilies of fuzzy measures are also polytopes. Studying the combinatorial structure of these polytopes arises as a natural problem that allows us to better understand the properties of the associated fuzzy measures. Knowing the combinatorial structure of these polytopes helps us to develop algorithms to generate points uniformly at random inside these polytopes. Generating points uniformly inside a polytope is a complex problem from both a theoretical and a computational point of view. Having algorithms that allow us to sample uniformly in polytopes associated to fuzzy measures allows us to solve many problems, among them the identi cation problem, i.e. estimate the fuzzy measure that underlies an observed data set...La presente tesis doctoral esta dedicada al estudio de distintas propiedades geometricas y combinatorias de politopos de medidas difusas. Las medidas difusas son una herramienta esencial puesto que generalizan el concepto de probabilidad. Esta mayor generalidad permite desarrollar aplicaciones en diversos campos, desde la Teoría de la Decision a laTeoría de Juegos. El conjunto formado por todas las medidas difusas sobre un referencial tiene estructura de politopo. De la misma forma, la mayora de las subfamilias mas relevantes de medidas difusas son tambien politopos. Estudiar la estructura combinatoria de estos politopos surge como un problema natural que nos permite comprender mejor las propiedades delas medidas difusas asociadas. Conocer la estructura combinatoria de estos politopos tambien nos ayuda a desarrollar algoritmos para generar aleatoria y uniformemente puntos dentro de estos politopos. Generar puntos de forma uniforme dentro de un politopo es un problema complejo desde el punto de vista tanto teorico como computacional. Disponer de algoritmos que nos permitan generar uniformemente en politopos asociados a medidas difusas nos permite resolver muchos problemas, entre ellos el problema de identificacion que trata de estimarla medida difusa que subyace a un conjunto de datos observado...Fac. de Ciencias MatemáticasTRUEunpu
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