143 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    University of Windsor Graduate Calendar 2023 Spring

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    https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1027/thumbnail.jp

    University of Windsor Graduate Calendar 2023 Winter

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    https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1026/thumbnail.jp

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Operational Research: methods and applications

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    This is the final version. Available on open access from Taylor & Francis via the DOI in this recordThroughout its history, Operational Research has evolved to include methods, models and algorithms that have been applied to a wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first summarises the up-to-date knowledge and provides an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion and used as a point of reference by a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes

    The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers

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    The hypersimplex Δk+1,n\Delta_{k+1,n} is the image of the positive Grassmannian Grk+1,n0Gr^{\geq 0}_{k+1,n} under the moment map. It is a polytope of dimension n1n-1 in Rn\mathbb{R}^n. Meanwhile, the amplituhedron An,k,2(Z)\mathcal{A}_{n,k,2}(Z) is the projection of the positive Grassmannian Grk,n0Gr^{\geq 0}_{k,n} into Grk,k+2Gr_{k,k+2} under a map Z~\tilde{Z} induced by a matrix ZMatn,k+2>0Z\in \text{Mat}_{n,k+2}^{>0}. Introduced in the context of scattering amplitudes, it is not a polytope, and has dimension 2k2k. Nevertheless, there seem to be remarkable connections between these two objects via T-duality, as was first noted by Lukowski--Parisi--Williams (LPW). In this paper we use ideas from oriented matroid theory, total positivity, and the geometry of the hypersimplex and positroid polytopes to obtain a deeper understanding of the amplituhedron. We show that the inequalities cutting out positroid polytopes -- images of positroid cells of Grk+1,n0Gr^{\geq 0}_{k+1,n} under the moment map -- translate into sign conditions characterizing the T-dual Grasstopes -- images of positroid cells of Grk,n0Gr^{\geq 0}_{k,n} under Z~\tilde{Z}. Moreover, we subdivide the amplituhedron into chambers, just as the hypersimplex can be subdivided into simplices, with both chambers and simplices enumerated by the Eulerian numbers. We prove the main conjecture of (LPW): a collection of positroid polytopes is a triangulation of Δk+1,n\Delta_{k+1, n} if and only if the collection of T-dual Grasstopes is a triangulation of An,k,2(Z)\mathcal{A}_{n,k,2}(Z) for all ZZ. Moreover, we prove Arkani-Hamed--Thomas--Trnka's conjectural sign-flip characterization of An,k,2(Z)\mathcal{A}_{n,k,2}(Z), and Lukowski--Parisi--Spradlin--Volovich's conjectures on m=2m=2 cluster adjacency and on generalized triangles (images of 2k2k-dimensional positroid cells which map injectively into An,k,2(Z)\mathcal{A}_{n,k,2}(Z)). Finally, we introduce new cluster structures in the amplituhedron.Comment: 72 pages, many figures, comments welcome. v4: Minor edits v3: Strengthened results on triangulations and realizability of amplituhedron sign chambers. v2: Results added to Section 11.4, minor edit

    Topology and monoid representations

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    The goal of this paper is to use topological methods to compute Ext\mathrm{Ext} between an irreducible representation of a finite monoid inflated from its group completion and one inflated from its group of units, or more generally coinduced from a maximal subgroup, via a spectral sequence that collapses on the E2E_2-page over fields of good characteristic. For von Neumann regular monoids in which Green's L\mathscr L- and J\mathscr J-relations coincide (e.g., left regular bands), the computation of Ext\mathrm{Ext} between arbitrary simple modules reduces to this case, and so our results subsume those of S. Margolis, F. Saliola, and B. Steinberg, Combinatorial topology and the global dimension of algebras arising in combinatorics, J. Eur. Math. Soc. (JEMS), 17, 3037-3080 (2015). Applications include computing Ext\mathrm{Ext} between arbitrary simple modules and computing a quiver presentation for the algebra of Hsiao's monoid of ordered GG-partitions (connected to the Mantaci-Reutenauer descent algebra for the wreath product GSnG\wr S_n). We show that this algebra is Koszul, compute its Koszul dual and compute minimal projective resolutions of all the simple modules using topology. These generalize the results of S. Margolis, F. V. Saliola, and B. Steinberg. Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry, Mem. Amer. Math. Soc., 274, 1-135, (2021). We also determine the global dimension of the algebra of the monoid of all affine transformations of a vector space over a finite field. We provide a topological characterization of when a monoid homomorphism induces a homological epimorphism of monoid algebras and apply it to semidirect products. Topology is used to construct projective resolutions of modules inflated from the group completion for sufficiently nice monoids

    Operational research:methods and applications

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    Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order

    Determinantal Sieving

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    We introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial P(X)P(X) on a set of variables X={x1,,xn}X=\{x_1,\ldots,x_n\} and a linear matroid M=(X,I)M=(X,\mathcal{I}) of rank kk, both over a field F\mathbb{F} of characteristic 2, in 2k2^k evaluations we can sieve for those terms in the monomial expansion of PP which are multilinear and whose support is a basis for MM. Alternatively, using 2k2^k evaluations of PP we can sieve for those monomials whose odd support spans MM. Applying this framework, we improve on a range of algebraic FPT algorithms, such as: 1. Solving qq-Matroid Intersection in time O(2(q2)k)O^*(2^{(q-2)k}) and qq-Matroid Parity in time O(2qk)O^*(2^{qk}), improving on O(4qk)O^*(4^{qk}) (Brand and Pratt, ICALP 2021) 2. TT-Cycle, Colourful (s,t)(s,t)-Path, Colourful (S,T)(S,T)-Linkage in undirected graphs, and the more general Rank kk (S,T)(S,T)-Linkage problem, all in O(2k)O^*(2^k) time, improving on O(2k+S)O^*(2^{k+|S|}) respectively O(2S+O(k2log(k+F)))O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))}) (Fomin et al., SODA 2023) 3. Many instances of the Diverse X paradigm, finding a collection of rr solutions to a problem with a minimum mutual distance of dd in time O(2r(r1)d/2)O^*(2^{r(r-1)d/2}), improving solutions for kk-Distinct Branchings from time 2O(klogk)2^{O(k \log k)} to O(2k)O^*(2^k) (Bang-Jensen et al., ESA 2021), and for Diverse Perfect Matchings from O(22O(rd))O^*(2^{2^{O(rd)}}) to O(2r2d/2)O^*(2^{r^2d/2}) (Fomin et al., STACS 2021) All matroids are assumed to be represented over a field of characteristic 2. Over general fields, we achieve similar results at the cost of using exponential space by working over the exterior algebra. For a class of arithmetic circuits we call strongly monotone, this is even achieved without any loss of running time. However, the odd support sieving result appears to be specific to working over characteristic 2
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