作者：
Morier-Genoud, SophieUniv Paris 06
Inst Math Jussieu Paris Rive Gauche Sorbonne Paris Cite Univ Paris DiderotCNRSSorbonne UnivUMR 7586 F-75005 Paris France
Frieze patterns of numbers, introduced in the early 1970s by Coxeter, are currently attracting much interest due to connections with the recent theory of cluster algebras. The present survey aims to review the origina...
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Frieze patterns of numbers, introduced in the early 1970s by Coxeter, are currently attracting much interest due to connections with the recent theory of cluster algebras. The present survey aims to review the original work of Coxeter and the new developments around the notion of frieze, focusing on the representation theoretic, geometric and combinatorial approaches.
The design performance of real-time operating gas turbine power plants (GTPPs) deteriorates in terms of efficiency, reliability, and commercial availability due to aging, rubbing, contamination, and other similar prob...
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The design performance of real-time operating gas turbine power plants (GTPPs) deteriorates in terms of efficiency, reliability, and commercial availability due to aging, rubbing, contamination, and other similar problems. A manager minimizes operation and maintenance (O&M) expenses, consociating about the function of its basic structure (i.e., layout and design), availability (maintenance aspects), operation efficiency (trained workforce), safety and security, and other regulatory elements. Understanding plant structure improves performance, economical design, maintenance planning, etc. For a better understanding of design, a technique comprising graph theory, matrix method, and combinatorics is developed to determine the performance of GTPP. Detailed methodology for developing a system structure graph, various system structure matrices, and their permanent functions are described for the GTPP. For accessible and appropriate performance analysis, GTPP is divided into seven sub-systems. Structural interconnections between seven sub-systems of GTPP are as per real-time GTPP. The gas turbine system structure is developed in the form of a digraph. Matrix representation corresponding to digraph representation is developed, which is processed with the help of combinatorics. With the help of a computer programming tool developed in C++ for calculating the permanent of a matrix, the result comes out as a numerical value called the GTPP index. The methodology applied in the present work can be incorporated with standard computer programming tools developed for the performance assessment. In this way, it is easy to assess the dynamic behavior of the gas turbine system, as the methodology of the present work can incorporate tangible and intangible factors. Results obtained from the present methods agree with the available information in the literature.
In this paper, we utilize the machinery of cluster algebras, quiver mutations, and brane tilings to study a variety of historical enumerative combinatorics questions all under one roof. Previous work (Zhang in Cluster...
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In this paper, we utilize the machinery of cluster algebras, quiver mutations, and brane tilings to study a variety of historical enumerative combinatorics questions all under one roof. Previous work (Zhang in Cluster variables and perfect matchings of subgraphs of the dP3 lattice, . , 2012;Leoni et al. in J Phys A Math Theor 47:474011, 2014), which arose during the second author's mentoriship of undergraduates, and more recently of both authors (Lai and Musiker in Commun Math Phys 356(3):823-881, 2017), analyzed the cluster algebra associated with the cone over dP3, the del Pezzo surface of degree 6 (CP2 blown up at three points). By investigating sequences of toric mutations, those occurring only at vertices with two incoming and two outgoing arrows, in this cluster algebra, we obtained a family of cluster variables that could be parameterized by Z3 and whose Laurent expansions had elegant combinatorial interpretations in terms of dimer partition functions (in most cases). While the earlier work (Lai and Musiker 2017;Zhang 2012;Leoni et al. 2014) focused exclusively on one possible initial seed for this cluster algebra, there are in total four relevant initial seeds (up to graph isomorphism). In the current work, we explore the combinatorics of the Laurent expansions from these other initial seeds and how this allows us to relate enumerations of perfect matchings on Dungeons to Dragons.
If G is a finite graph, a proper coloring of G is a way to color the vertices of the graph using n colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts th...
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If G is a finite graph, a proper coloring of G is a way to color the vertices of the graph using n colors so that no two vertices connected by an edge have the same color. (The celebrated four-color theorem asserts that if G is planar, then there is at least one proper coloring of G with four colors.) By a classical result of Birkhoff, the number of proper colorings of G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read's conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh's result was subsequently refined and generalized by Huh and Katz (also in 2012), again using substantial doses of algebraic geometry. Both papers in fact establish log-concavity of the coefficients, which is stronger than unimodality. The breakthroughs of the Huh and Huh-Katz papers left open the more general Rota-Welsh conjecture, where graphs are generalized to (not necessarily representable) matroids, and the chromatic polynomial of a graph is replaced by the characteristic polynomial of a matroid. The Huh and Huh-Katz techniques are not applicable in this level of generality, since there is no underlying algebraic geometry to which to relate the problem. But in 2015 Adiprasito, Huh, and Katz announced a proof of the Rota-Welsh conjecture based on a novel approach motivated by but not making use of any results from algebraic geometry. The authors first prove that the Rota-Welsh conjecture would follow from combinatorial analogues of the hard Lefschetz theorem and Hodge-Riemann relations in algebraic geometry. They then implement an elaborate inductive procedure to prove the combinatorial hard Lefschetz theorem and Hodge-Riemann relations using purely combinatorial arguments. We will survey these developments.
Rearrangements are discrete processes whereby discrete segments of DNA are deleted, replicated and inserted into novel positions. A sequence of such configurations, termed a rearrangement evolution, results in jumbled...
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Rearrangements are discrete processes whereby discrete segments of DNA are deleted, replicated and inserted into novel positions. A sequence of such configurations, termed a rearrangement evolution, results in jumbled DNA arrangements, frequently observed in cancer genomes. We introduce a method that allows us to precisely count these different evolutions for a range of processes including breakage-fusion-bridge-cycles, tandem-duplications, inverted-duplications, reversals, transpositions and deletions, showing that the space of rearrangement evolution is super-exponential in size. These counts assume the infinite sites model of unique breakpoint usage. (C) 2020 Elsevier Ltd. All rights reserved.
Given a group and a normal subgroup, we study the problem of choosing coset representatives with few "carries." The problem is closely linked to the emerging field of additive combinatorics. We explore this ...
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Given a group and a normal subgroup, we study the problem of choosing coset representatives with few "carries." The problem is closely linked to the emerging field of additive combinatorics. We explore this link and give a gentle introduction to some results and techniques of additive combinatorics.
In this work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation...
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In this work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model via the so-called height mapping and the nature of boundary conditions is unravelled. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented, and finite size effect analysis is performed to study surface and corner effects, leading to the extrapolation of the central charge of the model. The solution allows for exact calculations of monomer and dimer correlation functions in the discrete level and the scaling behavior can be inferred in order to find the set of scaling dimensions and compare to the bosonic theory which predicts particular features concerning corner behaviors. Finally, some combinatorial and numerical properties of partition functions with boundary monomers are discussed, proved and checked with enumeration algorithms. (C) 2015 The Author. Published by Elsevier B.V.
This work applies the ideas of Alekseev and Meinrenken's non commutative Chern-Weil theory to describe a completely combinatorial and constructive proof of the wheeling theorem. In this theory, the crux of the pro...
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This work applies the ideas of Alekseev and Meinrenken's non commutative Chern-Weil theory to describe a completely combinatorial and constructive proof of the wheeling theorem. In this theory, the crux of the proof is, essentially, the familiar demonstration that a characteristic class does not depend on the choice of connection made to construct it. To a large extent, this work may be viewed as an exposition of the details of some of Alekseev and Meinrenken's theory written for Kontsevich integral specialists. Our goal was a presentation with full combinatorial detail in the setting of Jacobi diagrams. To achieve this goal, certain key algebraic steps required replacement with substantially different combinatorial arguments.
We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We p...
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We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by Gross-Hacking-Keel-Kontsevich: for the flag variety the cone is the tropicalization of their superpotential while for Schubert varieties a restriction of the superpotential is necessary. We prove that the two cones are unimodularly equivalent. As a corollary of our combinatorial result we realize Caldero's tonic degenerations of Schubert varieties as GHKK-degeneration using cluster theory. (C) 2019 Elsevier Inc. All rights reserved.
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