2,511 results
Search Results
2. Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization
- Author
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Sander Gribling, Monique Laurent, David de Laat, Econometrics and Operations Research, and Research Group: Operations Research
- Subjects
Optimization problem ,General Mathematics ,Quantum correlation ,Dimension (graph theory) ,quantum graph parameters ,FOS: Physical sciences ,Quantum entanglement ,90C22 ,Squashed entanglement ,01 natural sciences ,90C26 ,81P40 ,81P45 ,0103 physical sciences ,polynomial optimization ,FOS: Mathematics ,0101 mathematics ,010306 general physics ,Mathematics - Optimization and Control ,Mathematics ,Discrete mathematics ,Semidefinite programming ,Quantum Physics ,Quantum discord ,Full Length Paper ,quantum correlations ,010102 general mathematics ,90C30 ,TheoryofComputation_GENERAL ,16. Peace & justice ,entanglement dimension ,05C15 ,Optimization and Control (math.OC) ,Quantum graph ,Quantum Physics (quant-ph) ,Software - Abstract
In this paper we study bipartite quantum correlations using techniques from tracial noncommutative polynomial optimization. We construct a hierarchy of semidefinite programming lower bounds on the minimal entanglement dimension of a bipartite correlation. This hierarchy converges to a new parameter: the minimal average entanglement dimension, which measures the amount of entanglement needed to reproduce a quantum correlation when access to shared randomness is free. For synchronous correlations, we show a correspondence between the minimal entanglement dimension and the completely positive semidefinite rank of an associated matrix. We then study optimization over the set of synchronous correlations by investigating quantum graph parameters. We unify existing bounds on the quantum chromatic number and the quantum stability number by placing them in the framework of tracial optimization. In particular, we show that the projective packing number, the projective rank, and the tracial rank arise naturally when considering tracial analogues of the Lasserre hierarchy for the stability and chromatic number of a graph. We also introduce semidefinite programming hierarchies converging to the commuting quantum chromatic number and commuting quantum stability number., Comment: 26 pages
- Published
- 2018
3. Generating subtour elimination constraints for the TSP from pure integer solutions
- Author
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Ulrich Pferschy and Rostislav Staněk
- Subjects
90C27 ,0209 industrial biotechnology ,Speedup ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,Travelling salesman problem ,Combinatorics ,Traveling salesman problem ,020901 industrial engineering & automation ,Integer ,Simple (abstract algebra) ,Euclidean geometry ,FOS: Mathematics ,Cluster analysis ,Mathematics - Optimization and Control ,Mathematics ,Discrete mathematics ,ILP solver ,Original Paper ,021103 operations research ,Subtour elimination constraint ,Complete graph ,Random Euclidean graph ,Optimization and Control (math.OC) ,Branch and cut ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
The traveling salesman problem (TSP) is one of the most prominent combinatorial optimization problems. Given a complete graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G = (V, E)$$\end{document}G=(V,E) and non-negative distances d for every edge, the TSP asks for a shortest tour through all vertices with respect to the distances d. The method of choice for solving the TSP to optimality is a branch and cut approach. Usually the integrality constraints are relaxed first and all separation processes to identify violated inequalities are done on fractional solutions. In our approach we try to exploit the impressive performance of current ILP-solvers and work only with integer solutions without ever interfering with fractional solutions. We stick to a very simple ILP-model and relax the subtour elimination constraints only. The resulting problem is solved to integer optimality, violated constraints (which are trivial to find) are added and the process is repeated until a feasible solution is found. In order to speed up the algorithm we pursue several attempts to find as many relevant subtours as possible. These attempts are based on the clustering of vertices with additional insights gained from empirical observations and random graph theory. Computational results are performed on test instances taken from the TSPLIB95 and on random Euclidean graphs.
- Published
- 2016
4. Multigraphs (only) satisfy a weak triangle removal lemma
- Author
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Raphael Yuster and Asaf Shapira
- Subjects
Discrete mathematics ,Lemma (mathematics) ,Simple graph ,Applied Mathematics ,Short paper ,Computer Science::Computational Geometry ,Omega ,Theoretical Computer Science ,Combinatorics ,Computational Theory and Mathematics ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Geometry and Topology ,05D99 ,ComputingMethodologies_COMPUTERGRAPHICS ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
The triangle removal lemma states that a simple graph with o(n^3) triangles can be made triangle-free by removing o(n^2) edges. It is natural to ask if this widely used result can be extended to multi-graphs (or equivalently, weighted graphs). In this short paper we rule out the possibility of such an extension by showing that there are multi-graphs with only n^{2+o(1)} triangles that are still far from being triangle-free. On the other hand, we show that for some g(n)=\omega(1), if a multi-graph (or weighted graph) has only g(n)n^2 triangles then it must be close to being triangle-free. The proof relies on variants of the Ruzsa-Szemer\'edi theorem.
- Published
- 2009
5. A weak 2-weight problem for the Poisson-Hermite semigroup
- Author
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Gustavo Garrigós
- Subjects
Discrete mathematics ,Hermite polynomials ,Semigroup ,Poisson distribution ,42C10, 35C15, 33C45, 40A10 ,symbols.namesake ,Cancellative semigroup ,Weight problem ,Mathematics - Analysis of PDEs ,Bicyclic semigroup ,FOS: Mathematics ,symbols ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
This survey is a slightly extended version of the lecture given by the author at the \emph{VI International Course of Mathematical Analysis in Andaluc\'\i a} (CIDAMA), in September 2014. Most results are contained (in a slightly less general setting) in the earlier paper [3] (Garrig\'os, Hartzstein, Signes, Torrea and Viviani, Pointwise convergence to initial data of heat and Laplace equations, Trans. Amer. Math. Soc. 368 (9) (2016), 6575-6600)., Comment: 17 pages. This is a preprint version of survey paper published in Advanced courses of mathematical analysis VI, 153-171, World Sci. Publ., Hackensack, NJ, 2017
- Published
- 2022
6. Some aspects on solving transportation problem
- Author
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R. Jana, A. K. Das, and Deepmala
- Subjects
Discrete mathematics ,northwest corner solution ,Class (set theory) ,Structure (category theory) ,Sample (statistics) ,Transportation theory ,assignment problem ,Management Science and Operations Research ,sample survey ,transportation problem ,weighted hungarian method ,Hungarian algorithm ,Optimization and Control (math.OC) ,lcsh:T58.6-58.62 ,Corner solution ,FOS: Mathematics ,lcsh:Management information systems ,Convex function ,Mathematics - Optimization and Control ,Assignment problem ,weighted könig-egerváry theorem ,Mathematics - Abstract
In this paper, we consider a class of transportation problems which arises in sample surveys and other areas of statistics. The associated cost matrices of these transportation problems are of special structure. We observe that the optimality of North West corner solution holds for the general problem where cost component is replaced by a convex function. We revisit assignment problem and present a weighted version of K$\ddot{\mbox{o}}$nig-Egerv$\acute{\mbox{a}}$ry theorem and Hungarian method. The weighted Hungarian method proposed in the paper can be used for solving transportation problem.
- Published
- 2020
7. Double exponential lower bounds for possible solutions in the Second Case of the Fermat Last Theorem
- Author
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Michael Th. Rassias and Preda Mihailescu
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Power series ,Fermat's Last Theorem ,Discrete mathematics ,Algebra and Number Theory ,Mathematics - Number Theory ,Mathematics::Number Theory ,010102 general mathematics ,Double exponential function ,Context (language use) ,Extension (predicate logic) ,01 natural sciences ,Upper and lower bounds ,Prime (order theory) ,010101 applied mathematics ,FOS: Mathematics ,Order (group theory) ,Number Theory (math.NT) ,0101 mathematics ,Mathematics - Abstract
In a recent paper, the first author provided some lower bounds to solutions of the equations of Fermat and Catalan, based on local power series developments at the ramified prime of a prime cyclotomic extension. Although both equations have in fact been proved not to have any unknown solutions, these improved bounds are interesting in the context of a new effective abc inequality announced in the paper \cite{MFHMP} based on Mochizuki's \cite{Mo}[IUT-IV, Theorem A]. In this paper we provide a strengthening of the lower bound for FLT2, which is necessary in order to take advantage of the best upper bounds for primes $p$ for which it was verified on a computer that FLT2 has no solutions., Submitted, under review
- Published
- 2021
8. Noncatastrophic convolutional codes over a finite ring
- Author
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Diego Napp, Raquel Pinto, Conceição Rocha, Universidad de Alicante. Departamento de Matemáticas, and Grupo de Álgebra y Geometría (GAG)
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FOS: Computer and information sciences ,Discrete mathematics ,Finite ring ,Polynomial ,Algebra and Number Theory ,Information Theory (cs.IT) ,Finite support convolutional codes ,Applied Mathematics ,Computer Science - Information Theory ,Order (ring theory) ,Convolutional codes ,Context (language use) ,Mathematics - Rings and Algebras ,Prime (order theory) ,Finite field ,Noncatastrophicity ,Rings and Algebras (math.RA) ,Convolutional code ,FOS: Mathematics ,Finite rings ,Generator matrix ,Mathematics - Abstract
Noncatastrophic encoders are an important class of polynomial generator matrices of convolutional codes. When these polynomials have coefficients in a finite field, these encoders have been characterized are being polynomial left prime matrices. In this paper we study the notion of noncatastrophicity in the context of convolutional codes when the polynomial matrices have entries in a finite ring. In particular, we need to introduce two different notion of primeness in order to fully characterize noncatastrophic encoders over the finite ring Z_{p^r}. The second part of the paper is devoted to investigate the notion of free and column distance in this context when the convolutional code is a free finitely generated Z_{p^r}-module. We introduce the notion of b-degree and provide new bounds on the free distances and column distance. We show that this class of convolutional codes is optimal with respect to the column distance and to the free distance if and only if its projection on Z_p is., 17 pages
- Published
- 2021
9. SIR epidemics and vaccination on random graphs with clustering
- Author
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Carolina Fransson and Pieter Trapman
- Subjects
Poisson distribution ,01 natural sciences ,Quantitative Biology::Other ,Communicable Diseases ,Models, Biological ,Article ,Clustering ,010305 fluids & plasmas ,Disease Outbreaks ,03 medical and health sciences ,symbols.namesake ,0103 physical sciences ,FOS: Mathematics ,Computer Graphics ,Quantitative Biology::Populations and Evolution ,Cluster Analysis ,Humans ,Computer Simulation ,Configuration model ,Cluster analysis ,030304 developmental biology ,Mathematics ,Branching process ,Discrete mathematics ,Random graph ,0303 health sciences ,Applied Mathematics ,Probability (math.PR) ,Vaccination ,Numerical Analysis, Computer-Assisted ,Computer Science::Social and Information Networks ,Models, Theoretical ,Agricultural and Biological Sciences (miscellaneous) ,Outcome (probability) ,Branching processes ,Modeling and Simulation ,symbols ,SIR epidemics ,Graph (abstract data type) ,Disease Susceptibility ,Epidemic model ,Basic reproduction number ,Mathematics - Probability - Abstract
In this paper we consider Susceptible \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow $$\end{document}→ Infectious \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow $$\end{document}→ Recovered (SIR) epidemics on random graphs with clustering. To incorporate group structure of the underlying social network, we use a generalized version of the configuration model in which each node is a member of a specified number of triangles. SIR epidemics on this type of graph have earlier been investigated under the assumption of homogeneous infectivity and also under the assumption of Poisson transmission and recovery rates. We extend known results from literature by relaxing the assumption of homogeneous infectivity both in individual infectivity and between different kinds of neighbours. An important special case of the epidemic model analysed in this paper is epidemics in continuous time with arbitrary infectious period distribution. We use branching process approximations of the spread of the disease to provide expressions for the basic reproduction number \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0, the probability of a major outbreak and the expected final size. In addition, the impact of random vaccination with a perfect vaccine on the final outcome of the epidemic is investigated. We find that, for this particular model, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_0$$\end{document}R0 equals the perfect vaccine-associated reproduction number. Generalizations to groups larger than three are discussed briefly.
- Published
- 2019
10. Periodic trajectories in P-time event graphs and the non-positive circuit weight problem
- Author
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Jörg Raisch, Jan Komenda, and Davide Zorzenon
- Subjects
FOS: Computer and information sciences ,Discrete mathematics ,Control and Optimization ,Discrete Mathematics (cs.DM) ,Linear programming ,Natural number ,Graph theory ,Systems and Control (eess.SY) ,Directed graph ,Upper and lower bounds ,Electrical Engineering and Systems Science - Systems and Control ,Optimization and Control (math.OC) ,Control and Systems Engineering ,Computer Science - Data Structures and Algorithms ,FOS: Electrical engineering, electronic engineering, information engineering ,FOS: Mathematics ,Data Structures and Algorithms (cs.DS) ,Time complexity ,Mathematics - Optimization and Control ,Parametric statistics ,Mathematics ,Event (probability theory) ,Computer Science - Discrete Mathematics - Abstract
P-time event graphs (P-TEGs) are specific timed discrete-event systems, in which the timing of events is constrained by intervals. An important problem is to check, for all natural numbers $d$, the existence of consistent $d$-periodic trajectories for a given P-TEG. In graph theory, the Proportional-Inverse-Constant-Non-positive Circuit weight Problem (PIC-NCP) consists in finding all the values of a parameter such that a particular parametric weighted directed graph does not contain circuits with positive weight. In a related paper, we have proposed a strongly polynomial algorithm that solves the PIC-NCP in lower worst-case complexity compared to other algorithms reported in literature. In the present paper, we show that the first problem can be formulated as an instance of the second; consequently, we prove that the same algorithm can be used to find $d$-periodic trajectories in P-TEGs. Moreover, exploiting the connection between the PIC-NCP and max-plus algebra we prove that, given a P-TEG, the existence of a consistent 1-periodic trajectory of a certain period is a necessary and sufficient condition for the existence of a consistent $d$-periodic trajectory of the same period, for any value of $d$., Minor corrections
- Published
- 2021
11. On a conjecture for $\ell$-torsion in class groups of number fields: from the perspective of moments
- Author
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Melanie Matchett Wood, Lillian B. Pierce, and Caroline L. Turnage-Butterbaugh
- Subjects
Pointwise ,Discrete mathematics ,Conjecture ,Mathematics - Number Theory ,General Mathematics ,Field (mathematics) ,Algebraic number field ,Elliptic curve ,Number theory ,Discriminant ,Bounded function ,FOS: Mathematics ,Number Theory (math.NT) ,Mathematics - Abstract
It is conjectured that within the class group of any number field, for every integer $\ell \geq 1$, the $\ell$-torsion subgroup is very small (in an appropriate sense, relative to the discriminant of the field). In nearly all settings, the full strength of this conjecture remains open, and even partial progress is limited. Significant recent progress toward average versions of the $\ell$-torsion conjecture has crucially relied on counts for number fields, raising interest in how these two types of question relate. In this paper we make explicit the quantitative relationships between the $\ell$-torsion conjecture and other well-known conjectures: the Cohen-Lenstra heuristics, counts for number fields of fixed discriminant, counts for number fields of bounded discriminant (or related invariants), and counts for elliptic curves with fixed conductor. All of these considerations reinforce that we expect the $\ell$-torsion conjecture is true, despite limited progress toward it. Our perspective focuses on the relation between pointwise bounds, averages, and higher moments, and demonstrates the broad utility of the "method of moments.", The paper arXiv:1709.09637v1 by the same authors was submitted for publication in a significantly shorter version; some of the excluded material appears in this new paper, which also proves further results. Version 2 updates some references, with corresponding improvements in Section 6. Version 3 has a few small changes to streamline exposition after referee process
- Published
- 2021
12. The Landis conjecture with sharp rate of decay
- Author
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Luca Rossi, Centre National de la Recherche Scientifique (CNRS), Centre d'Analyse et de Mathématique sociales (CAMS), and École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Discrete mathematics ,Conjecture ,Landis conjecture ,exponential decay ,exterior domain ,unique continuation ,radial operators ,General Mathematics ,010102 general mathematics ,Dimension (graph theory) ,01 natural sciences ,Domain (mathematical analysis) ,Elliptic operator ,Operator (computer programming) ,Mathematics - Analysis of PDEs ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,0101 mathematics ,Exponential decay ,[MATH]Mathematics [math] ,Constant (mathematics) ,Mathematics ,Sign (mathematics) ,Analysis of PDEs (math.AP) - Abstract
The so called Landis conjecture states that if a solution of the equation $$\Delta u+V(x)u=0$$ in an exterior domain decays faster than $e^{-\kappa|x|}$, for some $\kappa>\sqrt{\sup |V|}$, then it must be identically equal to $0$. This property can be viewed as a unique continuation at infinity (UCI) for solutions satisfying a suitable exponential decay. The Landis conjecture was disproved by Meshkov in the case of complex-valued functions, but it remained open in the real case. In the 2000s, several papers have addressed the issue of the UCI for linear elliptic operators with real coefficients. The results that have been obtained require some kind of sign condition, either on the solution or on the zero order coefficient of the equation. The Landis conjecture is still open nowadays in its general form. In the present paper, we start with considering a general (real) elliptic operator in dimension $1$. We derive the UCI property with a rate of decay $\kappa$ which is sharp when the coefficients of the operator are constant. In particular, we prove the Landis conjecture in dimension $1$, and we can actually reach the threshold value $\kappa=\sqrt{\sup |V|}$. Next, we derive the UCI property -- and then the Landis conjecture -- for radial operators in arbitrary dimension. Finally, with a different approach, we prove the same result for positive supersolutions of general elliptic equations.
- Published
- 2021
13. Efficient computation of the oriented chromatic number of recursively defined digraphs
- Author
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Dominique Komander, Marvin Lindemann, and Frank Gurski
- Subjects
FOS: Computer and information sciences ,Vertex (graph theory) ,Discrete mathematics ,Disjoint union ,Mathematics::Combinatorics ,General Computer Science ,Oriented coloring ,Parameterized complexity ,Theoretical Computer Science ,Greedy coloring ,Computer Science::Discrete Mathematics ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Computer Science - Data Structures and Algorithms ,FOS: Mathematics ,Topological sorting ,Mathematics - Combinatorics ,Data Structures and Algorithms (cs.DS) ,Combinatorics (math.CO) ,Chromatic scale ,Time complexity ,Mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In this paper we consider colorings of oriented graphs, i.e. digraphs without cycles of length 2. Given some oriented graph $G=(V,E)$, an oriented $r$-coloring for $G$ is a partition of the vertex set $V$ into $r$ independent sets, such that all the arcs between two of these sets have the same direction. The oriented chromatic number of $G$ is the smallest integer $r$ such that $G$ permits an oriented $r$-coloring. In this paper we consider the Oriented Chromatic Number problem on classes of recursively defined oriented graphs. Oriented co-graphs (short for oriented complement reducible graphs) can be recursively defined defined from the single vertex graph by applying the disjoint union and order composition. This recursive structure allows to compute an optimal oriented coloring and the oriented chromatic number in linear time. We generalize this result using the concept of perfect orderable graphs. Therefore, we show that for acyclic transitive digraphs every greedy coloring along a topological ordering leads to an optimal oriented coloring. Msp-digraphs (short for minimal series-parallel digraphs) can be defined from the single vertex graph by applying the parallel composition and series composition. We prove an upper bound of $7$ for the oriented chromatic number for msp-digraphs and we give an example to show that this is bound best possible. We apply this bound and the recursive structure of msp-digraphs to obtain a linear time solution for computing the oriented chromatic number of msp-digraphs. In order to generalize the results on computing the oriented chromatic number of special graph classes, we consider the parameterized complexity of the Oriented Chromatic Number problem by so-called structural parameters, which are measuring the difficulty of decomposing a graph into a special tree-structure, 25 pages. arXiv admin note: text overlap with arXiv:2006.13911
- Published
- 2020
14. Decreasing Minimization on M-convex Sets: Background and Structures
- Author
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András Frank and Kazuo Murota
- Subjects
Discrete mathematics ,General Mathematics ,90C27, 05C, 68R10 ,Orientation (graph theory) ,Type (model theory) ,Flow network ,Matroid ,Domain (mathematical analysis) ,Set (abstract data type) ,FOS: Mathematics ,Graph (abstract data type) ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Element (category theory) ,Software ,Mathematics - Abstract
The present work is the first member of a pair of papers concerning decreasingly-minimal (dec-min) elements of a set of integral vectors, where a vector is dec-min if its largest component is as small as possible, within this, the next largest component is as small as possible, and so on. This discrete notion, along with its fractional counterpart, showed up earlier in the literature under various names. The domain we consider is an M-convex set, that is, the set of integral elements of an integral base-polyhedron. A fundamental difference between the fractional and the discrete case is that a base-polyhedron has always a unique dec-min element, while the set of dec-min elements of an M-convex set admits a rich structure, described here with the help of a "canonical chain". As a consequence, we prove that this set arises from a matroid by translating the characteristic vectors of its bases with an integral vector. By relying on these characterizations, we prove that an element is dec-min if and only if the square-sum of its components is minimum, a property resulting in a new type of min-max theorems. The characterizations also give rise, as shown in the companion paper, to a strongly polynomial algorithm, and to several applications in the areas of resource allocation, network flow, matroid, and graph orientation problems, which actually provided a major motivation to the present investigations. In particular, we prove a conjecture on graph orientation., 39 pages This is a revised version of the first half of "A. Frank and K. Murota; Discrete decreasing minimization, PartI: Base-polyhedra with applications in network optimization" arXiv:1808.07600
- Published
- 2020
15. APPROXIMATING SMOOTH, MULTIVARIATE FUNCTIONS ON IRREGULAR DOMAINS
- Author
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Daan Huybrechs and Ben Adcock
- Subjects
Statistics and Probability ,Large class ,Multivariate statistics ,Polynomial ,STOCHASTIC COLLOCATION ,POLYNOMIAL-APPROXIMATION ,Mathematics, Applied ,DIMENSIONALITY ,010103 numerical & computational mathematics ,NIKOLSKII-TYPE INEQUALITIES ,01 natural sciences ,Theoretical Computer Science ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Numerical Analysis ,0101 mathematics ,Mathematical Physics ,Mathematics ,Discrete mathematics ,Algebra and Number Theory ,Science & Technology ,Frame (networking) ,ALGORITHMS ,Numerical Analysis (math.NA) ,ANALYTIC-FUNCTIONS ,010101 applied mathematics ,Computational Mathematics ,Physical Sciences ,CHAOS ,Geometry and Topology ,Hypercube ,Analysis ,EXTENSION - Abstract
In this paper, we introduce a method known aspolynomial frame approximationfor approximating smooth, multivariate functions defined on irregular domains in$d$dimensions, where$d$can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the truncated approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization, for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the correspondingNikolskii inequalityfor the domain. In particular, we show that the sample complexity for points drawn from the uniform measure is quadratic (up to a log factor) in the dimension of the polynomial space, independently of $d$, for a large class of nontrivial domains. This extends a well-known result for polynomial approximation in hypercubes.
- Published
- 2020
16. A new principle in the interpretability logic of all reasonable arithmetical theories
- Author
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Joost J. Joosten and Evan Goris
- Subjects
Mathematical theory ,Discrete mathematics ,Intersection ,Logic ,Semantics (computer science) ,FOS: Mathematics ,Arithmetic function ,Mathematics - Logic ,Logic (math.LO) ,Interpretability logic ,Mathematics - Abstract
The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper from 2011 revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by ${\textbf{IL}}({\rm All})$. In this paper we present a new principle $\sf R$ in ${\textbf{IL}}({\rm All})$. We show that $\sf R$ does not follow from the logic ${\textbf{IL}}{\sf P_0W^*}$ that contains all previously known principles. This is done by providing a modal incompleteness proof of ${\textbf{IL}}{\sf P_0W^*}$: showing that $\sf R$ follows semantically but not syntactically from ${\textbf{IL}}{\sf P_0W^*}$. Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with so-called Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated. After the modal results it is shown that the new principle $\sf R$ is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories.
- Published
- 2020
17. Reflexive coloring complexes for 3-edge-colorings of cubic graphs
- Author
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Bojan Mohar, Nathan Singer, and Fiachra Knox
- Subjects
Discrete mathematics ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,Edge (geometry) ,01 natural sciences ,Graph ,Theoretical Computer Science ,law.invention ,Combinatorics ,05C15 ,010201 computation theory & mathematics ,law ,Outerplanar graph ,Line graph ,0202 electrical engineering, electronic engineering, information engineering ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Cubic graph ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Mathematics - Abstract
Given a 3-colorable graph X , the 3-coloring complex B ( X ) is the graph whose vertices are all the independent sets which occur as color classes in some 3-coloring of X . Two color classes C , D ∈ V ( B ( X ) ) are joined by an edge if C and D appear together in a 3-coloring of X . The graph B ( X ) is 3-colorable. Graphs for which B ( B ( X ) ) is isomorphic to X are termed reflexive graphs. In this paper, we consider 3-edge-colorings of cubic graphs for which we allow half-edges. Then we consider the 3-coloring complexes of their line graphs. The main result of this paper is a surprising outcome that the line graph of any connected cubic triangle-free outerplanar graph is reflexive. We also exhibit some other interesting classes of reflexive line graphs.
- Published
- 2020
18. Arithmetic progressions in finite groups
- Author
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Marius Tărnăuceanu
- Subjects
Discrete mathematics ,General Mathematics ,Arithmetic progression ,Structure (category theory) ,FOS: Mathematics ,Group Theory (math.GR) ,Abelian group ,Element (category theory) ,Mathematics - Group Theory ,Mathematics - Abstract
In this paper, we describe the structure of finite groups whose element orders or proper (abelian) subgroup orders form an arithmetic progression of ratio [Formula: see text]. This extends the case [Formula: see text] studied in previous papers [R. Brandl and W. Shi, Finite groups whose element orders are consecutive integers, J. Algebra 143 (1991) 388–400; Y. Feng, Finite groups whose abelian subgroup orders are consecutive integers, J. Math. Res. Exp. 18 (1998) 503–506; W. Shi, Finite groups whose proper subgroup orders are consecutive integers, J. Math. Res. Exp. 14 (1994) 165–166].
- Published
- 2020
19. Quantitative absolute continuity of planar measures with two independent Alberti representations
- Author
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David Bate, Tuomas Orponen, and Department of Mathematics and Statistics
- Subjects
Discrete mathematics ,Lemma (mathematics) ,Lebesgue measure ,Plane (geometry) ,DERIVATIVES ,Applied Mathematics ,010102 general mathematics ,Metric Geometry (math.MG) ,Absolute continuity ,01 natural sciences ,010101 applied mathematics ,Planar ,Mathematics - Metric Geometry ,Mathematics - Classical Analysis and ODEs ,Bounded function ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Exponent ,111 Mathematics ,0101 mathematics ,Analysis ,Reverse holder inequality ,Mathematics - Abstract
We study measures $\mu$ on the plane with two independent Alberti representations. It is known, due to Alberti, Cs\"ornyei, and Preiss, that such measures are absolutely continuous with respect to Lebesgue measure. The purpose of this paper is to quantify the result of A-C-P. Assuming that the representations of $\mu$ are bounded from above, in a natural way to be defined in the introduction, we prove that $\mu \in L^{2}$. If the representations are also bounded from below, we show that $\mu$ satisfies a reverse H\"older inequality with exponent $2$, and is consequently in $L^{2 + \epsilon}$ by Gehring's lemma. A substantial part of the paper is also devoted to showing that both results stated above are optimal., Comment: 16 pages, 4 figures. v2: corrected typos, expanded introduction
- Published
- 2020
20. On The Maximum Order Complexity Of Thue-Morse And Rudin-Shapiro Sequences Along Polynomial Values
- Author
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Pierre Popoli, Institut Élie Cartan de Lorraine (IECL), Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS), and ANR-15-IDEX-0004,LUE,Isite LUE(2015)
- Subjects
[INFO.INFO-CC]Computer Science [cs]/Computational Complexity [cs.CC] ,Polynomial ,Open problem ,Cryptography ,010103 numerical & computational mathematics ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,Measure (mathematics) ,[INFO.INFO-CR]Computer Science [cs]/Cryptography and Security [cs.CR] ,Subsequence ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Mathematics - Combinatorics ,Order (group theory) ,Number Theory (math.NT) ,0101 mathematics ,Mathematics ,Discrete mathematics ,Sequence ,Mathematics - Number Theory ,business.industry ,010102 general mathematics ,General Medicine ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Bounded function ,Combinatorics (math.CO) ,business ,Computer Science::Formal Languages and Automata Theory - Abstract
Both the Thue-Morse and Rudin-Shapiro sequences are not suitable sequences for cryptography since their expansion complexity is small and their correlation measure of order 2 is large. These facts imply that these sequences are highly predictable despite the fact that they have a large maximum order complexity. Sun and Winterhof (2019) showed that the Thue-Morse sequence along squares keeps a large maximum order complexity. Since, by Christol's theorem, the expansion complexity of this rarefied sequence is no longer bounded, this provides a potentially better candidate for cryptographic applications. Similar results are known for the Rudin-Shapiro sequence and more general pattern sequences. In this paper we generalize these results to any polynomial subsequence (instead of squares) and thereby answer an open problem of Sun and Winterhof. We conclude this paper by some open problems., Comment: 11 pages
- Published
- 2020
21. Consistency and asymptotic normality of stochastic block models estimators from sampled data
- Author
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Mahendra Mariadassou and Timothée Tabouy
- Subjects
Statistics and Probability ,Missing data ,Block (permutation group theory) ,Asymptotic distribution ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,01 natural sciences ,010104 statistics & probability ,Stochastic block model ,Consistency (statistics) ,0502 economics and business ,FOS: Mathematics ,Asymptotic normality ,0101 mathematics ,Concentration inequality ,050205 econometrics ,Mathematics ,Discrete mathematics ,05 social sciences ,Sampling (statistics) ,Estimator ,Statistics, Probability and Uncertainty ,Stochastic Block Model ,Maximum likelihood - Abstract
Statistical analysis of network is an active research area and the literature counts a lot of papers concerned with network models and statistical analysis of networks. However, very few papers deal with missing data in network analysis and we reckon that, in practice, networks are often observed with missing values. In this paper we focus on the Stochastic Block Model with valued edges and consider a MCAR setting by assuming that every dyad (pair of nodes) is sampled identically and independently of the others with probability $\rho > 0$. We prove that maximum likelihood estimators and its variational approximations are consistent and asymptotically normal in the presence of missing data as soon as the sampling probability $\rho$ satisfies $\rho\gg\log(n)/n$., Comment: 32 pages, 0 figures. arXiv admin note: text overlap with arXiv:1704.06629
- Published
- 2020
22. Representations of the Vertex Reinforced Jump Process as a mixture of Markov processes on $\mathbb {Z}^{d}$ and infinite trees
- Author
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Thomas Gerard
- Subjects
Statistics and Probability ,Discrete mathematics ,Vertex (graph theory) ,Random field ,Martin boundary ,Probability (math.PR) ,Boundary (topology) ,Markov process ,31C35 ,Markov processes in random environment ,Tree (descriptive set theory) ,symbols.namesake ,Distribution (mathematics) ,60K37 ,Harmonic function ,reinforced processes ,FOS: Mathematics ,symbols ,Statistics, Probability and Uncertainty ,60J75 ,Jump process ,Mathematics - Probability ,Mathematics - Abstract
This paper concerns the Vertex Reinforced Jump Process (VRJP) and its representations as a Markov process in random environment. In [21], it was shown that the VRJP on finite graphs, under a certain time rescaling, has the distribution of a mixture of Markov jump processes. This representation was extended to infinite graphs in [23], by introducing a random potential $\beta $. In this paper, we show that all possible representations of the VRJP as a mixture of Markov processes can be expressed in a similar form as in [23], using the random field $\beta $ and harmonic functions for an associated operator $H_\beta $. This allows to show that the VRJP on $\mathbb {Z}^{d}$ (with certain initial conditions) has a unique representation, by proving that an associated Martin boundary is trivial. Moreover, on infinite trees, we construct a family of representations, that are all different when the VRJP is transient and the tree is $d$-regular (with $d\geq 3$).
- Published
- 2020
23. A geometric characterization of minimal codes and their asymptotic performance
- Author
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Alessandro Neri, Martino Borello, Gianira N. Alfarano, Universität Zürich [Zürich] = University of Zurich (UZH), Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 8 Vincennes-Saint-Denis (UP8)-Centre National de la Recherche Scientifique (CNRS)-Université Sorbonne Paris Nord, Geometry, arithmetic, algorithms, codes and encryption (GRACE), Laboratoire d'informatique de l'École polytechnique [Palaiseau] (LIX), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)
- Subjects
Discrete mathematics ,FOS: Computer and information sciences ,Algebra and Number Theory ,Computer Networks and Communications ,Applied Mathematics ,Computer Science - Information Theory ,Information Theory (cs.IT) ,020206 networking & telecommunications ,0102 computer and information sciences ,02 engineering and technology ,Characterization (mathematics) ,16. Peace & justice ,Blocking (statistics) ,01 natural sciences ,Microbiology ,Dimension (vector space) ,010201 computation theory & mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Field size ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,[MATH]Mathematics [math] ,Mathematics - Abstract
In this paper, we give a geometric characterization of minimal linear codes. In particular, we relate minimal linear codes to cutting blocking sets, introduced in a recent paper by Bonini and Borello. Using this characterization, we derive some bounds on the length and the distance of minimal codes, according to their dimension and the underlying field size. Furthermore, we show that the family of minimal codes is asymptotically good. Finally, we provide some geometrical constructions of minimal codes., Comment: 22 pages
- Published
- 2019
24. Guarantees for the Kronecker Fast Johnson-Lindenstrauss Transform Using a Coherence and Sampling Argument
- Author
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Stephen Becker and Osman Asif Malik
- Subjects
Kronecker product ,Discrete mathematics ,Numerical Analysis ,Algebra and Number Theory ,010102 general mathematics ,Structure (category theory) ,Sampling (statistics) ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,Coherence (statistics) ,15-02, 65F30 ,16. Peace & justice ,01 natural sciences ,symbols.namesake ,Dimension (vector space) ,Kronecker delta ,FOS: Mathematics ,symbols ,Discrete Mathematics and Combinatorics ,Embedding ,Geometry and Topology ,Mathematics - Numerical Analysis ,0101 mathematics ,Subspace topology ,Mathematics - Abstract
In the recent paper [Jin, Kolda & Ward, arXiv:1909.04801], it is proved that the Kronecker fast Johnson-Lindenstrauss transform (KFJLT) is, in fact, a Johnson-Lindenstrauss transform, which had previously only been conjectured. In this paper, we provide an alternative proof of this, for when the KFJLT is applied to Kronecker vectors, using a coherence and sampling argument. Our proof yields a different bound on the embedding dimension, which can be combined with the bound in the paper by Jin et al. to get a better bound overall. As a stepping stone to proving our result, we also show that the KFJLT is a subspace embedding for matrices with columns that have Kronecker product structure. Lastly, we compare the KFJLT to four other sketch techniques in numerical experiments on both synthetic and real-world data., Accepted to Linear Algebra and its Applications
- Published
- 2019
25. The Barwise-Schlipf Theorem
- Author
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James H. Schmerl and Ali Enayat
- Subjects
Discrete mathematics ,Mathematics::Logic ,03H15, 03C62 ,Second-order arithmetic ,Applied Mathematics ,General Mathematics ,Peano axioms ,FOS: Mathematics ,Mathematics - Logic ,Logic (math.LO) ,Mathematics - Abstract
In 1975 Barwise and Schlipf published a landmark paper whose main theorem asserts that a nonstandard model $\mathcal{M}$ of PA (Peano arithmetic) is recursively saturated iff $\mathcal{M}$ has an expansion that satisfies the subsystem $\Delta_1^1$-${\sf CA}_0$ of second order arithmetic. In this paper we identify a crucial error in the Barwise-Schlipf proof of the right-to-left direction of the theorem, and additionally, we offer a correct proof of the problematic direction., Comment: Some editorial changes have been made. To appear in PAMS
- Published
- 2019
26. On Data-Processing and Majorization Inequalities for f-Divergences with Applications
- Author
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Igal Sason
- Subjects
FOS: Computer and information sciences ,f-divergences ,Tunstall trees ,Distribution (number theory) ,Computer Science - Information Theory ,General Physics and Astronomy ,List decoding ,lcsh:Astrophysics ,02 engineering and technology ,Information theory ,Mathematical proof ,01 natural sciences ,Article ,010305 fluids & plasmas ,0103 physical sciences ,lcsh:QB460-466 ,hypothesis testing ,0202 electrical engineering, electronic engineering, information engineering ,Probability mass function ,FOS: Mathematics ,data-processing inequalities ,Majorization ,lcsh:Science ,Variable (mathematics) ,Mathematics ,Discrete mathematics ,Information Theory (cs.IT) ,Probability (math.PR) ,contraction coefficient ,020206 networking & telecommunications ,lcsh:QC1-999 ,Tree (data structure) ,majorization theory ,Rényi information measures ,lcsh:Q ,Tsallis entropy ,list decoding ,lcsh:Physics ,Mathematics - Probability - Abstract
This paper is focused on derivations of data-processing and majorization inequalities for $f$-divergences, and their applications in information theory and statistics. For the accessibility of the material, the main results are first introduced without proofs, followed by exemplifications of the theorems with further related analytical results, interpretations, and information-theoretic applications. One application refers to the performance analysis of list decoding with either fixed or variable list sizes; some earlier bounds on the list decoding error probability are reproduced in a unified way, and new bounds are obtained and exemplified numerically. Another application is related to a study of the quality of approximating a probability mass function, induced by the leaves of a Tunstall tree, by an equiprobable distribution. The compression rates of finite-length Tunstall codes are further analyzed for asserting their closeness to the Shannon entropy of a memoryless and stationary discrete source. Almost all the analysis is relegated to the appendices, which form a major part of this manuscript., This paper is published in the Entropy journal, vol. 21, no. 10, paper 1022, pages 1-80, October 21, 2019 (https://www.mdpi.com/1099-4300/21/10/1022)
- Published
- 2019
27. The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs
- Author
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Remco van der Hofstad, Sanchayan Sen, Shankar Bhamidi, Stochastic Operations Research, and Probability
- Subjects
Statistics and Probability ,Critical random graphs ,01 natural sciences ,Lévy process ,Giant component ,010104 statistics & probability ,FOS: Mathematics ,Mathematics - Combinatorics ,0101 mathematics ,60C05, 05C80 ,Fundamental class ,Mathematics ,Discrete mathematics ,Random graph ,Gromov-Hausdorff distance ,p-trees ,Gromov-weak topology ,010102 general mathematics ,Multiplicative function ,Probability (math.PR) ,Minkowski–Bouligand dimension ,Inhomogeneous continuum random trees ,16. Peace & justice ,Scaling limit ,$$\mathbf {p}$$p-trees ,Exponent ,Combinatorics (math.CO) ,Multiplicative coalescent ,Statistics, Probability and Uncertainty ,Analysis ,Mathematics - Probability - Abstract
One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array of random graph models with degree exponent $\tau\in (3,4)$, distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{(\tau-3)/(\tau-1)}$. In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of L\'evy processes "without replacement" [10], yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent $\tau\in (3,4)$ where edges are rescaled by $n^{-(\tau-3)/(\tau-1)}$ yielding the first rigorous proof of the above conjecture. The limits in this case are compact "tree-like" random fractals with finite fractal dimensions and with a dense collection of hubs (infinite degree vertices) a finite number of which are identified with leaves to form shortcuts. In a special case, we show that the Minkowski dimension of the limiting spaces equal $(\tau-2)/(\tau-3)$ a.s., in stark contrast to the Erd\H{o}s-R\'{e}nyi scaling limit whose Minkowski dimension is 2 a.s. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent $\tau\in (3,4)$., Comment: 71 pages, 5 figures, To appear in Probability Theory and Related Fields
- Published
- 2018
28. Improved convergence rates for Lasserre-type hierarchies of upper bounds for box-constrained polynomial optimization
- Author
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Etienne de Klerk, Monique Laurent, Roxana Hess, Tilburg University [Netherlands], Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Centrum Wiskunde & Informatica (CWI), Econometrics and Operations Research, Research Group: Operations Research, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), and Université de Toulouse (UT)
- Subjects
box-constrained global optimization ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Type (model theory) ,01 natural sciences ,Measure (mathematics) ,Theoretical Computer Science ,Combinatorics ,sum-of-squares polynomial ,polynomial optimization ,FOS: Mathematics ,0101 mathematics ,[MATH]Mathematics [math] ,Borel measure ,Mathematics - Optimization and Control ,Mathematics ,Discrete mathematics ,Polynomial (hyperelastic model) ,90C60, 90C56, 90C26 ,generalized eigenvalue problem ,021103 operations research ,Lebesgue measure ,Degree (graph theory) ,semidefinite programming ,Optimization and Control (math.OC) ,Probability distribution ,Jackson kernel ,semidenite programming ,Hypercube ,Software - Abstract
We consider the problem of minimizing a given $n$-variate polynomial $f$ over the hypercube $[-1,1]^n$. An idea introduced by Lasserre, is to find a probability distribution on $[-1,1]^n$ with polynomial density function $h$ (of given degree $r$) that minimizes the expectation $\int_{[-1,1]^n} f(x)h(x)d\mu(x)$, where $d\mu(x)$ is a fixed, finite Borel measure supported on $[-1,1]^n$. It is known that, for the Lebesgue measure $d\mu(x) = dx$, one may show an error bound $O(1/\sqrt{r})$ if $h$ is a sum-of-squares density, and an $O(1/r)$ error bound if $h$ is the density of a beta distribution. In this paper, we show an error bound of $O(1/r^2)$, if $d\mu(x) = \left(\prod_{i=1}^n \sqrt{1-x_i^2} \right)^{-1}dx$ (the well-known measure in the study of orthogonal polynomials), and $h$ has a Schm\"udgen-type representation with respect to $[-1,1]^n$, which is a more general condition than a sum of squares. The convergence rate analysis relies on the theory of polynomial kernels, and in particular on Jackson kernels. We also show that the resulting upper bounds may be computed as generalized eigenvalue problems, as is also the case for sum-of-squares densities., Comment: 21 pages, 3 figures, published in SIAM Journal on Optimization, 27(1):347-367, March 2017. This arXiv version is an updated version of the journal paper. Typos were corrected in the proofs of Lemmata 2.2 and 3.5 and Theorem 4.2
- Published
- 2017
29. Computability, orders, and solvable groups
- Author
-
Arman Darbinyan
- Subjects
Discrete mathematics ,Logic ,Group (mathematics) ,Computability ,Existential quantification ,010102 general mathematics ,Extension (predicate logic) ,Group Theory (math.GR) ,Mathematics - Logic ,01 natural sciences ,Undecidable problem ,010101 applied mathematics ,Philosophy ,Solvable group ,FOS: Mathematics ,Embedding ,Word problem (mathematics) ,0101 mathematics ,Logic (math.LO) ,Mathematics - Group Theory ,Mathematics - Abstract
The main objective of this paper is the following two results. (1) There exists a computable bi-orderable group that does not have a computable bi-ordering; (2) There exists a bi-orderable, two-generated recursively presented solvable group with undecidable word problem. Both of the groups can be found among two-generated solvable groups of derived length $3$. (1) answers a question posed by Downey and Kurtz; (2) answers a question posed by Bludov and Glass in Kourovka Notebook. One of the technical tools used to obtain the main results is a computational extension of an embedding theorem of B. Neumann that was studied by the author earlier. In this paper we also compliment that result and derive new corollaries that might be of independent interest., Added Subsection 1.1 which is a revised version of Section 2 from v1. Other noticeable revisions, especially, in Section 4
- Published
- 2019
30. Enclosings of decompositions of complete multigraphs in 2-edge-connected r -factorizations
- Author
-
John Asplund, Pierre Charbit, Carl Feghali, Dalton State College, University System of Georgia (USG), Institut de Recherche en Informatique Fondamentale (IRIF (UMR_8243)), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Networks, Graphs and Algorithms (GANG), Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)-Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), University of Bergen (UIB), This paper received partial support his paper received partial support by Fondation Sciences Mathématiques de Paris and the Research Council of Norway via the project CLASSIS, Grant Number 249994., and University of Bergen (UiB)
- Subjects
Discrete mathematics ,010102 general mathematics ,Multigraph ,[INFO.INFO-DS]Computer Science [cs]/Data Structures and Algorithms [cs.DS] ,0102 computer and information sciences ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,[INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG] ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,Factorization ,010201 computation theory & mathematics ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Partition (number theory) ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,05B99 - Abstract
A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G(1), \ldots , G(k)$. It is called an $r$-factorization if every $G(i)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a decomposition of $G$ is said to be enclosed in a decomposition of $H$ if, for every $1 \leq i \leq k$, $G(i)$ is a subgraph of $H(i)$. Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of $\lambda K_n$ to be enclosed in some $2$-edge-connected $r$-factorization of $\mu K_{m}$ for some range of values for the parameters $n$, $m$, $\lambda$, $\mu$, $r$: $r=2$, $\mu>\lambda$ and either $m \geq 2n-1$, or $m=2n-2$ and $\mu = 2$ and $\lambda=1$, or $n=3$ and $m=4$. We generalize their result to every $r \geq 2$ and $m \geq 2n - 2$. We also give some sufficient conditions for enclosing a given decomposition of $\lambda K_n$ in some $2$-edge-connected $r$-factorization of $\mu K_{m}$ for every $r \geq 3$ and $m = (2 - C)n$, where $C$ is a constant that depends only on $r$, $\lambda$ and~$\mu$., Comment: 17 pages; fixed the proof of Theorem 1.4 and other minor changes
- Published
- 2019
31. Minimal linear codes arising from blocking sets
- Author
-
Martino Borello and Matteo Bonini
- Subjects
Discrete mathematics ,FOS: Computer and information sciences ,Algebra and Number Theory ,Information Theory (cs.IT) ,Computer Science - Information Theory ,010102 general mathematics ,Context (language use) ,0102 computer and information sciences ,Mathematical proof ,Blocking (statistics) ,01 natural sciences ,Secret sharing ,Linear code ,Combinatorics ,010201 computation theory & mathematics ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Algebraic number ,Link (knot theory) ,Mathematics - Abstract
Minimal linear codes are algebraic objects which gained interest in the last twenty years, due to their link with Massey's secret sharing schemes. In this context, Ashikhmin and Barg provided a useful and a quite easy to handle sufficient condition for a linear code to be minimal, which has been applied in the construction of many minimal linear codes. In this paper, we generalize some recent constructions of minimal linear codes which are not based on Ashikhmin-Barg's condition. More combinatorial and geometric methods are involved in our proofs. In particular, we present a family of codes arising from particular blocking sets, which are well-studied combinatorial objects. In this context, we will need to define cutting blocking sets and to prove some of their relations with other notions in blocking sets' theory. At the end of the paper, we provide one explicit family of cutting blocking sets and related minimal linear codes, showing that infinitely many of its members do not satisfy the Ashikhmin-Barg's condition., 13 pages
- Published
- 2019
32. Sharp Bounds on the Runtime of the (1+1) EA via Drift Analysis and Analytic Combinatorial Tools
- Author
-
Hsien-Kuei Hwang and Carsten Witt
- Subjects
Discrete mathematics ,FOS: Computer and information sciences ,Logarithm ,Probability (math.PR) ,Inverse ,Computer Science - Neural and Evolutionary Computing ,Randomized search heuristics ,Drift analysis ,Function (mathematics) ,State (functional analysis) ,Binary logarithm ,Evolutionary computation ,Asymptotic methods ,Computer Science - Data Structures and Algorithms ,Benchmark (computing) ,FOS: Mathematics ,Data Structures and Algorithms (cs.DS) ,Neural and Evolutionary Computing (cs.NE) ,Asymptotic expansion ,(1+1) EA ,Mathematics - Probability ,Mathematics - Abstract
The expected running time of the classical (1+1) EA on the OneMax benchmark function has recently been determined by Hwang et al. (2018) up to additive errors of $O((\log n)/n)$. The same approach proposed there also leads to a full asymptotic expansion with errors of the form $O(n^{-K}\log n)$ for any $K>0$. This precise result is obtained by matched asymptotics with rigorous error analysis (or by solving asymptotically the underlying recurrences via inductive approximation arguments), ideas radically different from well-established techniques for the running time analysis of evolutionary computation such as drift analysis. This paper revisits drift analysis for the (1+1) EA on OneMax and obtains that the expected running time $E(T)$, starting from $\lceil n/2\rceil$ one-bits, is determined by the sum of inverse drifts up to logarithmic error terms, more precisely $$\sum_{k=1}^{\lfloor n/2\rfloor}\frac{1}{\Delta(k)} - c_1\log n \le E(T) \le \sum_{k=1}^{\lfloor n/2\rfloor}\frac{1}{\Delta(k)} - c_2\log n,$$ where $\Delta(k)$ is the drift (expected increase of the number of one-bits from the state of $n-k$ ones) and $c_1,c_2 >0$ are explicitly computed constants. This improves the previous asymptotic error known for the sum of inverse drifts from $\tilde{O}(n^{2/3})$ to a logarithmic error and gives for the first time a non-asymptotic error bound. Using standard asymptotic techniques, the difference between $E(T)$ and the sum of inverse drifts is found to be $(e/2)\log n+O(1)$., Comment: 33 pages; preprint of a paper that will be published in the proceedings of FOGA 2019; v2: minor corrections
- Published
- 2019
33. Biased random k-SAT
- Author
-
Klas Markström and Joel Larsson
- Subjects
Matching (graph theory) ,random constraint satisfaction problem ,General Mathematics ,0102 computer and information sciences ,Computer Science::Computational Complexity ,01 natural sciences ,random k-SAT ,Set (abstract data type) ,Combinatorics ,FOS: Mathematics ,Mathematics - Combinatorics ,60C05 ,Special case ,QA ,Coupon collector's problem ,QC ,Mathematics ,Discrete Mathematics ,Computer Sciences ,Singleton ,Applied Mathematics ,Probability (math.PR) ,Diskret matematik ,Computer Graphics and Computer-Aided Design ,Datavetenskap (datalogi) ,Cover (topology) ,phase transition ,010201 computation theory & mathematics ,ombinatorial probability ,Combinatorics (math.CO) ,Hypercube ,Constant (mathematics) ,Software ,Mathematics - Probability - Abstract
This thesis consists of the following papers.I J. Larsson, The Minimum Matching in Pseudo-dimension 0 0.The second paper lives in a more general setting. There we search for any cover of the ground set V, for general families F. Each set f ∈ F receives weight w(f) uniformly at random from [0,1]. The cost of a cover f_1,f_2,...f_m is then taken to be max_i w(f_i). This is equivalent (after a rescaling) to drawing sets from F at Poisson times, and the cost of a cover is the first time V is covered. This problem is known under a number of names, perhaps most famously the coupon collector problem. In the classical formulation, single elements of V are drawn, not sets. The classical coupon collector thus corresponds to the family F consisting of singleton sets, and we call the version allowing larger sets structured coupon collector problems. The main concern of this paper is to identify relevant properties of F that affect the covering time (i.e. minimal cost of a cover), and to provide (easily checkable) sufficient conditions for concentration of the covering time.For the third paper we narrow the scopes once more, and study the biased random k-SAT problem. The random k-SAT problem can be seen as a special case of the structured coupon collector, but a special case that has far richer structure than the generic case. The ground set is the hypercube Σ_n = {0, 1}^n, and the coupons are all the k-codimensional subcubes of Σ_n. We study a slight variation on this problem: subcubes are drawn with a constant bias towards 0, so that vertices in Σ_n with fewer 1's and more 0's are easier to cover.
- Published
- 2019
34. Random polynomials: central limit theorems for the real roots
- Author
-
Oanh Nguyen and Van Vu
- Subjects
Discrete mathematics ,Large class ,Real roots ,General Mathematics ,Probability (math.PR) ,Universality (philosophy) ,Asymptotic distribution ,Limiting ,Random polynomials ,FOS: Mathematics ,Random variable ,Mathematics - Probability ,Mathematics ,Central limit theorem - Abstract
The number of real roots has been a central subject in the theory of random polynomials and random functions since the fundamental papers of Littlewood-Offord and Kac in the 1940s. The main task here is to determine the limiting distribution of this random variable. In 1974, Maslova famously proved a central limit theorem (CLT) for the number of real roots of Kac polynomials. It has remained the only limiting theorem available for the number of real roots for more than four decades. In this paper, using a new approach, we derive a general CLT for the number of real roots of a large class of random polynomials with coefficients growing polynomially. Our result both generalizes and strengthens Maslova's theorem., some references were added and several proofs were elaborated
- Published
- 2019
35. Dismantlability, connectedness, and mixing in relational structures
- Author
-
Andrei A. Bulatov, Raimundo Briceño, Benoit Larose, and Víctor Dalmau
- Subjects
FOS: Computer and information sciences ,Computer Science - Logic in Computer Science ,Social connectedness ,Constraint satisfaction problem ,Duality (mathematics) ,Structure (category theory) ,FOS: Physical sciences ,0102 computer and information sciences ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,Mixing properties ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,Relational structure ,0101 mathematics ,Mixing (physics) ,Mathematical Physics ,Mathematics ,Discrete mathematics ,000 Computer science, knowledge, general works ,Probability (math.PR) ,010102 general mathematics ,Solution set ,Mathematical Physics (math-ph) ,Gibbs measure ,Logic in Computer Science (cs.LO) ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Homomorphism ,Computer Science ,Graph (abstract data type) ,Combinatorics (math.CO) ,08A70, 68Q87, 68R01, 82B20, 68R10, 05C15 ,Mathematics - Probability - Abstract
The Constraint Satisfaction Problem (CSP) and its counting counterpart appears under different guises in many areas of mathematics, computer science, and elsewhere. Its structural and algorithmic properties have demonstrated to play a crucial role in many of those applications. For instance, in the decision CSPs, structural properties of the relational structures involved---like, for example, dismantlability---and their logical characterizations have been instrumental for determining the complexity and other properties of the problem. Topological properties of the solution set such as connectedness are related to the hardness of CSPs over random structures. Additionally, in approximate counting and statistical physics, where CSPs emerge in the form of spin systems, mixing properties and the uniqueness of Gibbs measures have been heavily exploited for approximating partition functions and free energy. In spite of the great diversity of those features, there are some eerie similarities between them. These were observed and made more precise in the case of graph homomorphisms by Brightwell and Winkler, who showed that dismantlability of the target graph, connectedness of the set of homomorphisms, and good mixing properties of the corresponding spin system are all equivalent. In this paper we go a step further and demonstrate similar connections for arbitrary CSPs. This requires much deeper understanding of dismantling and the structure of the solution space in the case of relational structures, and new refined concepts of mixing introduced by Brice\~no. In addition, we develop properties related to the study of valid extensions of a given partially defined homomorphism, an approach that turns out to be novel even in the graph case. We also add to the mix the combinatorial property of finite duality and its logic counterpart, FO-definability, studied by Larose, Loten, and Tardif., Comment: 27 pages, full version of the paper accepted to ICALP 2019
- Published
- 2019
36. Rigidity theory for $C^*$-dynamical systems and the 'Pedersen Rigidity Problem', II
- Author
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Tron Omland, John Quigg, and Steven Kaliszewski
- Subjects
Discrete mathematics ,Exterior equivalences ,Pure mathematics ,Dynamical systems theory ,Mathematics::Operator Algebras ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Outer conjugacy ,Generalized fixed point algebra ,01 natural sciences ,Rigidity (electromagnetism) ,Crossed product ,Primary 46L55 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Locally compact space ,0101 mathematics ,Abelian group ,Rigidity theory ,Operator Algebras (math.OA) ,Crossed products ,Mathematics ,Conjugate - Abstract
This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen's theorem, which does hold for an arbitrary locally compact group $G$, saying that two actions $(A,\alpha)$ and $(B,\beta)$ of $G$ are outer conjugate if and only if the dual coactions $(A\rtimes_{\alpha}G,\widehat\alpha)$ and $(B\rtimes_{\beta}G,\widehat\beta)$ of $G$ are conjugate via an isomorphism that maps the image of $A$ onto the image of $B$ (inside the multiplier algebras of the respective crossed products). We do not know of any examples of a pair of non-outer-conjugate actions such that their dual coactions are conjugate, and our interest is therefore exploring the necessity of latter condition involving the images, and we have decided to use the term "Pedersen rigid" for cases where this condition is indeed redundant. There is also a related problem, concerning the possibility of a so-called equivariant coaction having a unique generalized fixed-point algebra, that we call "fixed-point rigidity". In particular, if the dual coaction of an action is fixed-point rigid, then the action itself is Pedersen rigid, and no example of non-fixed-point-rigid coaction is known., Comment: Minor revision. To appear in Internat. J. Math
- Published
- 2018
37. Multi-variate correlation and mixtures of product measures
- Author
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Tim Austin
- Subjects
Discrete mathematics ,FOS: Computer and information sciences ,Information Theory (cs.IT) ,Computer Science - Information Theory ,Probability (math.PR) ,Mathematics - Statistics Theory ,Mutual information ,Statistics Theory (math.ST) ,Theoretical Computer Science ,Artificial Intelligence ,Control and Systems Engineering ,Joint probability distribution ,Product (mathematics) ,Metric (mathematics) ,FOS: Mathematics ,Product measure ,60B99 (primary), 60G99, 62B10, 94A17 ,Total correlation ,Electrical and Electronic Engineering ,Random variable ,Software ,Mathematics - Probability ,Information Systems ,Dual total correlation ,Mathematics - Abstract
Total correlation (`TC') and dual total correlation (`DTC') are two classical ways to quantify the correlation among an $n$-tuple of random variables. They both reduce to mutual information when $n=2$. The first part of this paper sets up the theory of TC and DTC for general random variables, not necessarily finite-valued. This generality has not been exposed in the literature before. The second part considers the structural implications when a joint distribution $\mu$ has small TC or DTC. If $\mathrm{TC}(\mu) = o(n)$, then $\mu$ is close to a product measure according to a suitable transportation metric: this follows directly from Marton's classical transportation-entropy inequality. If $\mathrm{DTC}(\mu) = o(n)$, then the structural consequence is more complicated: $\mu$ is a mixture of a controlled number of terms, most of them close to product measures in the transportation metric. This is the main new result of the paper., Comment: 39 pages [v2:] Slight changes in presentation based on feedback from colleagues [v3, v4:] Small revisions during preparation for journal
- Published
- 2018
38. Computing Modular Data for Pointed Fusion Categories
- Author
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Angus Gruen and Scott Morrison
- Subjects
Discrete mathematics ,Code (set theory) ,Class (set theory) ,business.industry ,General Mathematics ,Dimension (graph theory) ,Center (category theory) ,Mathematics - Category Theory ,Modular design ,Homogeneous space ,Mathematics - Quantum Algebra ,FOS: Mathematics ,18D10 (Primary), 18-04 (Secondary) ,Quantum Algebra (math.QA) ,Category Theory (math.CT) ,Morita equivalence ,Quasi-Hopf algebra ,business ,Mathematics - Abstract
A formula for the modular data of $\mathcal{Z}(Vec^{\omega}G)$ was given by Coste, Gannon and Ruelle in arXiv:hep-th/0001158, but without an explicit proof for arbitrary 3-cocycles. This paper presents a derivation using the representation category of the quasi Hopf algebra $D^{\omega}G$. Further, we have written code to compute this modular data for many pairs of small finite groups and 3-cocycles. This code is optimised using Galois symmetries of the S and T matrices. We have posted a database of modular data for the Drinfeld center of every Morita equivalence class of pointed fusion categories of dimension less than 64., Comment: 28 pages, 4 figures, 7 pages of appendices. v2 update fixes the arXiv reference in the abstract. v3 update: due to recent work the size of the database was increased adding in the modular data for groups with order 48-63 inclusive. Additionally parts of the paper have been re-written based off referee suggestions
- Published
- 2018
39. Time-Bounded Influence Diffusion with Incentives
- Author
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Ugo Vaccaro, Adele A. Rescigno, Gennaro Cordasco, Luisa Gargano, Joseph G. Peters, M. Borokhovich, Cordasco, Gennaro, Gargano, Luisa, Peters, Joseph G., Rescigno, Adele A., and Vaccaro, Ugo
- Subjects
Discrete mathematics ,Social and Information Networks (cs.SI) ,FOS: Computer and information sciences ,Physics - Physics and Society ,General problem ,Computer Science (all) ,FOS: Physical sciences ,Computer Science - Social and Information Networks ,02 engineering and technology ,Physics and Society (physics.soc-ph) ,Graph ,Theoretical Computer Science ,Incentive ,020204 information systems ,Bounded function ,Computer Science - Data Structures and Algorithms ,0202 electrical engineering, electronic engineering, information engineering ,FOS: Mathematics ,Mathematics - Combinatorics ,020201 artificial intelligence & image processing ,Data Structures and Algorithms (cs.DS) ,Combinatorics (math.CO) ,Mathematics - Abstract
A widely studied model of influence diffusion in social networks represents the network as a graph $G=(V,E)$ with an influence threshold $t(v)$ for each node. Initially the members of an initial set $S\subseteq V$ are influenced. During each subsequent round, the set of influenced nodes is augmented by including every node $v$ that has at least $t(v)$ previously influenced neighbours. The general problem is to find a small initial set that influences the whole network. In this paper we extend this model by using \emph{incentives} to reduce the thresholds of some nodes. The goal is to minimize the total of the incentives required to ensure that the process completes within a given number of rounds. The problem is hard to approximate in general networks. We present polynomial-time algorithms for paths, trees, and complete networks., An extended abstract of this paper was presented at the 25th International Colloquium on Structural Information and Communication Complexity (Sirocco 2018) June 18-21, 2018 Ma'ale HaHamisha, Israel
- Published
- 2018
40. Sturm's theorem on the zeros of sums of eigenfunctions: Gelfand's strategy implemented
- Author
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Bernard Helffer, Pierre Bérard, Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Laboratoire de Mathématiques Jean Leray (LMJL), Université de Nantes - Faculté des Sciences et des Techniques, Université de Nantes (UN)-Université de Nantes (UN)-Centre National de la Recherche Scientifique (CNRS), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Nantes - UFR des Sciences et des Techniques (UN UFR ST), Université de Nantes (UN)-Université de Nantes (UN), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11)
- Subjects
General Mathematics ,FOS: Physical sciences ,Zeros of eigenfunction ,Interval (mathematics) ,Sturm theorem ,01 natural sciences ,Section (fiber bundle) ,Mathematics - Spectral Theory ,03 medical and health sciences ,0302 clinical medicine ,Mathematics - Analysis of PDEs ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,FOS: Mathematics ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,030212 general & internal medicine ,0101 mathematics ,Linear combination ,Sturm's theorem ,Spectral Theory (math.SP) ,Mathematical Physics ,Mathematics ,Discrete mathematics ,Operator (physics) ,010102 general mathematics ,Multiplicity (mathematics) ,Mathematical Physics (math-ph) ,Eigenfunction ,Nodal domain ,Courant nodal domain theorem ,MSC 2010: 35P99, 35Q99, 58J50 ,Slater determinant ,[MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP] ,Analysis of PDEs (math.AP) - Abstract
In the second section "Courant-Gelfand theorem" of his last published paper (Topological properties of eigenoscillations in mathematical physics, Proc. Steklov Institute Math. 273 (2011) 25--34), Arnold recounts Gelfand's strategy to prove that the zeros of any linear combination of the $n$ first eigenfunctions of the Sturm-Liouville problem $$-\, y"(s) + q(x)\, y(x) = \lambda\, y(x) \mbox{ in } ]0,1[\,, \mbox{ with } y(0)=y(1)=0\,,$$divide the interval into at most $n$ connected components, and concludes that "the lack of a published formal text with a rigorous proof \dots is still distressing." Inspired by Quantum mechanics, Gelfand's strategy consists in replacing the ana\-lysis of linear combinations of the $n$ first eigenfunctions by that of their Slater determinant which is the first eigenfunction of the associated $n$-particle operator acting on Fermions. In the present paper, we implement Gelfand's strategy, and give a complete proof of the above assertion. As a matter of fact, refining Gelfand's strategy, we prove a stronger property taking the multiplicity of zeros into account, a result which actually goes back to Sturm (1836)., Comment: Comments: One subsection deleted. References added. Minor corrections
- Published
- 2018
41. The components of the singular locus of a component of a Springer fiber over x^2 = 0
- Author
-
Anna Melnikov and Ronit Mansour
- Subjects
Discrete mathematics ,Algebra and Number Theory ,Computation ,010102 general mathematics ,Combinatorial algorithms ,01 natural sciences ,0103 physical sciences ,Singular component ,FOS: Mathematics ,Young tableau ,Mathematics - Combinatorics ,010307 mathematical physics ,Geometry and Topology ,Combinatorics (math.CO) ,0101 mathematics ,Locus (mathematics) ,05E10 ,Mathematics - Abstract
For $x\in End(K^n)$ satisfying $x^2 = 0$ let $F_x$ be the variety of full flags stable under the action of $x$ (Springer fiber over $x$). The full classification of the components of $F_x$ according to their smoothness was provided in a paper of Fresse-Melnikov in terms of both Young tableaux and link patterns. Moreover in a paper of Fresse the purely combinatorial algorithm to compute the singular locus of a singular components of $F_x$ is provided. However this algorithm involves the computation of the graph of the component, and the complexity of computations grows very quickly, so that in practice it is impossible to use it. In this paper, we construct another algorithm, derived from the algorithm of Fresse, providing all the components of the singular locus of a singular component of $F_x$ in terms of link patterns constructed straightforwardly from its link pattern., 34 pages
- Published
- 2018
42. On the complexity of quasiconvex integer minimization problem
- Author
-
Panos M. Pardalos, Dmitriy S. Malyshev, S. I. Veselov, D. V. Gribanov, N. Yu. Zolotykh, and A. Yu. Chirkov
- Subjects
Discrete mathematics ,FOS: Computer and information sciences ,021103 operations research ,Control and Optimization ,Optimization problem ,Applied Mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,Computational Complexity (cs.CC) ,Upper and lower bounds ,Oracle ,Computer Science Applications ,Exponential function ,Quasiconvex function ,Computer Science - Computational Complexity ,Conic section ,Optimization and Control (math.OC) ,FOS: Mathematics ,Ball (mathematics) ,Convex function ,Mathematics - Optimization and Control ,Mathematics - Abstract
In this paper, we consider the class of quasiconvex functions and its proper subclass of conic functions. The integer minimization problem of these functions is considered in the paper, assuming that an optimized function is defined by the comparison oracle. We will show that there is no a polynomial algorithm on $\log R$ to optimize quasiconvex functions in the ball of integer radius $R$ using only the comparison oracle. On the other hand, if an optimized function is conic, then we show that there is a polynomial on $\log R$ algorithm. We also present an exponential on the dimension lower bound for the oracle complexity of the conic function integer optimization problem. Additionally, we give examples of known problems that can be polynomially reduced to the minimization problem of functions in our classes., Some new proofs have been added. Some fixes are done
- Published
- 2018
43. Facial unique-maximum colorings of plane graphs with restriction on big vertices
- Author
-
Riste Škrekovski, Kacy Messerschmidt, and Bernard Lidický
- Subjects
Discrete mathematics ,Conjecture ,05C15, 05C10 ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,A* search algorithm ,Four color theorem ,Theoretical Computer Science ,law.invention ,Planar graph ,Vertex (geometry) ,Combinatorics ,symbols.namesake ,law ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,symbols ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,ComputingMethodologies_COMPUTERGRAPHICS ,MathematicsofComputing_DISCRETEMATHEMATICS ,Mathematics - Abstract
A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and G\"{o}ring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial unique-maximum coloring using four colors. This conjecture has been disproven for general plane graphs and it was shown that five colors suffice. In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially unique-maximum 4-colorable. This improves a previous result for subcubic plane graphs by Andova, Lidick\'y, Lu\v{z}ar, and \v{S}krekovski (2018). We conclude the paper by proposing some problems., Comment: 8 pages, 5 figures
- Published
- 2018
44. The DMT classification of real and quaternionic lattice codes
- Author
-
Roope Vehkalahti, Laura Luzzi, Equipes Traitement de l'Information et Systèmes (ETIS - UMR 8051), Ecole Nationale Supérieure de l'Electronique et de ses Applications (ENSEA)-Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Department of Communications and Networking [Aalto Univ], School of Electrical Engineering [Aalto Univ], Aalto University-Aalto University, and IEEE Information Theory Society
- Subjects
ta113 ,FOS: Computer and information sciences ,Discrete mathematics ,Mathematics - Number Theory ,Optimality criterion ,Computer Science - Information Theory ,Information Theory (cs.IT) ,020206 networking & telecommunications ,02 engineering and technology ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Corollary ,[INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT] ,Lattice (order) ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Division algebra ,Number Theory (math.NT) ,Quaternion ,Mathematics - Abstract
In this paper we consider space-time codes where the code-words are restricted to either real or quaternion matrices. We prove two separate diversity-multiplexing gain trade-off (DMT) upper bounds for such codes and provide a criterion for a lattice code to achieve these upper bounds. We also point out that lattice codes based on Q-central division algebras satisfy this optimality criterion. As a corollary this result provides a DMT classification for all Q-central division algebra codes that are based on standard embeddings., Comment: 6 pages, 1 figure. Conference paper submitted to the International Symposium on Information Theory 2018
- Published
- 2018
45. Dynkin games with Poisson random intervention times
- Author
-
Gechun Liang and Haodong Sun
- Subjects
Discrete mathematics ,Computer Science::Computer Science and Game Theory ,Control and Optimization ,Applied Mathematics ,ComputingMilieux_PERSONALCOMPUTING ,Poisson distribution ,Mathematical Finance (q-fin.MF) ,FOS: Economics and business ,symbols.namesake ,Optimization and Control (math.OC) ,Quantitative Finance - Mathematical Finance ,Intervention (counseling) ,Bellman equation ,symbols ,FOS: Mathematics ,91G40, 91G80, 60H30 ,Convertible bond ,QA ,Mathematics - Optimization and Control ,Mathematics ,Sequence (medicine) - Abstract
This paper introduces a new class of Dynkin games, where the two players are allowed to make their stopping decisions at a sequence of exogenous Poisson arrival times. The value function and the associated optimal stopping strategy are characterized by the solution of a backward stochastic differential equation. The paper further provides a replication strategy for the game and applies the model to study the optimal conversion and calling strategies of convertible bonds, and their asymptotics when the Poisson intensity goes to infinity.\ud \ud \ud
- Published
- 2018
46. Construction of Milnorian representations
- Author
-
Ilia Smilga
- Subjects
Discrete mathematics ,20G20, 20G05, 22E40, 20H15 ,Group (mathematics) ,Hyperbolic geometry ,010102 general mathematics ,Lie group ,Algebraic geometry ,Group Theory (math.GR) ,Space (mathematics) ,01 natural sciences ,Differential geometry ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Geometry and Topology ,Affine transformation ,0101 mathematics ,Representation Theory (math.RT) ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Mathematics ,Projective geometry - Abstract
We prove a partial converse to the main theorem of the author's previous paper "Proper affine actions: a sufficient criterion" (submitted; available at arXiv:1612.08942). More precisely, let $G$ be a semisimple real Lie group with a representation $\rho$ on a finite-dimensional real vector space $V$, that does not satisfy the criterion from the previous paper. Assuming that $\rho$ is irreducible and under some additional assumptions on $G$ and $\rho$, we then prove that there does not exist a group of affine transformations acting properly discontinuously on $V$ whose linear part is Zariski-dense in $\rho(G)$., Comment: This version differs from the previous one only by a reference that I added
- Published
- 2018
47. The cycle polynomial of a permutation group
- Author
-
Peter J. Cameron, Jason Semeraro, University of St Andrews. Pure Mathematics, and University of St Andrews. Centre for Interdisciplinary Research in Computational Algebra
- Subjects
Polynomial ,Mathematics(all) ,General problem ,Reciprocity ,NDAS ,Chromatic polynomial ,Theoretical Computer Science ,Combinatorics ,05C31, 20B05 ,Symmetric polynomial ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,QA Mathematics ,QA ,Mathematics ,Discrete mathematics ,Applied Mathematics ,Permutation group ,Graph ,Computational Theory and Mathematics ,Combinatorics (math.CO) ,Geometry and Topology ,Null graph ,Reciprocal - Abstract
The cycle polynomial of a finite permutation group $G$ is the generating function for the number of elements of $G$ with a given number of cycles: \[F_G(x) = \sum_{g\in G}x^{c(g)},\] where $c(g)$ is the number of cycles of $g$ on $\Omega$. In the first part of the paper, we develop basic properties of this polynomial, and give a number of examples. In the 1970s, Richard Stanley introduced the notion of reciprocity for pairs of combinatorial polynomials. We show that, in a considerable number of cases, there is a polynomial in the reciprocal relation to the cycle polynomial of $G$; this is the orbital chromatic polynomial of $\Gamma$ and $G$, where $\Gamma$ is a $G$-invariant graph, introduced by the first author, Jackson and Rudd. We pose the general problem of finding all such reciprocal pairs, and give a number of examples and characterisations: the latter include the cases where $\Gamma$ is a complete or null graph or a tree. The paper concludes with some comments on other polynomials associated with a permutation group., Comment: 16 pages
- Published
- 2018
48. Weak∗ fixed point property in ℓ1 and polyhedrality in Lindenstrauss spaces
- Author
-
Łukasz Piasecki, Roxana Popescu, Enrico Miglierina, and Emanuele Casini
- Subjects
General Mathematics ,ℓ1 space ,Nonexpansive mappings ,W∗-fixed point property ,Fixed-point property ,01 natural sciences ,Polyhedral spaces ,Lindenstrauss spaces ,FOS: Mathematics ,`1 space ,Extension of compact operators ,Stability of the w∗-fixed point property ,Mathematics (all) ,0101 mathematics ,w-fixed point property ,Mathematics ,Discrete mathematics ,Nonexpansive mappings, w-fixed point property, Stability of the w- fixed point property, Lindenstrauss spaces, Polyhedral spaces, `1 space, Extension of compact operators ,010102 general mathematics ,Settore MAT/05 - ANALISI MATEMATICA ,Functional Analysis (math.FA) ,Stability of the w- fixed point property ,010101 applied mathematics ,Mathematics - Functional Analysis ,47H10, 46B25, 46B45 - Abstract
The aim of this paper is to study the $w^*$-fixed point property for nonexpansive mappings in the duals of separable Lindenstrauss spaces by means of suitable geometrical properties of the dual ball. First we show that a property concerning the behaviour of a class of $w^*$-closed subsets of the dual sphere is equivalent to the $w^*$-fixed point property. Then, the main result of our paper shows an equivalence between another, stronger geometrical property of the dual ball and the stable $w^*$-fixed point property. The last geometrical notion was introduced by Fonf and Vesel\'{y} as a strengthening of the notion of polyhedrality. In the last section we show that also the first geometrical assumption that we have introduced can be related to a polyhedral concept for the predual space. Indeed, we give a hierarchical structure among various polyhedrality notions in the framework of Lindenstrauss spaces. Finally, as a by-product, we obtain an improvement of an old result about the norm-preserving compact extension of compact operators.
- Published
- 2018
49. Semi-Baxter and strong-Baxter: two relatives of the Baxter sequence
- Author
-
Andrew Rechnitzer, Veronica Guerrini, Mathilde Bouvel, and Simone Rinaldi
- Subjects
Discrete mathematics ,Sequence ,Mathematics::Combinatorics ,General Mathematics ,permutations ,010102 general mathematics ,0102 computer and information sciences ,enumerative combinatorics ,01 natural sciences ,Enumerative combinatorics ,Combinatorics ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,010201 computation theory & mathematics ,Mathematics::Quantum Algebra ,generating functions ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Mathematics - Abstract
In this paper, we enumerate two families of pattern-avoiding permutations: those avoiding the vincular pattern $2-41-3$, which we call semi-Baxter permutations, and those avoiding the vincular patterns $2-41-3$, $3-14-2$ and $3-41-2$, which we call strong-Baxter permutations. We call semi-Baxter numbers and strong-Baxter numbers the associated enumeration sequences. We prove that the semi-Baxter numbers enumerate in addition plane permutations (avoiding $2-14-3$). The problem of counting these permutations was open and has given rise to several conjectures, which we also prove in this paper. For each family (that of semi-Baxter -- or equivalently, plane -- and that of strong-Baxter permutations), we describe a generating tree, which translates into a functional equation for the generating function. For semi-Baxter permutations, it is solved using (a variant of) the kernel method: this gives an expression for the generating function while also proving its D-finiteness. From the obtained generating function, we derive closed formulas for the semi-Baxter numbers, a recurrence that they satisfy, as well as their asymptotic behavior. For strong-Baxter permutations, we show that their generating function is (a slight modification of) that of a family of walks in the quarter plane, which is known to be non D-finite., Comment: Version 3 incorporates changes suggested by a referee. Most important changes are that the paths sections have been removed and that the proof of the asymptotic equivalent has been simplified
- Published
- 2018
50. On the Complexity of Closest Pair via Polar-Pair of Point-Sets
- Author
-
C S Karthik, Bundit Laekhanukit, and Roee David
- Subjects
Computational Geometry (cs.CG) ,FOS: Computer and information sciences ,Discrete mathematics ,000 Computer science, knowledge, general works ,General Mathematics ,010102 general mathematics ,Metric Geometry (math.MG) ,0102 computer and information sciences ,Closest pair of points problem ,Computational Complexity (cs.CC) ,01 natural sciences ,Graph ,Combinatorics ,Computer Science - Computational Complexity ,Mathematics - Metric Geometry ,010201 computation theory & mathematics ,Computer Science ,FOS: Mathematics ,Polar ,Computer Science - Computational Geometry ,SPHERES ,0101 mathematics ,F.2.2 ,Computer Science::Databases ,Mathematics - Abstract
Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $\Delta$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$ such that $G$ can be represented by a collection of spheres (all of the same radius) in $\Delta$ is called the sphericity of $G$, and if the collection of spheres are non-overlapping, then the value $d$ is called the contact-dimension of $G$. In this paper, we study the sphericity and contact dimension of the complete bipartite graph $K_{n,n}$ in various $L^p$-metrics and consequently connect the complexity of the monochromatic closest pair and bichromatic closest pair problems., Comment: The paper was previously titled, "The Curse of Medium Dimension for Geometric Problems in Almost Every Norm"
- Published
- 2018
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