12,130 results
Search Results
2. On a paper of Erdös and Szekeres
- Author
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Mei-Chu Chang and Jean Bourgain
- Subjects
010101 applied mathematics ,Discrete mathematics ,Set (abstract data type) ,Partial differential equation ,Functional analysis ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,Analysis ,Mathematics - Abstract
Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products $$\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}$$ integers. In the discussion below, {a1,..., an} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.
- Published
- 2018
3. Correction and notes to the paper 'A classification of Artin–Schreier defect extensions and characterizations of defectless fields'
- Author
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Franz-Viktor Kuhlmann
- Subjects
Discrete mathematics ,Lemma (mathematics) ,14B05 ,General Mathematics ,13A18 ,010102 general mathematics ,12J10 ,Mistake ,Commutative Algebra (math.AC) ,Linearly disjoint ,Mathematics - Commutative Algebra ,01 natural sciences ,Primary 12J10, 13A18, Secondary 12J25, 12L12, 14B05 ,Field extension ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,12J25 ,12L12 ,Mathematics - Abstract
We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end, we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.
- Published
- 2019
4. Notes on the paper 'A note on pronormal p-subgroups of finite groups'
- Author
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Haoran Yu and Suli Liu
- Subjects
Discrete mathematics ,Lemma (mathematics) ,010505 oceanography ,General Mathematics ,010102 general mathematics ,0101 mathematics ,01 natural sciences ,0105 earth and related environmental sciences ,Mathematics - Abstract
In this short note, we show that Theorem 4.3 of Liu and Yu (Monatshefte Math 195:173–176, 2021) is a consequence of Lemma 2 of Ballester-Bolinches and Esteban-Romero (J Aust Math Soc 75:181–191, 2003).
- Published
- 2021
5. From coin tossing to rock-paper-scissors and beyond: a log-exp gap theorem for selecting a leader
- Author
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Hsien-Kuei Hwang, Yoshiaki Itoh, and Michael Fuchs
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Statistics and Probability ,Discrete mathematics ,Class (set theory) ,Coin flipping ,Group (mathematics) ,General Mathematics ,Variance (accounting) ,01 natural sciences ,Exponential function ,010104 statistics & probability ,Logarithmic mean ,Bounded function ,0103 physical sciences ,Gap theorem ,0101 mathematics ,Statistics, Probability and Uncertainty ,010306 general physics ,Mathematics - Abstract
A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.
- Published
- 2017
6. Local Restrictions from the Furst-Saxe-Sipser Paper
- Author
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Osamu Watanabe and Suguru Tamaki
- Subjects
Discrete mathematics ,Computational complexity theory ,Parity function ,True quantified Boolean formula ,Boolean circuit ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Bounded function ,Theory of computation ,Isomorphism ,0101 mathematics ,Boolean satisfiability problem ,Mathematics - Abstract
In their celebrated paper (Furst et al., Math. Syst. Theory 17(1), 13---27 (12)), Furst, Saxe, and Sipser used random restrictions to reveal the weakness of Boolean circuits of bounded depth, establishing that constant-depth and polynomial-size circuits cannot compute the parity function. Such local restrictions have played important roles and have found many applications in complexity analysis and algorithm design over the past three decades. In this article, we give a brief overview of two intriguing applications of local restrictions: the first one is for the Isomorphism Conjecture and the second one is for moderately exponential time algorithms for the Boolean formula satisfiability problem.
- Published
- 2016
7. Notes on the Paper 'On SS-Quasinormal and S-Quasinormally Embedded Subgroups of Finite Groups' of Shen et al
- Author
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Yuemei Mao, Xiaolan Yi, and Changwen Li
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Discrete mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Zhàng ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,GeneralLiterature_MISCELLANEOUS ,Mathematics - Abstract
We correct an error in the paper of Z. Shen, S. Li, and J. Zhang published in [4]. In addition, we give an answer to a question posed by the authors.
- Published
- 2018
8. Conference paper
- Author
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Marcin Wrochna, Stanislav Živný, Alex Brandts, Artur Czumaj, Anuj Dawar, and Emanuela Merelli
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FOS: Computer and information sciences ,Discrete Mathematics (cs.DM) ,G.2.1 ,0102 computer and information sciences ,Computational Complexity (cs.CC) ,Hardness of approximation ,01 natural sciences ,Theoretical Computer Science ,label cover ,68R05, 68Q17 ,0101 mathematics ,Algebraic number ,Time complexity ,Constraint satisfaction problem ,Mathematics ,Discrete mathematics ,F.2.2 ,010102 general mathematics ,Theory of computation → Problems, reductions and completeness ,16. Peace & justice ,promise constraint satisfaction ,algebraic approach ,PCSP ,Computer Science - Computational Complexity ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,Benchmark (computing) ,Theory of computation → Constraint and logic programming ,Cover (algebra) ,polymorphisms ,Computer Science - Discrete Mathematics - Abstract
While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin, Guruswami, and H\r{a}stad roved a result known as "$(2+\varepsilon)$-SAT is NP-hard" [FOCS'14/SICOMP'17]. They showed that the problem of distinguishing k-CNF formulas that are g-satisfiable (i.e. some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if $\frac{g}{k} < \frac{1}{2}$ and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus we give a dichotomy for a natural fragment of promise constraint satisfaction problems (PCSPs) on arbitrary finite domains. The hardness side is proved using the algebraic approach, via a new general NP-hardness criterion on polymorphisms of the problem, based on a gap version of the Layered Label Cover problem. We show that previously used criteria are insufficient -- the problem hence gives an interesting benchmark of algebraic techniques for proving hardness of approximation problems such as PCSPs., Comment: Full version of an ICALP 2020 paper
- Published
- 2019
- Full Text
- View/download PDF
9. Corrections and complements to my paper 'On a class of operator monotone functions of several variables'
- Author
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A. R. Mirotin
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,Finite-rank operator ,Compact operator ,Strongly monotone ,Shift operator ,01 natural sciences ,Semi-elliptic operator ,Algebra ,Pseudo-monotone operator ,Monotone polygon ,Multiplication operator ,0101 mathematics ,Mathematics - Published
- 2017
10. A correction to the paper 'Injective mappings and solvable vector fields of Euclidean spaces'
- Author
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José Ruidival dos Santos Filho and Francisco Braun
- Subjects
010101 applied mathematics ,Discrete mathematics ,Euclidean geometry ,Vector field ,Geometry and Topology ,Construct (python library) ,0101 mathematics ,01 natural sciences ,Injective function ,Topology (chemistry) ,Counterexample ,Mathematics - Abstract
We construct a counterexample that disproves the claim of Theorem 0.2 of the paper “Injective mappings and solvable vector fields of Euclidean spaces”, which appeared in Topology Appl. 136 (2004), 261–274.
- Published
- 2016
11. 'Graph Paper' Trace Characterizations of Functions of Finite Energy
- Author
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Robert S. Strichartz
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Discrete mathematics ,Plane (geometry) ,General Mathematics ,010102 general mathematics ,Voltage graph ,Mathematics::General Topology ,Graph paper ,01 natural sciences ,Sierpinski triangle ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Sobolev space ,Coxeter graph ,Sierpinski carpet ,0103 physical sciences ,String graph ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Analysis ,Mathematics - Abstract
We characterize functions of finite energy in the plane in terms of their traces on the lines that make up "graph paper" with squares of side length $mn$ for all $n$, and certain $\12-$order Sobolev norms on the graph paper lines. We also obtain analogous results for functions of finite energy on two classical fractals: the Sierpinski gasket and the Sierpinski carpet.
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- 2013
- Full Text
- View/download PDF
12. A Note on the Paper 'A Common Fixed Point Theorem with Applications'
- Author
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Jiangqian Zou and Hui Huang
- Subjects
Discrete mathematics ,Kantorovich inequality ,021103 operations research ,Control and Optimization ,Applied Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Theory of computation ,Common fixed point ,Log sum inequality ,Rearrangement inequality ,0101 mathematics ,Mathematical economics ,Common fixed point theorem ,Mathematics - Abstract
In this note, we give affirmative answers to two open problems posed by Agarwal et al. in the paper (J Optim Theory Appl 163(2):482---490, 2014).
- Published
- 2015
13. A note on Lavi’s paper 'A Ganzstellensatz for semi-algebraic sets and a boundedness criterion for rational functions'
- Author
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Tomasz Kowalczyk
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Model theory ,Discrete mathematics ,Computer Science::Computer Science and Game Theory ,Algebra and Number Theory ,valuation theory ,010102 general mathematics ,010103 numerical & computational mathematics ,Valuation theory ,Rational function ,01 natural sciences ,Algebra ,model theory ,Real algebraic geometry ,real algebraic geometry ,0101 mathematics ,Algebraic number ,Mathematics - Abstract
This article is intended to indicate and discuss some errors in N. Lavi’s paper “A Ganzstellensatz for semi-algebraic sets and a boundedness criterion for rational functions”.
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- 2018
14. Addendum to the paper 'Linearly topologized modules over a discrete valuation ring'
- Author
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Patricia Couto G. Mauro and Dinamérico P. Pombo
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Discrete mathematics ,General Mathematics ,010102 general mathematics ,Addendum ,0101 mathematics ,01 natural sciences ,Discrete valuation ring ,Linear equation ,Mathematics - Abstract
For any discrete valuation ring R, any R-linear mapping u from an R-module E into an R-module F and any \(y_0\in F\), a necessary and sufficient condition for the solvability of the equation \(u(x)=y_0\) is established, and an application of this result is presented.
- Published
- 2016
15. Some comments on the paper: Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959
- Author
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Michelle Pierri and Donal O'Regan
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Discrete mathematics ,Class (set theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Differential systems ,01 natural sciences ,010101 applied mathematics ,Controllability ,Algebra ,0101 mathematics ,Differential (mathematics) ,Mathematics - Abstract
The abstract results and applications presented in “Controllability of fractional neutral stochastic functional differential systems, Z. Angew. Math. Phys. 65 (2014), no. 5, 941–959, are not correct. Moreover, the class of differential control problems studied in [1] is not H-controllable.
- Published
- 2016
16. Cryptanalysis of a Public Key Cryptosystem Based on Diophantine Equations via Weighted LLL Reduction (Short Paper)
- Author
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Tsuyoshi Takagi, Shinya Okumura, Chengdong Tao, Momonari Kudo, and Jintai Ding
- Subjects
Discrete mathematics ,Degree (graph theory) ,Computer science ,business.industry ,Diophantine equation ,010102 general mathematics ,02 engineering and technology ,Computer security ,computer.software_genre ,01 natural sciences ,law.invention ,Reduction (complexity) ,Public-key cryptography ,law ,Linear cryptanalysis ,0202 electrical engineering, electronic engineering, information engineering ,Cryptosystem ,020201 artificial intelligence & image processing ,0101 mathematics ,business ,Cryptanalysis ,computer ,128-bit - Abstract
Okumura proposed a candidate of post-quantum cryptosystem based on Diophantine equations of degree increasing type (DEC). Sizes of public keys in DEC are small, e.g., 1,200 bits for 128 bit security, and it is a strongly desired property in post-quantum erea.
- Published
- 2016
17. Notes on Perelman’s papers
- Author
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Bruce Kleiner and John Lott
- Subjects
Mathematics - Differential Geometry ,three-manifold ,geometrization theorem ,53C21 ,Mathematical proof ,57M40 ,53C44 ,01 natural sciences ,Physics::Fluid Dynamics ,Perelman ,symbols.namesake ,Ricci flow ,0103 physical sciences ,FOS: Mathematics ,Mathematics::Metric Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Poincaré Conjecture ,Discrete mathematics ,Ricci flow with surgery ,Conjecture ,010102 general mathematics ,Hyperbolization theorem ,Mathematics::Geometric Topology ,57M50 ,Manifold ,long-term behaviour ,Differential Geometry (math.DG) ,Cover (topology) ,Poincaré conjecture ,symbols ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,Geometrization conjecture ,entropy formula - Abstract
These are detailed notes on Perelman's papers "The entropy formula for the Ricci flow and its geometric applications" and "Ricci flow with surgery on three-manifolds"., 216 pages, minor corrections made
- Published
- 2008
18. Cantor sets of arcs in decomposable local Siegel disk boundaries☆☆Portions of this paper were presented by the first author at the SE Sectional Meeting of the AMS (Gainesville, FL, March 1999), and at the Spring Topology and Dynamics Conference (Salt Lake City, UT, March 1999)
- Author
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Andrew O. Maner, Lex Oversteegen, and John C. Mayer
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Discrete mathematics ,Pure mathematics ,Mathematics::Dynamical Systems ,Decomposable continuum ,010102 general mathematics ,Mathematics::General Topology ,u-invariant ,Conformal map ,01 natural sciences ,Cantor set ,Siegel disk ,Monotone polygon ,Bounded function ,0103 physical sciences ,Embedding ,Interval (graph theory) ,Point (geometry) ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Tranche ,Mathematics - Abstract
In this paper we construct a family of circle-like continua, each admitting a finest monotone map onto S 1 such that there exists a subset of point inverses which is homeomorphic to the Cantor set cross an interval. We then show how to realize some members of this family as the boundaries ∂U of bounded irreducible local Siegel disks U . These boundaries are geometrically rigid in the following sense: there exist arbitrarily small periodic homeomorphisms of the sphere, conformal on U , which keep U invariant. The embedding portion of this paper follows a flexible construction of Herman. These results provide a partial answer to a question of Rogers and a complete answer to a question of Brechner, Guay, and Mayer.
- Published
- 2001
19. A Note on the Paper 'The Asymptotic Behavior of the Composition of Firmly Nonexpansive Mappings'
- Author
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David Ariza-Ruiz, Adriana Nicolae, and Genaro López-Acedo
- Subjects
Discrete mathematics ,021103 operations research ,Control and Optimization ,Applied Mathematics ,0211 other engineering and technologies ,010103 numerical & computational mathematics ,02 engineering and technology ,Management Science and Operations Research ,Composition (combinatorics) ,01 natural sciences ,Theory of computation ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this note we correct an error in the paper (Ariza-Ruiz et al. in J Optim Theory Appl 167:409–429, 2015).
- Published
- 2017
20. Correction of the paper 'Bicyclic graphs with extremal values of PI index'
- Author
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Qiuju Bian and Gang Ma
- Subjects
Discrete mathematics ,Bicyclic molecule ,Applied Mathematics ,010102 general mathematics ,Bicyclic graphs ,0102 computer and information sciences ,01 natural sciences ,Graph ,Combinatorics ,010201 computation theory & mathematics ,Pi ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Mathematics - Abstract
In Vukicevic and Stevanovic (2013), Theorem 2 is erroneous when m = 3 k + 1 for some integer k . The correct one is given now. That is, P I ( G ) ≥ m 2 − 3 m + 2 when m = 3 k + 1 for some integer k , where G is a connected bicyclic graph with m edges.
- Published
- 2016
21. Bounded arithmetic, proof complexity and two papers of Parikh
- Author
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Samuel R. Buss
- Subjects
Discrete mathematics ,Bounded arithmetic ,Exponentiation ,Unification ,Logic ,Proof complexity ,010102 general mathematics ,Feasibility ,0102 computer and information sciences ,Mathematical proof ,01 natural sciences ,Algebra ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Proof length ,010201 computation theory & mathematics ,Arithmetic circuit complexity ,0101 mathematics ,Probabilistically checkable proof ,Mathematics - Abstract
This article surveys R. Parikh's work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh's papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs.
- Published
- 1999
22. ϕ-contractive multivalued mappings in complex valued metric spaces and remarks on some recent papers
- Author
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Naval Singh, Vishal Joshi, and Deepak Singh
- Subjects
Discrete mathematics ,Cauchy sequence ,lcsh:Mathematics ,010102 general mathematics ,Complex valued ,Product metric ,common fixed point ,complex valued metric spaces ,multivalued mappings ,lcsh:QA1-939 ,complete complex valued metric spaces ,01 natural sciences ,Convex metric space ,010101 applied mathematics ,Algebra ,Metric space ,General Energy ,Section (category theory) ,Point (geometry) ,0101 mathematics ,Control (linguistics) ,Mathematics - Abstract
The purpose of this paper is twofold. Firstly, certain common fixed point theorems are established via $ \phi $-contractive multivalued mappings involving point-dependent control functions as coefficients in the framework of complex valued metric spaces. Our results improve and extend several results in the existing literature. Moreover, this section is equipped by some illustrative examples in support of our results. Secondly, we point out some slip-ups in the examples of some recent papers Ahmad, Klin-Eam, and Azam (2013) and Kutbi, Ahmad, Azam, and Al-Rawashdeh (2014) based on multivalued contractive mappings in complex valued metric spaces. Our observations are also authenticated with the aid of some appropriate examples. Some rectifications to correct the erratic examples are also suggested.
- Published
- 2016
23. Recurrent critical points and typical limit sets for conformal measures☆☆Portions of the paper were presented at the AMS Special Session on Low-Dimensional Dynamics in Milwaukee, Wisconsin, October 1997
- Author
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Lex Oversteegen, Alexander Blokh, and John C. Mayer
- Subjects
Discrete mathematics ,010102 general mathematics ,Julia set ,Conformal map ,Density theorem ,Lebesgue integration ,Postcritical set ,01 natural sciences ,Measure (mathematics) ,symbols.namesake ,0103 physical sciences ,symbols ,Complex dynamics ,Ergodic theory ,ω-limit set ,Point (geometry) ,010307 mathematical physics ,Geometry and Topology ,Limit (mathematics) ,0101 mathematics ,Conformal measure ,Mathematics - Abstract
For a rational f : C → C with a conformal measure μ we show that if there is a subset of the Julia set J(f) of positive μ -measure whose points are not eventual preimages of critical or parabolic points and have limit sets not contained in the union of the limit sets of recurrent critical points, then μ is non-atomic, μ(J(f))=1 , ω(x)=J(f) for μ -a.e. point x∈J(f) and f is conservative, ergodic and exact. The proof uses a version of the Lebesgue Density Theorem valid for Borel measures and conformal balls.
- Published
- 2000
24. Rebuttal of Donnelly's paper on the spectral gap
- Author
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Antoine Henrot, Mark S. Ashbaugh, Richard S. Laugesen, Department of Mathematics, University of Missouri Columbia, University of Missouri [Columbia] (Mizzou), University of Missouri System-University of Missouri System, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Robust control of infinite dimensional systems and applications (CORIDA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire de Mathématiques et Applications de Metz (LMAM), Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria), Department of Mathematics [Urbana], University of Illinois at Urbana-Champaign [Urbana], University of Illinois System-University of Illinois System, CORIDA, and Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Université Paul Verlaine - Metz (UPVM)-Centre National de la Recherche Scientifique (CNRS)-Inria Nancy - Grand Est
- Subjects
Discrete mathematics ,Sequence ,Conjecture ,General Mathematics ,010102 general mathematics ,Mathematics::History and Overview ,Mathematics::Spectral Theory ,01 natural sciences ,Domain (mathematical analysis) ,Computer Science::Computers and Society ,010101 applied mathematics ,symbols.namesake ,Physics::Popular Physics ,Dirichlet boundary condition ,Euclidean geometry ,symbols ,Calculus ,Convex body ,Quantitative Biology::Populations and Evolution ,[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP] ,Spectral gap ,0101 mathematics ,Mathematics ,Unit interval - Abstract
The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ2 − λ1 ≥ 3π for all convex euclidean domains of diameter 1, where λ1 and λ2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrodinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [9, 11]. This Schrodinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [8], and more recently in all dimensions by B. Andrews and J. Clutterbuck [1]. The proof in this journal by H. Donnelly [3] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature. Donnelly’s approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [5]. (For some history on this approach, and on the gap conjecture more generally, see the report on the AIM meeting “Low Eigenvalues of Laplace and Schrodinger Operators” [10], especially page 12 of the open problems list.) The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for
- Published
- 2011
25. Note on a paper by A. Granville and K. Soundararajan
- Author
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Jie Wu, Institut Élie Cartan de Nancy (IECN), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), and Wu, Jie
- Subjects
Discrete mathematics ,Algebra and Number Theory ,Distribution (number theory) ,010102 general mathematics ,Equal probability ,01 natural sciences ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,Combinatorics ,symbols.namesake ,Results on L(1,χ) ,\chi)$ ,0103 physical sciences ,symbols ,Results on $L(1 ,010307 mathematical physics ,11M20 ,0101 mathematics ,Random variable ,Euler product ,[MATH.MATH-NT] Mathematics [math]/Number Theory [math.NT] ,Mathematics - Abstract
In this note, we improve some results of Granville and Soundararajan on the distribution of values of the truncated random Euler product L ( 1 , X ; y ) : = ∏ p ⩽ y ( 1 − X ( p ) / p ) −1 , where the X ( p ) are independent random variables, taking the values ±1 with equal probability p / 2 ( p + 1 ) and 0 with probability 1 / ( p + 1 ) .
- Published
- 2007
26. Remarks on E. A. Rahmanov's paper 'on the asymptotics of the ratio of orthogonal polynomials'
- Author
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Paul Nevai and Attila Máté
- Subjects
Discrete mathematics ,Mathematics(all) ,Numerical Analysis ,Statement (logic) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Orthogonal polynomials ,0101 mathematics ,Analysis ,Mathematics ,Counterexample - Abstract
It is pointed out that the proof of the basic result of Rahmanov's paper has a serious gap. It is documented by original sources that a statement he relied on in the proof contains a misprint, and it is shown by a counterexample that this statement (with the misprint) is, in fact, false. A somewhat weaker statement is proved true.
- Published
- 1982
27. Some remarks on a paper of Kingman
- Author
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R. K. Getoor
- Subjects
Discrete mathematics ,Statistics and Probability ,Zero set ,Subordinator ,Applied Mathematics ,010102 general mathematics ,Markov process ,Fixed point ,01 natural sciences ,symbols.namesake ,010104 statistics & probability ,Probability theory ,Joint probability distribution ,symbols ,State space ,0101 mathematics ,Finite set ,Mathematics - Abstract
We illustrate a technique for computing certain integrals that arise in probability theory by giving a new derivation of a formula of Kingman. This formula contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t) = inf{s: X(t - s) = b} where X is a time homogeneous, continuous parameter, Markov process and b is a fixed point in its state space. We then extend this formula to the situation in which b is replaced by a finite set {b 1, …, b n }.
- Published
- 1974
28. 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
29. On sums of squares of primes and a k-th power of prime
- Author
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Zhixin Liu and Rui Zhang
- Subjects
Discrete mathematics ,010505 oceanography ,General Mathematics ,010102 general mathematics ,Short paper ,Sander ,01 natural sciences ,Prime (order theory) ,Power (physics) ,Integer ,Congruence (manifolds) ,0101 mathematics ,0105 earth and related environmental sciences ,Mathematics - Abstract
In this short paper, we consider the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of two squares of primes and a k-th power of prime for any integer $$k \ge 3$$ . Our results improve the recent results due to Brudern (in: Sander, Steuding, Steuding (eds) From arithmetic to zeta-functions, Springer, Cham 2016). The similar method can be also applied to some related questions in this direction, and this can improve the previous results.
- Published
- 2018
30. Selling reduction versus Niggli reduction for crystallographic lattices
- Author
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Lawrence C. Andrews, Herbert J. Bernstein, and Nicholas K. Sauter
- Subjects
Discrete mathematics ,Delaunay triangulation ,media_common.quotation_subject ,010102 general mathematics ,Delone ,Niggli ,010403 inorganic & nuclear chemistry ,Condensed Matter Physics ,01 natural sciences ,Biochemistry ,Research Papers ,0104 chemical sciences ,Inorganic Chemistry ,Reduction (complexity) ,unit-cell reduction ,Selling ,Structural Biology ,Simple (abstract algebra) ,General Materials Science ,Simplicity ,0101 mathematics ,Physical and Theoretical Chemistry ,Mathematics ,media_common ,Delaunay - Abstract
The unit-cell reduction described by Selling and used by Delone (Delaunay) is explained in a simple form., The unit-cell reduction described by Selling and used by Delone (whose early publications were under the spelling Delaunay) is explained in a simple form. The transformations needed to implement the reduction are listed. The simplicity of this reduction contrasts with the complexity of Niggli reduction.
- Published
- 2019
31. Biochemical and phylogenetic networks-I: hypertrees and corona products
- Author
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Krishnan Balasubramanian, A. Arul Shantrinal, T. M. Rajalaxmi, Indra Rajasingh, K. Jagadeesh Kumar, and R. Sundara Rajan
- Subjects
Discrete mathematics ,Eccentricity-based topological indices ,Topological indices of hypertrees ,Original Paper ,010304 chemical physics ,Degree (graph theory) ,Phylogenetic tree ,Biochemical networks ,Applied Mathematics ,010102 general mathematics ,General Chemistry ,01 natural sciences ,Corona (optical phenomenon) ,Product (mathematics) ,0103 physical sciences ,Corona product of graphs ,Graph (abstract data type) ,Mathematical modeling ,0101 mathematics ,Mathematics - Abstract
We have obtained graph-theoretically based topological indices for the characterization of certain graph theoretical networks of biochemical interest. We have derived certain distance, degree and eccentricity based topological indices for various k-level hypertrees and corona product of hypertrees. We have also pointed out errors in a previous study. The validity of our results is supported by computer codes for the respective indices. Several biochemical applications are pointed out.
- Published
- 2021
32. Romberg extrapolation for Euler summation-based cubature on regular regions
- Author
-
Willi Freeden and Christian Gerhards
- Subjects
Discrete mathematics ,Original Paper ,Extrapolation ,010103 numerical & computational mathematics ,01 natural sciences ,Romberg extrapolation ,Cubature ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Trapezoidal rule (differential equations) ,symbols.namesake ,Rate of convergence ,Simple (abstract algebra) ,Modeling and Simulation ,Romberg's method ,symbols ,General Earth and Planetary Sciences ,65D30 ,65B99 ,0101 mathematics ,Remainder ,Cube ,Euler summation ,Mathematics - Abstract
Romberg extrapolation is a long-known method to improve the convergence rate of the trapezoidal rule on intervals. For simple regions such as the cube \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,1]^q$$\end{document}[0,1]q it is directly transferable to cubature in q dimensions. In this paper, we formulate Romberg extrapolation for Euler summation-based cubature on arbitrary q-dimensional regular regions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}\subset \mathbb {R}^q$$\end{document}G⊂Rq and derive an explicit representation for the remainder term.
- Published
- 2017
33. A note on q-difference equations for Cigler’s polynomials
- Author
-
Jian Cao and Da-Wei Niu
- Subjects
Discrete mathematics ,Multilinear map ,Algebra and Number Theory ,Applied Mathematics ,010102 general mathematics ,Short paper ,Generating function ,Bilinear interpolation ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Classical orthogonal polynomials ,Difference polynomials ,Orthogonal polynomials ,0101 mathematics ,Analysis ,Mathematics - Abstract
Viewing from polynomials solutions of q-difference equations makes certain generating functions calculated easily. In this short paper, we build the relations between Cigler’s polynomials and q-difference equations. In addition, we deduce generating functions and bilinear generating functions for Cigler’s polynomials. More over, we gain multilinear and mixed generating functions for Cigler’s polynomials. At last, we gain U(n+1) type generating functions for Cigler’s polynomials.
- Published
- 2016
34. Exact testing with random permutations
- Author
-
Jesse Hemerik and Jelle J. Goeman
- Subjects
0301 basic medicine ,Statistics and Probability ,Conditional monte carlo ,Resampling ,Computer science ,Existential quantification ,Mathematics - Statistics Theory ,Statistics Theory (math.ST) ,01 natural sciences ,010104 statistics & probability ,03 medical and health sciences ,Permutation ,62G09 ,FOS: Mathematics ,Permutation test ,0101 mathematics ,Discrete mathematics ,Original Paper ,Mathematics::Combinatorics ,Nonparametric statistics ,Permutation group ,030104 developmental biology ,Nonparametric test ,Multiple comparisons problem ,Statistics, Probability and Uncertainty ,62G10 - Abstract
When permutation methods are used in practice, often a limited number of random permutations are used to decrease the computational burden. However, most theoretical literature assumes that the whole permutation group is used, and methods based on random permutations tend to be seen as approximate. There exists a very limited amount of literature on exact testing with random permutations, and only recently a thorough proof of exactness was given. In this paper, we provide an alternative proof, viewing the test as a “conditional Monte Carlo test” as it has been called in the literature. We also provide extensions of the result. Importantly, our results can be used to prove properties of various multiple testing procedures based on random permutations.
- Published
- 2018
35. A Remark on the Local Cohomology Modules of a Union of Disjoint Matroids
- Author
-
Cong Minh Nguyen and Minh Cong Nguyen
- Subjects
Discrete mathematics ,Ideal (set theory) ,Mathematics::Commutative Algebra ,General Mathematics ,010102 general mathematics ,Short paper ,0102 computer and information sciences ,Disjoint sets ,Local cohomology ,01 natural sciences ,Matroid ,Simplicial complex ,010201 computation theory & mathematics ,0101 mathematics ,Mathematics - Abstract
Let I be the Stanley–Reisner ideal of a simplicial complex Δ. In this short paper, we shall give a formula of vanishing of the local cohomology modules for S/I(r) in the case Δ is a union of disjoint matroids, where I(r) is the rth symbolic power of I. As an application, we will improve a previous result in Minh and Nakamura (Nagoya Math. J. 213, 127–140, 2014) for the k-Buchsbaumness of S/I(r).
- Published
- 2015
36. Lattice Polygons and the Number 2i + 7
- Author
-
Josef Schicho and Christian Haase
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Integer lattice ,Toric variety ,Graph paper ,Computer Science::Computational Geometry ,01 natural sciences ,Combinatorics ,Lattice (order) ,0103 physical sciences ,Polygon ,010307 mathematical physics ,0101 mathematics ,Algebraic number ,Invariant (mathematics) ,Mathematics - Abstract
0.1. How it all began. When the second author translated a result on algebraic sur faces into the language of lattice polygons using toric geometry, he got a simple inequality for lattice polygons. This inequality had originally been discovered by Scott [12]. The first author then found a third proof. Subsequently, both authors went through a phase of polygon addiction. Once you get started drawing lattice polygons on graph paper and discovering relations between their numerical invariants, it is not so easy to stop! (The gentle reader has been warned.) Thus, it was just unavoidable that the authors came up with new inequalities: Scott's inequality can be sharpened if one takes into account another invariant, which is de fined by peeling off the skins of the polygons like an onion (see Section 3).
- Published
- 2009
37. Moving vertices to make drawings plane
- Author
-
Goaoc, X., Kratochvil, J., Okamoto, Y., Shin, C.S., Wolff, A., Hong, S.K., Nishizeki, T., Quan, W., Effective Geometric Algorithms for Surfaces and Visibility (VEGAS), INRIA Lorraine, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS)-Université Henri Poincaré - Nancy 1 (UHP)-Université Nancy 2-Institut National Polytechnique de Lorraine (INPL)-Centre National de la Recherche Scientifique (CNRS), Department of Applied Mathematics (KAM) (KAM), Univerzita Karlova v Praze, Discrete Optimization Laboratory, Toyohashi University of Technology, Departement of Electronics and Information Engineering, Hankuk University of Foreign Studies, Faculteit Wiskunde & Informatica, Technische Universiteit Eindhoven (TU/e), and Algorithms
- Subjects
Computational Geometry (cs.CG) ,FOS: Computer and information sciences ,Discrete Mathematics (cs.DM) ,Planar straight-line graph ,Slope number ,0102 computer and information sciences ,Computational Complexity (cs.CC) ,[INFO.INFO-CG]Computer Science [cs]/Computational Geometry [cs.CG] ,01 natural sciences ,Combinatorics ,symbols.namesake ,Graph drawing ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Outerplanar graph ,Wheel graph ,Dominance drawing ,0101 mathematics ,Mathematics ,Discrete mathematics ,Book embedding ,010102 general mathematics ,Planar graph ,Computer Science - Computational Complexity ,010201 computation theory & mathematics ,symbols ,Computer Science - Computational Geometry ,Computer Science - Discrete Mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by moving some of the vertices. Let shift$(G,\delta)$ denote the minimum number of vertices that need to be moved to turn $\delta$ into a plane drawing of $G$. We show that shift$(G,\delta)$ is NP-hard to compute and to approximate, and we give explicit bounds on shift$(G,\delta)$ when $G$ is a tree or a general planar graph. Our hardness results extend to 1BendPointSetEmbeddability, a well-known graph-drawing problem., Comment: This paper has been merged with http://arxiv.org/abs/0709.0170
- Published
- 2008
38. Uniformity of saturated orthogonal arrays
- Author
-
Yu Tang and E Chen
- Subjects
Statistics and Probability ,Discrete mathematics ,Series (mathematics) ,Applied Mathematics ,05 social sciences ,01 natural sciences ,Prime (order theory) ,010104 statistics & probability ,0502 economics and business ,Statistical inference ,0101 mathematics ,Statistics, Probability and Uncertainty ,Orthogonal array ,050205 econometrics ,Mathematics - Abstract
Orthogonal array has been used in various fields, as it possesses attractive combinatorial properties. However, for a long time, combinatorially equivalent orthogonal arrays are thought to be indistinguishable, especially when an ANOVA model is established. Later on, some papers pointed out that permuting levels of orthogonal arrays will alter their statistical inference abilities. Criteria have been recommended for evaluating the different performance of orthogonal arrays. In this paper, a revised method is proposed to construct saturated orthogonal arrays when the level of the factors is an odd prime and properties of the related wrap-around L 2 -discrepancy will be investigated. Theoretical result shows that the wrap-around L 2 -discrepancies of the saturated orthogonal arrays constructed using the revised method are less than those of the original ones, and asymptotically attain the lower bounds. A series of numerical examples also confirms the effectiveness of the proposed method.
- Published
- 2022
39. The Birthday Problem and Zero-Error List Codes
- Author
-
Michelle Effros, Victoria Kostina, Michael Langberg, and Parham Noorzad
- Subjects
FOS: Computer and information sciences ,Discrete Mathematics (cs.DM) ,Distribution (number theory) ,Computer Science - Information Theory ,Computation ,Population ,List decoding ,Data_CODINGANDINFORMATIONTHEORY ,02 engineering and technology ,Library and Information Sciences ,Information theory ,01 natural sciences ,Birthday problem ,Rényi entropy ,010104 statistics & probability ,Encoding (memory) ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Mathematics - Combinatorics ,Entropy (information theory) ,0101 mathematics ,education ,Computer Science::Information Theory ,Mathematics ,Discrete mathematics ,education.field_of_study ,Information Theory (cs.IT) ,Codebook ,020206 networking & telecommunications ,Mutual information ,Expression (computer science) ,Computer Science Applications ,Constraint (information theory) ,Combinatorics (math.CO) ,Decoding methods ,Computer Science - Discrete Mathematics ,Information Systems - Abstract
As an attempt to bridge the gap between the probabilistic world of classical information theory and the combinatorial world of zero-error information theory, this paper studies the performance of randomly generated codebooks over discrete memoryless channels under a zero-error list-decoding constraint. This study allows the application of tools from one area to the other. Furthermore, it leads to an information-theoretic formulation of the birthday problem, which is concerned with the probability that in a given population, a fixed number of people have the same birthday. Due to the lack of a closed-form expression for this probability when the distribution of birthdays is not uniform, the resulting expression is not simple to analyze; in the information-theoretic formulation, however, the asymptotic behavior of this probability can be characterized exactly for all distributions., Comment: Extended version of paper presented at ISIT 2017 in Aachen
- Published
- 2021
40. Limits of multiplicative inhomogeneous random graphs and Lévy trees: limit theorems
- Author
-
Nicolas Broutin, Minmin Wang, and Thomas Duquesne
- Subjects
Statistics and Probability ,Random graph ,Discrete mathematics ,Continuum (topology) ,010102 general mathematics ,Multiplicative function ,Boundary (topology) ,01 natural sciences ,Lévy process ,010104 statistics & probability ,Metric space ,Mathematics::Probability ,Embedding ,Limit (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Analysis ,Mathematics - Abstract
We consider a natural model of inhomogeneous random graphs that extends the classical Erdős–Rényi graphs and shares a close connection with the multiplicative coalescence, as pointed out by Aldous (Ann Probab 25:812–854, 1997). In this model, the vertices are assigned weights that govern their tendency to form edges. It is by looking at the asymptotic distributions of the masses (sum of the weights) of the connected components of these graphs that Aldous and Limic (Electron J Probab 3:1–59, 1998) have identified the entrance boundary of the multiplicative coalescence, which is intimately related to the excursion lengths of certain Lévy-type processes. We, instead, look at the metric structure of these components and prove their Gromov–Hausdorff–Prokhorov convergence to a class of (random) compact measured metric spaces that have been introduced in a companion paper (Broutin et al. in Limits of multiplicative inhomogeneous random graphs and Lévy trees: the continuum graphs.arXiv:1804.05871, 2020). Our asymptotic regimes relate directly to the general convergence condition appearing in the work of Aldous and Limic. Our techniques provide a unified approach for this general “critical” regime, and relies upon two key ingredients: an encoding of the graph by some Lévy process as well as an embedding of its connected components into Galton–Watson forests. This embedding transfers asymptotically into an embedding of the limit objects into a forest of Lévy trees, which allows us to give an explicit construction of the limit objects from the excursions of the Lévy-type process. The mains results combined with the ones in the other paper allow us to extend and complement several previous results that had been obtained via model- or regime-specific proofs, for instance: the case of Erdős–Rényi random graphs obtained by Addario-Berry et al. (Probab Theory Relat Fields 152:367–406, 2012), theasymptotic homogeneouscase as studied by Bhamidi et al. (Probab Theory Relat Fields 169:565–641, 2017), or thepower-lawcase as considered by Bhamidi et al. (Probab Theory Relat Fields 170:387–474, 2018).
- Published
- 2021
41. Fractional partitions and conjectures of Chern–Fu–Tang and Heim–Neuhauser
- Author
-
Zack Tripp, Kathrin Bringmann, Larry Rolen, and Ben Kane
- Subjects
Discrete mathematics ,Conjecture ,010102 general mathematics ,Multiplicative function ,Context (language use) ,0102 computer and information sciences ,General Medicine ,Partition function (mathematics) ,01 natural sciences ,Range (mathematics) ,Future study ,010201 computation theory & mathematics ,Partition (number theory) ,0101 mathematics ,Mathematics - Abstract
Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern–Fu–Tang and Heim–Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov–Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern–Fu–Tang conjecture and to show the Heim–Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.
- Published
- 2021
42. Formal self duality
- Author
-
Robert Schüler and Lukas Kölsch
- Subjects
Discrete mathematics ,20C15, 20K01 ,Computer Networks and Communications ,Applied Mathematics ,010102 general mathematics ,Duality (mathematics) ,0102 computer and information sciences ,01 natural sciences ,Dual (category theory) ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Abelian group ,Connection (algebraic framework) ,Boolean function ,Mathematics - Abstract
We study the notion of formal self duality in finite abelian groups. Formal duality in finite abelian groups has been proposed by Cohn, Kumar, Reiher and Schürmann. In this paper we give a precise definition of formally self dual sets and discuss results from the literature in this perspective. Also, we discuss the connection to formally dual codes. We prove that formally self dual sets can be reduced to primitive formally self dual sets similar to a previously known result on general formally dual sets. Furthermore, we describe several properties of formally self dual sets. Also, some new examples of formally self dual sets are presented within this paper. Lastly, we study formally self dual sets of the form $\{(x,F(x)) \ : \ x\in {\mathbb {F}}_{2^{n}}\}$ { ( x , F ( x ) ) : x ∈ F 2 n } where F is a vectorial Boolean function mapping ${\mathbb {F}}_{2^{n}}$ F 2 n to ${\mathbb {F}}_{2^{n}}$ F 2 n .
- Published
- 2021
43. Projective shape analysis of contours and finite 3D configurations from digital camera images
- Author
-
Robert L. Paige, Victor Patrangenaru, K. David Yao, David Lester, and Mingfei Qiu
- Subjects
Statistics and Probability ,Discrete mathematics ,010102 general mathematics ,Lie group ,01 natural sciences ,Algebra ,010104 statistics & probability ,Digital image ,Data analysis ,Projective space ,0101 mathematics ,Statistics, Probability and Uncertainty ,Projective test ,Pencil (mathematics) ,Statistical hypothesis testing ,Shape analysis (digital geometry) ,Mathematics - Abstract
We analyze infinite dimensional projective shape data collected from digital camera images, focusing on two-sample hypothesis testing for both finite and infinite extrinsic mean configurations. The two sample test methodology is based on a Lie group technique that was derived by Crane and Patrangenaru (J Multivar Anal 102:225–237, 2011) and Qiu et al. (Neighborhood hypothesis testing for mean change on infinite dimensional Lie groups and 3D projective shape analysis of matched contours, 2015). In infinite dimensions, the equality of two extrinsic means is likely to be rejected, thus a neighborhood hypothesis is suitably tested, combining the ideas in these two papers with data analysis methods on Hilbert manifolds in Ellingson et al. (J Multivar Anal 122:317–333, 2013). In this manuscript, we apply these general results to the two sample problem for independent projective shapes of 3D facial configurations and for matched projective shapes of 2D and 3D contours. Digital images of 3D scenes are today at the fingertips of any statistician. Here and in the literature referenced in the in this paper, we provide a methodology for properly analyzing such data, when more pictures of a given scene are available.
- Published
- 2016
44. A localization method in Hamiltonian graph theory
- Author
-
Nikolay K. Khachatryan, Armen S. Asratian, and Jonas B. Granholm
- Subjects
Structure (category theory) ,0102 computer and information sciences ,01 natural sciences ,Theoretical Computer Science ,Combinatorics ,Set (abstract data type) ,symbols.namesake ,FOS: Mathematics ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,0101 mathematics ,05C45 ,Mathematics ,Series (mathematics) ,Discrete Mathematics ,Dirac (video compression format) ,010102 general mathematics ,Hamilton cycle ,Local conditions ,Infinite graphs ,Hamiltonian curve ,Diskret matematik ,Hamiltonian path ,Vertex (geometry) ,Computational Theory and Mathematics ,010201 computation theory & mathematics ,symbols ,Ball (bearing) ,Combinatorics (math.CO) ,Hamiltonian (control theory) - Abstract
The classical global criteria for the existence of Hamilton cycles only apply to graphs with large edge density and small diameter. In a series of papers Asratian and Khachatryan developed local criteria for the existence of Hamilton cycles in finite connected graphs, which are analogues of the classical global criteria due to Dirac (1952), Ore (1960), Jung (1978), and Nash-Williams (1971). The idea was to show that the global concept of Hamiltonicity can, under rather general conditions, be captured by local phenomena, using the structure of balls of small radii. (The ball of radius $r$ centered at a vertex $u$ is a subgraph of $G$ induced by the set of vertices whose distances from $u$ do not exceed $r$.) Such results are called localization theorems and present a possibility to extend known classes of finite Hamiltonian graphs. In this paper we formulate a general approach for finding localization theorems and use this approach to formulate local analogues of well-known results of Bauer et al. (1989), Bondy (1980), H\"aggkvist and Nicoghossian (1981), and Moon and Moser (1963). Finally we extend two of our results to infinite locally finite graphs and show that they guarantee the existence of Hamiltonian curves, introduced by K\"undgen, Li and Thomassen (2017)., Comment: 25 pages, 3 figures
- Published
- 2021
45. Virtual χ−y‐genera of Quot schemes on surfaces
- Author
-
Woonam Lim
- Subjects
Discrete mathematics ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,0101 mathematics ,Mathematics::Symplectic Geometry ,01 natural sciences ,14N99 (primary), 14J80 (secondary) ,Mathematics - Abstract
This paper studies the virtual $\chi_{-y}$-genera of Grothendieck's Quot schemes on surfaces, thus refining the calculations of the virtual Euler characteristics by Oprea-Pandharipande. We first prove a structural result expressing the equivariant virtual $\chi_{-y}$-genera of Quot schemes universally in terms of the Seiberg-Witten invariants. The formula is simpler for curve classes of Seiberg-Witten length $N$, which are defined in the paper. By way of application, we give complete answers in the following cases: (i) arbitrary surfaces for the zero curve class, (ii) relatively minimal elliptic surfaces for rational multiples of the fiber class, (iii) minimal surfaces of general type with $p_g>0$ for any curve classes. Furthermore, a blow up formula is obtained for curve classes of Seiberg-Witten length $N$. As a result of these calculations, we prove that the generating series of the virtual $\chi_{-y}$-genera are given by rational functions for all surfaces with $p_g>0$, addressing a conjecture of Oprea-Pandharipande. In addition, we study the reduced $\chi_{-y}$-genera for $K3$ surfaces and primitive curve classes with connections to the Kawai-Yoshioka formula., Comment: 48 pages. Reformulation of Theorem 1 for readability. Extra explanation for the mixed terms in section 2.2.3. Reference updates
- Published
- 2021
46. Quasiisometric mappings in the distance ratio metric in Banach spaces
- Author
-
Xiantao Wang, Tiantian Guan, Manzi Huang, and Saminathan Ponnusamy
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,010505 oceanography ,General Mathematics ,010102 general mathematics ,Distance ratio ,Metric (mathematics) ,Banach space ,0101 mathematics ,01 natural sciences ,Homeomorphism ,0105 earth and related environmental sciences ,Mathematics - Abstract
In Li et al. (J Aust Math Soc 97:383–390, 2014), the authors discussed two classes of mappings in Banach spaces, which are $$\varphi $$ -quasiisometric mappings in the distance ratio metric (briefly, $$\varphi $$ -DRQI mappings) and fully $$\varphi $$ -quasiisometric mappings in the distance ratio metric (briefly, $$\varphi $$ -FDRQI mappings), where $$\varphi :$$ $$[0,\infty )\rightarrow [0,\infty )$$ denotes a quasiisometric control function. In this paper, we continue this discussion by introducing a new class of mappings, that is, $$\varphi $$ -PDRQI mappings. The purpose of this paper is twofold. Let f be a homeomorphism between two proper domains in a Banach space. First, we prove that f being $$\varphi $$ -PDRQI is equivalent to f being $$\varphi $$ -FDRQI. Then, as an application of the obtained equivalent relation, we show that f being $$\varphi $$ -DRQI, f being $$\varphi $$ -PDRQI and f being $$\varphi $$ -FDRQI are equivalent to each other when the quasiisometric control functions have the form $$\varphi (t)=Mt$$ .
- Published
- 2021
47. Positive-definite modification of a covariance matrix by minimizing the matrix $$\ell_{\infty}$$ norm with applications to portfolio optimization
- Author
-
Johan Lim, Shota Katayama, Young-Geun Choi, and Seonghun Cho
- Subjects
0106 biological sciences ,Statistics and Probability ,Discrete mathematics ,Economics and Econometrics ,Covariance matrix ,Applied Mathematics ,Estimator ,Positive-definite matrix ,010603 evolutionary biology ,01 natural sciences ,010104 statistics & probability ,Matrix (mathematics) ,Positive definiteness ,Modeling and Simulation ,Norm (mathematics) ,Diagonal matrix ,Convex combination ,0101 mathematics ,Social Sciences (miscellaneous) ,Analysis ,Mathematics - Abstract
The covariance matrix, which should be estimated from the data, plays an important role in many multivariate procedures, and its positive definiteness (PDness) is essential for the validity of the procedures. Recently, many regularized estimators have been proposed and shown to be consistent in estimating the true matrix and its support under various structural assumptions on the true covariance matrix. However, they are often not PD. In this paper, we propose a simple modification to make a regularized covariance matrix be PD while preserving its support and the convergence rate. We focus on the matrix $$\ell_{\infty }$$ norm error in covariance matrix estimation because it could allow us to bound the error in the downstream multivariate procedure relying on it. Our proposal in this paper is an extension of the fixed support positive-definite (FSPD) modification by Choi et al. (2019) from spectral and Frobenius norms to the matrix $$\ell_{\infty }$$ norm. Like the original FSPD, we consider a convex combination between the initial estimator (the regularized covariance matrix without PDness) and a given form of the diagonal matrix minimize the $$\ell_{\infty }$$ distance between the initial estimator and the convex combination, and find a closed-form expression for the modification. We apply the procedure to the minimum variance portfolio (MVP) optimization problem and show that the vector $$\ell_{\infty }$$ error in the estimation of the optimal portfolio weight is bounded by the matrix $$\ell _{\infty }$$ error of the plug-in covariance matrix estimator. We illustrate the MVP results with S&P 500 daily returns data from January 1978 to December 2014.
- Published
- 2021
48. On the turnpike property with interior decay for optimal control problems
- Author
-
Martin Gugat
- Subjects
Discrete mathematics ,0209 industrial biotechnology ,Control and Optimization ,Partial differential equation ,Applied Mathematics ,010102 general mathematics ,Order (ring theory) ,Natural number ,02 engineering and technology ,State (functional analysis) ,Optimal control ,01 natural sciences ,020901 industrial engineering & automation ,Control and Systems Engineering ,Ordinary differential equation ,Signal Processing ,Integral element ,ddc:510 ,0101 mathematics ,Stationary state ,Mathematics - Abstract
In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval $$[0,\, T]$$ [ 0 , T ] and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals $$[0, \, T]$$ [ 0 , T ] give rise to a turnpike structure in the sense that for natural numbers n if T is sufficiently large, the contribution of the objective function from subintervals of [0, T] of the form $$\begin{aligned} {[}t - t/2^n,\; t + (T-t)/2^n] \end{aligned}$$ [ t - t / 2 n , t + ( T - t ) / 2 n ] is of the order $$1/\min \{t^n, (T-t)^n\}$$ 1 / min { t n , ( T - t ) n } . We also show that a similar result holds for $$\epsilon $$ ϵ -optimal solutions of the optimal control problems if $$\epsilon >0$$ ϵ > 0 is chosen sufficiently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied.
- Published
- 2021
49. On the Rates of Convergence in Central Limit Theorems for Compound Random Sums of Independent Random Variables
- Author
-
Tran Loc Hung
- Subjects
Discrete mathematics ,Independent identically distributed ,General Mathematics ,010102 general mathematics ,Term (logic) ,01 natural sciences ,010305 fluids & plasmas ,Normal distribution ,0103 physical sciences ,Convergence (routing) ,0101 mathematics ,Algebra over a field ,Random variable ,Mathematics ,Central limit theorem - Abstract
Since the appearance of Robbins’s paper (1948) the central limit theorem for a sum of a random number of independent identically distributed random variables is one of the most fundamental results in probability, and explains the appearance of the normal distribution in a whole host of diverse applications in mathematics, physics, biology and the social sciences. Compound random sums are extensions of classical random sums when the numbers of independent summands in sums are partial sums of independent identically distributed positive integer-valued random variables, assumed independent of summands of sums. The main aim of this paper is to introduce central limit theorems for normalized compound random sums of independent random variables and establish the convergence rates in types of small-o and large- $$\mathcal{O}$$ estimates, in term of Trotter-distance. The obtained results in this paper are extensions of several known ones.
- Published
- 2021
50. Cohn-Leavitt path algebras of bi-separated graphs
- Author
-
Mohan. R and B. N. Suhas
- Subjects
Discrete mathematics ,Algebra and Number Theory ,Mathematics::Operator Algebras ,Mathematics::Rings and Algebras ,010102 general mathematics ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Mathematics - Rings and Algebras ,010103 numerical & computational mathematics ,Common framework ,01 natural sciences ,Rings and Algebras (math.RA) ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Path (graph theory) ,FOS: Mathematics ,0101 mathematics ,Mathematics - Abstract
The purpose of this paper is to provide a common framework for studying various generalizations of Leavitt algebras and Leavitt path algebras. This paper consists of two parts. In part I we define Cohn-Leavitt path algebras of a new class of graphs with an additional structure called bi-separated graphs, which generalize the constructions of Leavitt path algebras of various types of graphs. We define and study the category \textbf{BSG} of bi-separated graphs with appropriate morphisms so that the functor which associates a bi-separated graph to its Cohn-Leavitt path algebra is continuous. We also characterize a full subcategory of \textbf{BSG} whose objects are direct limits of finite complete subobjects. We compute normal forms of these algebras and apply them to study some algebraic theoretic properties in terms of bi-separated graph-theoretic properties. In part II we specialize our attention to Cohn-Leavitt path algebras of a special class of bi-separated graphs called B-hypergraphs. We investigate their non-stable K-theory and show that the lattice of order-ideals of V-monoids of these algebras is determined by bi-separated graph-theoretic data. Using this information we study representations of Leavitt path algebras of regular hypergraphs and also find a matrix criterion for Leavitt path algebras of finite hypergraphs to have IBN property.
- Published
- 2021
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