556 results
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2. 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
3. Delone sets in ℝ3: Regularity Conditions
- Author
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N. P. Dolbilin
- Subjects
Statistics and Probability ,Discrete mathematics ,Euclidean space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Delone set ,01 natural sciences ,Identity (music) ,010305 fluids & plasmas ,Set (abstract data type) ,0103 physical sciences ,Homogeneous space ,Mathematics::Metric Geometry ,0101 mathematics ,Symmetry (geometry) ,Orbit (control theory) ,Link (knot theory) ,Mathematics - Abstract
A regular system is a Delone set in Euclidean space with a transitive group of symmetries or, in other words, the orbit of a crystallographic group. The local theory for regular systems, created by the geometric school of B. N. Delone, was aimed, in particular, to rigorously establish the “local-global-order” link, i.e., the link between the arrangement of a set around each of its points and symmetry/regularity of the set as a whole. The main result of this paper is a proof of the so-called 10R-theorem. This theorem asserts that identity of neighborhoods within a radius 10R of all points of a Delone set (in other words, an (r, R)-system) in 3D Euclidean space implies regularity of this set. The result was obtained and announced long ago independently by M. Shtogrin and the author of this paper. However, a detailed proof remains unpublished for many years. In this paper, we give a proof of the 10R-theorem. In the proof, we use some recent results of the author, which simplify the proof.
- Published
- 2020
4. An 𝐿^{𝑝} theory of sparse graph convergence I: Limits, sparse random graph models, and power law distributions
- Author
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Jennifer Chayes, Yufei Zhao, Henry Cohn, and Christian Borgs
- Subjects
Random graph ,Discrete mathematics ,Dense graph ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,01 natural sciences ,Power law ,Limit theory ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Equivalence (formal languages) ,Mathematics - Probability ,Mathematics - Abstract
We introduce and develop a theory of limits for sequences of sparse graphs based on $L^p$ graphons, which generalizes both the existing $L^\infty$ theory of dense graph limits and its extension by Bollob\'as and Riordan to sparse graphs without dense spots. In doing so, we replace the no dense spots hypothesis with weaker assumptions, which allow us to analyze graphs with power law degree distributions. This gives the first broadly applicable limit theory for sparse graphs with unbounded average degrees. In this paper, we lay the foundations of the $L^p$ theory of graphons, characterize convergence, and develop corresponding random graph models, while we prove the equivalence of several alternative metrics in a companion paper., Comment: 44 pages
- Published
- 2019
5. An extension of the Hermite–Hadamard inequality for convex and s-convex functions
- Author
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Péter Kórus
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,010103 numerical & computational mathematics ,Extension (predicate logic) ,01 natural sciences ,Iterated integrals ,Hermite–Hadamard inequality ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Convex function ,Mathematics - Abstract
The Hermite–Hadamard inequality was extended using iterated integrals by Retkes [Acta Sci Math (Szeged) 74:95–106, 2008]. In this paper we further extend the main results of the above paper for convex and also for s-convex functions in the second sense.
- Published
- 2019
6. A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth
- Author
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Jaroslav Nesetril, Patrice Ossona de Mendez, Computer Science Institute of Charles University [Prague] (IUUK), Charles University [Prague] (CU), Centre d'Analyse et de Mathématique sociales (CAMS), École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), Supported by grant ERCCZ LL-1201 and CE-ITI P202/12/G061, and by the European Associated Laboratory 'Structures in Combinatorics' (LEA STRUCO), Department of Applied Mathematics (KAM) (KAM), and Univerzita Karlova v Praze
- Subjects
Model theory ,General Mathematics ,Stone space ,0102 computer and information sciences ,Tree-depth ,01 natural sciences ,Graph ,Combinatorics ,Definable set ,Measurable graph ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Mathematics - Combinatorics ,Radon measures ,Relational structure ,0101 mathematics ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,Connected component ,Discrete mathematics ,Applied Mathematics ,010102 general mathematics ,Graph limits ,Colored ,010201 computation theory & mathematics ,Bounded function ,Standard probability space ,Combinatorics (math.CO) ,First-order logic ,Tuple ,Structural limits - Abstract
In this paper we introduce a general framework for the study of limits of relational structures in general and graphs in particular, which is based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this framework and that they naturally appear as "tractable cases" of a general theory. As an outcome of this, we provide extensions of known results. We believe that this put these into next context and perspective. For example, we prove that the sparse--dense dichotomy exactly corresponds to random free graphons. The second part of the paper is devoted to the study of sparse structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the convergence can be "almost" studied component-wise. We also propose the structure of limits objects for convergent sequences of sparse structures. Eventually, we consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded tree-depth, motivated by their role of elementary brick these graphs play in decompositions of sparse graphs, and give an explicit construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that every first-order definable set of tuples is measurable. This is an example of the general concept of {\em modeling} we introduce here. Our example is also the first "intermediate class" with explicitly defined limit structures., Comment: added journal reference
- Published
- 2020
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7. On symmetric linear diffusions
- Author
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Liping Li and Jiangang Ying
- Subjects
Discrete mathematics ,Representation theorem ,Dirichlet form ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Disjoint sets ,01 natural sciences ,Dirichlet distribution ,010104 statistics & probability ,symbols.namesake ,Closure (mathematics) ,symbols ,Interval (graph theory) ,Countable set ,0101 mathematics ,Mathematics - Abstract
The main purpose of this paper is to explore the structure of local and regular Dirichlet forms associated with symmetric one-dimensional diffusions, which are also called symmetric linear diffusions. Let ( E , F ) (\mathcal {E},\mathcal {F}) be a regular and local Dirichlet form on L 2 ( I , m ) L^2(I,m) , where I I is an interval and m m is a fully supported Radon measure on I I . We shall first present a complete representation for ( E , F ) (\mathcal {E},\mathcal {F}) , which shows that ( E , F ) (\mathcal {E},\mathcal {F}) lives on at most countable disjoint “effective" intervals with an “adapted" scale function on each interval, and any point outside these intervals is a trap of the one-dimensional diffusion. Furthermore, we shall give a necessary and sufficient condition for C c ∞ ( I ) C_c^\infty (I) being a special standard core of ( E , F ) (\mathcal {E},\mathcal {F}) and shall identify the closure of C c ∞ ( I ) C_c^\infty (I) in ( E , F ) (\mathcal {E},\mathcal {F}) when C c ∞ ( I ) C_c^\infty (I) is contained but not necessarily dense in F \mathcal {F} relative to the E 1 1 / 2 \mathcal {E}_1^{1/2} -norm. This paper is partly motivated by a result of Hamza’s that was stated in a theorem of Fukushima, Oshima, and Takeda’s and that provides a different point of view to this theorem. To illustrate our results, many examples are provided.
- Published
- 2018
8. Increasing the smoothness of vector and Hermite subdivision schemes
- Author
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Nira Dyn and Caroline Moosmüller
- Subjects
Limit of a function ,Discrete mathematics ,Hermite polynomials ,65D17, 65D05, 40A99 ,business.industry ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Scalar (mathematics) ,MathematicsofComputing_NUMERICALANALYSIS ,Numerical Analysis (math.NA) ,010103 numerical & computational mathematics ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,business ,Smoothing ,Mathematics ,Subdivision - Abstract
In this paper we suggest a method for transforming a vector subdivision scheme generating $C^{\ell}$ limits to another such scheme of the same dimension, generating $C^{\ell+1}$ limits. In scalar subdivision, it is well known that a scheme generating $C^{\ell}$ limit curves can be transformed to a new scheme producing $C^{\ell+1}$ limit curves by multiplying the scheme's symbol with the smoothing factor $\tfrac{z+1}{2}$. We extend this approach to vector and Hermite subdivision schemes, by manipulating symbols. The algorithms presented in this paper allow to construct vector (Hermite) subdivision schemes of arbitrarily high regularity from a convergent vector scheme (from a Hermite scheme whose Taylor scheme is convergent with limit functions of vanishing first component)., 28 pages, 4 figures. Corrected typos, updated contact information
- Published
- 2018
9. Redheffer type bounds for Bessel and modified Bessel functions of the first kind
- Author
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Árpád Baricz and Khaled Mehrez
- Subjects
Discrete mathematics ,Pure mathematics ,Hankel transform ,Cylindrical harmonics ,Bessel process ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Dirichlet eta function ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Bessel polynomials ,Struve function ,symbols ,Discrete Mathematics and Combinatorics ,Bessel's inequality ,0101 mathematics ,Bessel function ,Mathematics - Abstract
In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.
- Published
- 2018
10. Proof of the Achievability Conjectures for the General Stochastic Block Model
- Author
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Emmanuel Abbe and Colin Sandon
- Subjects
Discrete mathematics ,Conjecture ,Applied Mathematics ,General Mathematics ,020206 networking & telecommunications ,02 engineering and technology ,Belief propagation ,01 natural sciences ,Algebra ,010104 statistics & probability ,Operator (computer programming) ,Power iteration ,Stochastic block model ,0202 electrical engineering, electronic engineering, information engineering ,Order (group theory) ,0101 mathematics ,Cluster analysis ,Time complexity ,Mathematics - Abstract
In a paper that initiated the modern study of the stochastic block model (SBM), Decelle, Krzakala, Moore, and Zdeborova, backed by Mossel, Neeman, and Sly, conjectured that detecting clusters in the symmetric SBM in polynomial time is always possible above the Kesten-Stigum (KS) threshold, while it is possible to detect clusters information theoretically (i.e., not necessarily in polynomial time) below the KS threshold when the number of clusters k is at least 4. Massoulie, Mossel et al., and Bordenave, Lelarge, and Massoulie proved that the KS threshold is in fact efficiently achievable for k = 2, while Mossel et al. proved that it cannot be crossed at k = 2. The above conjecture remained open for k ≥ 3. This paper proves the two parts of the conjecture, further extending the results to general SBMs. For the efficient part, an approximate acyclic belief propagation (ABP) algorithm is developed and proved to detect communities for any k down to the KS threshold in quasi-linear time. Achieving this requires showing optimality of ABP in the presence of cycles, a challenge for message-passing algorithms. The paper further connects ABP to a power iteration method on a nonbacktracking operator of generalized order, formalizing the interplay between message passing and spectral methods. For the information-theoretic part, a nonefficient algorithm sampling a typical clustering is shown to break down the KS threshold at k = 4. The emerging gap is shown to be large in some cases, making the SBM a good case study for information-computation gaps. © 2017 Wiley Periodicals, Inc.
- Published
- 2017
11. On p-convergent Operators on Banach Lattices
- Author
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Elroy D. Zeekoei and Jan Fourie
- Subjects
Unbounded operator ,Discrete mathematics ,Mathematics::Functional Analysis ,Approximation property ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Spectrum (functional analysis) ,Finite-rank operator ,Compact operator ,01 natural sciences ,Strictly singular operator ,010101 applied mathematics ,Pseudo-monotone operator ,0101 mathematics ,C0-semigroup ,Mathematics - Abstract
The notion of a p-convergent operator on a Banach space was originally introduced in 1993 by Castillo and Sanchez in the paper entitled “Dunford–Pettis-like properties of continuous vector function spaces”. In the present paper we consider the p-convergent operators on Banach lattices, prove some domination properties of the same and consider their applications (together with the notion of a weak p-convergent operator, which we introduce in the present paper) to a study of the Schur property of order p. Also, the notion of a disjoint p-convergent operator on Banach lattices is introduced, studied and its applications to a study of the positive Schur property of order p are considered.
- Published
- 2017
12. On the Enumeration of Hypermaps Which are Self-Equivalent with Respect to Reversing the Colors of Vertices
- Author
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M. A. Deryagina
- Subjects
Statistics and Probability ,Connected component ,Discrete mathematics ,Mathematics::Combinatorics ,Applied Mathematics ,General Mathematics ,Riemann surface ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Graph ,symbols.namesake ,Colored ,010201 computation theory & mathematics ,Enumeration ,symbols ,Bipartite graph ,Bibliography ,Reversing ,0101 mathematics ,Mathematics - Abstract
A map (S,G) is a closed Riemann surface S with embedded graph G such that S \G is the disjoint union of connected components, called faces, each of which is homeomorphic to an open disk. Tutte began a systematic study of maps in the 1960s and contemporary authors are actively developing it. In the present paper, after recalling the concept of a circular map introduced by the author and Mednykh, a relationship between bipartite maps and circular maps is demonstrated via the concept of the duality of maps. In this way an enumeration formula for the number of bipartite maps with a given number of edges is obtained. A hypermap is a map whose vertices are colored black and white in such a way that every edge connects vertices of different colors. The hypermaps are also known as dessins d’enfants (or Grothendieck’s dessins). A hypermap is self-equivalent with respect to reversing the colors of vertices if it is equivalent to the hypermap obtained by reversing the colors of its vertices. The main result of the present paper is an enumeration formula for the number of unrooted hypermaps, regardless of genus, which have n edges and are self-equivalent with respect to reversing the colors of vertices. Bibliography: 13 titles.
- Published
- 2017
13. Some weak specification properties and strongly mixing
- Author
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Jiandong Yin, Tao Wang, and Qi Yan
- Subjects
010101 applied mathematics ,Discrete mathematics ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0101 mathematics ,Equivalence (formal languages) ,01 natural sciences ,Mathematics - Abstract
In this paper, the authors first construct a dynamical system which is strongly mixing but has no weak specification property. Then the authors introduce two new concepts which are called the quasi-weak specification property and the semi-weak specification property in this paper, respectively, and the authors prove the equivalence of quasi-weak specification property, semi-weak specification property and strongly mixing.
- Published
- 2017
14. A generalization of a graph theory Mertens’ theorem: Galois covering case
- Author
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Iwao Sato, Seiken Saito, and Takehiro Hasegawa
- Subjects
Mertens conjecture ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,Fundamental theorem of Galois theory ,010102 general mathematics ,Galois group ,0102 computer and information sciences ,01 natural sciences ,Embedding problem ,Combinatorics ,Differential Galois theory ,symbols.namesake ,010201 computation theory & mathematics ,Mertens function ,Mertens' theorems ,symbols ,Galois extension ,0101 mathematics ,Mathematics - Abstract
In 1874, Franz Mertens proved the so-called Mertens’ theorem, and in 1974, Kenneth S. Williams showed Mertens’ theorem associated with a character. In a previous paper, we presented a graph-theoretic analogue to Williams’ theorem. In this paper, we generalize our previous work in the sense that a character is extended to a representation. To our knowledge, a number-theoretic analogue to our result is not yet known. So, we expect that, by using our methods, it can be proven in the future.
- Published
- 2017
15. Structure Graphs of Rings: Definitions and First Results
- Author
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Aleksandar Lipkovski
- Subjects
Statistics and Probability ,Discrete mathematics ,Cayley graph ,Algebraic structure ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Directed graph ,01 natural sciences ,010101 applied mathematics ,Quadratic equation ,Vieta's formulas ,Branched covering ,0101 mathematics ,Commutative property ,Complex number ,Mathematics - Abstract
The quadratic Vieta formulas (x, y) ↦ (u, v) = (x + y, xy) in the complex geometry define a two-fold branched covering ℂ2 → ℂ2 ramified over the parabola u 2 = 4v. Thinking about topics considered in Arnold’s paper Topological content of the Maxwell theorem on multipole representation of spherical functions, I came to a very simple idea: in fact, these formulas describe the algebraic structure, i.e., addition and multiplication, of complex numbers. What if, instead of the field of complex numbers, we consider an arbitrary ring? Namely for an arbitrary ring A (commutative, with unity) consider the mapping Φ: A 2 → A 2 defined by the Vieta formulas (x, y) ↦ (u, v) = (x + y, xy). What kind of algebraic properties of the ring itself does this map reflect? At first, it is interesting to investigate the simplest finite rings A = ℤ m and A = ℤ k ×ℤ m . Recently, it has been very popular to consider graphs associated to rings (the zero-divisor graph, the Cayley graph, etc.). In the present paper, we study the directed graph defined by the Vieta mapping Φ.
- Published
- 2017
16. Sequential Analogues of the Lyapunov and Krein–Milman Theorems in Fréchet Spaces
- Author
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F. S. Stonyakin
- Subjects
Statistics and Probability ,Discrete mathematics ,Mathematics::Functional Analysis ,Dual space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Eberlein–Šmulian theorem ,Banach space ,Hahn–Banach theorem ,02 engineering and technology ,01 natural sciences ,Fréchet space ,Locally convex topological vector space ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Closed graph theorem ,0101 mathematics ,Open mapping theorem (functional analysis) ,Mathematics - Abstract
In this paper we develop the theory of anti-compact sets we introduced earlier. We describe the class of Frechet spaces where anti-compact sets exist. They are exactly the spaces that have a countable set of continuous linear functionals. In such spaces we prove an analogue of the Hahn–Banach theorem on extension of a continuous linear functional from the original space to a space generated by some anti-compact set. We obtain an analogue of the Lyapunov theorem on convexity and compactness of the range of vector measures, which establishes convexity and a special kind of relative weak compactness of the range of an atomless vector measure with values in a Frechet space possessing an anti-compact set. Using this analogue of the Lyapunov theorem, we prove the solvability of an infinite-dimensional analogue of the problem of fair division of resources. We also obtain an analogue of the Lyapunov theorem for nonadditive analogues of measures that are vector quasi-measures valued in an infinite-dimensional Frechet space possessing an anti-compact set. In the class of Frechet spaces possessing an anti-compact set, we obtain analogues of the Krein–Milman theorem on extreme points for convex bounded sets that are not necessarily compact. A special place is occupied by analogues of the Krein–Milman theorem in terms of extreme sequences introduced in the paper (the so-called sequential analogues of the Krein–Milman theorem).
- Published
- 2017
17. How does the core sit inside the mantle?
- Author
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Kathrin Skubch, Mihyun Kang, Oliver Cooley, and Amin Coja-Oghlan
- Subjects
FOS: Computer and information sciences ,Discrete Mathematics (cs.DM) ,General Mathematics ,05C80 ,Structure (category theory) ,0102 computer and information sciences ,Characterization (mathematics) ,01 natural sciences ,Mantle (geology) ,Combinatorics ,010104 statistics & probability ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Mathematics - Combinatorics ,0101 mathematics ,Mathematics ,Branching process ,Discrete mathematics ,Random graph ,Series (mathematics) ,Degree (graph theory) ,Applied Mathematics ,Probability (math.PR) ,Computer Graphics and Computer-Aided Design ,Tree (graph theory) ,Vertex (geometry) ,010201 computation theory & mathematics ,Bounded function ,Core (graph theory) ,Combinatorics (math.CO) ,Combinatorial theory ,Mathematics - Probability ,Software ,Computer Science - Discrete Mathematics - Abstract
The k-core, defined as the maximal subgraph of minimum degree at least k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [J Combin Theory Ser B 67 (1996), 111–151] determined the threshold dk for the appearance of an extensive k-core. The aim of the present paper is to describe how the k-core is “embedded” into the random graph in the following sense. Let k≥3 and fix d=np>dk. Colour each vertex that belongs to the k-core of G(n,p) in black and all remaining vertices in white. Here we derive a multi-type branching process that describes the local structure of this coloured random object as n tends to infinity. This generalises prior results on, e.g., the internal structure of the k-core. In the physics literature it was suggested to characterize the core by means of a message passing algorithm called Warning Propagation. Ibrahimi, Kanoria, Kraning and Montanari [Ann Appl Probab 25 (2015), 2743–2808] used this characterization to describe the 2-core of random hypergraphs. To derive our main result we use a similar approach. A key observation is that a bounded number of iterations of this algorithm is enough to give a good approximation of the k-core. Based on this the study of the k-core reduces to the analysis of Warning Propagation on a suitable Galton-Watson tree. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2017
- Published
- 2017
18. On $$\varvec{n}$$ n -norm preservers and the Aleksandrov conservative $$\varvec{n}$$ n -distance problem
- Author
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György Pál Gehér
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Order (ring theory) ,010103 numerical & computational mathematics ,01 natural sciences ,Surjective function ,Nonlinear system ,Transformation (function) ,Norm (mathematics) ,Distance problem ,Discrete Mathematics and Combinatorics ,Affine transformation ,0101 mathematics ,Unit (ring theory) ,Mathematics - Abstract
The goal of this paper is to point out that the results obtained in the recent papers (Chen and Song in Nonlinear Anal 72:1895–1901, 2010; Chu in J Math Anal Appl 327:1041–1045, 2007; Chu et al. in Nonlinear Anal 59:1001–1011, 2004a, J. Math Anal Appl 289:666–672, 2004b) can be seriously strengthened in the sense that we can significantly relax the assumptions of the main results so that we still get the same conclusions. In order to do this first, we prove that for $$n \ge 3$$ any transformation which preserves the n-norm of any n vectors is automatically plus-minus linear. This will give a re-proof of the well-known Mazur–Ulam-type result that every n-isometry is automatically affine ( $$n \ge 2$$ ) which was proven in several papers, e.g. in Chu et al. (Nonlinear Anal 70:1068–1074, 2009). Second, following the work of Rassias and Semrl (Proc Am Math Soc 118:919–925, 1993), we provide the solution of a natural Aleksandrov-type problem in n-normed spaces, namely, we show that every surjective transformation which preserves the unit n-distance in both directions ( $$n\ge 2$$ ) is automatically an n-isometry.
- Published
- 2017
19. On embedding certain partial orders into the P-points under Rudin-Keisler and Tukey reducibility
- Author
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Dilip Raghavan and Saharon Shelah
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Boolean algebra (structure) ,010102 general mathematics ,Ultrafilter ,Natural number ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,symbols ,Embedding ,Continuum (set theory) ,0101 mathematics ,Partially ordered set ,Continuum hypothesis ,Axiom ,Mathematics - Abstract
The study of the global structure of ultrafilters on the natural numbers with respect to the quasi-orders of Rudin-Keisler and Rudin-Blass reducibility was initiated in the 1970s by Blass, Keisler, Kunen, and Rudin. In a 1973 paper Blass studied the special class of P-points under the quasi-ordering of Rudin-Keisler reducibility. He asked what partially ordered sets can be embedded into the P-points when the P-points are equipped with this ordering. This question is of most interest under some hypothesis that guarantees the existence of many P-points, such as Martin’s axiom for σ \sigma -centered posets. In his 1973 paper he showed under this assumption that both ω 1 {\omega }_{1} and the reals can be embedded. Analogous results were obtained later for the coarser notion of Tukey reducibility. We prove in this paper that Martin’s axiom for σ \sigma -centered posets implies that the Boolean algebra P ( ω ) / FIN \mathcal {P}(\omega ) / \operatorname {FIN} equipped with its natural partial order can be embedded into the P-points both under Rudin-Keisler and Tukey reducibility. Consequently, the continuum hypothesis implies that every partial order of size at most continuum embeds into the P-points under both notions of reducibility.
- Published
- 2017
20. On the strong convergence forweighted sums of negatively superadditive dependent random variables
- Author
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Xuejun Wang, Lulu Zheng, and Wenzhi Yang
- Subjects
Discrete mathematics ,Superadditivity ,General Mathematics ,Similar distribution ,010103 numerical & computational mathematics ,01 natural sciences ,Exponential function ,010104 statistics & probability ,Convergence of random variables ,Negatively associated ,Convergence (routing) ,Dependent random variables ,Applied mathematics ,0101 mathematics ,Random variable ,Mathematics - Abstract
In this paper, we present some results on the complete convergence for arrays of rowwise negatively superadditive dependent (NSD, in short) random variables by using the Rosenthal-type maximal inequality, Kolmogorov exponential inequality and the truncation method. The results obtained in the paper extend the corresponding ones for weighted sums of negatively associated random variables with identical distribution to the case of arrays of rowwise NSD random variables without identical distribution.
- Published
- 2017
21. Harnessing the Bethe free energy†
- Author
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Amin Coja-Oghlan and Victor Bapst
- Subjects
FOS: Computer and information sciences ,Cavity method ,Discrete Mathematics (cs.DM) ,cavity method ,General Mathematics ,Belief propagation ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,regularity lemma ,FOS: Mathematics ,0101 mathematics ,Gibbs measure ,Research Articles ,random graphs ,Probability measure ,Random graph ,Discrete mathematics ,Belief Propagation ,Partition function (quantum field theory) ,000 Computer science, knowledge, general works ,Applied Mathematics ,Probability (math.PR) ,010102 general mathematics ,Computer Graphics and Computer-Aided Design ,Computer Science ,symbols ,Graph (abstract data type) ,Ising model ,05C80, 82B44 ,Mathematics - Probability ,Software ,Computer Science - Discrete Mathematics ,Research Article - Abstract
A wide class of problems in combinatorics, computer science and physics can be described along the following lines. There are a large number of variables ranging over a finite domain that interact through constraints that each bind a few variables and either encourage or discourage certain value combinations. Examples include the $k$-SAT problem or the Ising model. Such models naturally induce a Gibbs measure on the set of assignments, which is characterised by its partition function. The present paper deals with the partition function of problems where the interactions between variables and constraints are induced by a sparse random (hyper)graph. According to physics predictions, a generic recipe called the "replica symmetric cavity method" yields the correct value of the partition function if the underlying model enjoys certain properties [Krzkala et al., PNAS 2007]. Guided by this conjecture, we prove general sufficient conditions for the success of the cavity method. The proofs are based on a "regularity lemma" for probability measures on sets of the form $\Omega^n$ for a finite $\Omega$ and a large $n$ that may be of independent interest., Comment: This version replaces version 1 and the RANDOM 2015 version of the paper, which contained critical errors affecting the main results
- Published
- 2016
22. Mutual information decay for factors of i.i.d
- Author
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Viktor Harangi and Balázs Gerencsér
- Subjects
Independent and identically distributed random variables ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,Mutual information ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with factors of independent and identically distributed processes on the $d$ -regular tree for $d\geq 3$ . We study the mutual information of values on two given vertices. If the vertices are neighbors (i.e. their distance is $1$ ), then a known inequality between the entropy of a vertex and the entropy of an edge provides an upper bound for the (normalized) mutual information. In this paper we obtain upper bounds for vertices at an arbitrary distance $k$ , of order $(d-1)^{-k/2}$ . Although these bounds are sharp, we also show that an interesting phenomenon occurs here: for any fixed process, the rate of mutual information decay is much faster, essentially of order $(d-1)^{-k}$ .
- Published
- 2019
23. Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals
- Author
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Peter Kritzer, Henryk Woźniakowski, and Friedrich Pillichshammer
- Subjects
Statistics and Probability ,Discrete mathematics ,Numerical Analysis ,Polynomial ,Control and Optimization ,Algebra and Number Theory ,Logarithm ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hilbert space ,010103 numerical & computational mathematics ,01 natural sciences ,Exponential polynomial ,Exponential function ,Singular value ,symbols.namesake ,Tensor product ,Bounded function ,symbols ,0101 mathematics ,Mathematics - Abstract
We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an e -approximation for the d -variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on e − 1 . The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on e − 1 , we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential ( s , t ) -weak tractability with max ( s , t ) ≥ 1 . For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential ( s , t ) -weak tractability with max ( s , t ) 1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).
- Published
- 2020
24. Group actions and geometric combinatorics in 𝔽qd$\mathbb{F}_{q}^{d}$
- Author
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Michael Rudnev, Jonathan Pakianathan, Alex Iosevich, Derrick Hart, and Michael Bennett
- Subjects
Discrete mathematics ,Group action ,010201 computation theory & mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,0101 mathematics ,Geometric combinatorics ,01 natural sciences ,Mathematics - Abstract
In this paper we apply a group action approach to the study of Erdős–Falconer-type problems in vector spaces over finite fields and use it to obtain non-trivial exponents for the distribution of simplices. We prove that there exists s 0 ( d ) < d ${s_{0}(d) such that if E ⊂ 𝔽 q d ${E\subset{\mathbb{F}}_{q}^{d}}$ , d ≥ 2 ${d\geq 2}$ , with | E | ≥ C q s 0 ${|E|\geq Cq^{s_{0}}}$ , then | T d d ( E ) | ≥ C ′ q ( d + 1 2 ) ${|T^{d}_{d}(E)|\geq C^{\prime}q^{d+1\choose 2}}$ , where T k d ( E ) ${T^{d}_{k}(E)}$ denotes the set of congruence classes of k-dimensional simplices determined by k + 1 ${k+1}$ -tuples of points from E. Non-trivial exponents were previously obtained by Chapman, Erdogan, Hart, Iosevich and Koh [4] for T k d ( E ) ${T^{d}_{k}(E)}$ with 2 ≤ k ≤ d - 1 ${2\leq k\leq d-1}$ . A non-trivial result for T 2 2 ( E ) ${T^{2}_{2}(E)}$ in the plane was obtained by Bennett, Iosevich and Pakianathan [2]. These results are significantly generalized and improved in this paper. In particular, we establish the Wolff exponent 4 3 ${\frac{4}{3}}$ , previously established in [4] for the q ≡ 3 mod 4 ${q\equiv 3\textup{ mod }4}$ case to the case q ≡ 1 mod 4 ${q\equiv 1\textup{ mod }4}$ , and this results in a new sum-product type inequality. We also obtain non-trivial results for subsets of the sphere in 𝔽 q d ${{\mathbb{F}}_{q}^{d}}$ , where previous methods have yielded nothing. The key to our approach is a group action perspective which quickly leads to natural and effective formulae in the style of the classical Mattila integral from geometric measure theory.
- Published
- 2016
25. Uniqueness of difference-differential polynomials of entire functions sharing one value
- Author
-
Ashwini M. Hattikal and Renukadevi S. Dyavanal
- Subjects
010101 applied mathematics ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,Entire function ,010102 general mathematics ,Zhàng ,Multiplicity (mathematics) ,Uniqueness ,0101 mathematics ,01 natural sciences ,Nevanlinna theory ,Mathematics - Abstract
In this paper, we study the uniqueness of difference-differential polynomials of entire functions $f$ and $g$ sharing one value with counting multiplicity. In this paper we extend and generalize the results of X. Y. Zhang, J. F. Chen and W. C. Lin [17] L. Kai, L. Xin-ling and C. Ting-bin [7] and many others [2, 16].
- Published
- 2016
26. A note on tractability of multivariate analytic problems
- Author
-
Yongping Liu and Guiqiao Xu
- Subjects
Statistics and Probability ,Discrete mathematics ,Numerical Analysis ,Multivariate statistics ,Matching (statistics) ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Random variate ,Linear problem ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
In this paper we study d -variate approximation for weighted Korobov spaces in the worst-case setting. The considered algorithms use finitely many evaluations of arbitrary linear functionals. We give matching necessary and sufficient conditions for some notions of tractability in terms of two weight parameters. Our result is an affirmative answer to a problem which is left open in a recent paper of Kritzer, Pillichshammer and Wo?niakowski.
- Published
- 2016
27. To the History of the Appearance of the Notion of the ε-Entropy of an Authomorphism of a Lebesgue Space and (ε,T)-Entropy of a Dynamical System with Continuous Time
- Author
-
D. Z. Arov
- Subjects
Statistics and Probability ,Discrete mathematics ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Automorphism ,01 natural sciences ,Separable space ,Compact space ,0103 physical sciences ,Entropy (information theory) ,Standard probability space ,Ergodic theory ,010307 mathematical physics ,Invariant measure ,0101 mathematics ,Mathematics - Abstract
The paper is devoted to the master thesis on “information theory” which was written by the author in 1956–57. The topic was suggested by his advisor A. A. Bobrov (a student of A. Ya. Khinchin and A. N. Kolmogorov), and the thesis was written under the influence of lectures by N. I. Gavrilov (a student of I. G. Petrovskii) on the qualitative theory of differential equations, which included the statement of Birkhoff’s theorem for ergodic dynamical systems. In the thesis, the author used the concept of Shannon entropy in the study of ergodic dynamical systems f(p, t) in a separable compact metric space R with an invariant measure μ (where μ(R) = 1) and introduced the notion of the (ϵ, T)-entropy of a system as a quantitative characteristic of the degree of mixing. In the work, not only partitions of R were considered, but also partitions of the interval (−∞,∞) into subintervals of length T > 0. In particular, f(p, T) was regarded as an automorphism S of X = R, and the (ϵ, T)-entropy is essentially the e-entropy of S. But, despite some “oversights” in the definition of the (ϵ, T)-entropy and many years that have passed, the author decided to publish the corresponding chapter of the thesis in connection with the following: 1) There is a number of papers that refer to this work in the explanation of the history of the concept of Kolmogorov’s entropy. 2) Recently, B. M. Gurevich obtained new results on the ϵ-entropy hϵ(S), which show that for two ergodic automorphisms with equal finite entropies their ϵ-entropies also coincide for all ϵ, but, on the other hand, there are unexpected nonergodic automorphisms with equal finite entropies, but different ϵ-entropies for some ϵ. This shows that the concept of ϵ-entropy is of scientific value.
- Published
- 2016
28. Graph-Links: Nonrealizability, Orientation, and Jones Polynomial
- Author
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V. S. Safina and Denis Petrovich Ilyutko
- Subjects
Statistics and Probability ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Jones polynomial ,Bracket polynomial ,01 natural sciences ,Graph ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,MathematicsofComputing_DISCRETEMATHEMATICS ,Writhe ,Mathematics - Abstract
The present paper is devoted to graph-links with many components and consists of two parts. In the first part of the paper we classify vertices of a labeled graph according to the component they belong to. Using this classification, we construct an invariant of graph-links. This invariant shows that the labeled second Bouchet graph generates a nonrealizable graph-link. In the second part of the work we introduce the notion of an oriented graph-link. We define a writhe number for the oriented graph-link and we get an invariant of oriented graph-links, the Jones polynomial, by normalizing the Kauffman bracket with the writhe number.
- Published
- 2016
29. COMMON SOLUTION TO GENERALIZED MIXED EQUILIBRIUM PROBLEM AND FIXED POINT PROBLEM FOR A NONEXPANSIVE SEMIGROUP IN HILBERT SPACE
- Author
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Mohammad Farid, K. R. Kazmi, and Behzad Djafari-Rouhani
- Subjects
Discrete mathematics ,Sequence ,Semigroup ,Generalization ,Iterative method ,General Mathematics ,010102 general mathematics ,Hilbert space ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Fixed point problem ,Scheme (mathematics) ,Variational inequality ,symbols ,Applied mathematics ,0101 mathematics ,Mathematics - Abstract
In this paper, we introduce and study an explicit hybrid re- laxed extragradient iterative method to approximate a common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converges strongly to the common solution to generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solu- tion is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results presented in this paper are the supplement, improvement and generalization of the previously known results in this area.
- Published
- 2016
30. The continuum limit of the Kuramoto model on sparse random graphs
- Author
-
Georgi S. Medvedev
- Subjects
Random graph ,Discrete mathematics ,Dense graph ,Dynamical systems theory ,Applied Mathematics ,General Mathematics ,Kuramoto model ,010102 general mathematics ,FOS: Physical sciences ,Dynamical Systems (math.DS) ,Dynamical system ,01 natural sciences ,Nonlinear Sciences - Adaptation and Self-Organizing Systems ,010305 fluids & plasmas ,Convergence of random variables ,0103 physical sciences ,FOS: Mathematics ,Limit (mathematics) ,Continuum (set theory) ,0101 mathematics ,Mathematics - Dynamical Systems ,Adaptation and Self-Organizing Systems (nlin.AO) ,Mathematics ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
In this paper, we study convergence of coupled dynamical systems on convergent sequences of graphs to a continuum limit. We show that the solutions of the initial value problem for the dynamical system on a convergent graph sequence tend to that for the nonlocal diffusion equation on a unit interval, as the graph size tends to infinity. We improve our earlier results in [Arch. Ration. Mech. Anal., 21 (2014), pp. 781--803] and extend them to a larger class of graphs, which includes directed and undirected, sparse and dense, random and deterministic graphs. There are three main ingredients of our approach. First, we employ a flexible framework for incorporating random graphs into the models of interacting dynamical systems, which fits seamlessly with the derivation of the continuum limit. Next, we prove the averaging principle for approximating a dynamical system on a random graph by its deterministic (averaged) counterpart. The proof covers systems on sparse graphs and yields almost sure convergence on time intervals of order $\log n,$ where $n$ is the number of vertices. Finally, a Galerkin scheme is developed to show convergence of the averaged model to the continuum limit. The analysis of this paper covers the Kuramoto model of coupled phase oscillators on a variety of graphs including sparse Erd\H{o}s-R{\' e}nyi, small-world, and power law graphs., Comment: To appear in Communications in Mathematical Sciences
- Published
- 2018
- Full Text
- View/download PDF
31. Number of Jumps in Two-Sided First-Exit Problems for a Compound Poisson Process
- Author
-
Yi Lu, Can Jin, and Shuanming Li
- Subjects
Statistics and Probability ,Discrete mathematics ,Laplace transform ,General Mathematics ,010102 general mathematics ,Probability density function ,01 natural sciences ,Lévy process ,Exponential function ,010104 statistics & probability ,Compound Poisson process ,Applied mathematics ,Probability-generating function ,0101 mathematics ,Random variable ,Joint (geology) ,Mathematics - Abstract
In this paper, we study the joint Laplace transform and probability generating functions of two pairs of random variables: (1) the two-sided first-exit time and the number of claims by this time; (2) the two-sided smooth exit-recovery time and its associated number of claims. The joint transforms are expressed in terms of the so-called doubly-killed scale function that is defined in this paper. We also find explicit expressions for the joint density function of the two-sided first-exit time and the number of claims by this time. Numerical examples are presented for exponential claims.
- Published
- 2015
32. Almost Diagonal Matrices and Besov-Type Spaces Based on Wavelet Expansions
- Author
-
Markus Weimar
- Subjects
Discrete mathematics ,Sequence ,Function space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Order (ring theory) ,Context (language use) ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,Diagonal matrix ,Nabla symbol ,0101 mathematics ,Biorthogonal wavelet ,Analysis ,Mathematics - Abstract
This paper is concerned with problems in the context of the theoretical foundation of adaptive algorithms for the numerical treatment of operator equations. It is well-known that the analysis of such schemes naturally leads to function spaces of Besov type. But, especially when dealing with equations on non-smooth manifolds, the definition of these spaces is not straightforward. Nevertheless, motivated by applications, recently Besov-type spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ on certain two-dimensional, patchwise smooth surfaces were defined and employed successfully. In the present paper, we extend this definition (based on wavelet expansions) to a quite general class of d-dimensional manifolds and investigate some analytical properties of the resulting quasi-Banach spaces. In particular, we prove that different prominent constructions of biorthogonal wavelet systems $$\Psi $$ on domains or manifolds $$\Gamma $$ which admit a decomposition into smooth patches actually generate the same Besov-type function spaces $$B^\alpha _{\Psi ,q}(L_p(\Gamma ))$$ , provided that their univariate ingredients possess a sufficiently large order of cancellation and regularity. For this purpose, a theory of almost diagonal matrices on related sequence spaces $$b^\alpha _{p,q}(\nabla )$$ of Besov type is developed.
- Published
- 2015
33. Meyniel's conjecture holds for random graphs
- Author
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Nicholas C. Wormald and Paweł Prałat
- Subjects
Random graph ,Discrete mathematics ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,16. Peace & justice ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Graph ,Combinatorics ,010201 computation theory & mathematics ,Random regular graph ,Almost surely ,0101 mathematics ,Absolute constant ,Software ,Connectivity ,Mathematics - Abstract
In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most C|VG|. In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph Gn,p, which improves upon existing results showing that asymptotically almost surely the cop number of Gn,p is Onlogn provided that pni¾?2+elogn for some e>0. We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion-type properties. This will also be used in a separate paper on random d-regular graphs, where we show that the conjecture holds asymptotically almost surely when d=dni¾?3. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 396-421, 2016
- Published
- 2015
34. Existence of nontrivial solution for Schrödinger–Poisson systems with indefinite steep potential well
- Author
-
Juntao Sun, Yuanze Wu, and Tsung-fang Wu
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,General Physics and Astronomy ,Lambda ,Poisson distribution ,01 natural sciences ,Omega ,010101 applied mathematics ,symbols.namesake ,symbols ,0101 mathematics ,Schrödinger's cat ,Mathematics - Abstract
In this paper, we study a class of nonlinear Schrodinger–Poisson systems with indefinite steep potential well: $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} -\Delta u+V_{\lambda }(x)u+K(x)\phi u=|u|^{p-2}u &{} \text { in }\mathbb {R}^{3},\\ -\Delta \phi =K\left( x\right) u^{2} &{} \ \text {in }\mathbb {R}^{3}, \end{array} \right. \end{aligned}$$ where $$30$$ and $$ K(x)\ge 0$$ for all $$x\in \mathbb {R}^{3}$$ . We require that $$a\in C( \mathbb {R}^{3}) $$ is nonnegative and has a potential well $$\Omega _{a}$$ , namely $$a(x)\equiv 0$$ for $$x\in \Omega _{a}$$ and $$a(x)>0$$ for $$x\in \mathbb {R}^{3}\setminus \overline{\Omega _{a}}$$ . Unlike most other papers on this problem, we allow that $$b\in C(\mathbb {R}^{3}) $$ is unbounded below and sign-changing. By introducing some new hypotheses on the potentials and applying the method of penalized functions, we obtain the existence of nontrivial solutions for $$\lambda $$ sufficiently large. Furthermore, the concentration behavior of the nontrivial solution is also described as $$\lambda \rightarrow \infty $$ .
- Published
- 2017
35. Gowers norms control diophantine inequalities
- Author
-
Aled Walker
- Subjects
Gowers norms ,General Mathematics ,Möbius function ,System of linear equations ,01 natural sciences ,Task (project management) ,Integer ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,Möbius orthogonality ,Number Theory (math.NT) ,0101 mathematics ,Mathematics ,Pseudorandom number generator ,Discrete mathematics ,11B30 ,Algebra and Number Theory ,Mathematics - Number Theory ,Applied Mathematics ,Diophantine equation ,010102 general mathematics ,11D75 ,Linear inequality ,Bounded function ,diophantine inequalities ,010307 mathematical physics ,Combinatorics (math.CO) ,11J25 ,generalised von Neumann theorem - Abstract
A central tool in the study of systems of linear equations with integer coefficients is the Generalised von Neumann Theorem of Green and Tao. This theorem reduces the task of counting the weighted solutions of these equations to that of counting the weighted solutions for a particular family of forms, the Gowers norms $\Vert f \Vert_{U^{s+1}[N]}$ of the weight $f$. In this paper we consider systems of linear inequalities with real coefficients, and show that the number of solutions to such weighted diophantine inequalities may also be bounded by Gowers norms. Furthermore, we provide a necessary and sufficient condition for a system of real linear forms to be governed by Gowers norms in this way. We present applications to cancellation of the M\"{o}bius function over certain sequences. The machinery developed in this paper can be adapted to the case in which the weights are unbounded but suitably pseudorandom, with applications to counting the number of solutions to diophantine inequalities over the primes. Substantial extra difficulties occur in this setting, however, and we have prepared a separate paper on these issues., Comment: 75 pages. Reworked introduction based on referee comments. To appear in Algebra & Number Theory
- Published
- 2017
36. The Fifth Moment of modular L-functions
- Author
-
Matthew P. Young and Eren Mehmet Kıral
- Subjects
Discrete mathematics ,Mathematics - Number Theory ,business.industry ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Modular design ,01 natural sciences ,Prime (order theory) ,Moment (mathematics) ,FOS: Mathematics ,Number Theory (math.NT) ,0101 mathematics ,business ,Mathematics - Abstract
We prove a sharp bound on the fifth moment of modular L-functions of fixed small weight, and large prime level., Comment: v1: 81 pages. v3: 65 pages. We have extracted some material which now appears in two shorter, stand-alone papers. The calculations of Kloosterman sums and Fourier coefficients of Eisenstein series are in one new paper, and the stationary phase results are in a new paper with Ian Petrow as an additional co-author. This version also implements minor corrections and improvements to exposition
- Published
- 2017
37. Degenerate random environments
- Author
-
Thomas S. Salisbury and Mark Holmes
- Subjects
Random graph ,Discrete mathematics ,Percolation critical exponents ,Random field ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Random function ,Random element ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Directed percolation ,Combinatorics ,010104 statistics & probability ,FOS: Mathematics ,Random compact set ,Continuum percolation theory ,0101 mathematics ,Mathematics - Probability ,Software ,Mathematics - Abstract
We consider connectivity properties of certain i.i.d. random environments on i¾?d, where at each location some steps may not be available. Site percolation and oriented percolation are examples of such environments. In these models, one of the quantities most often studied is the random set of vertices that can be reached from the origin by following a connected path. More generally, for the models we consider, multiple different types of connectivity are of interest, including: the set of vertices that can be reached from the origin; the set of vertices from which the origin can be reached; the intersection of the two. As with percolation models, many of the models we consider admit, or are expected to admit phase transitions. Among the main results of the paper is a proof of the existence of phase transitions for some two-dimensional models that are non-monotone in their underlying parameter, and an improved bound on the critical value for oriented site percolation on the triangular lattice. The connectivity of the random directed graphs provides a foundation for understanding the asymptotic properties of random walks in these random environments, which we study in a second paper. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 111-137, 2014
- Published
- 2012
38. ON THE GERBER-SHIU PENALTY FUNCTION FOR THE COMPOUND BINOMIAL RISK MODEL WITH DELAYED CLAIMS
- Author
-
Y. Liu, Z. Bao, and H. Liu
- Subjects
Discrete mathematics ,050208 finance ,Binomial (polynomial) ,Applied Mathematics ,General Mathematics ,05 social sciences ,Zero (complex analysis) ,Function (mathematics) ,Expression (computer science) ,01 natural sciences ,010104 statistics & probability ,Risk model ,0502 economics and business ,Applied mathematics ,Renewal equation ,Penalty method ,0101 mathematics ,Mathematics - Abstract
In this paper we consider the Gerber-Shiu penalty function in the compound binomial risk model with time-correlated claims. It is assumed that each main claim will induce a by-claim but the occurrence of the by-claim may be delayed with a certain probability. Formulas for the probability generating function of the penalty function are obtained, together with the expression for the penalty function with zero initial surplus. Then we show that the penalty function satisfies a defective renewal equation. When the claims follow geometric distributions, explicit expression for the Gerber-Shiu function is derived. Numerical example is provided to illustrate the application of the result discussed in the paper.
- Published
- 2016
39. On A Novel Eccentricity-Based Invariant Of A Graph
- Author
-
Kexiang Xu, Ayse Dilek Maden, and Kinkar Ch. Das
- Subjects
Discrete mathematics ,Graph center ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,01 natural sciences ,Graph ,Vertex (geometry) ,Combinatorics ,010201 computation theory & mathematics ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Abstract
In this paper, for the purpose of measuring the non-self-centrality extent of non-selfcentered graphs, a novel eccentricity-based invariant, named as non-self-centrality number (NSC number for short), of a graph G is defined as follows: \(N\left( G \right) = \sum\nolimits_{{v_i}{v_j} \in V\left( G \right)} {|{e_i} - {e_j}|} \) where the summation goes over all the unordered pairs of vertices in G and ei is the eccentricity of vertex vi in G, whereas the invariant will be called third Zagreb eccentricity index if the summation only goes over the adjacent vertex pairs of graph G. In this paper, we determine the lower and upper bounds on N(G) and characterize the corresponding graphs at which the lower and upper bounds are attained. Finally we propose some attractive research topics for this new invariant of graphs.
- Published
- 2016
40. BMO-Type Norms Related to the Perimeter of Sets
- Author
-
Jean Bourgain, Haim Brezis, Alessio Figalli, Luigi Ambrosio, Ambrosio, Luigi, Bourgain, Jean, Brezis, Haim, and Figalli, Alessio
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Isotropy ,Mathematics::Analysis of PDEs ,Type (model theory) ,Characterization (mathematics) ,16. Peace & justice ,01 natural sciences ,Functional Analysis (math.FA) ,010101 applied mathematics ,Perimeter ,Mathematics - Functional Analysis ,Settore MAT/05 - Analisi Matematica ,Norm (mathematics) ,FOS: Mathematics ,Mathematics::Metric Geometry ,0101 mathematics ,Mathematics - Abstract
In this paper we consider an isotropic variant of the BMO-type norm recently introduced (Bourgain, Brezis, and Mironescu, 2015). We prove that, when considering characteristic functions of sets, this norm is related to the perimeter. A byproduct of our analysis is a new characterization of the perimeter of sets in terms of this norm, independent of the theory of distributions. In this paper we consider an isotropic variant of the BMO-type norm recently introduced (Bourgain, Brezis, and Mironescu, 2015). We prove that, when considering characteristic functions of sets, this norm is related to the perimeter. A byproduct of our analysis is a new characterization of the perimeter of sets in terms of this norm, independent of the theory of distributions.
- Published
- 2016
41. A curvature-free 𝐿𝑜𝑔(2𝑘-1) theorem
- Author
-
Florent Balacheff and Louis Merlin
- Subjects
Discrete mathematics ,Lemma (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,16. Peace & justice ,Curvature ,Mathematics::Geometric Topology ,01 natural sciences ,Volume entropy ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
This paper presents a curvature-free version of the Log ( 2 k − 1 ) \text {Log}(2k-1) Theorem of Anderson, Canary, Culler, and Shalen [J. Differential Geometry 44 (1996), pp. 738–782]. It generalizes a result by Hou [J. Differential Geometry 57 (2001), no. 1, pp. 173–193] and its proof is rather straightforward once we know the work by Lim [Trans. Amer. Math. Soc. 360 (2008), no. 10, pp. 5089–5100] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the Collar Lemma in all dimensions.
- Published
- 2023
42. Generalized tractability for multivariate problems Part I
- Author
-
Henryk Woźniakowski and Michael Gnewuch
- Subjects
Discrete mathematics ,Statistics and Probability ,Polynomial ,Numerical Analysis ,Algebra and Number Theory ,Control and Optimization ,Information-based complexity ,General Mathematics ,media_common.quotation_subject ,Applied Mathematics ,010103 numerical & computational mathematics ,Function (mathematics) ,Space (mathematics) ,Infinity ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Set (abstract data type) ,Tensor product ,Bounded function ,0101 mathematics ,Mathematics ,media_common - Abstract
Many papers study polynomial tractability for multivariate problems. Let n(@?,d) be the minimal number of information evaluations needed to reduce the initial error by a factor of @? for a multivariate problem defined on a space of d-variate functions. Here, the initial error is the minimal error that can be achieved without sampling the function. Polynomial tractability means that n(@?,d) is bounded by a polynomial in @?^-^1 and d and this holds for all (@?^-^1,d)@?[1,~)xN. In this paper we study generalized tractability by verifying when n(@?,d) can be bounded by a power of T(@?^-^1,d) for all (@?^-^1,d)@[email protected], where @W can be a proper subset of [1,~)xN. Here T is a tractability function, which is non-decreasing in both variables and grows slower than exponentially to infinity. In this article we consider the set @W=[1,~)x{1,2,...,d^*}@?[1,@?"0^-^1)xN for some d^*>=1 and @?"[email protected]?(0,1). We study linear tensor product problems for which we can compute arbitrary linear functionals as information evaluations. We present necessary and sufficient conditions on T such that generalized tractability holds for linear tensor product problems. We show a number of examples for which polynomial tractability does not hold but generalized tractability does.
- Published
- 2007
- Full Text
- View/download PDF
43. Decomposing a Graph Into Expanding Subgraphs
- Author
-
Asaf Shapira and Guy Moshkovitz
- Subjects
General Mathematics ,Symmetric graph ,0102 computer and information sciences ,01 natural sciences ,law.invention ,Combinatorics ,symbols.namesake ,010104 statistics & probability ,law ,Line graph ,FOS: Mathematics ,Mathematics - Combinatorics ,Cograph ,0101 mathematics ,Complement graph ,Mathematics ,Universal graph ,Forbidden graph characterization ,Discrete mathematics ,Applied Mathematics ,010102 general mathematics ,Computer Graphics and Computer-Aided Design ,Planar graph ,010201 computation theory & mathematics ,symbols ,Regular graph ,Combinatorics (math.CO) ,Software ,MathematicsofComputing_DISCRETEMATHEMATICS - Abstract
A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that any graph is close to being the disjoint union of expanders. Our goal in this paper is to show that in several of the instantiations of the above approach, the quantitative bounds that were obtained are essentially best possible. Three examples of our results are the following: A classical result of Lipton, Rose and Tarjan from 1979 states that if F is a hereditary family of graphs and every graph in F has a vertex separator of size n/(logn)1+o(1), then every graph in F has O(n) edges. We construct a hereditary family of graphs with vertex separators of size n/(logn)1−o(1) such that not all graphs in the family have O(n) edges. Trevisan and Arora-Barak-Steurer have recently shown that given a graph G, one can remove only 1% of its edges to obtain a graph in which each connected component has good expansion properties. We show that in both of these decomposition results, the expansion properties they guarantee are essentially best possible, even when one is allowed to remove 99% of G's edges. Sudakov and the second author have recently shown that every graph with average degree d contains an n-vertex subgraph with average degree at least (1−o(1))d and vertex expansion 1/(logn)1+o(1). We show that one cannot guarantee a better vertex expansion even if allowing the average degree to be O(1). The above results are obtained as corollaries of a new family of graphs which we construct in this paper. These graphs have a super-linear number of edges and nearly logarithmic girth, yet each of their subgraphs has (optimally) poor expansion properties.
- Published
- 2014
44. On characterizing hypergraph regularity
- Author
-
Penny Haxell, V. Rödl, Yulia Dementieva, and Brendan Nagle
- Subjects
Discrete mathematics ,Hypergraph ,Lemma (mathematics) ,Mathematics::Combinatorics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Graph theory ,Szemerédi regularity lemma ,0102 computer and information sciences ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Graph ,Extremal graph theory ,Combinatorics ,010201 computation theory & mathematics ,Bipartite graph ,Verifiable secret sharing ,0101 mathematics ,Software ,Mathematics - Abstract
Szemeredi's Regularity Lemma is a well-known and powerful tool in modern graph theory. This result led to a number of interesting applications, particularly in extremal graph theory. A regularity lemma for 3-uniform hypergraphs developed by Frankl and Rodl [8] allows some of the Szemeredi Regularity Lemma graph applications to be extended to hypergraphs. An important development regarding Szemeredi's Lemma showed the equivalence between the property of e-regularity of a bipartite graph G and an easily verifiable property concerning the neighborhoods of its vertices (Alon et al. [1]; cf. [6]). This characterization of e-regularity led to an algorithmic version of Szemeredi's lemma [1]. Similar problems were also considered for hypergraphs. In [2], [9], [13], and [18], various descriptions of quasi-randomness of k-uniform hypergraphs were given. As in [1], the goal of this paper is to find easily verifiable conditions for the hypergraph regularity provided by [8]. The hypergraph regularity of [8] renders quasi-random "blocks of hyperedges" which are very sparse. This situation leads to technical difficulties in its application. Moreover, as we show in this paper, some easily verifiable conditions analogous to those considered in [2] and [18] fail to be true in the setting of [8]. However, we are able to find some necessary and sufficient conditions for this hypergraph regularity. These conditions enable us to design an algorithmic version of a hypergraph regularity lemma in [8]. This algorithmic version is presented by the authors in [5].
- Published
- 2002
45. Random Matrices and Erasure Robust Frames
- Author
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Yang Wang
- Subjects
FOS: Computer and information sciences ,Discrete mathematics ,42C15 ,Information Theory (cs.IT) ,Applied Mathematics ,General Mathematics ,Data erasure ,Computer Science - Information Theory ,010102 general mathematics ,010103 numerical & computational mathematics ,01 natural sciences ,Restricted isometry property ,Combinatorics ,Normal distribution ,Singular value ,Redundancy (information theory) ,Erasure ,0101 mathematics ,Random matrix ,Condition number ,Analysis ,Mathematics - Abstract
Data erasure can often occur in communication. Guarding against erasures involves redundancy in data representation. Mathematically this may be achieved by redundancy through the use of frames. One way to measure the robustness of a frame against erasures is to examine the worst case condition number of the frame with a certain number of vectors erased from the frame. The term numerically erasure-robust frames was introduced in Fickus and Mixon (Linear Algebra Appl 437:1394–1407, 2012) to give a more precise characterization of erasure robustness of frames. In the paper the authors established that random frames whose entries are drawn independently from the standard normal distribution can be robust against up to approximately 15 % erasures, and asked whether there exist frames that are robust against erasures of more than 50 %. In this paper we show that with very high probability random frames are, independent of the dimension, robust against erasures as long as the number of remaining vectors is at least \(1+\delta _0\) times the dimension for some \(\delta _0>0\). This is the best possible result, and it also implies that the proportion of erasures can be arbitrarily close to 1 while still maintaining robustness. Our result depends crucially on a new estimate for the smallest singular value of a rectangular random matrix with independent standard normal entries.
- Published
- 2014
46. Complexity of Weighted Approximation over Rd
- Author
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Henryk Woźniakowski and G. W. Wasilkowski
- Subjects
Statistics and Probability ,Discrete mathematics ,Numerical Analysis ,Weight function ,Control and Optimization ,Algebra and Number Theory ,Applied Mathematics ,General Mathematics ,Approximation algorithm ,integration ,020206 networking & telecommunications ,010103 numerical & computational mathematics ,02 engineering and technology ,01 natural sciences ,Continuation ,worst case complexity ,Approximation error ,Norm (mathematics) ,weighted multivariate approximation ,0202 electrical engineering, electronic engineering, information engineering ,Worst-case complexity ,Uniform boundedness ,Partial derivative ,0101 mathematics ,Mathematics - Abstract
We study approximation of multivariate functions defined over Rd. We assume that all rth order partial derivatives of the functions considered are continuous and uniformly bounded. Approximation algorithms U(f) only use the values of f or its partial derivatives up to order r. We want to recover the function f with small error measured in a weighted Lq norm with a weight function ρ. We study the worst case (information) complexity which is equal to the minimal number of function and derivative evaluations needed to obtain error ε. We provide necessary and sufficient conditions in terms of the weight ρ and the parameters q and r for the weighted approximation problem to have finite complexity. We also provide conditions guaranteeing that the complexity is of the same order as the complexity of the classical approximation problem over a finite domain. Since the complexity of the weighted integration problem is equivalent to the complexity of the weighted approximation problem with q=1, the results of this paper also hold for weighted integration. This paper is a continuation of [7], where weighted approximation over R was studied.
- Published
- 2001
47. Approximating Fixed Points of Weakly Contracting Mappings
- Author
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K. Sikorski, Leonid Khachiyan, and Z. Huang
- Subjects
Discrete mathematics ,Statistics and Probability ,Numerical Analysis ,Algebra and Number Theory ,Control and Optimization ,Iterative method ,General Mathematics ,Applied Mathematics ,Dimension (graph theory) ,010103 numerical & computational mathematics ,Function (mathematics) ,Fixed point ,01 natural sciences ,Ellipsoid ,Upper and lower bounds ,010101 applied mathematics ,Combinatorics ,Approximation error ,Simple (abstract algebra) ,0101 mathematics ,Mathematics - Abstract
We consider the problem of approximating fixed points of contractive functions whose contraction factor is close to 1. In a previous paper (1993, K. Sikorski et al. , J. Complexity 9 , 181–200), we proved that for the absolute error criterion, the upper bound on the number of function evaluations to compute e -approximations is O( n 3 (ln(1/ e )+ln(1/(1− q ))+ln n )) in the worst case, where 0 q n is the dimension of the problem. This upper bound is achieved by the circumscribed ellipsoid (CE) algorithm combined with a dimensional deflation process. In this paper we present an inscribed ellipsoid (IE) algorithm that enjoys O( n (ln(1/ e )+ln(1/(1− q )))) bound. For q close to 1, the IE algorithm thus runs in many fewer iterations than the simple iteration method, that requires ⌈ln(1/ e )/ln(1/ q )⌉ function evaluations. Our analysis also implies that: (1) The dimensional deflation procedure in the CE algorithm is not necessary and that the resulting “plain” CE algorithm enjoys O( n 2 (log(1/ e )+log(1/(1− q )))) upper bound on the number of function evaluations. (2) The IE algorithm solves the problem in the residual sense, i.e., computes x such that ‖ f ( x )− x ‖⩽ δ , with O( n ln(1/ δ )) function evaluations for every q ⩽1.
- Published
- 1999
- Full Text
- View/download PDF
48. On q-poly-Bernoulli numbers arising from combinatorial interpretations
- Author
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José L. Ramírez and Beáta Bényi
- Subjects
Discrete mathematics ,Recall ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,021107 urban & regional planning ,02 engineering and technology ,01 natural sciences ,Ask price ,Discrete Mathematics and Combinatorics ,Natural (music) ,0101 mathematics ,Bernoulli number ,Mathematics - Abstract
In this paper we present several natural q-analogues of the poly-Bernoulli numbers arising in combinatorial contexts. We also recall some related analytical results and ask for combinatorial interpretations.
- Published
- 2021
49. Connections between the Support and Linear Independence of Refinable Distributions
- Author
-
Jianzhong Wang and David K. Ruch
- Subjects
Discrete mathematics ,Mathematics(all) ,Numerical Analysis ,General Mathematics ,Multiresolution analysis ,Applied Mathematics ,010102 general mathematics ,Regular polygon ,01 natural sciences ,010101 applied mathematics ,Dilation (metric space) ,Distribution (mathematics) ,If and only if ,Linear independence ,0101 mathematics ,Independence (probability theory) ,Analysis ,Mathematics ,Integer (computer science) - Abstract
The purpose of this paper is to study the relationships between the support of a refinable distributionφand the global and local linear independence of the integer translates ofφ. It has been shown elsewhere that a compactly supported distributionφhas globally independent integer translates if and only ifφhas minimal convex support. However, such a distribution may have “holes” in its support. By insisting thatφ∈L2(R) and generates a multiresolution analysis, Lemarié and Malgouyres have ensured that no such holes can occur. In this article we generalize this result to refinable distributions. We also give a result on the local linear independence of the integer translates ofφ. We work with integer dilation factorN⩾2 throughout this paper.
- Published
- 1998
- Full Text
- View/download PDF
50. The Monge problem in $ {\mathbb{R}^d} $ : Variations on a theme
- Author
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Thierry Champion, Luigi De Pascale, Institut de Mathématiques de Toulon - EA 2134 (IMATH), and Université de Toulon (UTLN)
- Subjects
Statistics and Probability ,Discrete mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,01 natural sciences ,010101 applied mathematics ,Norm (mathematics) ,Bibliography ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,0101 mathematics ,[MATH]Mathematics [math] ,Algorithm ,ComputingMilieux_MISCELLANEOUS ,Mathematics ,General norm - Abstract
In a recent paper, the authors proved that, under natural assumptions on the first marginal, the Monge problem in $ {\mathbb{R}^d} $ for the cost given by a general norm admits a solution. Although the basic idea of the proof is simple, it involves some complex technical results. Here we will give a proof of the result in the simpler case of a uniformly convex norm, and we will also use very recent results by Ahmad, Kim, and McCann. This allows us to reduce the technical burdens while still giving the main ideas of the general proof. The proof of the density of the transport set in the particular case considered in this paper is original. Bibliography: 22 titles.
- Published
- 2012
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