969 results
Search Results
2. An improvement on Furstenberg’s intersection problem
- Author
-
Han Yu
- Subjects
Combinatorics ,Intersection ,Applied Mathematics ,General Mathematics ,Bounded function ,010102 general mathematics ,Dimension (graph theory) ,Zero (complex analysis) ,0101 mathematics ,Invariant (mathematics) ,Dynamical system (definition) ,01 natural sciences ,Mathematics - Abstract
In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim A 2 + dim A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .
- Published
- 2021
3. Degrees of Enumerations of Countable Wehner-Like Families
- Author
-
I. Sh. Kalimullin and M. Kh. Faizrahmanov
- Subjects
Statistics and Probability ,Class (set theory) ,Degree (graph theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Spectrum (topology) ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,Enumeration ,Countable set ,Family of sets ,0101 mathematics ,Turing ,computer ,Finite set ,computer.programming_language ,Mathematics - Abstract
This paper is a survey of results on countable families with natural degree spectra. These results were obtained by a modification of the methodology proposed by Wechner, who first found a family of sets with the spectrum consisting precisely of nonzero Turing degrees. Based on this method, many researchers obtained examples of families with other natural spectra. In addition, in this paper we extend these results and present new examples of natural spectra. In particular, we construct a family of finite sets with the spectrum consisting of exactly non-K-trivial degrees and also we find new sufficient conditions on $$ {\Delta}_2^0 $$ -degree a, which guarantees that the class {x : x ≰ a} is the degree spectrum of some family. Finally, we give a survey of our recent results on the degree spectra of α-families, where α is an arbitrary computable ordinal.
- Published
- 2021
4. On graphs with equal total domination and Grundy total domination numbers
- Author
-
Tilen Marc, Tim Kos, Tanja Dravec, and Marko Jakovac
- Subjects
Sequence ,Domination analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,Characterization (mathematics) ,01 natural sciences ,Vertex (geometry) ,Combinatorics ,Dominating set ,Chordal graph ,Bipartite graph ,Discrete Mathematics and Combinatorics ,Projective plane ,0101 mathematics ,Mathematics - Abstract
A sequence $$(v_1,\ldots ,v_k)$$ of vertices in a graph G without isolated vertices is called a total dominating sequence if every vertex $$v_i$$ in the sequence totally dominates at least one vertex that was not totally dominated by $$\{v_1,\ldots , v_{i-1}\}$$ and $$\{v_1,\ldots ,v_k\}$$ is a total dominating set of G. The length of a shortest such sequence is the total domination number of G ( $$\gamma _{t}(G)$$ ), while the length of a longest such sequence is the Grundy total domination number of G ( $$\gamma _{gr}^t(G)$$ ). In this paper we study graphs with equal total and Grundy total domination numbers. We characterize bipartite graphs with both total and Grundy total dominations number equal to 4, and show that there is no connected chordal graph G with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=4$$ . The main result of the paper is a characterization of bipartite graphs with $$\gamma _{t}(G)=\gamma _{gr}^t(G)=6$$ proved by establishing a surprising correspondence between the existence of such graphs and a classical but still open problem of the existence of certain finite projective planes.
- Published
- 2021
5. Simplest Test for the Three-Dimensional Dynamical Inverse Problem (The BC-Method)
- Author
-
Mikhail I. Belishev, N. A. Karazeeva, and A. S. Blagoveshchensky
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,Operator (physics) ,010102 general mathematics ,Boundary (topology) ,Function (mathematics) ,Inverse problem ,Positive function ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,Nabla symbol ,0101 mathematics ,Dynamical system (definition) ,Realization (systems) ,Mathematics - Abstract
A dynamical system $$ {\displaystyle \begin{array}{ll}{u}_{tt}-\Delta u-\nabla 1\mathrm{n}\;\rho \cdot \nabla u=0& in\kern0.6em {\mathrm{\mathbb{R}}}_{+}^3\times \left(0,T\right),\\ {}{\left.u\right|}_{t=0}={\left.{u}_t\right|}_{t=0}=0& in\kern0.6em \overline{{\mathrm{\mathbb{R}}}_{+}^3},\\ {}{\left.{u}_z\right|}_{z=0}=f& for\kern0.36em 0\le t\le T,\end{array}} $$ is under consideration, where ρ = ρ(x, y, z) is a smooth positive function; f = f(x, y, t) is a boundary control; u = uf (x, y, z, t) is a solution. With the system one associates a response operator R : f ↦ uf|z = 0. The inverse problem is to recover the function ρ via the response operator. A short representation of the local version of the BC-method, which recovers ρ via the data given on a part of the boundary, is provided. If ρ is constant, the forward problem is solved in explicit form. In the paper, the corresponding representations for the solutions and response operator are derived. A way to use them for testing the BC-algorithm, which solves the inverse problem, is outlined. The goal of the paper is to extend the circle of the BC-method users, who are interested in numerical realization of methods for solving inverse problems.
- Published
- 2021
6. On some universal Morse–Sard type theorems
- Author
-
Alba Roviello, Adele Ferone, Mikhail V. Korobkov, Ferone, A., Korobkov, M. V., and Roviello, A.
- Subjects
Uncertainty principle ,Dubovitskii-Federer theorems ,Near critical ,Morse-Sard theorem ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Algebraic geometry ,Morse code ,Sobolev-Lorentz mapping ,Holder mapping ,01 natural sciences ,law.invention ,Sobolev space ,Combinatorics ,law ,0103 physical sciences ,010307 mathematical physics ,Differentiable function ,Bessel potential space ,0101 mathematics ,Critical set ,Mathematics - Abstract
The classical Morse–Sard theorem claims that for a mapping v : R n → R m + 1 of class C k the measure of critical values v ( Z v , m ) is zero under condition k ≥ n − m . Here the critical set, or m-critical set is defined as Z v , m = { x ∈ R n : rank ∇ v ( x ) ≤ m } . Further Dubovitskiĭ in 1957 and independently Federer and Dubovitskiĭ in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the C k category. Here we formulate and prove a bridge theorem that includes all the above results as particular cases: namely, if a function v : R n → R d belongs to the Holder class C k , α , 0 ≤ α ≤ 1 , then for every q > m the identity H μ ( Z v , m ∩ v − 1 ( y ) ) = 0 holds for H q -almost all y ∈ R d , where μ = n − m − ( k + α ) ( q − m ) . Intuitively, the sense of this bridge theorem is very close to Heisenberg's uncertainty principle in theoretical physics: the more precise is the information we receive on measure of the image of the critical set, the less precisely the preimages are described, and vice versa. The result is new even for the classical C k -case (when α = 0 ); similar result is established for the Sobolev classes of mappings W p k ( R n , R d ) with minimal integrability assumptions p = max ( 1 , n / k ) , i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some N-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also crucially use very deep Y. Yomdin's entropy estimates of near critical values for polynomials (based on algebraic geometry tools).
- Published
- 2020
7. Packing colorings of subcubic outerplanar graphs
- Author
-
Nicolas Gastineau, Olivier Togni, Boštjan Brešar, Faculty of Natural Sciences and Mathematics [Maribor], University of Maribor, Laboratoire d'Informatique de Bourgogne [Dijon] (LIB), Université de Bourgogne (UB), and Togni, Olivier
- Subjects
05C15, 05C12, 05C70 ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,010103 numerical & computational mathematics ,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM] ,01 natural sciences ,Graph ,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO] ,Combinatorics ,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM] ,Integer ,Outerplanar graph ,Bounded function ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,FOS: Mathematics ,Bipartite graph ,Mathematics - Combinatorics ,Discrete Mathematics and Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Invariant (mathematics) ,ComputingMilieux_MISCELLANEOUS ,Mathematics - Abstract
Given a graph $G$ and a nondecreasing sequence $S=(s_1,\ldots,s_k)$ of positive integers, the mapping $c:V(G)\longrightarrow \{1,\ldots,k\}$ is called an $S$-packing coloring of $G$ if for any two distinct vertices $x$ and $y$ in $c^{-1}(i)$, the distance between $x$ and $y$ is greater than $s_i$. The smallest integer $k$ such that there exists a $(1,2,\ldots,k)$-packing coloring of a graph $G$ is called the packing chromatic number of $G$, denoted $\chi_{\rho}(G)$. The question of boundedness of the packing chromatic number in the class of subcubic (planar) graphs was investigated in several earlier papers; recently it was established that the invariant is unbounded in the class of all subcubic graphs. In this paper, we prove that the packing chromatic number of any 2-connected bipartite subcubic outerplanar graph is bounded by $7$. Furthermore, we prove that every subcubic triangle-free outerplanar graph has a $(1,2,2,2)$-packing coloring, and that there exists a subcubic outerplanar graph with a triangle that does not admit a $(1,2,2,2)$-packing coloring. In addition, there exists a subcubic triangle-free outerplanar graph that does not admit a $(1,2,2,3)$-packing coloring. A similar dichotomy is shown for bipartite outerplanar graphs: every such graph admits an $S$-packing coloring for $S=(1,3,\ldots,3)$, where $3$ appears $\Delta$ times ($\Delta$ being the maximum degree of vertices), and this property does not hold if one of the integers $3$ is replaced by $4$ in the sequence $S$., Comment: 24 pages
- Published
- 2020
8. Bernoulliness of when is an irrational rotation: towards an explicit isomorphism
- Author
-
Christophe Leuridan
- Subjects
Rational number ,Lebesgue measure ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Diophantine approximation ,01 natural sciences ,Irrational rotation ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,Bernoulli scheme ,Isomorphism ,0101 mathematics ,Real number ,Unit interval ,Mathematics - Abstract
Let $\unicode[STIX]{x1D703}$ be an irrational real number. The map $T_{\unicode[STIX]{x1D703}}:y\mapsto (y+\unicode[STIX]{x1D703})\!\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ from the unit interval $\mathbf{I}= [\!0,1\![$ (endowed with the Lebesgue measure) to itself is ergodic. In a short paper [Parry, Automorphisms of the Bernoulli endomorphism and a class of skew-products. Ergod. Th. & Dynam. Sys.16 (1996), 519–529] published in 1996, Parry provided an explicit isomorphism between the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift when $\unicode[STIX]{x1D703}$ is extremely well approximated by the rational numbers, namely, if $$\begin{eqnarray}\inf _{q\geq 1}q^{4}4^{q^{2}}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ A few years later, Hoffman and Rudolph [Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2)156 (2002), 79–101] showed that for every irrational number, the measure-preserving map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ is isomorphic to the unilateral dyadic Bernoulli shift. Their proof is not constructive. In the present paper, we relax notably Parry’s condition on $\unicode[STIX]{x1D703}$: the explicit map provided by Parry’s method is an isomorphism between the map $[T_{\unicode[STIX]{x1D703}},\text{Id}]$ and the unilateral dyadic Bernoulli shift whenever $$\begin{eqnarray}\inf _{q\geq 1}q^{4}~\text{dist}(\unicode[STIX]{x1D703},q^{-1}\mathbb{Z})=0.\end{eqnarray}$$ This condition can be relaxed again into $$\begin{eqnarray}\inf _{n\geq 1}q_{n}^{3}~(a_{1}+\cdots +a_{n})~|q_{n}\unicode[STIX]{x1D703}-p_{n}| where $[0;a_{1},a_{2},\ldots ]$ is the continued fraction expansion and $(p_{n}/q_{n})_{n\geq 0}$ the sequence of convergents of $\Vert \unicode[STIX]{x1D703}\Vert :=\text{dist}(\unicode[STIX]{x1D703},\mathbb{Z})$. Whether Parry’s map is an isomorphism for every $\unicode[STIX]{x1D703}$ or not is still an open question, although we expect a positive answer.
- Published
- 2020
9. On the Structure of a 3-Connected Graph. 2
- Author
-
D. V. Karpov
- Subjects
Statistics and Probability ,Hypergraph ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Type (model theory) ,01 natural sciences ,010305 fluids & plasmas ,Set (abstract data type) ,Combinatorics ,0103 physical sciences ,Decomposition (computer science) ,Graph (abstract data type) ,0101 mathematics ,Connectivity ,Hyperbolic tree ,Mathematics - Abstract
In this paper, the structure of relative disposition of 3-vertex cutsets in a 3-connected graph is studied. All such cutsets are divided into structural units – complexes of flowers, of cuts, of single cutsets, and trivial complexes. The decomposition of the graph by a complex of each type is described in detail. It is proved that for any two complexes C1 and C2 of a 3-connected graph G there is a unique part of the decomposition of G by C1 that contains C2. The relative disposition of complexes is described with the help of a hypertree T (G) – a hypergraph any cycle of which is a subset of a certain hyperedge. It is also proved that each nonempty part of the decomposition of G by the set of all of its 3-vertex cutsets is either a part of the decomposition of G by one of the complexes or corresponds to a hyperedge of T (G). This paper can be considered as a continuation of studies begun in the joint paper by D. V. Karpov and A. V. Pastor “On the structure of a 3-connected graph,” published in 2011. Bibliography: 10 titles.
- Published
- 2020
10. On Tetravalent Vertex-Transitive Bi-Circulants
- Author
-
Sha Qiao and Jin-Xin Zhou
- Subjects
Transitive relation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Cyclic group ,0102 computer and information sciences ,Automorphism ,01 natural sciences ,Graph ,Vertex (geometry) ,Combinatorics ,010201 computation theory & mathematics ,0101 mathematics ,Mathematics - Abstract
A graph Γ is called a bi-circulant if it admits a cyclic group as a group of automorphisms acting semiregularly on the vertices of Γ with two orbits. The characterization of tetravalent edgetransitive bi-circulants was given in several recent papers. In this paper, a classification is given of connected tetravalent vertex-transitive bi-circulants of order twice an odd integer.
- Published
- 2020
11. Products of Commutators on a General Linear Group Over a Division Algebra
- Author
-
Nikolai Gordeev and E. A. Egorchenkova
- Subjects
Statistics and Probability ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Center (category theory) ,General linear group ,Field (mathematics) ,01 natural sciences ,010305 fluids & plasmas ,Combinatorics ,0103 physical sciences ,Division algebra ,0101 mathematics ,Word (group theory) ,Mathematics - Abstract
The word maps $$ \tilde{w}:\kern0.5em {\mathrm{GL}}_m{(D)}^{2k}\to {\mathrm{GL}}_n(D) $$ and $$ \tilde{w}:\kern0.5em {D}^{\ast 2k}\to {D}^{\ast } $$ for a word $$ w=\prod \limits_{i=1}^k\left[{x}_i,{y}_i\right], $$ where D is a division algebra over a field K, are considered. It is proved that if $$ \tilde{w}\left({D}^{\ast 2k}\right)=\left[{D}^{\ast },{D}^{\ast}\right], $$ then $$ \tilde{w}\left({\mathrm{GL}}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right), $$ where En(D) is the subgroup of GLn(D), generated by transvections, and Z(En(D)) is its center. Furthermore if, in addition, n > 2, then $$ \tilde{w}\left({E}_n(D)\right)\supset {E}_n(D)\backslash Z\left({E}_n(D)\right). $$ The proof of the result is based on an analog of the “Gauss decomposition with prescribed semisimple part” (introduced and studied in two papers of the second author with collaborators) in the case of the group GLn(D), which is also considered in the present paper.
- Published
- 2019
12. Exceptional sets for sums of almost equal prime cubes
- Author
-
Mengdi Wang
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Waring–Goldbach problem ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Prime (order theory) ,Mathematics - Abstract
In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.
- Published
- 2019
13. Ultrametric properties for valuation spaces of normal surface singularities
- Author
-
Evelia R. García Barroso, Patrick Popescu-Pampu, Pedro Daniel González Pérez, and Matteo Ruggiero
- Subjects
Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_GENERAL ,Block (permutation group theory) ,14B05, 14J17, 32S25 ,Intersection number ,Function (mathematics) ,01 natural sciences ,Linear subspace ,Combinatorics ,Mathematics - Algebraic Geometry ,Tree (descriptive set theory) ,Singularity ,FOS: Mathematics ,0101 mathematics ,Normal surface ,Algebraic Geometry (math.AG) ,Ultrametric space ,Mathematics - Abstract
Let $L$ be a fixed branch -- that is, an irreducible germ of curve -- on a normal surface singularity $X$. If $A,B$ are two other branches, define $u_L(A,B) := \dfrac{(L \cdot A) \: (L \cdot B)}{A \cdot B}$, where $A \cdot B$ denotes the intersection number of $A$ and $B$. Call $X$ arborescent if all the dual graphs of its resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of P{\l}oski by proving that whenever $X$ is arborescent, the function $u_L$ is an ultrametric on the set of branches on $X$ different from $L$. In the present paper we prove that, conversely, if $u_L$ is an ultrametric, then $X$ is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on $X$, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which $u_L$ is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing $L$ to be an arbitrary semivaluation on $X$ and by defining $u_L$ on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if $X$ is arborescent, and without any restriction on $X$ we exhibit special subspaces of the space of semivaluations in restriction to which $u_L$ is still an ultrametric., Comment: 50 pages, 14 figures. Final version
- Published
- 2019
14. Reproducing kernel orthogonal polynomials on the multinomial distribution
- Author
-
Robert C. Griffiths and Persi Diaconis
- Subjects
Numerical Analysis ,Stationary distribution ,Markov chain ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Poisson kernel ,010103 numerical & computational mathematics ,Kravchuk polynomials ,01 natural sciences ,Combinatorics ,symbols.namesake ,Kernel (statistics) ,Orthogonal polynomials ,symbols ,Test statistic ,Multinomial distribution ,0101 mathematics ,Analysis ,Mathematics - Abstract
Diaconis and Griffiths (2014) study the multivariate Krawtchouk polynomials orthogonal on the multinomial distribution. In this paper we derive the reproducing kernel orthogonal polynomials Q n ( x , y ; N , p ) on the multinomial distribution which are sums of products of orthonormal polynomials in x and y of fixed total degree n = 0 , 1 , … , N . The Poisson kernel ∑ n = 0 N ρ n Q n ( x , y ; N , p ) arises naturally from a probabilistic argument. An application to a multinomial goodness of fit test is developed, where the chi-squared test statistic is decomposed into orthogonal components which test the order of fit. A new duplication formula for the reproducing kernel polynomials in terms of the 1-dimensional Krawtchouk polynomials is derived. The duplication formula allows a Lancaster characterization of all reversible Markov chains with a multinomial stationary distribution whose eigenvectors are multivariate Krawtchouk polynomials and where eigenvalues are repeated within the same total degree. The χ 2 cutoff time, and total variation cutoff time is investigated in such chains. Emphasis throughout the paper is on a probabilistic understanding of the polynomials and their applications, particularly to Markov chains.
- Published
- 2019
15. Convergence of Solutions of General Dispersive Equations Along Curve
- Author
-
Yong Ding and Yaoming Niu
- Subjects
Combinatorics ,010104 statistics & probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Maximal operator ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, the authors give the local L2 estimate of the maximal operator $$S_{\phi ,\gamma }^ * $$ of the operator family {St,ϕ, γ} defined initially by $${S_{t,\phi ,\gamma }}f(x): = {{\rm{e}}^{{\rm{i}}\,t\phi (\sqrt { - \Delta } )}}f(\gamma (x,t)) = {(2\pi )^{ - 1}}\int_\mathbb{R} {{{\rm{e}}^{{\rm{i}}\gamma (x,t) \cdot \xi + {\rm{i}}\,t\phi ({\rm{|}}\xi {\rm{|}})}}} \hat f(\xi ){\rm{d}}\xi ,\;\;\;\;\;\;\;\;f \in {\cal S}(\mathbb{R}),$$ which is the solution (when {itn} = 1) of the following dispersive equations (*) along a curve {itγ}: $$\left\{ {\matrix{ {{\rm{i}}{\partial _t}u + \phi (\sqrt { - {\rm{\Delta }}} )u = 0,} \hfill \;\;\;\;\; {(x,t) \in \mathbb{R}{^n} \times \mathbb{R},} \hfill \cr {u(x,0) = f(x),} \hfill \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; {f \in {\cal S}({\mathbb{R}^n}),} \hfill \cr } } \right.$$ where {itϕ}: ℝ+ → ℝ satisfies some suitable conditions and $$\phi (\sqrt { - {\rm{\Delta }}} )$$ is a pseudo-differential operator with symbol {itϕ}(∣{itξ}∣). As a consequence of the above result, the authors give the pointwise convergence of the solution (when {itn} = 1) of the equation (*) along curve {itγ}. Moreover, a global {itL}2 estimate of the maximal operator $$S_{\phi ,\gamma }^ * $$ is also given in this paper.
- Published
- 2019
16. Comparison of probabilistic and deterministic point sets on the sphere
- Author
-
Peter J. Grabner and T. A. Stepanyuk
- Subjects
Unit sphere ,Numerical Analysis ,Sequence ,Applied Mathematics ,General Mathematics ,Existential quantification ,010102 general mathematics ,Probabilistic logic ,Sampling (statistics) ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Point (geometry) ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t -designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s -energy of sequences of well separated t -designs on the unit sphere S d ⊂ R d + 1 , d ≥ 2 . The case d = 2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d ≥ 2 , there exists a constant c d , such that for every N > c d t d there exists a well-separated spherical t -design on S d with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t -designs such that t and N are related by N ≍ t d .
- Published
- 2019
17. On the existence of optimal meshes in every convex domain on the plane
- Author
-
András Kroó
- Subjects
Numerical Analysis ,Polynomial ,Conjecture ,Degree (graph theory) ,Plane (geometry) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Polytope ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Cardinality ,Polygon mesh ,0101 mathematics ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper we study the so called optimal polynomial meshes for domains in K ⊂ R d , d ≥ 2 . These meshes are discrete point sets Y n of cardinality c n d which have the property that ‖ p ‖ K ≤ A ‖ p ‖ Y n for every polynomial p of degree at most n with a constant A > 1 independent of n . It was conjectured earlier that optimal polynomial meshes exist in every convex domain. This statement was previously shown to hold for polytopes and C 2 like domains. In this paper we give a complete affirmative answer to the above conjecture when d = 2 .
- Published
- 2019
18. Permutations of zero-sumsets in a finite vector space
- Author
-
Giovanni Falcone, Marco Pavone, Giovanni Falcone, and Marco Pavone
- Subjects
permutations of zero-sums ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_GENERAL ,Zero (complex analysis) ,Subset sum ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,subset sum problem ,Settore MAT/05 - Analisi Matematica ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Subset sum problem ,Settore MAT/03 - Geometria ,0101 mathematics ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Vector space ,Mathematics - Abstract
In this paper, we consider a finite-dimensional vector space 𝒫 {{\mathcal{P}}} over the Galois field GF ( p ) {\operatorname{GF}(p)} , with p being an odd prime, and the family ℬ k x {{\mathcal{B}}_{k}^{x}} of all k-sets of elements of 𝒫 {\mathcal{P}} summing up to a given element x. The main result of the paper is the characterization, for x = 0 {x=0} , of the permutations of 𝒫 {\mathcal{P}} inducing permutations of ℬ k 0 {{\mathcal{B}}_{k}^{0}} as the invertible linear mappings of the vector space 𝒫 {\mathcal{P}} if p does not divide k, and as the invertible affinities of the affine space 𝒫 {\mathcal{P}} if p divides k. The same question is answered also in the case where the elements of the k-sets are required to be all nonzero, and, in fact, the two cases prove to be intrinsically inseparable.
- Published
- 2021
19. On exponential bases and frames with non-linear phase functions and some applications
- Author
-
Chun-Kit Lai, Jean-Pierre Gabardo, and Vignon Oussa
- Subjects
Mathematics::Functional Analysis ,42C15 ,Degree (graph theory) ,Lebesgue measure ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,020206 networking & telecommunications ,02 engineering and technology ,Type (model theory) ,Lambda ,01 natural sciences ,Orthogonal basis ,Functional Analysis (math.FA) ,Combinatorics ,Mathematics - Functional Analysis ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Orthonormal basis ,Locally compact space ,0101 mathematics ,Borel measure ,Analysis ,Mathematics - Abstract
In this paper, we study the spectrality and frame-spectrality of exponential systems of the type $$E(\Lambda ,\varphi ) = \{e^{2\pi i \lambda \cdot \varphi (x)}: \lambda \in \Lambda \}$$ where the phase function $$\varphi $$ is a Borel measurable which is not necessarily linear. A complete characterization of pairs $$(\Lambda ,\varphi )$$ for which $$E(\Lambda ,\varphi )$$ is an orthogonal basis or a frame for $$L^{2}(\mu )$$ is obtained. In particular, we show that the middle-third Cantor measures and the unit disc, each admits an orthogonal basis with a certain non-linear phase. Under a natural regularity condition on the phase functions, when $$\mu $$ is the Lebesgue measure on [0, 1] and $$\Lambda = {{\mathbb {Z}}},$$ we show that only the standard phase functions $$\varphi (x) = \pm x$$ are the only possible functions that give rise to orthonormal bases. Surprisingly, however we prove that there exist a greater degree of flexibility, even for continuously differentiable phase functions in higher dimensions. For instance, we were able to describe a large class of functions $$\varphi $$ defined on $${{\mathbb {R}}}^{d}$$ such that the system $$E(\Lambda ,\varphi )$$ is an orthonormal basis for $$L^{2}[0,1]^{d}$$ when $$d\ge 2.$$ Moreover, we discuss how our results apply to the discretization problem of unitary representations of locally compact groups for the construction of orthonormal bases. Finally, we conclude the paper by stating several open problems.
- Published
- 2020
20. Domains Without Dense Steklov Nodal Sets
- Author
-
Jeffrey Galkowski and Oscar P. Bruno
- Subjects
Applied Mathematics ,General Mathematics ,Open problem ,010102 general mathematics ,Sigma ,Mathematics::Spectral Theory ,Eigenfunction ,01 natural sciences ,Omega ,Combinatorics ,0103 physical sciences ,010307 mathematical physics ,Ball (mathematics) ,0101 mathematics ,Analysis ,Eigenvalues and eigenvectors ,Mathematics - Abstract
This article concerns the asymptotic geometric character of the nodal set of the eigenfunctions of the Steklov eigenvalue problem $$\begin{aligned} -\Delta \phi _{\sigma _j}=0,\quad \hbox { on }\,\,\Omega ,\quad \partial _\nu \phi _{\sigma _j}=\sigma _j \phi _{\sigma _j}\quad \hbox { on }\,\,\partial \Omega \end{aligned}$$-Δϕσj=0,onΩ,∂νϕσj=σjϕσjon∂Ωin two-dimensional domains $$\Omega $$Ω. In particular, this paper presents a dense family $$\mathcal {A}$$A of simply-connected two-dimensional domains with analytic boundaries such that, for each $$\Omega \in \mathcal {A}$$Ω∈A, the nodal set of the eigenfunction $$\phi _{\sigma _j}$$ϕσj “is not dense at scale $$\sigma _j^{-1}$$σj-1”. This result addresses a question put forth under “Open Problem 10” in Girouard and Polterovich (J Spectr Theory 7(2):321–359, 2017). In fact, the results in the present paper establish that, for domains $$\Omega \in \mathcal {A}$$Ω∈A, the nodal sets of the eigenfunctions $$\phi _{\sigma _j}$$ϕσj associated with the eigenvalue $$\sigma _j$$σj have starkly different character than anticipated: they are not dense at any shrinking scale. More precisely, for each $$\Omega \in \mathcal {A}$$Ω∈A there is a value $$r_1>0$$r1>0 such that for each j there is $$x_j\in \Omega $$xj∈Ω such that $$\phi _{\sigma _j}$$ϕσj does not vanish on the ball of radius $$r_1$$r1 around $$x_j$$xj.
- Published
- 2020
21. Bochner–Riesz Means of Morrey Functions
- Author
-
Jie Xiao and David R. Adams
- Subjects
Pointwise ,Mathematics::Functional Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,020206 networking & telecommunications ,02 engineering and technology ,Lambda ,01 natural sciences ,Combinatorics ,0202 electrical engineering, electronic engineering, information engineering ,0101 mathematics ,Analysis ,Mathematics - Abstract
This paper concerns both norm estimation and pointwise approximation for the Bochner–Riesz means of an arbitrary Morrey function on $${{{{\mathbb {R}}}}^n}$$—Theorems 1.1 and 1.2 for $$L^{p,\lambda }({{{{\mathbb {R}}}}^n})$$—thereby generalizing the corresponding results for $$L^p({{{{\mathbb {R}}}}^n})$$ in Stein (Acta Math 100:93–147, 1958) and Carbery et al. (J Lond Math Soc 38:513–524, 1988). As a side note, this paper also establishes Lemma 4.1 of Tomas–Stein type—if $$f\in L^{p,\lambda }({{{{\mathbb {R}}}}^n})$$ under $$ 2^{-1}(n+1)
- Published
- 2020
22. A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth
- Author
-
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
- Full Text
- View/download PDF
23. On Riesz Means of the Coefficients of Epstein’s Zeta Functions
- Author
-
O. M. Fomenko
- Subjects
Statistics and Probability ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Generating function ,Type (model theory) ,01 natural sciences ,Omega ,010305 fluids & plasmas ,Riemann zeta function ,Combinatorics ,symbols.namesake ,Riesz mean ,0103 physical sciences ,symbols ,0101 mathematics ,Mathematics - Abstract
Let rk(n) denote the number of lattice points on a k-dimensional sphere of radius $$ \sqrt{n} $$ . The generating function $$ {\zeta}_k(s)=\sum \limits_{n=1}^{\infty }{r}_k(n){n}^{-s},\kern0.5em k\ge 2, $$ is Epstein’s zeta function. The paper considers the Riesz mean of the type $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{1}{\Gamma \left(\rho +1\right)}\sum \limits_{n\le x}{\left(x-n\right)}^{\rho }{r}_3(n), $$ where ρ > 0; the error term Δρ(x; ζ3) is defined by $$ {D}_{\rho}\left(x;{\zeta}_3\right)=\frac{\uppi^{3/2}{x}^{\rho +3/2}}{\Gamma \left(\rho +5/2\right)}+\frac{x^{\rho }}{\Gamma \left(\rho +1\right)}{\zeta}_3(0)+{\Delta}_{\rho}\left(x;{\zeta}_3\right). $$ K. Chandrasekharan and R. Narasimhan (1962, MR25#3911) proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\Big({x}^{1/2+\rho /2\Big)}& \left(\rho >1\right),\\ {}{\Omega}_{\pm}\left({x}^{1/2+\rho /2}\right)& \left(\rho \ge 0\right).\end{array}} $$ In the present paper, it is proved that $$ {\Delta}_{\rho}\left(x;{\zeta}_3\right)=\Big\{{\displaystyle \begin{array}{ll}O\left(x\log x\right)& \left(\rho =1\right),\\ {}O\left({x}^{2/3+\rho /3+\varepsilon}\right)& \left(1/2
- Published
- 2018
24. Quadratic Interaction Estimate for Hyperbolic Conservation Laws: an Overview
- Author
-
Stefano Modena
- Subjects
Statistics and Probability ,Conservation law ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Prime (order theory) ,Interaction time ,Combinatorics ,Quadratic equation ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation laws $$ \left\{\begin{array}{l}{u}_t+f{(u)}_x=0,\\ {}u\left(t=0\right)={u}_0(x),\end{array}\right. $$ where u : [0, ∞) × ℝ → ℝn, f : ℝn → ℝn is strictly hyperbolic, and Tot.Var.(u0) ≪ 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form $$ \sum \limits_{t_j\;\mathrm{interaction}\ \mathrm{time}}\frac{\left|\sigma \left({\alpha}_j\right)-\sigma \left({\alpha}_j^{\prime}\right)\right|\left|{\alpha}_j\right|\left|{\alpha}_j^{\prime}\right|}{\left|{\alpha}_j\right|+\left|{\alpha}_j^{\prime}\right|}\le C(f)\mathrm{Tot}.\mathrm{Var}.{\left({u}_0\right)}^2, $$ where αj and $$ {\alpha}_j^{\prime } $$ are the wavefronts interacting at the interaction time tj, σ(·) is the speed, |·| denotes the strength, and C(f) is a constant depending only on f (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form). The aim of this paper is to provide the reader with a proof for such a quadratic estimate in a simplified setting, in which: • all the main ideas of the construction are presented; • all the technicalities of the proof in the general setting [8] are avoided.
- Published
- 2018
25. Regularity of Maximum Distance Minimizers
- Author
-
Yana Teplitskaya
- Subjects
Statistics and Probability ,Class (set theory) ,Applied Mathematics ,General Mathematics ,Pipeline (computing) ,010102 general mathematics ,01 natural sciences ,Steiner tree problem ,010101 applied mathematics ,Set (abstract data type) ,Combinatorics ,symbols.namesake ,Compact space ,Tangent lines to circles ,symbols ,Hausdorff measure ,Point (geometry) ,0101 mathematics ,Mathematics - Abstract
We study properties of sets having the minimum length (one-dimensional Hausdorff measure) in the class of closed connected sets Σ ⊂ ℝ2 satisfying the inequality max yϵM dist (y, Σ) ≤ r for a given compact set M ⊂ ℝ2 and given r > 0. Such sets play the role of the shortest possible pipelines arriving at a distance at most r to every point of M where M is the set of customers of the pipeline. In this paper, it is announced that every maximum distance minimizer is a union of finitely many curves having one-sided tangent lines at every point. This shows that a maximum distance minimizer is isotopic to a finite Steiner tree even for a “bad” compact set M, which distinguishes it from a solution of the Steiner problem (an example of a Steiner tree with infinitely many branching points can be found in a paper by Paolini, Stepanov, and Teplitskaya). Moreover, the angle between these lines at each point of a maximum distance minimizer is at least 2π/3. Also, we classify the behavior of a minimizer Σ in a neighborhood of any point of Σ. In fact, all the results are proved for a more general class of local minimizers.
- Published
- 2018
26. On the invariance equation for two-variable weighted nonsymmetric Bajraktarević means
- Author
-
Zsolt Páles and Amr Zakaria
- Subjects
Weight function ,39B12, 39.35, 26E60 ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Hyperbolic function ,Zero (complex analysis) ,010103 numerical & computational mathematics ,01 natural sciences ,Combinatorics ,Monotone polygon ,Természettudományok ,Mathematics - Classical Analysis and ODEs ,Functional equation ,Discrete Mathematics and Combinatorics ,Matematika- és számítástudományok ,0101 mathematics ,Mathematics ,Variable (mathematics) - Abstract
The purpose of this paper is to investigate the invariance of the arithmetic mean with respect to two weighted Bajraktarevic means, i.e., to solve the functional equation $$\begin{aligned} \left( \frac{f}{g}\right) ^{\!\!-1}\!\!\left( \frac{tf(x)+sf(y)}{tg(x)+sg(y)}\right) +\left( \frac{h}{k}\right) ^{\!\!-1}\!\!\left( \frac{sh(x)+th(y)}{sk(x)+tk(y)}\right) =x+y \qquad (x,y\in I), \end{aligned}$$ where $$f,g,h,k:I\rightarrow \mathbb {R}$$ are unknown continuous functions such that g, k are nowhere zero on I, the ratio functions f / g, h / k are strictly monotone on I, and $$t,s\in \mathbb {R}_+$$ are constants different from each other. By the main result of this paper, the solutions of the above invariance equation can be expressed either in terms of hyperbolic functions or in terms of trigonometric functions and an additional weight function. For the necessity part of this result, we will assume that $$f,g,h,k:I\rightarrow \mathbb {R}$$ are four times continuously differentiable.
- Published
- 2018
27. An approximation principle for congruence subgroups
- Author
-
Tobias Finis and Erez Lapid
- Subjects
20E18, 20G25 ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Group Theory (math.GR) ,01 natural sciences ,Prime (order theory) ,Combinatorics ,Conjugacy class ,Group scheme ,0103 physical sciences ,Lie algebra ,Simply connected space ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Mathematics - Group Theory ,Group theory ,Mathematics ,Congruence subgroup - Abstract
The motivating question of this paper is roughly the following: given a group scheme $G$ over $\mathbb{Z}_p$, $p$ prime, with semisimple generic fiber $G_{\mathbb{Q}_p}$, how far are open subgroups of $G(\mathbb{Z}_p)$ from subgroups of the form $X(\mathbb{Z}_p)\mathbf{K}_p(p^n)$, where $X$ is a subgroup scheme of $G$ and $\mathbf{K}_p(p^n)$ is the principal congruence subgroup $\operatorname{Ker} (G(\mathbb{Z}_p)\rightarrow G(\mathbb{Z}/p^n\mathbb{Z}))$? More precisely, we will show that for $G_{\mathbb{Q}_p}$ simply connected there exist constants $J\ge1$ and $\varepsilon>0$, depending only on $G$, such that any open subgroup of $G (\mathbb{Z}_p)$ of level $p^n$ admits an open subgroup of index $\le J$ which is contained in $X(\mathbb{Z}_p)\mathbf{K}_p(p^{\lceil \varepsilon n\rceil})$ for some proper connected algebraic subgroup $X$ of $G$ defined over $\mathbb{Q}_p$. Moreover, if $G$ is defined over $\mathbb{Z}$, then $\varepsilon$ and $J$ can be taken independently of $p$. We also give a correspondence between natural classes of $\mathbb{Z}_p$-Lie subalgebras of $\mathfrak{g}_{\mathbb{Z}_p}$ and of closed subgroups of $G(\mathbb{Z}_p)$ that can be regarded as a variant over $\mathbb{Z}_p$ of Nori's results on the structure of finite subgroups of $\operatorname{GL}(N_0,\mathbb{F}_p)$ for large $p$. As an application we give a bound for the volume of the intersection of a conjugacy class in the group $G (\hat{\mathbb{Z}}) = \prod_p G (\mathbb{Z}_p)$, for $G$ defined over $\mathbb{Z}$, with an arbitrary open subgroup. In a future paper, this result will be applied to the limit multiplicity problem for arbitrary congruence subgroups of the arithmetic lattice $G (\mathbb{Z})$., fixed a few inaccuracies and made some stylistic changes to appear in JEMS
- Published
- 2018
28. Hodge theory in combinatorics
- Author
-
Matthew Baker
- Subjects
Polynomial ,Sequence ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Algebraic geometry ,Chromatic polynomial ,01 natural sciences ,Matroid ,Unimodality ,Combinatorics ,Mathematics - Algebraic Geometry ,010104 statistics & probability ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Algebraic Geometry (math.AG) ,Characteristic polynomial ,Mathematics - Abstract
George Birkhoff proved in 1912 that the number of proper colorings of a finite graph G with n colors is a polynomial in n, called the chromatic polynomial of G. Read conjectured in 1968 that for any graph G, the sequence of absolute values of coefficients of the chromatic polynomial is unimodal: it goes up, hits a peak, and then goes down. Read's conjecture was proved by June Huh in a 2012 paper making heavy use of methods from algebraic geometry. Huh's result was subsequently refined and generalized by Huh and Katz, again using substantial doses of algebraic geometry. Both papers in fact establish log-concavity of the coefficients, which is stronger than unimodality. The breakthroughs of Huh and Huh-Katz left open the more general Rota-Welsh conjecture where graphs are generalized to (not necessarily representable) matroids and the chromatic polynomial of a graph is replaced by the characteristic polynomial of a matroid. The Huh and Huh-Katz techniques are not applicable in this level of generality, since there is no underlying algebraic geometry to which to relate the problem. But in 2015 Adiprasito, Huh, and Katz announced a proof of the Rota-Welsh conjecture based on a novel approach motivated by but not making use of any results from algebraic geometry. The authors first prove that the Rota-Welsh conjecture would follow from combinatorial analogues of the Hard Lefschetz Theorem and Hodge-Riemann relations in algebraic geometry. They then implement an elaborate inductive procedure to prove the combinatorial Hard Lefschetz Theorem and Hodge-Riemann relations using purely combinatorial arguments. We will survey these developments., Comment: 22 pages. This is an expository paper to accompany my lecture at the 2017 AMS Current Events Bulletin. v2: Numerous minor corrections
- Published
- 2017
29. A generalization of a graph theory Mertens’ theorem: Galois covering case
- Author
-
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
30. Minimum spanning acycle and lifetime of persistent homology in the Linial-Meshulam process
- Author
-
Yasuaki Hiraoka and Tomoyuki Shirai
- Subjects
Frieze ,Spanning tree ,Persistent homology ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Minimum weight ,0102 computer and information sciences ,Minimum spanning tree ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Graph ,Combinatorics ,010201 computation theory & mathematics ,FOS: Mathematics ,Mathematics - Combinatorics ,Algebraic Topology (math.AT) ,Mathematics - Algebraic Topology ,Combinatorics (math.CO) ,0101 mathematics ,Mathematics - Probability ,Software ,Mathematics - Abstract
This paper studies a higher dimensional generalization of Frieze's $\zeta(3)$-limit theorem in the Erd\"os-R\'enyi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to $\zeta(3)$ as the number of vertices goes to infinity. In this paper, we study the $d$-Linial-Meshulam process as a model for random simplicial complexes, where $d=1$ corresponds to the Erd\"os-R\'enyi graph process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in $O(n^{d-1})$., Comment: 24 pages
- Published
- 2017
31. Bounded Remainder Sets
- Author
-
V. G. Zhuravlev
- Subjects
Statistics and Probability ,Sequence ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Torus ,Space (mathematics) ,01 natural sciences ,Bounded operator ,010101 applied mathematics ,Combinatorics ,Distribution function ,Bounded function ,0101 mathematics ,Remainder ,Klein bottle ,Mathematics - Abstract
The paper considers the category ( $$ \mathcal{T} $$ , S, X) consisting of mappings S : $$ \mathcal{T} $$ −→ $$ \mathcal{T} $$ of spaces $$ \mathcal{T} $$ with distinguished subsets X ⊂ $$ \mathcal{T} $$ . Let rX (i, x0) be the distribution function of points of an S-orbit x0, x1 = S(x0), . . . , xi−1 = Si−1(x0) getting into X, and let δX (i, x0) be the deviation defined by the equation rX (i, x0) = aX i + δX (i, x0), where aX i is the average value. If δX (i, x0) = O(1), then such sets X are called bounded remainder sets. In the paper, bounded remainder sets X are constructed in the following cases: (1) the space $$ \mathcal{T} $$ is the circle, torus, or the Klein bottle; (2) the map S is a rotation of the circle, a shift or an exchange mapping of the torus; (3) X is a fixed subset X ⊂ $$ \mathcal{T} $$ or a sequence of subsets depending on the iteration number i = 0, 1, 2, . . .. Bibliography: 27 titles.
- Published
- 2017
32. On the Lattice of Subvarieties of the Wreath Product of the Variety of Semilattices and the Variety of Semigroups with Zero Multiplication
- Author
-
A. V. Tishchenko
- Subjects
Statistics and Probability ,Semigroup ,High Energy Physics::Lattice ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Integer lattice ,01 natural sciences ,Upper and lower bounds ,Exponential function ,010101 applied mathematics ,Combinatorics ,Wreath product ,Lattice (order) ,0101 mathematics ,Mathematics - Abstract
It is known that the monoid wreath product of any two semigroup varieties that are atoms in the lattice of all semigroup varieties may have a finite as well as an infinite lattice of subvarieties. If this lattice is finite, then as a rule it has at most eleven elements. This was proved in a paper of the author in 2007. The exclusion is the monoid wreath product Sl w N 2 of the variety of semilattices and the variety of semigroups with zero multiplication. The number of elements of the lattice L(Sl w N 2) of subvarieties of Sl w N 2 is still unknown. In our paper, we show that the lattice L(Sl w N 2) contains no less than 33 elements. In addition, we give some exponential upper bound of the cardinality of this lattice.
- Published
- 2017
33. On embedding certain partial orders into the P-points under Rudin-Keisler and Tukey reducibility
- Author
-
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
34. Universal covering Calabi–Yau manifolds of the Hilbert schemes of $n$ points of Enriques surfaces
- Author
-
Taro Hayashi
- Subjects
Degree (graph theory) ,Covering space ,Applied Mathematics ,General Mathematics ,Enriques surface ,010102 general mathematics ,Automorphism ,01 natural sciences ,Manifold ,Combinatorics ,Hilbert scheme ,0103 physical sciences ,Calabi–Yau manifold ,010307 mathematical physics ,Isomorphism ,0101 mathematics ,Mathematics - Abstract
Throughout this paper, we work over ${\mathbb C}$, and $n$ is an integer such that $n\geq 2$. For an Enriques surface $E$, let $E^{[n]}$ be the Hilbert scheme of $n$ points of $E$. By Oguiso and Schr\"oer, $E^{[n]}$ has a Calabi-Yau manifold $X$ as the universal covering space, $\pi :X\rightarrow E^{[n]}$ of degree $2$. The purpose of this paper is to investigate a relationship of the small deformation of $E^{[n]}$ and that of $X$ $({\rm Theorem}\ 1.1)$, the natural automorphism of $E^{[n]}$ $({\rm Theorem}\,1.2)$, and count the number of isomorphism classes of the Hilbert schemes of $n$ points of Enriques surfaces which has $X$ as the universal covering space when we fix one $X$ $({\rm Theorem}\,1.3)$.
- Published
- 2017
35. Polynomial control on weighted stability bounds and inversion norms of localized matrices on simple graphs
- Author
-
Qiquan Fang, Chang Eon Shin, and Qiyu Sun
- Subjects
Polynomial (hyperelastic model) ,Mathematics::Functional Analysis ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematics - Operator Algebras ,Inverse ,010103 numerical & computational mathematics ,Muckenhoupt weights ,01 natural sciences ,Functional Analysis (math.FA) ,Combinatorics ,Mathematics - Functional Analysis ,Matrix (mathematics) ,Bounded function ,Banach algebra ,Beurling algebra ,FOS: Mathematics ,0101 mathematics ,Operator Algebras (math.OA) ,Operator norm ,Analysis ,Mathematics - Abstract
The (un)weighted stability for some matrices on a graph is one of essential hypotheses in time-frequency analysis and applied harmonic analysis. In the first part of this paper, we show that for a localized matrix in a Beurling algebra, its weighted stabilities for different exponents and Muckenhoupt weights are equivalent to each other, and reciprocal of its optimal lower stability bound for one exponent and weight is controlled by a polynomial of reciprocal of its optimal lower stability bound for another exponent and weight. Banach algebras of matrices with certain off-diagonal decay is of great importance in many mathematical and engineering fields, and its inverse-closed property can be informally interpreted as localization preservation. Let $${{\mathcal {B}}}(\ell ^p_w)$$ be the Banach algebra of bounded linear operators on the weighted sequence space $$\ell ^p_w$$ on a graph. In the second part of this paper, we prove that Beurling algebras of localized matrices on a connected simple graph are inverse-closed in $${{\mathcal {B}}}(\ell ^p_w)$$ for all $$1\le p
- Published
- 2019
36. Some Results of the Theory of Exponential R-Groups
- Author
-
M. G. Amaglobeli and T. Bokelavadze
- Subjects
Statistics and Probability ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,State (functional analysis) ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Mathematics::Group Theory ,Nilpotent ,Euclidean domain ,0101 mathematics ,Variety (universal algebra) ,Algebraic number ,Nilpotent group ,Abelian group ,Mathematics - Abstract
This paper is devoted to the study of groups from the category M of R-power groups. We examine problems on the commutation of the tensor completion with basic group operations and on the exactness of the tensor completion. Moreover, we introduce the notion of a variety and obtain a description of abelian varieties and some results on nilpotent varieties of A-groups. We prove the hypothesis on irreducible coordinate groups of algebraic sets for the nilpotent R-groups of nilpotency class 2, where R is a Euclidean ring. We state that the analog to the Lyndon result for the free groups (see [10]) holds in this case, whereas the analog to the Myasnikov–Kharlampovich result fails.The paper is dedicated to partial R-power groups which are embeddable to their A-tensor completions. The free R-groups and free R-products are described with usual group-theoretical free constructions.
- Published
- 2016
37. The asymptotic distribution of symbols on diagonals of random weighted staircase tableaux
- Author
-
Amanda Lohss
- Subjects
Conjecture ,Distribution (number theory) ,Applied Mathematics ,General Mathematics ,Probability (math.PR) ,010102 general mathematics ,Diagonal ,Asymptotic distribution ,0102 computer and information sciences ,Asymmetric simple exclusion process ,Poisson distribution ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,Connection (mathematics) ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,FOS: Mathematics ,symbols ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Mathematics - Probability ,Software ,Mathematics - Abstract
Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, Hitczenko and Janson proved that distribution of parameters on the first diagonal is asymptotically normal. In addition, they conjectured that other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds upon those results to prove the conjecture for the kth diagonal where k is fixed. In particular, we prove that the distribution of the number of α's (β's) on the kth diagonal, k > 1, is asymptotically Poisson with parameter 1/2. In addition, we prove that symbols on the kth diagonal are asymptotically independent and thus, collectively follow the Poisson distribution with parameter 1. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 795–818, 2016
- Published
- 2016
38. ON SEMIDERIVATIONS IN 3-PRIME NEAR-RINGS
- Author
-
Mohammad Ashraf and Abdelkarim Boua
- Subjects
Pure mathematics ,Near-ring ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Multiplicative function ,Zero (complex analysis) ,Center (category theory) ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Combinatorics ,Product (mathematics) ,Domain (ring theory) ,0101 mathematics ,Commutative property ,Mathematics - Abstract
In the present paper, we expand the domain of work on theconcept of semiderivations in 3-prime near-rings through the study ofstructure and commutativity of near-rings admitting semiderivations sat-isfying certain differential identities. Moreover, several examples havebeen provided at places which show that the assumptions in the hypothe-ses of various theorems are not altogether superfluous. 1. IntroductionThroughout this paper, N is a zero-symmetric left near ring. A near ringN is called zero symmetric if 0x= 0 for all x∈ N (recall that in a left nearring x0 = 0 for all x∈ N). N is called 3-prime if xNy = {0} implies x= 0or y = 0. The symbol Z(N) will represent the multiplicative center of N,that is, Z(N) = {x∈ N | xy= yxfor all y∈ N}.For any x,y∈ N; as usual[x,y] = xy−yxand x◦y= xy+yxwill denote the well-known Lie product andJordan product, respectively. Recall that N is called 2-torsion free if 2x= 0implies x= 0 for all x∈ N. For terminologies concerning near-rings we referto G. Pilz [7].An additive mapping d: N → N is said to be a derivation if d(xy) = xd(y)+d(x)yforall x,y∈ N, orequivalently, asnotedin [8], that d(xy) = d(x)y+xd(y)for all x,y ∈ N. An additive mapping d: N → N is called semiderivation ifthere exists a function g : N → N such that d(xy) = xd(y) + d(x)g(y) =g(x)d(y)+d(x)yand d(g(x)) = g(d(x)) for all x,y∈ N.Obviously, any deriva-tion is a semiderivation, but the converse is not true in general (see [6]). Therehas been a greatdeal of workconcerning derivations in near-rings(see [1, 2, 4, 5]where further references can be found). In this paper, we study the commuta-tivity of addition and multiplication of near-rings. Two well-known results forderivations in near-rings have been generalized for semiderivation. In fact, ourresults generalize some theorems obtained by the authors together with Rajiin [1].
- Published
- 2016
39. Derangements in subspace actions of finite classical groups
- Author
-
Robert M. Guralnick and Jason Fulman
- Subjects
Classical group ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Codimension ,01 natural sciences ,Combinatorics ,Group of Lie type ,010201 computation theory & mathematics ,Simple group ,Bounded function ,Classification of finite simple groups ,CA-group ,0101 mathematics ,Subspace topology ,Mathematics - Abstract
This is the third in a series of papers in which we prove a conjecture of Boston and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.
- Published
- 2016
40. Classification of tile digit sets as product-forms
- Author
-
Chun-Kit Lai, Ka-Sing Lau, and Hui Rao
- Subjects
Polynomial (hyperelastic model) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Order (ring theory) ,01 natural sciences ,Prime (order theory) ,010101 applied mathematics ,Combinatorics ,Matrix (mathematics) ,Tree (descriptive set theory) ,Integer ,Product (mathematics) ,0101 mathematics ,Cyclotomic polynomial ,Mathematics - Abstract
Let $A$ be an expanding matrix on ${\Bbb R}^s$ with integral entries. A fundamental question in the fractal tiling theory is to understand the structure of the digit set ${\mathcal D}\subset{\Bbb Z}^s$ so that the integral self-affine set $T(A,\mathcal D)$ is a translational tile on ${\Bbb R}^s$. In our previous paper, we classified such tile digit sets ${\mathcal D}\subset{\Bbb Z}$ by expressing the mask polynomial $P_{\mathcal D}$ into product of cyclotomic polynomials. In this paper, we first show that a tile digit set in ${\Bbb Z}^s$ must be an integer tile (i.e. ${\mathcal D}\oplus{\mathcal L} = {\Bbb Z}^s$ for some discrete set ${\mathcal L}$). This allows us to combine the technique of Coven and Meyerowitz on integer tiling on ${\Bbb R}^1$ together with our previous results to characterize explicitly all tile digit sets ${\mathcal D}\subset {\Bbb Z}$ with $A = p^{\alpha}q$ ($p, q$ distinct primes) as {\it modulo product-form} of some order, an advance of the previously known results for $A = p^\alpha$ and $pq$.
- Published
- 2016
41. On the strong divergence of Hilbert transform approximations and a problem of Ul’yanov
- Author
-
Holger Boche and Volker Pohl
- Subjects
Numerical Analysis ,Sequence ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,020206 networking & telecommunications ,02 engineering and technology ,01 natural sciences ,Combinatorics ,symbols.namesake ,Uniform norm ,Subsequence ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Hilbert transform ,0101 mathematics ,Divergence (statistics) ,Finite set ,Fourier series ,Analysis ,Mathematics - Abstract
This paper studies the approximation of the Hilbert transform f ? = H f of continuous functions f with continuous conjugate f ? based on a finite number of samples. It is known that every sequence { H N f } N ? N which approximates f ? from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejer means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ? N there are functions f such that ? H N f ? ∞ exceeds any given bound for any given number of consecutive indices N .As an application, the later result is used to investigate a problem associated with a question of Ul'yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.
- Published
- 2016
42. Graph-Links: Nonrealizability, Orientation, and Jones Polynomial
- Author
-
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
43. On Algorithmic Methods of Analysis of Two-Colorings of Hypergraphs
- Author
-
A. V. Lebedeva
- Subjects
Statistics and Probability ,Combinatorics ,Hypergraph ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Upper and lower bounds ,Mathematics ,Vertex (geometry) - Abstract
This paper deals with an extremal problem concerning hypergraph colorings. Let k be an integer. The problem is to find the value m k (n) equal to the minimum number of edges in an n-uniform hypergraph not admitting two-colorings of the vertex set such that every edge of the hypergraph contains at least k vertices of each color. In this paper, we obtain upper bounds of m k (n) for small k and n, the exact value of m 4(8), and a lower bound for m 3(7).
- Published
- 2016
44. Random triangles in random graphs
- Author
-
Annika Heckel
- Subjects
Random graph ,Hypergraph ,Mathematics::Combinatorics ,Binomial (polynomial) ,Matching (graph theory) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Coupling (probability) ,01 natural sciences ,Computer Graphics and Computer-Aided Design ,05C80, 05C70, 05C65 ,Combinatorics ,Constant factor ,010104 statistics & probability ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Software ,Mathematics - Abstract
In a recent paper, Oliver Riordan shows that for $r \ge 4$ and $p$ up to and slightly larger than the threshold for a $K_r$-factor, the hypergraph formed by the copies of $K_r$ in $G(n,p)$ contains a copy of the binomial random hypergraph $H=H_r(n,\pi)$ with $\pi \sim p^{r \choose 2}$. For $r=3$, he gives a slightly weaker result where the density in the random hypergraph is reduced by a constant factor. Recently, Jeff Kahn announced an asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem for all $r \ge 3$. With Riordan's result, this immediately implies an asymptotically sharp bound for the threshold of a $K_r$-factor in $G(n,p)$ for $r \ge 4$. In this note, we resolve the missing case $r=3$ by modifying Riordan's argument. This means that Kahn's result also implies a sharp bound for triangle factors in $G(n,p)$., Comment: 4 pages; minor corrections, and updated references to new version of Oliver Riordan's paper
- Published
- 2018
45. Sparse generalised polynomials
- Author
-
Jakub Konieczny and Jakub Byszewski
- Subjects
Polynomial ,Sequence ,Fibonacci number ,Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Value (computer science) ,Dynamical Systems (math.DS) ,Function (mathematics) ,01 natural sciences ,Combinatorics ,Set (abstract data type) ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,IP set ,Number Theory (math.NT) ,Combinatorics (math.CO) ,010307 mathematical physics ,Mathematics - Dynamical Systems ,0101 mathematics ,Primary: 37A45, 05D10, 28D05, Secondary: 37B05, 11J54, 11J70, 11J71 ,Mathematics - Abstract
We investigate generalised polynomials (i.e. polynomial-like expressions involving the use of the floor function) which take the value $0$ on all integers except for a set of density $0$. Our main result is that the set of integers where a sparse generalised polynomial takes non-zero value cannot contain a translate of an IP set. We also study some explicit constructions, and show that the characteristic functions of the Fibonacci and Tribonacci numbers are given by generalised polynomails. Finally, we show that any sufficiently sparse $\{0,1\}$-valued sequence is given by a generalised polynomial. (This paper is essentially the first half of our earlier submission arXiv:1610.03900 [math.NT]. Because the material in arXiv:1610.03900 [math.NT] touches upon many different subjects, we believe it is preferable to split it into two independent papers.), 28 pages. arXiv admin note: substantial text overlap with arXiv:1610.03900
- Published
- 2018
46. Combinatorial cost: a coarse setting
- Author
-
Tom Kaiser
- Subjects
Property (philosophy) ,Group (mathematics) ,Mathematics::Operator Algebras ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Metric Geometry (math.MG) ,Group Theory (math.GR) ,01 natural sciences ,Combinatorics ,Mathematics - Metric Geometry ,If and only if ,FOS: Mathematics ,Graph (abstract data type) ,Property a ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,0101 mathematics ,Mathematics - Group Theory ,Mathematics - Abstract
The main inspiration for this paper is a paper by Elek where he introduces combinatorial cost for graph sequences. We show that having cost equal to 1 and hyperfiniteness are coarse invariants. We also show `cost-1' for box spaces behaves multiplicatively when taking subgroups. We show that graph sequences coming from Farber sequences of a group have property A if and only if the group is amenable. The same is true for hyperfiniteness. This generalises a theorem by Elek. Furthermore we optimise this result when Farber sequences are replaced by sofic approximations. In doing so we introduce a new concept: property almost-A., 20 pages. Comments welcome
- Published
- 2017
47. Meyniel's conjecture holds for random graphs
- Author
-
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
48. Stanley depth and Stanley support-regularity of monomial ideals
- Author
-
Yi-Huang Shen
- Subjects
Monomial ,Mathematics::Commutative Algebra ,Alexander duality ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Monomial ideal ,0102 computer and information sciences ,Square-free integer ,01 natural sciences ,Upper and lower bounds ,Combinatorics ,010201 computation theory & mathematics ,0101 mathematics ,Algebra over a field ,Quotient ,Mathematics - Abstract
Lyubeznik introduced the concept of size of a monomial ideal and showed that the size of a monomial ideal increased by \(1\) is a lower bound for its depth. Associated with this, Herzog, Popescu and Vladoiu showed that the size increased by \(1\) is simultaneously a lower bound for its Stanley depth. It turns out that the splitting-variables technique in their paper is particularly suited for comparing the Stanley depth and size of squarefree monomial ideals. In this paper, we generalize this technique and investigate the quotient \(I/J\) when \(J\subsetneq I\) are general monomial ideals. As an application, we consider lower bounds for Stanley depth of quotients of monomial ideals under various conditions. Through Alexander duality, we also consider upper bounds for the Stanley support-regularity of edge ideals of clutters. Meanwhile, recently, Katthan and Seyed Fakhari established another lower bound for depth and Stanley depth of \(I/J\) in terms of the lcm number. Using the splitting-variables technique, we provide an alternate proof of their result.
- Published
- 2015
49. Combinatorial topology and the global dimension of algebras arising in combinatorics
- Author
-
Franco Saliola, Benjamin Steinberg, and Stuart W. Margolis
- Subjects
Monoid ,General Mathematics ,0211 other engineering and technologies ,Structure (category theory) ,Group Theory (math.GR) ,02 engineering and technology ,Combinatorial topology ,01 natural sciences ,Representation theory ,Global dimension ,Combinatorics ,05E10, 16E10, 16G10, 52C35, 05E45 ,FOS: Mathematics ,Mathematics - Combinatorics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics ,Applied Mathematics ,Flag (linear algebra) ,010102 general mathematics ,Spectrum (functional analysis) ,021107 urban & regional planning ,Mathematics - Rings and Algebras ,Cohomology ,Rings and Algebras (math.RA) ,Combinatorics (math.CO) ,Mathematics - Group Theory ,Mathematics - Representation Theory - Abstract
In a highly influential paper, Bidigare, Hanlon and Rockmore showed that a number of popular Markov chains are random walks on the faces of a hyperplane arrangement. Their analysis of these Markov chains took advantage of the monoid structure on the set of faces. This theory was later extended by Brown to a larger class of monoids called left regular bands. In both cases, the representation theory of these monoids played a prominent role. In particular, it was used to compute the spectrum of the transition operators of the Markov chains and to prove diagonalizability of the transition operators. In this paper, we establish a close connection between algebraic and combinatorial invariants of a left regular band: we show that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. For instance, we show that the global dimension of these algebras is bounded above by the Leray number of the associated order complex. Conversely, we associate to every flag complex a left regular band whose algebra has global dimension precisely the Leray number of the flag complex., Comment: Appendix removed
- Published
- 2015
50. The long-Time behavior of 3-dimensional Ricci flow on certain topologies
- Author
-
Richard H. Bamler
- Subjects
Pure mathematics ,Class (set theory) ,Conjecture ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Ricci flow ,Curvature ,Network topology ,01 natural sciences ,Mathematics::Geometric Topology ,Pure Mathematics ,Manifold ,Combinatorics ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
In this paper we analyze the long-time behavior of 3-dimensional Ricci flow with surgery. We prove that under the topological condition that the initial manifold only has non-aspherical or hyperbolic components in its geometric decomposition, there are only finitely many surgeries and the curvature is bounded by C t - 1 $t^{-1}$ for large t. This proves a conjecture of Perelman for this class of initial topologies. The proof of this fact illustrates the fundamental ideas that are used in the subsequent papers of the author.
- Published
- 2017
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.