Displaying similar documents to “Some Parity Statistics in Integer Partitions”

Subspaces of L p , p > 2, determined by partitions and weights

Dale E. Alspach, Simei Tong (2003)

Studia Mathematica

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Many of the known complemented subspaces of L p have realizations as sequence spaces. In this paper a systematic approach to defining these spaces which uses partitions and weights is introduced. This approach gives a unified description of many well known complemented subspaces of L p . It is proved that the class of spaces with such norms is stable under (p,2) sums. By introducing the notion of an envelope norm, we obtain a necessary condition for a Banach sequence space with norm given...

Ramsey partitions and proximity data structures

Manor Mendel, Assaf Naor (2007)

Journal of the European Mathematical Society

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This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion.We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (also known as the non-linear version of the isomorphic Dvoretzky...

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

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Given a bounded open set Ω in n (or in a Riemannian manifold) and a partition of Ω by k open sets D j , we consider the quantity 𝚖𝚊𝚡 j λ ( D j ) where λ ( D j ) is the ground state energy of the Dirichlet realization of the Laplacian in D j . If we denote by k ( Ω ) the infimum over all the k -partitions of 𝚖𝚊𝚡 j λ ( D j ) , a minimal k -partition is then a partition which realizes the infimum. When k = 2 , we find the two nodal domains of a second eigenfunction, but the analysis of higher k ’s is non trivial and quite interesting. In this...

Linear combinations of partitions of unity with restricted supports

Christian Richter (2002)

Studia Mathematica

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Given a locally finite open covering of a normal space X and a Hausdorff topological vector space E, we characterize all continuous functions f: X → E which admit a representation f = C a C φ C with a C E and a partition of unity φ C : C subordinate to . As an application, we determine the class of all functions f ∈ C(||) on the underlying space || of a Euclidean complex such that, for each polytope P ∈ , the restriction f | P attains its extrema at vertices of P. Finally, a class of extremal functions on the...

A note on representing dowling geometries by partitions

František Matúš, Aner Ben-Efraim (2020)

Kybernetika

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We prove that a rank 3 Dowling geometry of a group H is partition representable if and only if H is a Frobenius complement. This implies that Dowling group geometries are secret-sharing if and only if they are multilinearly representable.

On the q -Pell sequences and sums of tails

Alexander E. Patkowski (2017)

Czechoslovak Mathematical Journal

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We examine the q -Pell sequences and their applications to weighted partition theorems and values of L -functions. We also put them into perspective with sums of tails. It is shown that there is a deeper structure between two-variable generalizations of Rogers-Ramanujan identities and sums of tails, by offering examples of an operator equation considered in a paper published by the present author. The paper starts with the classical example offered by Ramanujan and studied by previous...

On simple partitions of [ κ ] κ

David Asperó (2003)

Fundamenta Mathematicae

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For every uncountable regular cardinal κ, every κ-Borel partition of the space of all members of [ κ ] κ whose enumerating function does not have fixed points has a homogeneous club.

Extension of point-finite partitions of unity

Haruto Ohta, Kaori Yamazaki (2006)

Fundamenta Mathematicae

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A subspace A of a topological space X is said to be P γ -embedded ( P γ (point-finite)-embedded) in X if every (point-finite) partition of unity α on A with |α| ≤ γ extends to a (point-finite) partition of unity on X. The main results are: (Theorem A) A subspace A of X is P γ (point-finite)-embedded in X iff it is P γ -embedded and every countable intersection B of cozero-sets in X with B ∩ A = ∅ can be separated from A by a cozero-set in X. (Theorem B) The product A × [0,1] is P γ (point-finite)-embedded...

Pressure and recurrence

Véronique Maume-Deschamps, Bernard Schmitt, Mariusz Urbański, Anna Zdunik (2003)

Fundamenta Mathematicae

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We deal with a subshift of finite type and an equilibrium state μ for a Hölder continuous function. Let αⁿ be the partition into cylinders of length n. We compute (in particular we show the existence of the limit) l i m n n - 1 l o g j = 0 τ ( x ) μ ( α ( T j ( x ) ) ) , where α ( T j ( x ) ) is the element of the partition containing T j ( x ) and τₙ(x) is the return time of the trajectory of x to the cylinder αⁿ(x).

Partition ideals below ω

P. Dodos, J. Lopez-Abad, S. Todorcevic (2012)

Fundamenta Mathematicae

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Motivated by an application to the unconditional basic sequence problem appearing in our previous paper, we introduce analogues of the Laver ideal on ℵ₂ living on index sets of the form [ k ] ω and use this to refine the well-known high-dimensional polarized partition relation for ω of Shelah.

Partitioning planar graph of girth 5 into two forests with maximum degree 4

Min Chen, André Raspaud, Weifan Wang, Weiqiang Yu (2024)

Czechoslovak Mathematical Journal

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Given a graph G = ( V , E ) , if we can partition the vertex set V into two nonempty subsets V 1 and V 2 which satisfy Δ ( G [ V 1 ] ) d 1 and Δ ( G [ V 2 ] ) d 2 , then we say G has a ( Δ d 1 , Δ d 2 ) -partition. And we say G admits an ( F d 1 , F d 2 ) -partition if G [ V 1 ] and G [ V 2 ] are both forests whose maximum degree is at most d 1 and d 2 , respectively. We show that every planar graph with girth at least 5 has an ( F 4 , F 4 ) -partition.

On general Franklin systems

Gevorkyan Gegham, Kamont Anna

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AbstractWe study general Franklin systems, i.e. systems of orthonormal piecewise linear functions corresponding to quasi-dyadic sequences of partitions of [0,1]. The following problems are treated: unconditionality of the general Franklin basis in L p , 1 < p < ∞, and H p , 1/2 < p ≤ 1; equivalent conditions for the unconditional convergence of the Franklin series in L p for 0< p ≤ 1; relation between Haar and Franklin series with identical coefficients; characterization of the spaces...

Indestructible colourings and rainbow Ramsey theorems

Lajos Soukup (2009)

Fundamenta Mathematicae

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We show that if a colouring c establishes ω₂ ↛ [(ω₁:ω)]² then c establishes this negative partition relation in each Cohen-generic extension of the ground model, i.e. this property of c is Cohen-indestructible. This result yields a negative answer to a question of Erdős and Hajnal: it is consistent that GCH holds and there is a colouring c:[ω₂]² → 2 establishing ω₂ ↛ [(ω₁:ω)]₂ such that some colouring g:[ω₁]² → 2 does not embed into c. It is also consistent that 2 ω is arbitrarily large,...

Duality of Schramm-Loewner evolutions

Julien Dubédat (2009)

Annales scientifiques de l'École Normale Supérieure

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In this note, we prove a version of the conjectured duality for Schramm-Loewner Evolutions, by establishing exact identities in distribution between some boundary arcs of chordal SLE κ , κ &gt; 4 , and appropriate versions of SLE κ ^ , κ ^ = 16 / κ .

A note on representation functions with different weights

Zhenhua Qu (2016)

Colloquium Mathematicae

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For any positive integer k and any set A of nonnegative integers, let r 1 , k ( A , n ) denote the number of solutions (a₁,a₂) of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. Let k,l ≥ 2 be two distinct integers. We prove that there exists a set A ⊆ ℕ such that both r 1 , k ( A , n ) = r 1 , k ( A , n ) and r 1 , l ( A , n ) = r 1 , l ( A , n ) hold for all n ≥ n₀ if and only if log k/log l = a/b for some odd positive integers a,b, disproving a conjecture of Yang. We also show that for any set A ⊆ ℕ satisfying r 1 , k ( A , n ) = r 1 , k ( A , n ) for all n ≥ n₀, we have r 1 , k ( A , n ) as n → ∞.

Representation functions with different weights

Quan-Hui Yang (2014)

Colloquium Mathematicae

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For any given positive integer k, and any set A of nonnegative integers, let r 1 , k ( A , n ) denote the number of solutions of the equation n = a₁ + ka₂ with a₁,a₂ ∈ A. We prove that if k,l are multiplicatively independent integers, i.e., log k/log l is irrational, then there does not exist any set A ⊆ ℕ such that both r 1 , k ( A , n ) = r 1 , k ( A , n ) and r 1 , l ( A , n ) = r 1 , l ( A , n ) hold for all n ≥ n₀. We also pose a conjecture and two problems for further research.

Matchings in complete bipartite graphs and the r -Lah numbers

Gábor Nyul, Gabriella Rácz (2021)

Czechoslovak Mathematical Journal

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We give a graph theoretic interpretation of r -Lah numbers, namely, we show that the r -Lah number n k r counting the number of r -partitions of an ( n + r ) -element set into k + r ordered blocks is just equal to the number of matchings consisting of n - k edges in the complete bipartite graph with partite sets of cardinality n and n + 2 r - 1 ( 0 k n , r 1 ). We present five independent proofs including a direct, bijective one. Finally, we close our work with a similar result for r -Stirling numbers of the second kind. ...

Cambrian fans

Nathan Reading, David E. Speyer (2009)

Journal of the European Mathematical Society

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For a finite Coxeter group W and a Coxeter element c of W ; the c -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W . Its maximal cones are naturally indexed by the c -sortable elements of W . The main result of this paper is that the known bijection cl c between c -sortable elements and c -clusters induces a combinatorial isomorphism of fans. In particular, the c -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for...

A Deformed Quon Algebra

Hery Randriamaro (2019)

Communications in Mathematics

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The quon algebra is an approach to particle statistics in order to provide a theory in which the Pauli exclusion principle and Bose statistics are violated by a small amount. The quons are particles whose annihilation and creation operators obey the quon algebra which interpolates between fermions and bosons. In this paper we generalize these models by introducing a deformation of the quon algebra generated by a collection of operators a i , k , ( i , k ) * × [ m ] , on an infinite dimensional vector space satisfying...

Comparison between two types of large sample covariance matrices

Guangming Pan (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Let { X i j } , i , j = , be a double array of independent and identically distributed (i.i.d.) real random variables with E X 11 = μ , E | X 11 - μ | 2 = 1 and E | X 11 | 4 l t ; . Consider sample covariance matrices (with/without empirical centering) 𝒮 = 1 n j = 1 n ( 𝐬 j - 𝐬 ¯ ) ( 𝐬 j - 𝐬 ¯ ) T and 𝐒 = 1 n j = 1 n 𝐬 j 𝐬 j T , where 𝐬 ¯ = 1 n j = 1 n 𝐬 j and 𝐬 j = 𝐓 n 1 / 2 ( X 1 j , ... , X p j ) T with ( 𝐓 n 1 / 2 ) 2 = 𝐓 n , non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of 𝒮 and 𝐒 are different as n with p / n approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the...

Complete pairs of coanalytic sets

Jean Saint Raymond (2007)

Fundamenta Mathematicae

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Let X be a Polish space, and let C₀ and C₁ be disjoint coanalytic subsets of X. The pair (C₀,C₁) is said to be complete if for every pair (D₀,D₁) of disjoint coanalytic subsets of ω ω there exists a continuous function f : ω ω X such that f - 1 ( C ) = D and f - 1 ( C ) = D . We give several explicit examples of complete pairs of coanalytic sets.

( m , r ) -central Riordan arrays and their applications

Sheng-Liang Yang, Yan-Xue Xu, Tian-Xiao He (2017)

Czechoslovak Mathematical Journal

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For integers m > r 0 , Brietzke (2008) defined the ( m , r ) -central coefficients of an infinite lower triangular matrix G = ( d , h ) = ( d n , k ) n , k as d m n + r , ( m - 1 ) n + r , with n = 0 , 1 , 2 , , and the ( m , r ) -central coefficient triangle of G as G ( m , r ) = ( d m n + r , ( m - 1 ) n + k + r ) n , k . It is known that the ( m , r ) -central coefficient triangles of any Riordan array are also Riordan arrays. In this paper, for a Riordan array G = ( d , h ) with h ( 0 ) = 0 and d ( 0 ) , h ' ( 0 ) 0 , we obtain the generating function of its ( m , r ) -central coefficients and give an explicit representation for the ( m , r ) -central Riordan array G ( m , r ) in terms of the Riordan array G ....