Some Parity Statistics in Integer Partitions

Aubrey Blecher; Toufik Mansour; Augustine O. Munagi

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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 $ℝ^{n}$ (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 $ℒ_{k} (Ω)$ 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 $f = \sum_{C \in} a_{C} φ_{C}$ with $a_{C} \in E$ and a partition of unity $φ_{C} : C \in$ 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 $\geq 3$ 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) $l i m_{n \to \infty} n^{- 1} l o g \sum_{j = 0}^{τ ₙ (x)} μ (α ⁿ (T^{j} (x)))$ , where $α ⁿ (T^{j} (x))$ 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 $ℵ_{ω}$

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 ${[ℵ_{k}]}^{ω}$ and use this to refine the well-known high-dimensional polarized partition relation for $ℵ_{ω}$ 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 $Δ (G [V_{1}]) \leq d_{1}$ and $Δ (G [V_{2}]) \leq d_{2}$ , 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 $2^{ω ₁}$ 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 ${SLE}_{κ}$ , $κ > 4$ , and appropriate versions of ${SLE}_{\hat{κ}}$ , $\hat{κ} = 16 / κ$ .

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 $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ and $r_{1, l} (A, n) = r_{1, l} (ℕ ∖ A, n)$ 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 $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ for all n ≥ n₀, we have $r_{1, k} (A, n) \to \infty$ 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 $r_{1, k} (A, n) = r_{1, k} (ℕ ∖ A, n)$ and $r_{1, l} (A, n) = r_{1, l} (ℕ ∖ A, n)$ 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 ${⌊\binom{n}{k}⌋}_{r}$ 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 $n + 2 r - 1$ ( $0 \leq k \leq n$ , $r \geq 1$ ). 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 $a_{i, k}$ , $(i, k) \in ℕ^{*} \times [m]$ , 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 ${X_{i j}}$ , $i, j = \dots$ , 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 $E | X_{11} |^{4} l t; \infty$ . Consider sample covariance matrices (with/without empirical centering) $𝒮 = \frac{1}{n} \sum_{j = 1}^{n} (𝐬_{j} - \bar{𝐬}) {(𝐬_{j} - \bar{𝐬})}^{T}$ and $𝐒 = \frac{1}{n} \sum_{j = 1}^{n} 𝐬_{j} 𝐬_{j}^{T}$ , where $\bar{𝐬} = \frac{1}{n} \sum_{j = 1}^{n} 𝐬_{j}$ 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 $n \to \infty$ 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 $f : ω^{ω} \to X$ such that $f^{- 1} (C ₀) = D ₀$ and $f^{- 1} (C ₁) = D ₁$ . We give several explicit examples of complete pairs of coanalytic sets.