Ramsey partitions and proximity data structures

Manor Mendel; Assaf Naor

Displaying similar documents to “Ramsey partitions and proximity data structures”

Extension of point-finite partitions of unity

Haruto Ohta, Kaori Yamazaki (2006)

Fundamenta Mathematicae

Similarity:

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...

On a magnetic characterization of spectral minimal partitions

Bernard Helffer, Thomas Hoffmann-Ostenhof (2013)

Journal of the European Mathematical Society

Similarity:

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...

Isometric embeddings of a class of separable metric spaces into Banach spaces

Sophocles K. Mercourakis, Vassiliadis G. Vassiliadis (2018)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let $(M, d)$ be a bounded countable metric space and $c > 0$ a constant, such that $d (x, y) + d (y, z) - d (x, z) \geq c$ , for any pairwise distinct points $x, y, z$ of $M$ . For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $ℓ_{\infty}$ .

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

Similarity:

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 almost everywhere differentiability of the metric projection on closed sets in $l^{p} (ℝ^{n})$ , $2 < p < \infty$

Tord Sjödin (2018)

Czechoslovak Mathematical Journal

Similarity:

Let $F$ be a closed subset of $ℝ^{n}$ and let $P (x)$ denote the metric projection (closest point mapping) of $x \in ℝ^{n}$ onto $F$ in $l^{p}$ -norm. A classical result of Asplund states that $P$ is (Fréchet) differentiable almost everywhere (a.e.) in $ℝ^{n}$ in the Euclidean case $p = 2$ . We consider the case $2 < p < \infty$ and prove that the $i$ th component $P_{i} (x)$ of $P (x)$ is differentiable a.e. if $P_{i} (x) \neq x_{i}$ and satisfies Hölder condition of order $1 / (p - 1)$ if $P_{i} (x) = x_{i}$ .

Metric unconditionality and Fourier analysis

Stefan Neuwirth (1998)

Studia Mathematica

Similarity:

We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property (umap) in terms of “block unconditionality”. Then we focus on translation invariant subspaces $L_{E}^{p} ()$ and $C_{E} ()$ of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces $p_{E} ()$ , p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between $L_{E}^{p} ()$ ...

About $w c s$ -covers and $w c s^{*}$ -networks on the Vietoris hyperspace $ℱ (X)$

Luong Quoc Tuyen, Ong V. Tuyen, Phan D. Tuan, Nguzen X. Truc (2023)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

We study some generalized metric properties on the hyperspace $ℱ (X)$ of finite subsets of a space $X$ endowed with the Vietoris topology. We prove that $X$ has a point-star network consisting of (countable) $w c s$ -covers if and only if so does $ℱ (X)$ . Moreover, $X$ has a sequence of $w c s$ -covers with property $(P)$ which is a point-star network if and only if so does $ℱ (X)$ , where $(P)$ is one of the following properties: point-finite, point-countable, compact-finite, compact-countable, locally finite, locally countable....

On the characterization of harmonic functions with initial data in Morrey space

Bo Li, Jinxia Li, Bolin Ma, Tianjun Shen (2024)

Czechoslovak Mathematical Journal

Similarity:

Let $(X, d, μ)$ be a metric measure space satisfying the doubling condition and an $L^{2}$ -Poincaré inequality. Consider the nonnegative operator $ℒ$ generalized by a Dirichlet form on $X$ . We will show that a solution $u$ to $(- \partial_{t}^{2} + ℒ) u = 0$ on $X \times ℝ_{+}$ satisfies an $α$ -Carleson condition if and only if $u$ can be represented as the Poisson integral of the operator $ℒ$ with the trace in the generalized Morrey space $L^{2, α} (X)$ , where $α$ is a nonnegative function defined on a class of balls in $X$ . This result extends the analogous characterization...

Indestructible colourings and rainbow Ramsey theorems

Lajos Soukup (2009)

Fundamenta Mathematicae

Similarity:

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,...

Generalized Lebesgue points for Sobolev functions

Nijjwal Karak (2017)

Czechoslovak Mathematical Journal

Similarity:

In many recent articles, medians have been used as a replacement of integral averages when the function fails to be locally integrable. A point $x$ in a metric measure space $(X, d, μ)$ is called a generalized Lebesgue point of a measurable function $f$ if the medians of $f$ over the balls $B (x, r)$ converge to $f (x)$ when $r$ converges to $0$ . We know that almost every point of a measurable, almost everywhere finite function is a generalized Lebesgue point and the same is true for every point of a continuous function....

On the tangency of sets in generalized metric spaces for certain functions of the class $F_{ρ}^{*}$

T. Konik (1991)

Matematički Vesnik

Similarity:

Wasserstein metric and subordination

Philippe Clément, Wolfgang Desch (2008)

Studia Mathematica

Similarity:

Let $(X, d_{X})$ , $(Ω, d_{Ω})$ be complete separable metric spaces. Denote by (X) the space of probability measures on X, by $W_{p}$ the p-Wasserstein metric with some p ∈ [1,∞), and by $_{p} (X)$ the space of probability measures on X with finite Wasserstein distance from any point measure. Let $f : Ω \to_{p} (X)$ , $ω \mapsto f_{ω}$ , be a Borel map such that f is a contraction from $(Ω, d_{Ω})$ into $(_{p} (X), W_{p})$ . Let ν₁,ν₂ be probability measures on Ω with $W_{p} (ν ₁, ν ₂)$ finite. On X we consider the subordinated measures $μ_{i} = \int_{Ω} f_{ω} d ν_{i} (ω)$ . Then $W_{p} (μ ₁, μ ₂) \leq W_{p} (ν ₁, ν ₂)$ . As an application we show that the solution measures $ϱ_{α} (t)$ ...

Matchings in complete bipartite graphs and the $r$ -Lah numbers

Gábor Nyul, Gabriella Rácz (2021)

Czechoslovak Mathematical Journal

Similarity:

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. ...

Theoretical analysis for $ℓ_{1}$ - $ℓ_{2}$ minimization with partial support information

Haifeng Li, Leiyan Guo (2025)

Applications of Mathematics

Similarity:

We investigate the recovery of $k$ -sparse signals using the $ℓ_{1}$ - $ℓ_{2}$ minimization model with prior support set information. The prior support set information, which is believed to contain the indices of nonzero signal elements, significantly enhances the performance of compressive recovery by improving accuracy, efficiency, reducing complexity, expanding applicability, and enhancing robustness. We assume $k$ -sparse signals $𝐱$ with the prior support $T$ which is composed of $g$ true indices and $b$ wrong...

On path-quasar Ramsey numbers

Binlong Li, Bo Ning (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

Let $G_{1}$ and $G_{2}$ be two given graphs. The Ramsey number $R (G_{1}, G_{2})$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_{1}$ or $\bar{G}$ contains a $G_{2}$ . Parsons gave a recursive formula to determine the values of $R (P_{n}, K_{1, m})$ , where $P_{n}$ is a path on $n$ vertices and $K_{1, m}$ is a star on $m + 1$ vertices. In this note, we study the Ramsey numbers $R (P_{n}, K_{1} \lor F_{m})$ , where $F_{m}$ is a linear forest on $m$ vertices. We determine the exact values of $R (P_{n}, K_{1} \lor F_{m})$ for the cases $m \leq n$ and $m \geq 2 n$ , and for the case that $F_{m}$ has no odd component. Moreover, we...

Cambrian fans

Nathan Reading, David E. Speyer (2009)

Journal of the European Mathematical Society

Similarity:

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...

Best constants for the isoperimetric inequality in quantitative form

Marco Cicalese, Gian Paolo Leonardi (2013)

Journal of the European Mathematical Society

Similarity:

We prove some results in the context of isoperimetric inequalities with quantitative terms. In the $2$ -dimensional case, our main contribution is a method for determining the optimal coefficients $c_{1}, ..., c_{m}$ in the inequality $δ P (E) \geq \sum_{k = 1}^{m} c_{k} α {(E)}^{k} + o (α {(E)}^{m})$ , valid for each Borel set $E$ with positive and finite area, with $δ P (E)$ and $α (E)$ being, respectively, the $𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑑𝑒𝑓𝑖𝑐𝑖𝑡$ and the $𝐹𝑟𝑎𝑒𝑛𝑘𝑒𝑙𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦$ of $E$ . In $n$ dimensions, besides proving existence and regularity properties of minimizers for a wide class of $𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑎𝑡𝑖𝑣𝑒𝑖𝑠𝑜𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑟𝑖𝑐𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡𝑠$ including the lower semicontinuous extension of $\frac{δ P (E)}{α {(E)}^{2}}$ , we...