Negative / Null Result ReportOpen accessMathematics
Alexander Tolmachev · 2024 · arXiv
Determining the maximal density $m_1(\mathbb{R}^2)$ of planar sets without unit distances is a fundamental problem in combinatorial geometry. This paper investigates lower bounds for this quantity. We introduce a novel approach to estimating $m_1(\mathbb{R}^2)$ by reformulating the problem as a Maximal Independent Set (MIS) problem on graphs constructed from flat torus, focusing on periodic sets with respect to two non-collinear vectors. Our experimental results, supported by theoretical justifications of proposed method, demonstrate that for a sufficiently wide range of parameters this approa
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Nicolas Burq, Nicolas Camps, Mickaël Latocca et al. · 2024 · arXiv
We consider the Wick ordered cubic Schrödinger equation (NLS) posed on the two-dimensional sphere, with initial data distributed according to a Gaussian measure. We show that the second Picard iteration does not improve the regularity of the initial data in the scale of the classical Sobolev spaces. This is in sharp contrast with the Wick ordered NLS on the two-dimensional tori, a model for which we know from the work of Bourgain that the second Picard iteration gains one half derivative. Our proof relies on identifying a singular part of the nonlinearity. We show that this singular part is re
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Gilles Chardon · 2014 · arXiv
The scattering of waves by obstacles in a 2D setting is considered, in particular the computation of the scattered field via the collocation or the least-squares methods. In the case of multiple scattering by smooth obstacles, we prove that the scattered field can be uniformly approximated by sums of multipoles. For a unique obstacle, the choice of the number of points and their positions for the estimation of the error on the border of the scatterer is studied, showing the benefit of using a non-uniform distribution of points dependent on the scatterer and the approximation scheme. In general
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Nicolas Dupin · 2019 · arXiv
The discrete unit commitment problem with min-stop ramping constraints optimizes the daily production of thermal power plants (coal, gas, fuel units). For this problem, compact Integer Linear Programming (ILP) formulations have been designed to solve exactly small instances and heuristically real-size instances. This paper investigates whether Dantzig-Wolfe reformulation allows to improve the previous exact method and matheuristics. The extended ILP formulation is presented with the column generation algorithm to solve its linear relaxation. The experimental results show that the Dantzig-Wolfe
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Martina Zizza · 2024 · arXiv
In this paper we examine the discrete Shnirelman's inequality [Shnirelman A., 1985], which relates the $L^2$-distance of two discrete configurations of a fluid to the $L^1_tL^2_x$-norm of the vector field connecting them. Our proof is inspired by [Shnirelman A., 1985], where it was obtained $α=\frac{1}{64}$ in dimension $ν=2$, while here we get $α\geq\frac{2}{7}$. Moreover we prove that $α\geq\frac{1}{ν+1}$ for any dimension $ν\geq 3$. We point out that, even if this does not improve the bound in the continuous version, where it was proved that $α\geq\frac{2}{4+ν}$, with $ν\geq 3$, our bound i
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Aihua Xia, Fuxi Zhang · 2011 · arXiv
Random events in space and time often exhibit a locally dependent structure. When the events are very rare and dependent structure is not too complicated, various studies in the literature have shown that Poisson and compound Poisson processes can provide adequate approximations. However, the accuracy of approximations does not improve or may even deteriorate when the mean number of events increases. In this paper, we investigate an alternative family of approximating point processes and establish Stein's method for their approximations. We prove two theorems to accommodate respectively the po
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Peter Bürgisser · 2022 · arXiv
Consider a system $f_1(x)=0,\ldots,f_n(x)=0$ of $n$ random real polynomials in $n$ variables, where each $f_i$ has a prescribed set of exponent vectors described by a set $A_i \subseteq \mathbb{Z}^n$ of cardinality $t_i$, whose convex hull is denoted $P_i$. Assuming that the coefficients of the $f_i$ are independent standard Gaussian, we prove that the expected number of zeros of the random system in the positive orthant is at most $(2π)^{-\frac{n}{2}} V_0 (t_1-1)\ldots (t_n-1)$. Here $V_0$ denotes the number of vertices of the Minkowski sum $P_1+\ldots + P_n$. However, this bound does not imp
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Karunia Putra Wijaya, Dipo Aldila · 2017 · arXiv
Investigating the seasonality of disease incidences is very important in disease surveillance in regions with periodical climatic patterns. In lieu of the paradigm about disease incidences varying seasonally in line with meteorology, this work seeks to determine how well standard epidemic models can capture such seasonality for better forecasts and optimal futuristic interventions. Once incidence data are assimilated by a periodic model, asymptotic analysis in relation to the long-term behavior of the disease occurrences can be performed using the classical Floquet theory, which explains the s
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Alessandro Arsie, Emilio Frazzoli · 2006 · arXiv
In this paper we consider a class of dynamic vehicle routing problems, in which a number of mobile agents in the plane must visit target points generated over time by a stochastic process. It is desired to design motion coordination strategies in order to minimize the expected time between the appearance of a target point and the time it is visited by one of the agents. We propose control strategies that, while making minimal or no assumptions on communications between agents, provide the same level of steady-state performance achieved by the best known decentralized strategies. In other words
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Rujie Zhu, Xiaodong Xu, Stanisław Radziszowski · 2015 · arXiv
Let $Δ_s=R(K_3,K_s)-R(K_3,K_{s-1})$, where $R(G,H)$ is the Ramsey number of graphs $G$ and $H$ defined as the smallest $n$ such that any edge coloring of $K_n$ with two colors contains $G$ in the first color or $H$ in the second color. In 1980, Erdős and Sós posed some questions about the growth of $Δ_s$. The best known concrete bounds on $Δ_s$ are $3 \le Δ_s \le s$, and they have not improved since the stating of the problem. In this paper we present some constructions, which imply in particular that $R(K_3,K_s) \ge R(K_3,K_{s-1}-e) + 4$. This does not improve the lower bound of 3 on $Δ_s$, b
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Shibdas Roy, Ian R. Petersen, Elanor H. Huntington · 2015 · arXiv
We study a coherent-classical estimation scheme for a class of linear quantum systems, where the estimator is a mixed quantum-classical system that may or may not involve coherent feedback. We show that when the quantum plant or the quantum part of the estimator (coherent controller) is an annihilation operator only system, coherent-classical estimation without coherent feedback can provide no improvement over purely-classical estimation. Otherwise, coherent-classical estimation without feedback can be better than classical-only estimation for certain homodyne detector angles, although the for
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Courtney Paquette, Elliot Paquette · 2021 · arXiv
We analyze a class of stochastic gradient algorithms with momentum on a high-dimensional random least squares problem. Our framework, inspired by random matrix theory, provides an exact (deterministic) characterization for the sequence of loss values produced by these algorithms which is expressed only in terms of the eigenvalues of the Hessian. This leads to simple expressions for nearly-optimal hyperparameters, a description of the limiting neighborhood, and average-case complexity. As a consequence, we show that (small-batch) stochastic heavy-ball momentum with a fixed momentum parameter pr
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David Conlon, Lior Gishboliner, Yevgeny Levanzov et al. · 2019 · arXiv
Let $f(n,v,e)$ denote the maximum number of edges in a $3$-uniform hypergraph not containing $e$ edges spanned by at most $v$ vertices. One of the most influential open problems in extremal combinatorics then asks, for a given number of edges $e \geq 3$, what is the smallest integer $d=d(e)$ so that $f(n,e+d,e) = o(n^2)$? This question has its origins in work of Brown, Erdős and Sós from the early 70's and the standard conjecture is that $d(e)=3$ for every $e \geq 3$. The state of the art result regarding this problem was obtained in 2004 by Sárközy and Selkow, who showed that $f(n,e + 2 + \lf
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D. R. Heath-Brown · 2026 · arXiv
We show that there are $O(B^{3/5-3/1555+\ep})$ triples $(x,y,z)$ of square-full integesr up to $B$ satisfying the equation $x+y=z$ for any fixed $\ep>0$. This is the first improvement over the `easy' exponent $3/5$, given by Browning and Van Valckenborgh. One new tool is a strong uniform bound for the counting function for equations $aX^3+bY^3=cZ^3$.
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John Rozmarynowycz, Seungki Kim · 2023 · arXiv
We report the finding of the new upper bound on the lowest positive integer $x$ for which the Mertens conjecture \begin{equation*} \left| \sum_{1 \leq n \leq x} μ(n) \right| < \sqrt{x} \end{equation*} fails to hold: $x < \exp(1.017 \times 10^{29})$, an improvement over previously known $\exp(1.59 \times 10^{40})$ due to Kotnik and te Riele [7].
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Francesco Orabona, Ryan D'Orazio · 2025 · arXiv
The Polyak stepsize has been proven to be a fundamental stepsize in convex optimization, giving near optimal gradient descent rates across a wide range of assumptions. The universality of the Polyak stepsize has also inspired many stochastic variants, with theoretical guarantees and strong empirical performance. Despite the many theoretical results, our understanding of the convergence properties and shortcomings of the Polyak stepsize or its variants is both incomplete and fractured across different analyses. We propose a new, unified, and simple perspective for the Polyak stepsize and its va
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Jose Moncayo, Pedro H. Zambrano · 2023 · arXiv
We investigate different set-theoretic constructions in Residuated Logic based on Fitting's work on Intuitionistic Set Theory. We start by stating some results concerning constructible sets within valued models of Set Theory. We present two distinct constructions of the constructible universe: $\mathfrak{L}^{\mathbb{Q}}$ and $\mathbb{L}^{\mathbb{Q}}$, and show that they are isomorphic to V (the classical von Neumann universe) and L (the classical Gödel constructible universe), respectively. Even though lattice-valued models are the natural way to study non-classical Set Theory (e.g., Intuition
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Jan A. Bergstra, Inge Bethke · 2015 · arXiv
$\mathbb{Q}_0$ - the involutive meadow of the rational numbers - is the field of the rational numbers where the multiplicative inverse operation is made total by imposing $0^{-1}=0$. In this note, we prove that $\mathbb{Q}_0$ cannot be specified by the usual axioms for meadows augmented by a finite set of axioms of the form $(1+ \cdots +1+x^2)\cdot (1+ \cdots +1 +x^2)^{-1}=1$.
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J. Silverio Martinez-Baena, Salvador Villegas · 2024 · arXiv
In the regularity theory of solutions to elliptic partial differential equations often the concept of stability plays the role of a sufficient condition for smoothness. It is a natural question to ask if this holds true for nonstable but finite Morse index solutions. We provide a negative answer showing the existence of sequences of solutions with radial Morse index equal to 1 for which regularity estimates can not be satisfied.
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Paweł Pasteczka · 2013 · arXiv
In the present paper we are going to prove some necessary condition for a mean to be Hardy. This condition is then applied to completely characterize the Hardy property among the Gini means.
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Grigor Sargsyan · 2021 · arXiv
We show that in extender models there are no generic embeddings with critical point $ω_1$ that resemble the stationary tower at the second Woodin cardinal.
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Paweł Pasteczka · 2021 · arXiv
We establish the test which allows to show that a mean does not admit a weak-Hardy property. As a result we prove that Hardy and weak-Hardy properties are equivalent in the class of homogeneous, symmetric, repetition invariant, and Jensen concave mean on $\mathbb{R}_+$. More precisely, for every mean $\mathscr{M} \colon \bigcup_{n=1}^\infty \mathbb{R}_+^n \to \mathbb{R}$ as above, the inequality $$\mathscr{M}(a_1)+\mathscr{M}(a_1,a_2)+\dots<\infty$$ holds for all $a \in \ell^1(\mathbb{R}_+)$ if and only if there exists a positive, real constant $C$ (depending only on $\mathscr{M}$) such that $
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