Basel Problem

Karol Pąk; Artur Korniłowicz

Displaying similar documents to “Basel Problem”

Basel Problem – Preliminaries

Artur Korniłowicz, Karol Pąk (2017)

Formalized Mathematics

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In the article we formalize in the Mizar system [4] preliminary facts needed to prove the Basel problem [7, 1]. Facts that are independent from the notion of structure are included here.

Tight bounds for the dihedral angle sums of a pyramid

Sergey Korotov, Lars Fredrik Lund, Jon Eivind Vatne (2023)

Applications of Mathematics

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We prove that eight dihedral angles in a pyramid with an arbitrary quadrilateral base always sum up to a number in the interval $(3 π, 5 π)$ . Moreover, for any number in $(3 π, 5 π)$ there exists a pyramid whose dihedral angle sum is equal to this number, which means that the lower and upper bounds are tight. Furthermore, the improved (and tight) upper bound $4 π$ is derived for the class of pyramids with parallelogramic bases. This includes pyramids with rectangular bases, often used in finite element mesh generation...

A generalization of the Keller-Segel system to higher dimensions from a structural viewpoint

Fujie, Kentarou, Senba, Takasi

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We consider initial boundary problems of a two-chemical substances chemotaxis system. In the four-dimensional setting, it was shown that solutions exist globally in time and remain bounded if the total mass is less than ${(8 π)}^{2}$ , whereas the solution emanating from some initial data of large magnitude may blows up. This result can be regarded as a generalization of the well-known $8 π$ problem in the Keller–Segel system to higher dimensions. We will compare mathematical structures of the Keller–Segel...

When $Min {(G)}^{- 1}$ has a clopen $π$ -base

Ramiro Lafuente-Rodriguez, Warren Wm. McGovern (2021)

Mathematica Bohemica

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It is our aim to contribute to the flourishing collection of knowledge centered on the space of minimal prime subgroups of a given lattice-ordered group. Specifically, we are interested in the inverse topology. In general, this space is compact and $T_{1}$ , but need not be Hausdorff. In 2006, W. Wm. McGovern showed that this space is a boolean space (i.e. a compact zero-dimensional and Hausdorff space) if and only if the $l$ -group in question is weakly complemented. A slightly weaker topological...

One-dimensional symmetry of periodic minimizers for a mean field equation

Chang-Shou Lin, Marcello Lucia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation $Δ u + ρ (\frac{e^{u}}{\int_{T} e^{u}} - \frac{1}{| T |}) = 0, \int_{T} u = 0 .$ It is well-known that the associated energy functional admits a minimizer for each $ρ \leq 8 π$ . The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting $λ_{1} (T)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $ρ \leq min {8 π, λ_{1} (T) | T |}$ . Our results hold more generally for solutions that...

Copies of $l_{p}^{n}$ ’s uniformly in the spaces $Π_{2} (C [0, 1], X)$ and $Π_{1} (C [0, 1], X)$

Dumitru Popa (2017)

Czechoslovak Mathematical Journal

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We study the presence of copies of $l_{p}^{n}$ ’s uniformly in the spaces $Π_{2} (C [0, 1], X)$ and $Π_{1} (C [0, 1], X)$ . By using Dvoretzky’s theorem we deduce that if $X$ is an infinite-dimensional Banach space, then $Π_{2} (C [0, 1], X)$ contains $λ \sqrt{2}$ -uniformly copies of $l_{\infty}^{n}$ ’s and $Π_{1} (C [0, 1], X)$ contains $λ$ -uniformly copies of $l_{2}^{n}$ ’s for all $λ > 1$ . As an application, we show that if $X$ is an infinite-dimensional Banach space then the spaces $Π_{2} (C [0, 1], X)$ and $Π_{1} (C [0, 1], X)$ are distinct, extending the well-known result that the spaces $Π_{2} (C [0, 1], X)$ and $𝒩 (C [0, 1], X)$ are distinct.

Higgs bundles and representation spaces associated to morphisms

Indranil Biswas, Carlos Florentino (2015)

Archivum Mathematicum

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Let $G$ be a connected reductive affine algebraic group defined over the complex numbers, and $K \subset G$ be a maximal compact subgroup. Let $X$ , $Y$ be irreducible smooth complex projective varieties and $f : X \to Y$ an algebraic morphism, such that $π_{1} (Y)$ is virtually nilpotent and the homomorphism $f_{*} : π_{1} (X) \to π_{1} (Y)$ is surjective. Define $\begin{matrix} ℛ^{f} (π_{1} (X), G) & = {ρ \in Hom (π_{1} (X), G) ∣ A \circ ρ factors through f_{*}}, \\ ℛ^{f} (π_{1} (X), K) & = {ρ \in Hom (π_{1} (X), K) ∣ A \circ ρ factors through f_{*}}, \end{matrix}$ where $A : G \to GL (Lie (G))$ is the adjoint action. We prove that the geometric invariant theoretic quotient $ℛ^{f} (π_{1} (X, x_{0}), G) / / G$ admits a deformation retraction to $ℛ^{f} (π_{1} (X, x_{0}), K) / K$ . We also show that the space of conjugacy classes of $n$ almost commuting...

On a question of T. Sheil-Small regarding valency of harmonic maps

Daoud Bshouty, Abdallah Lyzzaik (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form $f (e^{i t}) = e^{i φ (t)}$ , $0 \leq t \leq 2 π$ where $φ$ is a continuously non-decreasing function that satisfies $φ (2 π) - φ (0) = 2 N π$ , assume every value finitely many times in the disc?

GUST E-Foundry Font Projects

Bogusław Jackowski, Piotr Strzelczyk, Piotr Pianowski (2016)

Zpravodaj Československého sdružení uživatelů TeXu

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In this paper, we present a summary of our work on the TeX Gyre fonts. We also discuss inconsistencies in the existing fonts and encodings and show how the problems are solved in the TeX Gyre fonts.

$H^{2}$ convergence of solutions of a biharmonic problem on a truncated convex sector near the angle $π$

Abdelkader Tami, Mounir Tlemcani (2021)

Applications of Mathematics

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We consider a biharmonic problem $Δ^{2} u_{ω} = f_{ω}$ with Navier type boundary conditions $u_{ω} = Δ u_{ω} = 0$ , on a family of truncated sectors $Ω_{ω}$ in $ℝ^{2}$ of radius $r$ , $0 < r < 1$ and opening angle $ω$ , $ω \in (2 π / 3, π]$ when $ω$ is close to $π$ . The family of right-hand sides ${(f_{ω})}_{ω \in (2 π / 3, π]}$ is assumed to depend smoothly on $ω$ in $L^{2} (Ω_{ω})$ . The main result is that $u_{ω}$ converges to $u_{π}$ when $ω \to π$ with respect to the $H^{2}$ -norm. We can also show that the $H^{2}$ -topology is optimal for such a convergence result.

A note on the $Π$ -property of some subgroups of finite groups

Zhengtian Qiu, Guiyun Chen, Jianjun Liu (2024)

Czechoslovak Mathematical Journal

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Let $H$ be a subgroup of a finite group $G$ . We say that $H$ satisfies the $Π$ -property in $G$ if for any chief factor $L / K$ of $G$ , $| G / K : N_{G / K} (H K / K \cap L / K) |$ is a $π (H K / K \cap L / K)$ -number. We obtain some criteria for the $p$ -supersolubility or $p$ -nilpotency of a finite group and extend some known results by concerning some subgroups that satisfy the $Π$ -property.

On $Π$ -property of some maximal subgroups of Sylow subgroups of finite groups

Zhengtian Qiu, Jianjun Liu, Guiyun Chen (2023)

Czechoslovak Mathematical Journal

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Let $H$ be a subgroup of a finite group $G$ . We say that $H$ satisfies the $Π$ -property in $G$ if for any chief factor $L / K$ of $G$ , $| G / K : N_{G / K} (H K / K \cap L / K) |$ is a $π (H K / K \cap L / K)$ -number. We study the influence of some $p$ -subgroups of $G$ satisfying the $Π$ -property on the structure of $G$ , and generalize some known results.

A Note on Badiale's Characterization of the $q$ -Gamma Functions

Marino Badiale, Francis J. Sullivan (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si precisano alcuni risultati del lavoro accennato nel titolo.

On $^{γ (\cdot, \cdot)}$ -subdifferentiable and [+γ]-subdifferentiable Functions

S. Rolewicz (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

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