Displaying similar documents to “On blocks of arithmetic progressions with equal products”

Real and complex analytic sets. The relevance of Segre varieties

Klas Diederich, Emmanuel Mazzilli (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let X n n be a closed real-analytic subset and put 𝒜 : = { z X A X , germ of a complex-analytic set, z A , dim z A > 0 } This article deals with the question of the structure of 𝒜 . In the main result a natural proof is given for the fact, that 𝒜 always is closed. As a main tool an interesting relation between complex analytic subsets of X of positive dimension and the Segre varieties of X is proved and exploited.

Topological properties of two-dimensional number systems

Shigeki Akiyama, Jörg M. Thuswaldner (2000)

Journal de théorie des nombres de Bordeaux

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In the two dimensional real vector space 2 one can define analogs of the well-known q -adic number systems. In these number systems a matrix M plays the role of the base number q . In the present paper we study the so-called fundamental domain of such number systems. This is the set of all elements of 2 having zero integer part in their “ M -adic” representation. It was proved by Kátai and Környei, that is a compact set and certain translates of it form a tiling of the 2 . We construct...

Explicit cogenerators for the homotopy category of projective modules over a ring

Amnon Neeman (2011)

Annales scientifiques de l'École Normale Supérieure

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Let R be a ring. In two previous articles [12, 14] we studied the homotopy category 𝐊 ( R - Proj ) of projective R -modules. We produced a set of generators for this category, proved that the category is 1 -compactly generated for any ring R , and showed that it need not always be compactly generated, but is for sufficiently nice R . We furthermore analyzed the inclusion j ! : 𝐊 ( R - Proj ) 𝐊 ( R - Flat ) and the orthogonal subcategory 𝒮 = 𝐊 ( R - Proj ) . And we even showed that the inclusion 𝒮 𝐊 ( R - Flat ) has a right adjoint; this forces some natural map to be...

The tangent function and power residues modulo primes

Zhi-Wei Sun (2023)

Czechoslovak Mathematical Journal

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Let p be an odd prime, and let a be an integer not divisible by p . When m is a positive integer with p 1 ( mod 2 m ) and 2 is an m th power residue modulo p , we determine the value of the product k R m ( p ) ( 1 + tan ( π a k / p ) ) , where R m ( p ) = { 0 < k < p : k is an m th power residue modulo p } . In particular, if p = x 2 + 64 y 2 with x , y , then k R 4 ( p ) 1 + tan π a k p = ( - 1 ) y ( - 2 ) ( p - 1 ) / 8 .

The new properties of the theta functions

Stefan Czekalski (2013)

Annales mathématiques Blaise Pascal

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It is shown, that the function H ( x ) = k = - e - k 2 x satisfies the relation H ( x ) = n = 0 ( 2 π ) 2 n ( 2 n ) ! H ( n ) ( x ) .

Integrability for very weak solutions to boundary value problems of p -harmonic equation

Hongya Gao, Shuang Liang, Yi Cui (2016)

Czechoslovak Mathematical Journal

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The paper deals with very weak solutions u θ + W 0 1 , r ( Ω ) , max { 1 , p - 1 } < r < p < n , to boundary value problems of the p -harmonic equation - div ( | u ( x ) | p - 2 u ( x ) ) = 0 , x Ω , u ( x ) = θ ( x ) , x Ω . ( * ) We show that, under the assumption θ W 1 , q ( Ω ) , q > r , any very weak solution u to the boundary value problem ( * ) is integrable with u θ + L weak q * ( Ω ) for q < n , θ + L weak τ ( Ω ) for q = n and any τ < , θ + L ( Ω ) for q > n , provided that r is sufficiently close to p .

The lattice of ideals of a numerical semigroup and its Frobenius restricted variety associated

Maria Angeles Moreno-Frías, José Carlos Rosales (2024)

Mathematica Bohemica

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Let Δ be a numerical semigroup. In this work we show that 𝒥 ( Δ ) = { I { 0 } : I is an ideal of Δ } is a distributive lattice, which in addition is a Frobenius restricted variety. We give an algorithm which allows us to compute the set 𝒥 a ( Δ ) = { S 𝒥 ( Δ ) : max ( Δ S ) = a } for a given a Δ . As a consequence, we obtain another algorithm that computes all the elements of 𝒥 ( Δ ) with a fixed genus.

Existence of weak solutions for elliptic Dirichlet problems with variable exponent

Sungchol Kim, Dukman Ri (2023)

Mathematica Bohemica

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This paper presents several sufficient conditions for the existence of weak solutions to general nonlinear elliptic problems of the type - div a ( x , u , u ) + b ( x , u , u ) = 0 in Ω , u = 0 on Ω , where Ω is a bounded domain of n , n 2 . In particular, we do not require strict monotonicity of the principal part a ( x , z , · ) , while the approach is based on the variational method and results of the variable exponent function spaces.