Proportionally modular diophantine inequalities

We study the sets of nonnegative solutions of Diophantine inequalities of the form with a,b and c positive integers. These sets are numerical semigroups, which we study and characterize.     

Systems of proportionally modular Diophantine inequalities

The set of integer solutions to the inequality ax mod b≤c x is a numerical semigroup. We study numerical semigroups that are intersections of these numerical semigroups. Recently it has been shown that this class of numerical semigroups coincides with the class of numerical semigroups having a Toms decomposition. The first author was (partially) supported by the Centro de Matemática da Universidade do Porto (CMUP), financed by FCT (Portugal) through the programmes POCTI and POSI, with national and European Community structural funds. The last three authors are supported by the project MTM2004-01446 and FEDER funds. The authors would like to thank the referee for her/his comments and suggestions.     

Fundamental Gaps in Numerical Semigroups with Respect to Their Multiplicity

For a numerical semigroup, we introduce the concept of a fundamental gap with respect tothe multiplicity of the semigroup.The semigroup is fully determined by its multiplicity and these gaps.We study the case when a set of non-negative integers is the set of fundamental gaps with respect to themultiplicity of a numerical semigroup.Numerical semigroups with maximum and mininmm number ofthis kind of gaps are described.     

Fundamental Gaps in Numerical Semigroups with Respect to Their Multiplicity

For a numerical semigroup, we introduce the concept of a fundamental gap with respect to the multiplicity of the semigroup. The semigroup is fully determined by its multiplicity and these gaps.We study the case when a set of non-negative integers is the set of fundamental gaps with respect to the multiplicity of a numerical semigroup, Numerical semigroups with maximum and minimum number of this kind of gaps are described.     

On two Diophantine inequalities over primes

Let $1<c<37∕18,c\ne 2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in (N,2N]$, the Diophantine inequality $|p_1^c+p_2^c+p_3^c?R|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3$. Moreover, we also investigate the problem of six primes and prove that the Diophantine inequality $|p_1^c+p_2^c+p_3^c+p_4^c+p_5^c+p_6^c?N|<{log}^{?1}N$ is solvable in primes $p_1,p_2,p_3,p_4,p_5,p_6$ for sufficiently large real number $N$.     

On additive semigroups starting parts

We present geometrical arguments suggesting that the part of the segment {0,1,…,N−1} covered by the additive semigroup generated by (a,b,c) between 0 and the Frobenius number N(a,b,c) should exceed λ V for some constant λ (which might be 1/3 or even more).      

Numerical semigroups with many gaps of the same parity

Let g _e (S) (respectively, g _o (S)) be the number of even (respectively, odd) gaps of a numerical semigroup S. In this work we study and characterize the numerical semigroups S that verify 2|g _e (S)−g _o (S)|+1∈S. As a consequence we will see that every numerical semigroup can be represented by means of a numerical semigroup with maximal embedding dimension with all its minimal generators odd. The first author is supported by the project MTM2007-62346 and FEDER funds. The authors want to thank P.A. García-Sánchez and the referee for their comments and suggestions.     

Numerical semigroups: Apéry sets and Hilbert series

Let a ₁,…,a _n be relatively prime positive integers, and let S be the semigroup consisting of all non-negative integer linear combinations of a ₁,…,a _n . In this paper, we focus our attention on AA-semigroups, that is semigroups being generated by almost arithmetic progressions. After some general considerations, we give a characterization of the symmetric AA-semigroups. We also present an efficient method to determine an Apéry set and the Hilbert series of an AA-semigroup. Dedicated to the memory of Ernst S. Selmer (1920–2006), whose calculations revealed the “Selmer group”.     

Constructing Almost Symmetric Numerical Semigroups from Irreducible Numerical Semigroups

Every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with its same Frobenius number.     

Harnack and HWI inequalities on infinite-dimensional spaces

In this paper, the dimensional-free Harnack inequalities are established on infinite-dimensional spaces. More precisely, we establish Harnack inequalities for heat semigroup on based loop group and for Ornstein-Uhlenbeck semigroup on the abstract Wiener space. As an application, we establish the HWI inequality on the abstract Wiener space, which contains three important quantities in one inequality, the relative entropy “H”, Wasserstein distance “W”, and Fisher information “I”.     

Generalized Bebiano-Lemos-Providência inequalities and their reverses

We improve Bebiano-Lemos-Providência inequality: For A,B?0

     

Integer solutions to decomposable form inequalities

This paper obtains a result on the finiteness of the number of integer solutions to decomposable form inequalities. Let k be a number field and let F(X₁,...,X_m) be a non-degenerate decomposable form with coefficients in k. We prove that, for every finite set of places S of k containing the archimedean places of k, for each real number and for each constant c>0, the inequality

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一个素变数丢番图不等式

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一个素变数丢番图不等式

本文证明了如果１＜ｃ＜２３７／２１４,则对于充分大的Ｎ和ε≥Ｎ－１／５（２３７／２１４－ｃ）＋ｖ（ｖ＞０）表达式Ｄ（Ｎ）：＝Σ｜ｐｃ１＋ｐｃ２＋ｐｃ３－Ｎ｜＜－εｌｏｇｐ１ｌｏｇｐ２ｌｏｇｐ３ 关于素变数ｐ１,ｐ２,ｐ３有渐近公式,改进了文献［１］的结果１＜Ｃ＜１．１．     

一个二元丢番图不等式

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