Displaying similar documents to “The lattice of ideals of a numerical semigroup and its Frobenius restricted variety associated”

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 covariety of perfect numerical semigroups with fixed Frobenius number

María Ángeles Moreno-Frías, José Carlos Rosales (2024)

Czechoslovak Mathematical Journal

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Let S be a numerical semigroup. We say that h S is an isolated gap of S if { h - 1 , h + 1 } S . A numerical semigroup without isolated gaps is called a perfect numerical semigroup. Denote by m ( S ) the multiplicity of a numerical semigroup S . A covariety is a nonempty family 𝒞 of numerical semigroups that fulfills the following conditions: there exists the minimum of 𝒞 , the intersection of two elements of 𝒞 is again an element of 𝒞 , and S { m ( S ) } 𝒞 for all S 𝒞 such that S min ( 𝒞 ) . We prove that the set 𝒫 ( F ) = { S : S is a perfect numerical semigroup...

On a sequence formed by iterating a divisor operator

Bellaouar Djamel, Boudaoud Abdelmadjid, Özen Özer (2019)

Czechoslovak Mathematical Journal

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Let be the set of positive integers and let s . We denote by d s the arithmetic function given by d s ( n ) = ( d ( n ) ) s , where d ( n ) is the number of positive divisors of n . Moreover, for every , m we denote by δ s , , m ( n ) the sequence d s ( d s ( ... d s ( d s ( n ) + ) + ... ) + ) m -times = d s ( n ) for m = 1 , d s ( d s ( n ) + ) for m = 2 , d s ( d s ( d s ( n ) + ) + ) for m = 3 , We present classical and nonclassical notes on the sequence ( δ s , , m ( n ) ) m 1 , where , n , s are understood as parameters.

On Kneser solutions of the n -th order nonlinear differential inclusions

Martina Pavlačková (2019)

Czechoslovak Mathematical Journal

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The paper deals with the existence of a Kneser solution of the n -th order nonlinear differential inclusion x ( n ) ( t ) - A 1 ( t , x ( t ) , ... , x ( n - 1 ) ( t ) ) x ( n - 1 ) ( t ) - ... - A n ( t , x ( t ) , ... , x ( n - 1 ) ( t ) ) x ( t ) for a.a. t [ a , ) , where a ( 0 , ) , and A i : [ a , ) × n , i = 1 , ... , n , are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.

Green's generic syzygy conjecture for curves of even genus lying on a K3 surface

Claire Voisin (2002)

Journal of the European Mathematical Society

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We consider the generic Green conjecture on syzygies of a canonical curve, and particularly the following reformulation thereof: For a smooth projective curve C of genus g in characteristic 0, the condition Cliff C > l is equivalent to the fact that K g - l ' - 2 , 1 ( C , K C ) = 0 , l ' l . We propose a new approach, which allows up to prove this result for generic curves C of genus g ( C ) and gonality gon(C) in the range g ( C ) 3 + 1 gon(C) g ( C ) 2 + 1 .

Persistence of Coron’s solution in nearly critical problems

Monica Musso, Angela Pistoia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider the problem - Δ u = u N + 2 N - 2 + λ in Ω ε ω , u &gt; 0 in Ω ε ω , u = 0 on Ω ε ω , where Ω and ω are smooth bounded domains in N , N 3 , ε &gt; 0 and λ . We prove that if the size of the hole ε goes to zero and if, simultaneously, the parameter λ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Jacques Giacomoni, Ian Schindler, Peter Takáč (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We investigate the following quasilinear and singular problem, t o 2 . 7 c m - Δ p u = λ u δ + u q in Ω ; u | Ω = 0 , u &gt; 0 in Ω , t o 2 . 7 c m (P) where Ω is an open bounded domain with smooth boundary, 1 &lt; p &lt; , p - 1 &lt; q p * - 1 , λ &gt; 0 , and 0 &lt; δ &lt; 1 . As usual, p * = N p N - p if 1 &lt; p &lt; N , p * ( p , ) is arbitrarily large if p = N , and p * = if p &gt; N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 0 1 , p ( Ω ) . While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle...

Traceability in { K 1 , 4 , K 1 , 4 + e } -free graphs

Wei Zheng, Ligong Wang (2019)

Czechoslovak Mathematical Journal

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A graph G is called { H 1 , H 2 , , H k } -free if G contains no induced subgraph isomorphic to any graph H i , 1 i k . We define σ k = min i = 1 k d ( v i ) : { v 1 , , v k } is an independent set of vertices in G . In this paper, we prove that (1) if G is a connected { K 1 , 4 , K 1 , 4 + e } -free graph of order n and σ 3 ( G ) n - 1 , then G is traceable, (2) if G is a 2-connected { K 1 , 4 , K 1 , 4 + e } -free graph of order n and | N ( x 1 ) N ( x 2 ) | + | N ( y 1 ) N ( y 2 ) | n - 1 for any two distinct pairs of non-adjacent vertices { x 1 , x 2 } , { y 1 , y 2 } of G , then G is traceable, i.e., G has a Hamilton path, where K 1 , 4 + e is a graph obtained by joining a pair of non-adjacent vertices in a K 1 , 4 .

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 .

On the Anderson-Badawi ω R [ X ] ( I [ X ] ) = ω R ( I ) conjecture

Peyman Nasehpour (2016)

Archivum Mathematicum

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Let R be a commutative ring with an identity different from zero and n be a positive integer. Anderson and Badawi, in their paper on n -absorbing ideals, define a proper ideal I of a commutative ring R to be an n -absorbing ideal of R , if whenever x 1 x n + 1 I for x 1 , ... , x n + 1 R , then there are n of the x i ’s whose product is in I and conjecture that ω R [ X ] ( I [ X ] ) = ω R ( I ) for any ideal I of an arbitrary ring R , where ω R ( I ) = min { n : I is an n -absorbing ideal of R } . In the present paper, we use content formula techniques to prove that their conjecture is true, if one of the following...

Positive solutions for concave-convex elliptic problems involving p ( x ) -Laplacian

Makkia Dammak, Abir Amor Ben Ali, Said Taarabti (2022)

Mathematica Bohemica

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We study the existence and nonexistence of positive solutions of the nonlinear equation - Δ p ( x ) u = λ k ( x ) u q ± h ( x ) u r in Ω , u = 0 on Ω where Ω N , N 2 , is a regular bounded open domain in N and the p ( x ) -Laplacian Δ p ( x ) u : = div ( | u | p ( x ) - 2 u ) is introduced for a continuous function p ( x ) > 1 defined on Ω . The positive parameter λ induces the bifurcation phenomena. The study of the equation (Q) needs generalized Lebesgue and Sobolev spaces. In this paper, under suitable assumptions, we show that some variational methods still work. We use them to prove the existence of positive...

Sufficient Conditions for Integrability of Distortion Function Kf 1

Costantino Capozzoli (2009)

Bollettino dell'Unione Matematica Italiana

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Assume that Ω , Ω are planar domains and f : Ω onto Ω is a homeomorphism belonging to Sobolev space W loc 1 , 1 ( Ω ; 2 ) with finite distortion. We prove that if the distortion function K f of f satisfies the condition dist EXP ( K f , L ) < 1 , then the distortion function K f - 1 of f - 1 belongs to L loc 1 ( Ω ) . We show that this result is sharp in sense that the conclusion fails if dist EXP ( K f , L ) = 1 . Moreover, we prove that if the distortion function K f satisfies the condition dist EXP ( K f , L ) = λ for some λ > 0 , then K f - 1 belongs to L loc p ( Ω ) for every p ( 0 , 1 2 λ ) . As special case of this result we show that if...

The algebra of the subspace semigroup of M ( q )

Jan Okniński (2002)

Colloquium Mathematicae

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The semigroup S = S ( M ( q ) ) of subspaces of the algebra M ( q ) of 2 × 2 matrices over a finite field q is studied. The ideal structure of S, the regular -classes of S and the structure of the complex semigroup algebra ℂ[S] are described.