Displaying similar documents to “On minimal spectrum of multiplication lattice modules”

Equivalent conditions for the validity of the Helmholtz decomposition of Muckenhoupt A p -weighted L p -spaces

Ryôhei Kakizawa (2018)

Czechoslovak Mathematical Journal

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We discuss the validity of the Helmholtz decomposition of the Muckenhoupt A p -weighted L p -space ( L w p ( Ω ) ) n for any domain Ω in n , n , n 2 , 1 < p < and Muckenhoupt A p -weight w A p . Set p ' : = p / ( p - 1 ) and w ' : = w - 1 / ( p - 1 ) . Then the Helmholtz decomposition of ( L w p ( Ω ) ) n and ( L w ' p ' ( Ω ) ) n and the variational estimate of L w , π p ( Ω ) and L w ' , π p ' ( Ω ) are equivalent. Furthermore, we can replace L w , π p ( Ω ) and L w ' , π p ' ( Ω ) by L w , σ p ( Ω ) and L w ' , σ p ' ( Ω ) , respectively. The proof is based on the reflexivity and orthogonality of L w , π p ( Ω ) and L w , σ p ( Ω ) and the Hahn-Banach theorem. As a corollary of our main result, we obtain the extrapolation...

Ramsey numbers for trees II

Zhi-Hong Sun (2021)

Czechoslovak Mathematical Journal

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Let r ( G 1 , G 2 ) be the Ramsey number of the two graphs G 1 and G 2 . For n 1 n 2 1 let S ( n 1 , n 2 ) be the double star given by V ( S ( n 1 , n 2 ) ) = { v 0 , v 1 , ... , v n 1 , w 0 , w 1 , ... , w n 2 } and E ( S ( n 1 , n 2 ) ) = { v 0 v 1 , ... , v 0 v n 1 , v 0 w 0 , w 0 w 1 , ... , w 0 w n 2 } . We determine r ( K 1 , m - 1 , S ( n 1 , n 2 ) ) under certain conditions. For n 6 let T n 3 = S ( n - 5 , 3 ) , T n ' ' = ( V , E 2 ) and T n ' ' ' = ( V , E 3 ) , where V = { v 0 , v 1 , ... , v n - 1 } , E 2 = { v 0 v 1 , ... , v 0 v n - 4 , v 1 v n - 3 , v 1 v n - 2 , v 2 v n - 1 } and E 3 = { v 0 v 1 , ... , v 0 v n - 4 , v 1 v n - 3 , v 2 v n - 2 , v 3 v n - 1 } . We also obtain explicit formulas for r ( K 1 , m - 1 , T n ) , r ( T m ' , T n ) ( n m + 3 ) , r ( T n , T n ) , r ( T n ' , T n ) and r ( P n , T n ) , where T n { T n ' ' , T n ' ' ' , T n 3 } , P n is the path on n vertices and T n ' is the unique tree with n vertices and maximal degree n - 2 .

On Fourier asymptotics of a generalized Cantor measure

Bérenger Akon Kpata, Ibrahim Fofana, Konin Koua (2010)

Colloquium Mathematicae

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Let d be a positive integer and μ a generalized Cantor measure satisfying μ = j = 1 m a j μ S j - 1 , where 0 < a j < 1 , j = 1 m a j = 1 , S j = ρ R + b j with 0 < ρ < 1 and R an orthogonal transformation of d . Then ⎧1 < p ≤ 2 ⇒ ⎨ s u p r > 0 r d ( 1 / α ' - 1 / p ' ) ( J x r | μ ̂ ( y ) | p ' d y ) 1 / p ' D ρ - d / α ' , x d , ⎩ p = 2 ⇒ infr≥1 rd(1/α’-1/2) (∫J₀r|μ̂(y)|² dy)1/2 ≥ D₂ρd/α’ , where J x r = i = 1 d ( x i - r / 2 , x i + r / 2 ) , α’ is defined by ρ d / α ' = ( j = 1 m a j p ) 1 / p and the constants D₁ and D₂ depend only on d and p.

On continuous self-maps and homeomorphisms of the Golomb space

Taras O. Banakh, Jerzy Mioduszewski, Sławomir Turek (2018)

Commentationes Mathematicae Universitatis Carolinae

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The Golomb space τ is the set of positive integers endowed with the topology τ generated by the base consisting of arithmetic progressions { a + b n : n 0 } with coprime a , b . We prove that the Golomb space τ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set Π of prime numbers is a dense metrizable subspace of τ , and each homeomorphism h of τ has the following properties: h ( 1 ) = 1 , h ( Π ) = Π , Π h ( x ) = h ( Π x ) , and h ( x ) = h ( x ) for all x . Here x : = { x n : n } and Π x denotes the set of...

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

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Given an integer n 3 , let u 1 , ... , u n be pairwise coprime integers 2 , 𝒟 a family of nonempty proper subsets of { 1 , ... , n } with “enough” elements, and ε a function 𝒟 { ± 1 } . Does there exist at least one prime q such that q divides i I u i - ε ( I ) for some I 𝒟 , but it does not divide u 1 u n ? We answer this question in the positive when the u i are prime powers and ε and 𝒟 are subjected to certain restrictions. We use the result to prove that, if ε 0 { ± 1 } and A is a set of three or more primes that contains all prime divisors of any...

A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations

Manabu Naito (2024)

Mathematica Bohemica

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The half-linear differential equation ( | u ' | α sgn u ' ) ' = α ( λ α + 1 + b ( t ) ) | u | α sgn u , t t 0 , is considered, where α and λ are positive constants and b ( t ) is a real-valued continuous function on [ t 0 , ) . It is proved that, under a mild integral smallness condition of b ( t ) which is weaker than the absolutely integrable condition of b ( t ) , the above equation has a nonoscillatory solution u 0 ( t ) such that u 0 ( t ) e - λ t and u 0 ' ( t ) - λ e - λ t ( t ), and a nonoscillatory solution u 1 ( t ) such that u 1 ( t ) e λ t and u 1 ' ( t ) λ e λ t ( t ).

Bigraphic pairs with a realization containing a split bipartite-graph

Jian Hua Yin, Jia-Yun Li, Jin-Zhi Du, Hai-Yan Li (2019)

Czechoslovak Mathematical Journal

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Let K s , t be the complete bipartite graph with partite sets { x 1 , ... , x s } and { y 1 , ... , y t } . A split bipartite-graph on ( s + s ' ) + ( t + t ' ) vertices, denoted by SB s + s ' , t + t ' , is the graph obtained from K s , t by adding s ' + t ' new vertices x s + 1 , ... , x s + s ' , y t + 1 , ... , y t + t ' such that each of x s + 1 , ... , x s + s ' is adjacent to each of y 1 , ... , y t and each of y t + 1 , ... , y t + t ' is adjacent to each of x 1 , ... , x s . Let A and B be nonincreasing lists of nonnegative integers, having lengths m and n , respectively. The pair ( A ; B ) is potentially SB s + s ' , t + t ' -bigraphic if there is a simple bipartite graph containing SB s + s ' , t + t ' (with s + s ' vertices x 1 , ... , x s + s ' in the part of size m ...

A new characterization of symmetric group by NSE

Azam Babai, Zeinab Akhlaghi (2017)

Czechoslovak Mathematical Journal

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Let G be a group and ω ( G ) be the set of element orders of G . Let k ω ( G ) and m k ( G ) be the number of elements of order k in G . Let nse ( G ) = { m k ( G ) : k ω ( G ) } . Assume r is a prime number and let G be a group such that nse ( G ) = nse ( S r ) , where S r is the symmetric group of degree r . In this paper we prove that G S r , if r divides the order of G and r 2 does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.

Differences of two semiconvex functions on the real line

Václav Kryštof, Luděk Zajíček (2016)

Commentationes Mathematicae Universitatis Carolinae

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It is proved that real functions on which can be represented as the difference of two semiconvex functions with a general modulus (or of two lower C 1 -functions, or of two strongly paraconvex functions) coincide with semismooth functions on (i.e. those locally Lipschitz functions on for which f + ' ( x ) = lim t x + f + ' ( t ) and f - ' ( x ) = lim t x - f - ' ( t ) for each x ). Further, for each modulus ω , we characterize the class D S C ω of functions on which can be written as f = g - h , where g and h are semiconvex with modulus C ω (for some C > 0 ) using a new...

Several quantitative characterizations of some specific groups

A. Mohammadzadeh, Ali Reza Moghaddamfar (2017)

Commentationes Mathematicae Universitatis Carolinae

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Let G be a finite group and let π ( G ) = { p 1 , p 2 , ... , p k } be the set of prime divisors of | G | for which p 1 < p 2 < < p k . The Gruenberg-Kegel graph of G , denoted GK ( G ) , is defined as follows: its vertex set is π ( G ) and two different vertices p i and p j are adjacent by an edge if and only if G contains an element of order p i p j . The degree of a vertex p i in GK ( G ) is denoted by d G ( p i ) and the k -tuple D ( G ) = ( d G ( p 1 ) , d G ( p 2 ) , ... , d G ( p k ) ) is said to be the degree pattern of G . Moreover, if ω π ( G ) is the vertex set of a connected component of GK ( G ) , then the largest ω -number which divides | G | , is...

On the recognizability of some projective general linear groups by the prime graph

Masoumeh Sajjadi (2022)

Commentationes Mathematicae Universitatis Carolinae

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Let G be a finite group. The prime graph of G is a simple graph Γ ( G ) whose vertex set is π ( G ) and two distinct vertices p and q are joined by an edge if and only if G has an element of order p q . A group G is called k -recognizable by prime graph if there exist exactly k nonisomorphic groups H satisfying the condition Γ ( G ) = Γ ( H ) . A 1-recognizable group is usually called a recognizable group. In this problem, it was proved that PGL ( 2 , p α ) is recognizable, if p is an odd prime and α > 1 is odd. But for even α , only...

Representation functions for binary linear forms

Fang-Gang Xue (2024)

Czechoslovak Mathematical Journal

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Let be the set of integers, 0 the set of nonnegative integers and F ( x 1 , x 2 ) = u 1 x 1 + u 2 x 2 be a binary linear form whose coefficients u 1 , u 2 are nonzero, relatively prime integers such that u 1 u 2 ± 1 and u 1 u 2 - 2 . Let f : 0 { } be any function such that the set f - 1 ( 0 ) has asymptotic density zero. In 2007, M. B. Nathanson (2007) proved that there exists a set A of integers such that r A , F ( n ) = f ( n ) for all integers n , where r A , F ( n ) = | { ( a , a ' ) : n = u 1 a + u 2 a ' : a , a ' A } | . We add the structure of difference for the binary linear form F ( x 1 , x 2 ) .