Displaying similar documents to “Submultiplicative functions and operator inequalities”

Ramsey numbers for trees II

Zhi-Hong Sun (2021)

Czechoslovak Mathematical Journal

Similarity:

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

Similarity:

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.

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

Ryôhei Kakizawa (2018)

Czechoslovak Mathematical Journal

Similarity:

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...

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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

Similarity:

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 ).

Differences of two semiconvex functions on the real line

Václav Kryštof, Luděk Zajíček (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

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...

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

Similarity:

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

Similarity:

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.

Prime ideal factorization in a number field via Newton polygons

Lhoussain El Fadil (2021)

Czechoslovak Mathematical Journal

Similarity:

Let K be a number field defined by an irreducible polynomial F ( X ) [ X ] and K its ring of integers. For every prime integer p , we give sufficient and necessary conditions on F ( X ) that guarantee the existence of exactly r prime ideals of K lying above p , where F ¯ ( X ) factors into powers of r monic irreducible polynomials in 𝔽 p [ X ] . The given result presents a weaker condition than that given by S. K. Khanduja and M. Kumar (2010), which guarantees the existence of exactly r prime ideals of K lying above p ....

Groups satisfying the two-prime hypothesis with a composition factor isomorphic to PSL 2 ( q ) for q 7

Mark L. Lewis, Yanjun Liu, Hung P. Tong-Viet (2018)

Czechoslovak Mathematical Journal

Similarity:

Let G be a finite group and write cd ( G ) for the degree set of the complex irreducible characters of G . The group G is said to satisfy the two-prime hypothesis if for any distinct degrees a , b cd ( G ) , the total number of (not necessarily different) primes of the greatest common divisor gcd ( a , b ) is at most 2 . We prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to PSL 2 ( q ) for q 7 .

Admissible spaces for a first order differential equation with delayed argument

Nina A. Chernyavskaya, Lela S. Dorel, Leonid A. Shuster (2019)

Czechoslovak Mathematical Journal

Similarity:

We consider the equation - y ' ( x ) + q ( x ) y ( x - ϕ ( x ) ) = f ( x ) , x , where ϕ and q ( q 1 ) are positive continuous functions for all x and f C ( ) . By a solution of the equation we mean any function y , continuously differentiable everywhere in , which satisfies the equation for all x . We show that under certain additional conditions on the functions ϕ and q , the above equation has a unique solution y , satisfying the inequality y ' C ( ) + q y C ( ) c f C ( ) , where the constant c ( 0 , ) does not depend on the choice of f .

Representation functions for binary linear forms

Fang-Gang Xue (2024)

Czechoslovak Mathematical Journal

Similarity:

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 ) .

Generalized reverse derivations and commutativity of prime rings

Shuliang Huang (2019)

Communications in Mathematics

Similarity:

Let R be a prime ring with center Z ( R ) and I a nonzero right ideal of R . Suppose that R admits a generalized reverse derivation ( F , d ) such that d ( Z ( R ) ) 0 . In the present paper, we shall prove that if one of the following conditions holds: (i) F ( x y ) ± x y Z ( R ) , (ii) F ( [ x , y ] ) ± [ F ( x ) , y ] Z ( R ) , (iii) F ( [ x , y ] ) ± [ F ( x ) , F ( y ) ] Z ( R ) , (iv) F ( x y ) ± F ( x ) F ( y ) Z ( R ) , (v) [ F ( x ) , y ] ± [ x , F ( y ) ] Z ( R ) , (vi) F ( x ) y ± x F ( y ) Z ( R ) for all x , y I , then R is commutative.

Lower bounds for Schrödinger operators in H¹(ℝ)

Ronan Pouliquen (1999)

Studia Mathematica

Similarity:

We prove trace inequalities of type | | u ' | | L 2 2 + j k j | u ( a j ) | 2 λ | | u | | L 2 2 where u H 1 ( ) , under suitable hypotheses on the sequences a j j and k j j , with the first sequence increasing and the second bounded.