Displaying similar documents to “Bigraphic pairs with a realization containing a split bipartite-graph”

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.

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 .

A compactness result in thin-film micromagnetics and the optimality of the Néel wall

Radu Ignat, Felix Otto (2008)

Journal of the European Mathematical Society

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In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for S 1 -valued maps m ' (the magnetization) of two variables x ' : E ε ( m ' ) = ε | ' · m ' | 2 d x ' + 1 2 | ' | - 1 / 2 ' · m ' 2 d x ' . We are interested in the behavior of minimizers as ε 0 . They are expected to be S 1 -valued maps m ' of vanishing distributional divergence ' · m ' = 0 , so that appropriate boundary conditions enforce line discontinuities. For finite ε > 0 , these line discontinuities are approximated by smooth transition layers, the so-called Néel...

Duality of matrix-weighted Besov spaces

Svetlana Roudenko (2004)

Studia Mathematica

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We determine the duals of the homogeneous matrix-weighted Besov spaces p α q ( W ) and p α q ( W ) which were previously defined in [5]. If W is a matrix A p weight, then the dual of p α q ( W ) can be identified with p ' - α q ' ( W - p ' / p ) and, similarly, [ p α q ( W ) ] * p ' - α q ' ( W - p ' / p ) . Moreover, for certain W which may not be in the A p class, the duals of p α q ( W ) and p α q ( W ) are determined and expressed in terms of the Besov spaces p ' - α q ' ( A Q - 1 ) and p ' - α q ' ( A Q - 1 ) , which we define in terms of reducing operators A Q Q associated with W. We also develop the basic theory of these reducing operator Besov spaces....

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...

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...

Elementary operators on Banach algebras and Fourier transform

Miloš Arsenović, Dragoljub Kečkić (2006)

Studia Mathematica

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We consider elementary operators x j = 1 n a j x b j , acting on a unital Banach algebra, where a j and b j are separately commuting families of generalized scalar elements. We give an ascent estimate and a lower bound estimate for such an operator. Additionally, we give a weak variant of the Fuglede-Putnam theorem for an elementary operator with strongly commuting families a j and b j , i.e. a j = a j ' + i a j ' ' ( b j = b j ' + i b j ' ' ), where all a j ' and a j ' ' ( b j ' and b j ' ' ) commute. The main tool is an L¹ estimate of the Fourier transform of a certain class...

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 ).

Comaximal graph of C ( X )

Mehdi Badie (2016)

Commentationes Mathematicae Universitatis Carolinae

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In this article we study the comaximal graph Γ 2 ' C ( X ) of the ring C ( X ) . We have tried to associate the graph properties of Γ 2 ' C ( X ) , the ring properties of C ( X ) and the topological properties of X . Radius, girth, dominating number and clique number of the Γ 2 ' C ( X ) are investigated. We have shown that 2 Rad Γ 2 ' C ( X ) 3 and if | X | > 2 then girth Γ 2 ' C ( X ) = 3 . We give some topological properties of X equivalent to graph properties of Γ 2 ' C ( X ) . Finally we have proved that X is an almost P -space which does not have isolated points if and only if C ( X ) is an almost...

Prime ideal factorization in a number field via Newton polygons

Lhoussain El Fadil (2021)

Czechoslovak Mathematical Journal

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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 ....

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...

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 ) .

On the least almost-prime in arithmetic progression

Jinjiang Li, Min Zhang, Yingchun Cai (2023)

Czechoslovak Mathematical Journal

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Let 𝒫 r denote an almost-prime with at most r prime factors, counted according to multiplicity. Suppose that a and q are positive integers satisfying ( a , q ) = 1 . Denote by 𝒫 2 ( a , q ) the least almost-prime 𝒫 2 which satisfies 𝒫 2 a ( mod q ) . It is proved that for sufficiently large q , there holds 𝒫 2 ( a , q ) q 1 . 8345 . This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range 1 . 845 in place of 1 . 8345 .

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

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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 .

Maximum bipartite subgraphs in H -free graphs

Jing Lin (2022)

Czechoslovak Mathematical Journal

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Given a graph G , let f ( G ) denote the maximum number of edges in a bipartite subgraph of G . Given a fixed graph H and a positive integer m , let f ( m , H ) denote the minimum possible cardinality of f ( G ) , as G ranges over all graphs on m edges that contain no copy of H . In this paper we prove that f ( m , θ k , s ) 1 2 m + Ω ( m ( 2 k + 1 ) / ( 2 k + 2 ) ) , which extends the results of N. Alon, M. Krivelevich, B. Sudakov. Write K k ' and K t , s ' for the subdivisions of K k and K t , s . We show that f ( m , K k ' ) 1 2 m + Ω ( m ( 5 k - 8 ) / ( 6 k - 10 ) ) and f ( m , K t , s ' ) 1 2 m + Ω ( m ( 5 t - 1 ) / ( 6 t - 2 ) ) , improving a result of Q. Zeng, J. Hou. We also give lower bounds on...