Displaying similar documents to “Earnshaw’s Theorem in Electrostatics and a Conditional Converse to Dirichlet’s Theorem”

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...

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 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...

Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

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Let Ω be a bounded domain of class C 2 in N and let K be a compact subset of Ω . Assume that q ( N + 1 ) / ( N 1 ) and denote by U K the maximal solution of Δ u + u q = 0 in Ω which vanishes on Ω K . We obtain sharp upper and lower estimates for U K in terms of the Bessel capacity C 2 / q , q ' and prove that U K is σ -moderate. In addition we describe the precise asymptotic behavior of U K at points σ K , which depends on the “density” of K at σ , measured in terms of the capacity C 2 / q , q ' .

Admissible spaces for a first order differential equation with delayed argument

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

Czechoslovak Mathematical Journal

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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 .

Variations on a question concerning the degrees of divisors of x n - 1

Lola Thompson (2014)

Journal de Théorie des Nombres de Bordeaux

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In this paper, we examine a natural question concerning the divisors of the polynomial x n - 1 : “How often does x n - 1 have a divisor of every degree between 1 and n ?” In a previous paper, we considered the situation when x n - 1 is factored in [ x ] . In this paper, we replace [ x ] with 𝔽 p [ x ] , where p is an arbitrary-but-fixed prime. We also consider those n where this condition holds for all p .

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...

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 ) .

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 ).

Recognition of some families of finite simple groups by order and set of orders of vanishing elements

Maryam Khatami, Azam Babai (2018)

Czechoslovak Mathematical Journal

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Let G be a finite group. An element g G is called a vanishing element if there exists an irreducible complex character χ of G such that χ ( g ) = 0 . Denote by Vo ( G ) the set of orders of vanishing elements of G . Ghasemabadi, Iranmanesh, Mavadatpour (2015), in their paper presented the following conjecture: Let G be a finite group and M a finite nonabelian simple group such that Vo ( G ) = Vo ( M ) and | G | = | M | . Then G M . We answer in affirmative this conjecture for M = S z ( q ) , where q = 2 2 n + 1 and either q - 1 , q - 2 q + 1 or q + 2 q + 1 is a prime number, and M = F 4 ( q ) , where...

Nonvanishing of a certain Bernoulli number and a related topic

Humio Ichimura (2013)

Acta Arithmetica

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Let p = 1 + 2 e + 1 q be an odd prime number with q an odd integer. Let δ (resp. φ) be an odd (resp. even) Dirichlet character of conductor p and order 2 e + 1 (resp. order d φ dividing q), and let ψₙ be an even character of conductor p n + 1 and order pⁿ. We put χ = δφψₙ, whose value is contained in K = ( ζ ( p - 1 ) p ) . It is well known that the Bernoulli number B 1 , χ is not zero, which is shown in an analytic way. In the extreme cases d φ = 1 and q, we show, in an algebraic and elementary manner, a stronger nonvanishing result: T r n / 1 ( ξ B 1 , χ ) 0 for any...

On the distribution of ( k , r ) -integers in Piatetski-Shapiro sequences

Teerapat Srichan (2021)

Czechoslovak Mathematical Journal

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A natural number n is said to be a ( k , r ) -integer if n = a k b , where k > r > 1 and b is not divisible by the r th power of any prime. We study the distribution of such ( k , r ) -integers in the Piatetski-Shapiro sequence { n c } with c > 1 . As a corollary, we also obtain similar results for semi- r -free integers.

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.

Local-global divisibility of rational points in some commutative algebraic groups

Roberto Dvornicich, Umberto Zannier (2001)

Bulletin de la Société Mathématique de France

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Let 𝒜 be a commutative algebraic group defined over a number field  k . We consider the following question:A complete answer for the case of the multiplicative group 𝔾 m is classical. We study other instances and in particular obtain an affirmative answer when r is a prime and  𝒜 is either an elliptic curve or a torus of small dimension with respect to r . Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when r is a prime. ...

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...

A Diophantine inequality with four squares and one k th power of primes

Quanwu Mu, Minhui Zhu, Ping Li (2019)

Czechoslovak Mathematical Journal

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Let k 5 be an odd integer and η be any given real number. We prove that if λ 1 , λ 2 , λ 3 , λ 4 , μ are nonzero real numbers, not all of the same sign, and λ 1 / λ 2 is irrational, then for any real number σ with 0 < σ < 1 / ( 8 ϑ ( k ) ) , the inequality | λ 1 p 1 2 + λ 2 p 2 2 + λ 3 p 3 2 + λ 4 p 4 2 + μ p 5 k + η | < max 1 j 5 p j - σ has infinitely many solutions in prime variables p 1 , p 2 , , p 5 , where ϑ ( k ) = 3 × 2 ( k - 5 ) / 2 for k = 5 , 7 , 9 and ϑ ( k ) = [ ( k 2 + 2 k + 5 ) / 8 ] for odd integer k with k 11 . This improves a recent result in W. Ge, T. Wang (2018).