Displaying similar documents to “Arithmetic theory of harmonic numbers (II)”

On the dimension of p -harmonic measure in space

John L. Lewis, Kaj Nyström, Andrew Vogel (2013)

Journal of the European Mathematical Society

Similarity:

Let Ω n , n 3 , and let p , 1 < p < , p 2 , be given. In this paper we study the dimension of p -harmonic measures that arise from non-negative solutions to the p -Laplace equation, vanishing on a portion of Ω , in the setting of δ -Reifenberg flat domains. We prove, for p n , that there exists δ ˜ = δ ˜ ( p , n ) > 0 small such that if Ω is a δ -Reifenberg flat domain with δ < δ ˜ , then p -harmonic measure is concentrated on a set of σ -finite H n 1 -measure. We prove, for p n , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of p -harmonic...

The harmonic Cesáro and Copson operators on the spaces L p ( ) , 1 ≤ p ≤ 2

Ferenc Móricz (2002)

Studia Mathematica

Similarity:

The harmonic Cesàro operator is defined for a function f in L p ( ) for some 1 ≤ p < ∞ by setting ( f ) ( x ) : = x ( f ( u ) / u ) d u for x > 0 and ( f ) ( x ) : = - - x ( f ( u ) / u ) d u for x < 0; the harmonic Copson operator ℂ* is defined for a function f in L ¹ l o c ( ) by setting * ( f ) ( x ) : = ( 1 / x ) x f ( u ) d u for x ≠ 0. The notation indicates that ℂ and ℂ* are adjoint operators in a certain sense. We present rigorous proofs of the following two commuting relations: (i) If f L p ( ) for some 1 ≤ p ≤ 2, then ( ( f ) ) ( t ) = * ( f ̂ ) ( t ) a.e., where f̂ denotes the Fourier transform of f. (ii) If f L p ( ) for some 1 < p ≤ 2, then...

Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li, Saminathan Ponnusamy (2016)

Czechoslovak Mathematical Journal

Similarity:

We consider the class 0 of sense-preserving harmonic functions f = h + g ¯ defined in the unit disk | z | < 1 and normalized so that h ( 0 ) = 0 = h ' ( 0 ) - 1 and g ( 0 ) = 0 = g ' ( 0 ) , where h and g are analytic in the unit disk. In the first part of the article we present two classes 𝒫 H 0 ( α ) and 𝒢 H 0 ( β ) of functions from 0 and show that if f 𝒫 H 0 ( α ) and F 𝒢 H 0 ( β ) , then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters α and β are satisfied. In the second part we study the harmonic sections...

On the integral representation of finely superharmonic functions

Abderrahim Aslimani, Imad El Ghazi, Mohamed El Kadiri (2019)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset U of a Brelot 𝒫 -harmonic space Ω with countable base of open subsets and satisfying the axiom D . When Ω satisfies the hypothesis of uniqueness, we define the Martin boundary of U and the Martin kernel K and we obtain the integral representation of invariant functions by using the kernel K . As an application of the integral representation we extend to the cone...

Some characterizations of harmonic Bloch and Besov spaces

Xi Fu, Bowen Lu (2016)

Czechoslovak Mathematical Journal

Similarity:

The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic ω - α -Bloch space and characterize it in terms of ω ( ( 1 - | x | 2 ) β ( 1 - | y | 2 ) α - β ) | f ( x ) - f ( y ) x - y | and ω ( ( 1 - | x | 2 ) β ( 1 - | y | 2 ) α - β ) | f ( x ) - f ( y ) | x | y - x ' | where ω is a majorant. Similar results are extended to harmonic little ω - α -Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G. Ren, U. Kähler (2005).

A Weighted Eigenvalue Problems Driven by both p ( · ) -Harmonic and p ( · ) -Biharmonic Operators

Mohamed Laghzal, Abdelouahed El Khalil, Abdelfattah Touzani (2021)

Communications in Mathematics

Similarity:

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p ( · ) -Harmonic and p ( · ) -biharmonic operators Δ p ( x ) 2 u - Δ p ( x ) u = λ w ( x ) | u | q ( x ) - 2 u in Ω , u W 2 , p ( · ) ( Ω ) W 0 1 , p ( · ) ( Ω ) , is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces L p ( · ) ( Ω ) and W m , p ( · ) ( Ω ) .

On power integral bases for certain pure number fields defined by x 18 - m

Lhoussain El Fadil (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

Let K = ( α ) be a number field generated by a complex root α of a monic irreducible polynomial f ( x ) = x 18 - m , m 1 , is a square free rational integer. We prove that if m 2 or 3 ( mod 4 ) and m ¬ 1 ( mod 9 ) , then the number field K is monogenic. If m 1 ( mod 4 ) or m 1 ( mod 9 ) , then the number field K is not monogenic.

Modular symbols, Eisenstein series, and congruences

Jay Heumann, Vinayak Vatsal (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let E and f be an Eisenstein series and a cusp form, respectively, of the same weight k 2 and of the same level N , both eigenfunctions of the Hecke operators, and both normalized so that a 1 ( f ) = a 1 ( E ) = 1 . The main result we prove is that when E and f are congruent mod a prime 𝔭 (which we take in this paper to be a prime of ¯ lying over a rational prime p &gt; 2 ), the algebraic parts of the special values L ( E , χ , j ) and L ( f , χ , j ) satisfy congruences mod the same prime. More explicitly, we prove that, under certain conditions, ...

On the Lucas sequence equations Vₙ = kVₘ and Uₙ = kUₘ

Refik Keskin, Zafer Şiar (2013)

Colloquium Mathematicae

Similarity:

Let P and Q be nonzero integers. The sequences of generalized Fibonacci and Lucas numbers are defined by U₀ = 0, U₁ = 1 and U n + 1 = P U - Q U n - 1 for n ≥ 1, and V₀ = 2, V₁ = P and V n + 1 = P V - Q V n - 1 for n ≥ 1, respectively. In this paper, we assume that P ≥ 1, Q is odd, (P,Q) = 1, Vₘ ≠ 1, and V r 1 . We show that there is no integer x such that V = V r V x ² when m ≥ 1 and r is an even integer. Also we completely solve the equation V = V V r x ² for m ≥ 1 and r ≥ 1 when Q ≡ 7 (mod 8) and x is an even integer. Then we show that when P ≡ 3 (mod 4) and...

Natural pseudodistances between closed surfaces

Pietro Donatini, Patrizio Frosini (2007)

Journal of the European Mathematical Society

Similarity:

Let us consider two closed surfaces , 𝒩 of class C 1 and two functions ϕ : , ψ : 𝒩 of class C 1 , called measuring functions. The natural pseudodistance d between the pairs ( , ) , ( 𝒩 , ψ ) is defined as the infimum of Θ ( f ) : = max P | ϕ ( P ) ψ ( f ( P ) ) | as f varies in the set of all homeomorphisms from onto 𝒩 . In this paper we prove that the natural pseudodistance equals either | c 1 c 2 | , 1 2 | c 1 c 2 | , or 1 3 | c 1 c 2 | , where c 1 and c 2 are two suitable critical values of the measuring functions. This shows that a previous relation between the natural pseudodistance and...

On the characterization of harmonic functions with initial data in Morrey space

Bo Li, Jinxia Li, Bolin Ma, Tianjun Shen (2024)

Czechoslovak Mathematical Journal

Similarity:

Let ( X , d , μ ) be a metric measure space satisfying the doubling condition and an L 2 -Poincaré inequality. Consider the nonnegative operator generalized by a Dirichlet form on X . We will show that a solution u to ( - t 2 + ) u = 0 on X × + satisfies an α -Carleson condition if and only if u can be represented as the Poisson integral of the operator with the trace in the generalized Morrey space L 2 , α ( X ) , where α is a nonnegative function defined on a class of balls in X . This result extends the analogous characterization...

Deformations of Metrics and Biharmonic Maps

Aicha Benkartab, Ahmed Mohammed Cherif (2020)

Communications in Mathematics

Similarity:

We construct biharmonic non-harmonic maps between Riemannian manifolds ( M , g ) and ( N , h ) by first making the ansatz that ϕ : ( M , g ) ( N , h ) be a harmonic map and then deforming the metric on N by h ˜ α = α h + ( 1 - α ) d f d f to render ϕ biharmonic, where f is a smooth function with gradient of constant norm on ( N , h ) and α ( 0 , 1 ) . We construct new examples of biharmonic non-harmonic maps, and we characterize the biharmonicity of some curves on Riemannian manifolds.

Sharp Weak-Type Inequality for the Haar System, Harmonic Functions and Martingales

Adam Osękowski (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

Let ( h k ) k 0 be the Haar system on [0,1]. We show that for any vectors a k from a separable Hilbert space and any ε k [ - 1 , 1 ] , k = 0,1,2,..., we have the sharp inequality | | k = 0 n ε k a k h k | | W ( [ 0 , 1 ] ) 2 | | k = 0 n a k h k | | L ( [ 0 , 1 ] ) , n = 0,1,2,..., where W([0,1]) is the weak- L space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound | | Y | | W ( Ω ) 2 | | X | | L ( Ω ) , where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

On the potential theory of some systems of coupled PDEs

Abderrahim Aslimani, Imad El Ghazi, Mohamed El Kadiri, Sabah Haddad (2016)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In this paper we study some potential theoretical properties of solutions and super-solutions of some PDE systems (S) of type L 1 u = - μ 1 v , L 2 v = - μ 2 u , on a domain D of d , where μ 1 and μ 2 are suitable measures on D , and L 1 , L 2 are two second order linear differential elliptic operators on D with coefficients of class 𝒞 . We also obtain the integral representation of the nonnegative solutions and supersolutions of the system (S) by means of the Green kernels and Martin boundaries associated with L 1 and L 2 , and...

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...