Displaying similar documents to “Harmonic functions in four variables with rational and algebraic p 4 -associates”

Landau's theorem for p-harmonic mappings in several variables

Sh. Chen, S. Ponnusamy, X. Wang (2012)

Annales Polonici Mathematici

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A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation Δ p f = 0 , where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function f : Ω m is said to be p-harmonic in Ω if each component function f i (i∈ 1,...,m) of f = ( f , . . . , f m ) is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...

Composite rational functions expressible with few terms

Clemens Fuchs, Umberto Zannier (2012)

Journal of the European Mathematical Society

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We consider a rational function f which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number of terms. Then we look at the possible decompositions f ( x ) = g ( h ( x ) ) , where g , h are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of g is bounded only in terms of (and we provide explicit bounds). This supports and quantifies...

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

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

Communications in Mathematics

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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 ) .

A note on Sierpiński's problem related to triangular numbers

Maciej Ulas (2009)

Colloquium Mathematicae

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We show that the system of equations t x + t y = t p , t y + t z = t q , t x + t z = t r , where t x = x ( x + 1 ) / 2 is a triangular number, has infinitely many solutions in integers. Moreover, we show that this system has a rational three-parameter solution. Using this result we show that the system t x + t y = t p , t y + t z = t q , t x + t z = t r , t x + t y + t z = t s has infinitely many rational two-parameter solutions.

On the dimension of p -harmonic measure in space

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

Journal of the European Mathematical Society

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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...

On a question of T. Sheil-Small regarding valency of harmonic maps

Daoud Bshouty, Abdallah Lyzzaik (2012)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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The aim of this work is to answer positively a more general question than the following which is due to T. Sheil-Small: Does the harmonic extension in the open unit disc of a mapping f from the unit circle into itself of the form f ( e i t ) = e i φ ( t ) , 0 t 2 π where φ is a continuously non-decreasing function that satisfies φ ( 2 π ) - φ ( 0 ) = 2 N π , assume every value finitely many times in the disc?

On the integral representation of finely superharmonic functions

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

Commentationes Mathematicae Universitatis Carolinae

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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...

Harmonie reflections

Lieven Vanhecke, Maria-Elena Vazquez-Abal (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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We study local reflections ϕ σ with respect to a curve σ in a Riemannian manifold and prove that σ is a geodesic if ϕ σ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if ϕ σ is harmonic for all geodesies σ .

Some characterizations of harmonic Bloch and Besov spaces

Xi Fu, Bowen Lu (2016)

Czechoslovak Mathematical Journal

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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).

Deformations of Metrics and Biharmonic Maps

Aicha Benkartab, Ahmed Mohammed Cherif (2020)

Communications in Mathematics

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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

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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.

Rational points on curves

Michael Stoll (2011)

Journal de Théorie des Nombres de Bordeaux

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This is an extended version of an invited lecture I gave at the Journées Arithmétiques in St. Étienne in July 2009. We discuss the state of the art regarding the problem of finding the set of rational points on a (smooth projective) geometrically integral curve  C over  . The focus is on practical aspects of this problem in the case that the genus of  C is at least  2 , and therefore the set of rational points is finite.

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

Ferenc Móricz (2002)

Studia Mathematica

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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...

Algebraic independence of the values at algebraic points of a class of functions considered by Mahler

N. Ch. Wass

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This thesis is concerned with the problem of determining a measure of algebraic independence for a particular m-tuple θ₁,..., θ m of complex numbers. Specifically, let K be a number field and let f₁(z),..., f m ( z ) be elements of K[[z]] algebraically independent over K(z) satisfying equations of the form(*) f j ( z b ) = i = 1 m f i ( z ) a i j ( z ) + b j ( z ) (j = i,...,m)for b ≥ 2, a i j ( z ) , b j ( z ) in K(z). Suppose finally that α ∈ K is such that 0 < |α| < 1, the f j ( z ) converge at z = α and the a i j ( z ) , b j ( z ) are analytic at z = α , α b , α b ² , . . . Then the θ i = f i ( α ) are algebraically independent...

Manin’s and Peyre’s conjectures on rational points and adelic mixing

Alex Gorodnik, François Maucourant, Hee Oh (2008)

Annales scientifiques de l'École Normale Supérieure

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Let X be the wonderful compactification of a connected adjoint semisimple group G defined over a number field K . We prove Manin’s conjecture on the asymptotic (as T ) of the number of K -rational points of X of height less than T , and give an explicit construction of a measure on X ( 𝔸 ) , generalizing Peyre’s measure, which describes the asymptotic distribution of the rational points 𝐆 ( K ) on X ( 𝔸 ) . Our approach is based on the mixing property of L 2 ( 𝐆 ( K ) 𝐆 ( 𝔸 ) ) which we obtain with a rate of convergence. ...

Arithmetic theory of harmonic numbers (II)

Zhi-Wei Sun, Li-Lu Zhao (2013)

Colloquium Mathematicae

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For k = 1,2,... let H k denote the harmonic number j = 1 k 1 / j . In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime p > 3 we have k = 1 p - 1 ( H k ) / ( k 2 k ) 7 / 24 p B p - 3 ( m o d p ² ) , k = 1 p - 1 ( H k , 2 ) / ( k 2 k ) - 3 / 8 B p - 3 ( m o d p ) , and k = 1 p - 1 ( H ² k , 2 n ) / ( k 2 n ) ( 6 n + 1 2 n - 1 + n ) / ( 6 n + 1 ) p B p - 1 - 6 n ( m o d p ² ) for any positive integer n < (p-1)/6, where B₀,B₁,B₂,... are Bernoulli numbers, and H k , m : = j = 1 k 1 / ( j m ) .

L p harmonic 1 -form on submanifold with weighted Poincaré inequality

Xiaoli Chao, Yusha Lv (2018)

Czechoslovak Mathematical Journal

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We deal with complete submanifolds with weighted Poincaré inequality. By assuming the submanifold is δ -stable or has sufficiently small total curvature, we establish two vanishing theorems for L p harmonic 1 -forms, which are extensions of the results of Dung-Seo and Cavalcante-Mirandola-Vitório.

Multiplicatively dependent triples of Tribonacci numbers

Carlos Alexis Ruiz Gómez, Florian Luca (2015)

Acta Arithmetica

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We consider the Tribonacci sequence T : = T n n 0 given by T₀ = 0, T₁ = T₂ = 1 and T n + 3 = T n + 2 + T n + 1 + T n for all n ≥ 0, and we find all triples of Tribonacci numbers which are multiplicatively dependent.

Injectivity of sections of convex harmonic mappings and convolution theorems

Liulan Li, Saminathan Ponnusamy (2016)

Czechoslovak Mathematical Journal

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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...

Harmonie reflections

Lieven Vanhecke, Maria-Elena Vazquez-Abal (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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We study local reflections ϕ σ with respect to a curve σ in a Riemannian manifold and prove that σ is a geodesic if ϕ σ is a harmonic map. Moreover, we prove that the Riemannian manifold has constant curvature if and only if ϕ σ is harmonic for all geodesies σ .