• Title/Summary/Keyword: n-harmonic

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SOME RESULTS ON STABLE f-HARMONIC MAPS

  • Embarka, Remli;Cherif, Ahmed Mohammed
    • Communications of the Korean Mathematical Society
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    • v.33 no.3
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    • pp.935-942
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    • 2018
  • In this paper, we prove that any stable f-harmonic map from sphere ${\mathbb{S}}^n$ to Riemannian manifold (N, h) is constant, where f is a smooth positive function on ${\mathbb{S}}^n{\times}N$ satisfying one condition with n > 2. We also prove that any stable f-harmonic map ${\varphi}$ from a compact Riemannian manifold (M, g) to ${\mathbb{S}}^n$ (n > 2) is constant where, in this case, f is a smooth positive function on $M{\times}{\mathbb{S}}^n$ satisfying ${\Delta}^{{\mathbb{S}}^n}(f){\circ}{\varphi}{\leq}0$.

ON THE DENOMINATORS OF 𝜀-HARMONIC NUMBERS

  • Wu, Bing-Ling;Yan, Xiao-Hui
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.6
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    • pp.1383-1392
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    • 2020
  • Let Hn be the n-th harmonic number and let νn be its denominator. Shiu proved that there are infinitely many positive integers n with νn = νn+1. Recently, Wu and Chen proved that the set of positive integers n with νn = νn+1 has density one. They also proved that the same result is true for the denominators of alternating harmonic numbers. In this paper, we prove that the result is true for the denominators of 𝜀-harmonic numbers, where 𝜀 = {𝜀i}i=1 is a pure recurring sequence with 𝜀i ∈ {-1, 1}.

On the Property of Harmonic Vector Field on the Sphere S2n+1

  • Han, Dongsoong
    • Honam Mathematical Journal
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    • v.25 no.1
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    • pp.163-172
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    • 2003
  • In this paper we study the property of harmonic vector fields. We call a vector fields ${\xi}$ harmonic if it is a harmonic map from the manifold into its tangent bundle with the Sasaki metric. We show that the characteristic polynomial of operator $A={\nabla}{\xi}\;in\;S^{2n+1}\;is\;(x^2+1)^n$.

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ON THE SPECTRAL GEOMETRY FOR THE JACOBI OPERATORS OF HARMONIC MAPS INTO PRODUCT MANIFOLDS

  • Kang, Tae-Ho;Ki, U-Hang;Pak, Jin-Suk
    • Journal of the Korean Mathematical Society
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    • v.34 no.2
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    • pp.483-500
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    • 1997
  • We investigate the geometric properties reflected by the spectra of the Jacobi operator of a harmonic map when the target manifold is a Riemannian product manifold or a Kaehlerian product manifold. And also we study the spectral characterization of Riemannian sumersions when the target manifold is $S^n \times S^n$ or $CP^n \times CP^n$.

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ON THE p-ADIC VALUATION OF GENERALIZED HARMONIC NUMBERS

  • Cagatay Altuntas
    • Bulletin of the Korean Mathematical Society
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    • v.60 no.4
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    • pp.933-955
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    • 2023
  • For any prime number p, let J(p) be the set of positive integers n such that the numerator of the nth harmonic number in the lowest terms is divisible by this prime number p. We consider an extension of this set to the generalized harmonic numbers, which are a natural extension of the harmonic numbers. Then, we present an upper bound for the number of elements in this set. Moreover, we state an explicit condition to show the finiteness of our set, together with relations to Bernoulli and Euler numbers.

ON F-HARMONIC MAPS AND CONVEX FUNCTIONS

  • Kang, Tae-Ho
    • East Asian mathematical journal
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    • v.19 no.2
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    • pp.165-171
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    • 2003
  • We show that any F-harmonic map from a compact manifold M to N is necessarily constant if N possesses a strictly-convex function, and prove 'Liouville type theorems' for F-harmonic maps. Finally, when the target manifold is the real line, we get a result for F-subharmonic functions.

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HARMONIC MORPHISMS AND STABLE MINIMAL SUBMANIFOLDS

  • Choi, Gundon;Yun, Gabjin
    • Korean Journal of Mathematics
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    • v.14 no.1
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    • pp.13-33
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    • 2006
  • In this article, we study the relations of horizontally conformal maps and harmonic morphisms with the stability of minimal fibers. Let ${\varphi}:(M^n,g){\rightarrow}(N^m,h)$ be a horizontally conformal submersion. There is a tensor T measuring minimality or totally geodesics of fibers of ${\varphi}$. We prove that if T is parallel and the horizontal distribution is integrable, then any minimal fiber of ${\varphi}$ is volume-stable. As a corollary, we obtain that any fiber of a submersive harmonic morphism whose fibers are totally geodesics and the horizontal distribution is integrable is volume-stable. As a consequence, we obtain if ${\varphi}:(M^n,g){\rightarrow}(N^2,h)$ is a submersive harmonic morphism of minimal fibers from a compact Riemannian manifold M into a surface N, T is parallel and the horizontal distribution is integrable, then ${\varphi}$ is energy-stable.

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STABLE f-HARMONIC MAPS ON SPHERE

  • CHERIF, AHMED MOHAMMED;DJAA, MUSTAPHA;ZEGGA, KADDOUR
    • Communications of the Korean Mathematical Society
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    • v.30 no.4
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    • pp.471-479
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    • 2015
  • In this paper, we prove that any stable f-harmonic map ${\psi}$ from ${\mathbb{S}}^2$ to N is a holomorphic or anti-holomorphic map, where N is a $K{\ddot{a}}hlerian$ manifold with non-positive holomorphic bisectional curvature and f is a smooth positive function on the sphere ${\mathbb{S}}^2$with Hess $f{\leq}0$. We also prove that any stable f-harmonic map ${\psi}$ from sphere ${\mathbb{S}}^n$ (n > 2) to Riemannian manifold N is constant.

HARMONIC CONJUGATES OF WEIGHTED HARMONIC BERGMAN FUNCTIONS ON HALF-SPACES

  • Nam, Kye-Sook;Yi, Heung-Su
    • Communications of the Korean Mathematical Society
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    • v.18 no.3
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    • pp.449-457
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    • 2003
  • On the setting of the upper half-space of the Euclidean space $R^{n}$, we show that to each weighted harmonic Bergman function $u\;\epsilon\;b^p_{\alpha}$, there corresponds a unique conjugate system ($upsilon$_1,…, $upsilon_{n-1}$) of u satisfying $upsilon_j{\epsilon}\;b^p_{\alpha}$ with an appropriate norm bound.

LIOUVILLE TYPE THEOREM FOR p-HARMONIC MAPS II

  • Jung, Seoung Dal
    • Communications of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.155-161
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    • 2014
  • Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that $Ric^M{\geq}-\frac{4(p-1)}{p^2}{\mu}_0$ at all $x{\in}M$ and Vol(M) is infinite, where ${\mu}_0$ > 0 is the infimum of the spectrum of the Laplacian acting on $L^2$-functions on M. Then any p-harmonic map ${\phi}:M{\rightarrow}N$ of finite p-energy is constant Also, we study Liouville type theorem for p-harmonic morphism.