• Title/Summary/Keyword: Laplace operator

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BI-ROTATIONAL HYPERSURFACE SATISFYING ∆IIIx =𝒜x IN 4-SPACE

  • Guler, Erhan;Yayli, Yusuf;Hacisalihoglu, Hasan Hilmi
    • Honam Mathematical Journal
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    • v.44 no.2
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    • pp.219-230
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    • 2022
  • We examine the bi-rotational hypersurface x = x(u, v, w) with the third Laplace-Beltrami operator in the four dimensional Euclidean space 𝔼4. Giving the i-th curvatures of the hypersurface x, we obtain the third Laplace-Beltrami operator of the bi-rotational hypersurface satisfying ∆IIIx =𝒜x for some 4 × 4 matrix 𝒜.

Remarks on volterra equations in Banach spaces

  • Kim, Mi-Hi
    • Communications of the Korean Mathematical Society
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    • v.12 no.4
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    • pp.1039-1064
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    • 1997
  • Existence and Uniqueness for Volterra equations (VE) with a weak regularity assumption on A, the relative closedness of A are investigaed by means of the Laplace transform theory. Also, (VE) are studied by means of the method of convoluted solution operator families.

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MONOTONICITY OF THE FIRST EIGENVALUE OF THE LAPLACE AND THE p-LAPLACE OPERATORS UNDER A FORCED MEAN CURVATURE FLOW

  • Mao, Jing
    • Journal of the Korean Mathematical Society
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    • v.55 no.6
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    • pp.1435-1458
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    • 2018
  • In this paper, we would like to give an answer to Problem 1 below issued firstly in [17]. In fact, by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced mean curvature flow considered here, we can obtain that the first eigenvalues of the Laplace and the p-Laplace operators are monotonic under this flow. Surprisingly, during this process, we get an interesting byproduct, that is, without any complicate constraint, we can give lower bounds for the first nonzero closed eigenvalue of the Laplacian provided additionally the second fundamental form of the initial hypersurface satisfies a pinching condition.

APPLICATIONS OF THE REPRODUCING KERNEL THEORY TO INVERSE PROBLEMS

  • Saitoh, Saburou
    • Communications of the Korean Mathematical Society
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    • v.16 no.3
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    • pp.371-383
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    • 2001
  • In this survey article, we shall introduce the applications of the theory of reproducing kernels to inverse problems. At the same time, we shall present some operator versions of our fundamental general theory for linear transforms in the framework of Hilbert spaces.

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ON THE WEAK LIMIT THEOREMS FOR GEOMETRIC SUMMATIONS OF INDEPENDENT RANDOM VARIABLES TOGETHER WITH CONVERGENCE RATES TO ASYMMETRIC LAPLACE DISTRIBUTIONS

  • Hung, Tran Loc
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.6
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    • pp.1419-1443
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    • 2021
  • The asymmetric Laplace distribution arises as a limiting distribution of geometric summations of independent and identically distributed random variables with finite second moments. The main purpose of this paper is to study the weak limit theorems for geometric summations of independent (not necessarily identically distributed) random variables together with convergence rates to asymmetric Laplace distributions. Using Trotter-operator method, the orders of approximations of the distributions of geometric summations by the asymmetric Laplace distributions are established in term of the "large-𝒪" and "small-o" approximation estimates. The obtained results are extensions of some known ones.

WEIGHTED LEBESGUE NORM INEQUALITIES FOR CERTAIN CLASSES OF OPERATORS

  • Song, Hi Ja
    • Korean Journal of Mathematics
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    • v.14 no.2
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    • pp.137-160
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    • 2006
  • We describe the weight functions for which Hardy's inequality of nonincreasing functions is satisfied. Further we characterize the pairs of weight functions $(w,v)$ for which the Laplace transform $\mathcal{L}f(x)={\int}^{\infty}_0e^{-xy}f(y)dy$, with monotone function $f$, is bounded from the weighted Lebesgue space $L^p(w)$ to $L^q(v)$.

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A NEW QUARTERNIONIC DIRAC OPERATOR ON SYMPLECTIC SUBMANIFOLD OF A PRODUCT SYMPLECTIC MANIFOLD

  • Rashmirekha Patra;Nihar Ranjan Satapathy
    • Korean Journal of Mathematics
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    • v.32 no.1
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    • pp.83-95
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    • 2024
  • The Quaternionic Dirac operator proves instrumental in tackling various challenges within spectral geometry processing and shape analysis. This work involves the introduction of the quaternionic Dirac operator on a symplectic submanifold of an exact symplectic product manifold. The self adjointness of the symplectic quaternionic Dirac operator is observed. This operator is verified for spin ${\frac{1}{2}}$ particles. It factorizes the Hodge Laplace operator on the symplectic submanifold of an exact symplectic product manifold. For achieving this a new complex structure and an almost quaternionic structure are formulated on this exact symplectic product manifold.