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  • Title/Summary/Keyword: weak regularity

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ON MIXED PRESSURE-VELOCITY REGULARITY CRITERIA FOR THE 3D MICROPOLAR EQUATIONS IN LORENTZ SPACES

  • Kim, Jae-Myoung;Kim, Jaewoo
    • Journal of the Chungcheong Mathematical Society
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    • v.34 no.1
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    • pp.85-92
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    • 2021
  • In present paper, inspired by the recently paper [1], we give the mixed pressure-velocity regular criteria in view of Lorentz spaces for weak solutions to 3D micropolar equations in a half space. Precisely, if (0.1) ${\frac{P}{(e^{-{\mid}x{\mid}^2}+{\mid}u{\mid})^{\theta}}{\in}L^p(0,T;L^{q,{\infty}}({\mathbb{R}}^3_+))$, p, q < ∞, and (0.2) 2p+3q=2θ, 0 ≤ θ ≤ 1, then (u, w) is regular on (0, T].

THE WEAK F-REGULARITY OF COHEN-MACAULAY LOCAL RINGS

  • Cho, Y.H.;Moon, M.I.
    • Bulletin of the Korean Mathematical Society
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    • v.28 no.2
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    • pp.175-180
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    • 1991
  • In [3], [4] and [5], Hochster and Huneke introduced the notions of the tight closure of an ideal and of the weak F-regularity of a ring. This notion enabled us to give new proofs of many results in commutative algebra. A regular ring is known to be F-regular, and a Gorenstein local ring is proved to be F-regular provided that one ideal generated by a system of parameters (briefly s.o.p.) is tightly closed. In fact, a Gorenstein local ring is weakly F-regular if and only if there exists a system of parameters ideal which is tightly closed [3]. But we do not know whether this fact is true or not if a ring is not Gorenstein, in particular, a ring is a Cohen Macaulay (briefly C-M) local ring. In this paper, we will prove this in the case of an 1-dimensional C-M local ring. For this, we study the F-rationality and the normality of the ring. And we will also prove that a C-M local ring is to be Gorenstein under some additional condition about the tight closure.

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Note on the Codimension Two Splitting Problem

  • Matsumoto, Yukio
    • Kyungpook Mathematical Journal
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    • v.59 no.3
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    • pp.563-589
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    • 2019
  • Let W and V be manifolds of dimension m + 2, M a locally flat submanifold of V whose dimension is m. Let f:WV be a homotopy equivalence. The problem we study in this paper is the following: When is f homotopic to another homotopy equivalence g:WV such that g is transverse regular along M and such that gg1(M):g1(M)M is a simple homotopy equivalence? Lˊopez de Medrano (1970) called this problem the weak h-regularity problem. We solve this problem applying the codimension two surgery theory developed by the author (1973). We will work in higher dimensions, assuming that m5.

Remarks on Fixed Point Theorems of Non-Lipschitzian Self-mappings

  • Kim, Tae-Hwa;Jeon, Byung-Ik
    • Kyungpook Mathematical Journal
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    • v.45 no.3
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    • pp.433-443
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    • 2005
  • In 1994, Lim-Xu asked whether the Maluta's constant D(X) < 1 implies the fixed point property for asymptotically nonexpansive mappings and gave a partial solution for this question under an additional assumption for T, i.e., weakly asymptotic regularity of T. In this paper, we shall prove that the result due to Lim-Xu is also satisfied for more general non-Lipschitzian mappings in reflexive Banach spaces with weak uniform normal structure. Some applications of this result are also added.

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GRADIENT TYPE ESTIMATES FOR LINEAR ELLIPTIC SYSTEMS FROM COMPOSITE MATERIALS

  • Youchan Kim;Pilsoo Shin
    • Journal of the Korean Mathematical Society
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    • v.60 no.3
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    • pp.635-682
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    • 2023
  • In this paper, we consider linear elliptic systems from composite materials where the coefficients depend on the shape and might have the discontinuity between the subregions. We derive a function which is related to the gradient of the weak solutions and which is not only locally piecewise Hölder continuous but locally Hölder continuous. The gradient of the weak solutions can be estimated by this derived function and we also prove the local piecewise gradient Hölder continuity which was obtained by the previous results.

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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A NOTE ON TIGHT CLOSURE AND FROBENIUS MAP

  • Moon, Myung-In
    • Journal of the Korean Mathematical Society
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    • v.34 no.1
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    • pp.13-21
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    • 1997
  • In recent years M. Hochster and C. Huneke introduced the notions of tight closure of an ideal and of the weak F-regularity of a ring of positive prime characteristic. Here 'F' stands for Frobenius. This notion enabled us to play an important role in a commutative ring theory, and other related topics.

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EXISTENCE, MULTIPLICITY AND REGULARITY OF SOLUTIONS FOR THE FRACTIONAL p-LAPLACIAN EQUATION

  • Kim, Yun-Ho
    • Journal of the Korean Mathematical Society
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    • v.57 no.6
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    • pp.1451-1470
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    • 2020
  • We are concerned with the following elliptic equations: $$\{(-{\Delta})^s_pu={\lambda}f(x,u)\;{\text{in {\Omega}}},\\u=0\;{\text{on {\mathbb{R}}^N{\backslash}{\Omega}},$$ where λ are real parameters, (-∆)sp is the fractional p-Laplacian operator, 0 < s < 1 < p < + ∞, sp < N, and f : Ω × ℝ → ℝ satisfies a Carathéodory condition. By applying abstract critical point results, we establish an estimate of the positive interval of the parameters λ for which our problem admits at least one or two nontrivial weak solutions when the nonlinearity f has the subcritical growth condition. In addition, under adequate conditions, we establish an apriori estimate in L(Ω) of any possible weak solution by applying the bootstrap argument.

PARAMETER CHANGE TEST FOR NONLINEAR TIME SERIES MODELS WITH GARCH TYPE ERRORS

  • Lee, Jiyeon;Lee, Sangyeol
    • Journal of the Korean Mathematical Society
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    • v.52 no.3
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    • pp.503-522
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    • 2015
  • In this paper, we consider the problem of testing for a parameter change in nonlinear time series models with GARCH type errors. We introduce two types of cumulative sum (CUSUM) tests: estimates-based and residual-based tests. It is shown that under regularity conditions, their limiting null distributions are the sup of independent Brownian bridges. A simulation study is conducted for illustration.