• 대한수학회지
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• 제39권2호
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• pp.237-252
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• 2002

We study the compatibility conditions and the existence of solutions or overdetermined PDE systems that admit complete prolongation. For a complete system of order k there exists a submanifold of the ($\kappa$-1)st jet space of unknown functions that is the largest possible set on which the initial conditions of ($\kappa$-1)st order may take values. There exists a unique solution for any initial condition that belongs to this set if and only if the complete system satisfies the compatibility conditions on the initial data set. We prove by applying the Frobenius theorem to a Pfaffian differential system associated with the complete prolongation.

• 호남수학학술지
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• 제45권2호
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• pp.248-268
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• 2023

As a generalization of conformal semi-invariant submersion, conformal slant submersion and conformal semi-slant submersion, in this paper we study conformal hemi-slant submersion from Kenmotsu manifold onto a Riemannian manifold. The necessary and sufficient conditions for the integrability and totally geodesicness of distributions are discussed. Moreover, we have obtained sufficient condition for a conformal hemi-slant submersion to be a homothetic map. The condition for a total manifold of the submersion to be twisted product is studied, followed by other decomposition theorems.

• 대한수학회지
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• 제43권4호
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• pp.717-732
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• 2006

In this paper, we study both slant 3nd semi-slant sub-manifolds of an almost product Riemannian manifold. We give characterization theorems for slant and semi-slant submanifolds and investigate special class of slant submanifolds which are product version of Kaehlerian slant submanifold. We also obtain integrability conditions for the distributions which are involved in the definition of a semi-slant submanifold. Finally, we prove a theorem on the geometry of leaves of distributions under a condition.

• 대한수학회보
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• 제38권3호
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• pp.605-610
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• 2001

It is shown that for each 1 < p < 2 there exist identically distributed uncorrelated random variables $X_n\; with\;E({$\mid$X_1$\mid$}^p)\;<\;{\infty}$, not fulfilling the weak law of large numbers (WLLN). If, however, the random variables are moreover non-negative, the weaker integrability condition $E(X_1\;log\;X_1)\;<\;{\infty}$ already guarantees the strong law of large numbers.

• 대한수학회보
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• 제50권3호
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• pp.951-962
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• 2013

We introduce semi-slant submersions from almost Hermitian manifolds onto Riemannian manifolds as a generalization of slant submersions, semi-invariant submersions, anti-invariant submersions, etc. We obtain characterizations, investigate the integrability of distributions and the geometry of foliations, etc. We also find a condition for such submersions to be harmonic. Moreover, we give lots of examples.

• 한국지능시스템학회논문지
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• 제14권7호
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• pp.852-856
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• 2004

이 논문에서는 퍼지 랜덤 변수의 합에 대한 약한 대수의 법칙을 일반화로서, 컴팩트 일양 적분 가능한 수준 연속 퍼지 랜덤 변수의 가중 합이 약 수렴하기 위한 동치 조건을 구하였다.

• 대한수학회보
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• 제53권2호
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• pp.441-460
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• 2016

We introduce the notions of h-v-semi-slant submersions and almost h-v-semi-slant submersions from almost quaternionic Hermitian manifolds onto Riemannian manifolds. We obtain characterizations, investigate the integrability of distributions, the geometry of foliations, and a decomposition theorem. We find a condition for such submersions to be totally geodesic. We also obtain an inequality of a h-v-semi-slant submersion in terms of squared mean curvature, scalar curvature, and h-v-semi-slant angle. Finally, we give examples of such maps.

• 대한수학회논문집
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• 제36권1호
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• pp.173-187
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• 2021

In this paper, we deal with the notion of a v-semi-slant submersion from an almost Hermitian manifold onto a Riemannian manifold. We investigate the integrability of distributions, the geometry of foliations, and a decomposition theorem. Given such a map with totally umbilical fibers, we have a condition for the fibers of the map to be minimal. We also obtain an inequality of a proper v-semi-slant submersion in terms of squared mean curvature, scalar curvature, and a v-semi-slant angle. Moreover, we give some examples of such maps and some open problems.

• 대한수학회보
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• 제46권5호
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• pp.931-940
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• 2009

Given a system of smooth 1-forms $\theta$ = ($\theta^1$,...,$\theta^s$) on a smooth manifold $M^m$, we give a necessary and sufficient condition for M to be foliated by integral manifolds of dimension n, n $\leq$ p := m - s, and construct an integrable supersystem ($\theta,\eta$) by finding additional 1-forms $\eta$ = ($\eta^1$,...,$\eta^{p-n}$). We also give a necessary and sufficient condition for M to be foliated by reduced submanifolds of dimension n, n $\geq$ p, and construct an integrable subsystem ($d\rho^1$,...,$d\rho^{m-n}$) by finding a system of first integrals $\rho=(\rho^1$,...,$\rho^{m-n})$. The special case n = p is the Frobenius theorem on involutivity.

• 대한수학회지
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• 제58권5호
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• pp.1109-1129
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• 2021

This paper considers an anisotropic polytropic infiltration equation with a source term $$u_t={\sum\limits_{i=1}^{N}}{\frac{{\partial}}{{\partial}x_i}}\(a_1(x){\mid}u{\mid}^{{\alpha}_i}{\mid}u_{x_i}{\mid}^{p_i-2}u_{x_i}\)+f(x,t,u)$$, where p_i > 1, α_i > 0, a_i(x) ≥ 0. The existence of weak solution is proved by parabolically regularized method. Based on local integrability $u_{x_i}{\in}W_{loc}^{1,p_i}(\Omega)$, the stability of weak solutions is proved without boundary value condition by the weak characteristic function method. One of the essential characteristics of an anisotropic equation different from an isotropic equation is found originally.