• 호남수학학술지
• /
• 제41권3호
• /
• pp.595-607
• /
• 2019

We consider quasigroups $Q({\Gamma})$ obtained as certain double covers of the symmetric group $S_3$ of degree 3, for directed graphs ${\Gamma}$ on the vertex set $S_3$. We completely characterize the strong compatibility of elements of $Q({\Gamma})$ for any quasigroup nonuniform homogeneous space of degree 4. For such homogeneous spaces, we classify all the strong and weak compatibility graphs of $Q({\Gamma})$.

• 대한수학회논문집
• /
• 제30권3호
• /
• pp.253-267
• /
• 2015

Given a complex submanifoldM of the projective space $\mathbb{P}$(T), the hyperplane system R on M characterizes the projective embedding of M into $\mathbb{P}$(T) in the following sense: for any two nondegenerate complex submanifolds $M{\subset}\mathbb{P}$(T) and $M^{\prime}{\subset}\mathbb{P}$(T'), there is a projective linear transformation that sends an open subset of M onto an open subset of M' if and only if (M,R) is locally equivalent to (M', R'). Se-ashi developed a theory for the differential invariants of these types of systems of linear differential equations. In particular, the theory applies to systems of linear differential equations that have symbols equivalent to the hyperplane systems on nondegenerate equivariant embeddings of compact Hermitian symmetric spaces. In this paper, we extend this result to hyperplane systems on nondegenerate equivariant embeddings of homogeneous spaces of the first kind.

• 대한수학회보
• /
• 제49권5호
• /
• pp.1101-1129
• /
• 2012

We study the geometry of the images of 1-dimensional homogeneous submersions for each of the model spaces X of the eight 3-dimensional geometries. In particular, We shall calculate the group of isometries and the curvatures of the base surfaces for each of the model spaces of 3-dimensional geometries, with respect to every closed subgroup of the isometries of X acting freely.

• 충청수학회지
• /
• 제25권1호
• /
• pp.35-42
• /
• 2012

In this paper we first introduce a Carleson inequality and study the generalized form of Carleson inequality in the context of spaces of homogeneous type. The previous inequality is known to play important roles in harmonic analysis.

• 대한수학회지
• /
• 제54권1호
• /
• pp.117-139
• /
• 2017

This paper presents a systematic study for theoretical aspects of a unified approach to the abstract relative Fourier transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor. Let H be a locally compact group, K be a locally compact Abelian (LCA) group, and ${\theta}:H{\rightarrow}Aut(K)$ be a continuous homomorphism. Let $G_{\theta}=H{\ltimes}_{\theta}K$ be the semi-direct product of H and K with respect to ${\theta}$ and $G_{\theta}/H$ be the canonical homogeneous space (left coset space) of $G_{\theta}$. We introduce the notions of relative dual homogeneous space and also abstract relative Fourier transform over $G_{\theta}/H$. Then we study theoretical properties of this approach.

• 대한수학회지
• /
• 제41권1호
• /
• pp.157-174
• /
• 2004

In this work we discuss "plank problems" for complex Banach spaces and in particular for the classical $L^{p}(\mu)$ spaces. In the case $1\;{\leq}\;p\;{\leq}\;2$ we obtain optimal results and for finite dimensional complex Banach spaces, in a special case, we have improved an early result by K. Ball [3]. By using these results, in some cases we are able to find best possible lower bounds for the norms of homogeneous polynomials which are products of linear forms. In particular, we give an estimate in the case of a real Hilbert space which seems to be a difficult problem. We have also obtained some results on the so-called n-th (linear) polarization constant of a Banach space which is an isometric property of the space. Finally, known polynomial inequalities have been derived as simple consequences of various results related to plank problems.

• 충청수학회지
• /
• 제23권3호
• /
• pp.453-461
• /
• 2010

We first introduce a space of homogeneous type X, and develop the theory of the tent spaces on the generalized upper half-space $X{\times}(0,{\infty})$. The goal of this paper is to study that every element of the tent spaces $T_{\Omega}^{p}$($X{\times}(0,{\infty})$, $0, can be decomposed into elementary particles which are called "atoms."

• 대한수학회보
• /
• 제32권2호
• /
• pp.281-285
• /
• 1995

The complete classification of the simply connected four dimensional naturally reductive homogeneous spaces is given in [3].

• 대한수학회지
• /
• 제53권2호
• /
• pp.315-329
• /
• 2016

In the setting of continuous logic, we study atomless probability spaces and atomless random variable structures. We characterize ${\kappa}$-saturated atomless probability spaces and ${\kappa}$-saturated atomless random variable structures for every infinite cardinal ${\kappa}$. Moreover, ${\kappa}$-saturated and strongly ${\kappa}$-homogeneous atomless probability spaces and ${\kappa}$-saturated and strongly ${\kappa}$-homogeneous atomless random variable structures are characterized for every infinite cardinal ${\kappa}$. For atomless probability spaces, we prove that ${\aleph}_1$-saturation is equivalent to Hoover-Keisler saturation. For atomless random variable structures whose underlying probability spaces are Hoover-Keisler saturated, we prove several equivalent conditions.

• Korean Journal of Mathematics
• /
• 제16권1호
• /
• pp.85-90
• /
• 2008

We consider Fourier multiplier operators whose symbols satisfy a generalization of $H{\ddot{o}}rmander^{\prime}s$ condition and establish their Sobolev-type mapping properties on the homogeneous Besov-Lipschitz spaces by making use of a continuous characterization of Besov-Lipschitz spaces. As an application, we derive Sobolev-type imbedding theorem.