• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.275-286
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• 2010

The algebraic structure of the natural integral cohomology operations is explored by means of examples. We decompose the generators of the groups $H^m(\mathbb{Z},\;n)$ with $2\;{\leq}\;n\;{\leq}\;7$ and $2\;{\leq}\;m\;{\leq}\;13$ into the operations of cup products, cross-cap products and compositions. Examination of these decompositions and comparison with other possible generators demonstrates the existence of relations between integral operations that have withheld formulation. The calculated groups and generators are collected in a table for practical reference.

• Bulletin of the Korean Mathematical Society
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• v.45 no.2
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• pp.355-363
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• 2008

For a Boolean rank 1 matrix $A=ab^t$, we define the perimeter of A as the number of nonzero entries in both a and b. The perimeter of an $m{\times}n$ Boolean matrix A is the minimum of the perimeters of the rank-1 decompositions of A. In this article we characterize the linear operators that preserve the perimeters of Boolean matrices.

• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.1989-2000
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• 2013

We show that for any given closed orientable 3-manifold M with a Heegaard surface of genus g, any positive integers b and n, there exists a knot K in M which admits a (g, b)-bridge splitting of distance greater than n with respect to the Heegaard surface except for (g, b) = (0, 1), (0, 2).

• Journal of the Korean Mathematical Society
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• v.36 no.6
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• pp.1075-1090
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• 1999

This paper deals with Littlewood-Paley type estimates of the Besov spaces {{{{ { B}`_{p,q } ^{$\alpha$ } }}}} on the d-dimensional unit cube for 0< p,q<$\infty$ by two certain classes. These classes are including biorthogonal wavelet systems or dual multiscale systems but not necessarily obtained as the dilates or translates of certain fixed functions. The main assumptions are local supports of both classes, sufficient smoothness for one class, and sufficiently many vanishing moments for the other class. With these estimates, we characterize the Besov spaces by coefficient norms of decompositions with respect to biorthogonal wavelet systems on the cube.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.149-163
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• 2018

In recent years, the study of slice Dirac operators has attracted more and more attention in the literature. In this paper, Almansitype decompositions for null solutions to the iterated slice Dirac operator and the generalized slice Dirac operator are obtained without a star-like domain centered at the origin. As applications, we investigate Riquier type problems and Dirichlet type problems in the theory of slice monogenic functions.

• Publications of The Korean Astronomical Society
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• v.7 no.1
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• pp.107-148
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• 1992

This paper presents a cosmological perturbation analysis in a Newtonian framework, using the Newtonian multi component version of the relativistic covariant equations. This work considers the fully nonlinear evolution of the perturbations, and is generalized to multicomponent systems and imperfect fluids. Known nonlinear solutions are presented in a general framework. Quasi-nonlinear analysis, considering both the compressible and rotational modes, is presented, including cases already known in the literature. The Fourier space representation of the conservation equations is also derived in a general context, with various decompositions of the velocity field. Commonly accepted cosmogonical frameworks are critically examined in the context of nonlinear evolution. This work may be regarded as the Newtonian counterpart of a recently presented general relativistic covariant formulation.

• East Asian mathematical journal
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• v.29 no.3
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• pp.337-347
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• 2013

For an even integer m at least 4 and any positive integer $t$, it is shown that the complete equipartite graph $K_{km(2t)}$ can be decomposed into edge-disjoint gregarious m-cycles for any positive integer ${\kappa}$ under the condition satisfying ${\frac{{(m-1)}^2+3}{4m}}$ < ${\kappa}$. Here it will be called a gregarious cycle if the cycle has at most one vertex from each partite set.

• Journal of the Korean Mathematical Society
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• v.33 no.1
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• pp.145-154
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• 1996

In the study of manifold decompositions, a central theme is to understand the source manifold taking advantage of the informations of a base space and a decomposition. The concepts of both Hurewicz fibrations and cell-like maps have played very important roles for investigating the mutual relations of three objects. But it is somewhat restrictive for a decomposition map to be cell-like because its inverse images must have trivial shapes.

• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.589-599
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• 2012

An element of a ring is called strongly nil clean provided that it can be written as the sum of an idempotent and a nilpotent element that commute. We characterize, in this article, the strongly nil cleanness of $2{\times}2$ and $3{\times}3$ matrices over $R[x]/(x^2-1)$ where $R$ is a commutative local ring with characteristic 2. Matrix decompositions over fields are derived as special cases.

• Journal of the Korean Mathematical Society
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• v.45 no.1
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• pp.205-219
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• 2008

It is shown that if $A{\geq}0$ and T are two bounded linear operators on a complex Hilbert space H satisfying the inequality $T^*\;AT{\leq}A$ and the condition $AT=A^{1/2}TA^{1/2}$, then there exists the maximum reducing subspace for A and $A^{1/2}T$ on which the equality $T^*\;AT=A$ is satisfied. We concretely express this subspace in two ways, and as applications, we derive certain decompositions for quasinormal contractions. Also, some facts concerning the quasi-isometries are obtained.