• Journal of the Korean Mathematical Society
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• v.40 no.2
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• pp.169-177
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• 2003

Recently, K. Watanabe Showed that the Hilbert-Kunz multiplicity of a toric ring is a rational number. In this paper we give an explicit formula to compute the Hilbert-Kunz multiplicity of two-dimensional toric rings. This formula also shows that the Hilbert-Kunz multiplicity of a two-dimensional non-regular toric ring is at least 3/2.

• Journal of the Korean Mathematical Society
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• v.34 no.2
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• pp.453-467
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• 1997

A transitive permutation representation of a group G is said to be multiplicity-free if all of its irreducible constituents are distinct. The character corresponding to the action is called the permutation character, given by $(1_H)^G$, where H is the stabilizer of a point. Multiplicity-free permutation characters are of interest in the study of centralizer algebras and distance-transitive graphs, and all finite simple groups are known to have such characters. In this article, we extend to the alternating groups the result of J. Saxl who determined the multiplicity-free permutation representations of the symmetric groups. We classify all subgroups H for which $(1_H)^An, n > 18$, is multiplicity-free.

• Journal of the Korean Mathematical Society
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• v.37 no.5
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• pp.735-749
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• 2000

The multiplicity if periodic sloutions of semilinear dissipative hyperbolic equations is treated

• Communications of the Korean Mathematical Society
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• v.12 no.4
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• pp.921-933
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• 1997

An Ambrosetti-Prodi type multiplicity result for periodic-Dirichlet problem to semilinear parabolic equation is treated.

• Journal of the Korean Mathematical Society
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• v.38 no.4
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• pp.853-881
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• 2001

Multiplicity for doubly-periodic solutions and Dirichlet-Periodic solutions are treated.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.5 no.2
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• pp.1-6
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• 2001

Multiplicity of nonlinear second order nonlinear ordinary differential equations will be discussed.

• Bulletin of the Korean Mathematical Society
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• v.57 no.5
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• pp.1251-1258
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• 2020

This paper establishes a formula changing the ring from a Noetherian local ring A of dimension d > 0 containing the residue field k to the polynomial ring in d variables k[X₁, X₂, …, X_d] for mixed multiplicities. And as consequences, we get a formula for the multiplicity of Rees rings and formulas for mixed multiplicities and the multiplicity of Rees rings of quotient rings of A by highest dimensional associated prime ideals of A.

• International Journal of Computer Science & Network Security
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• v.21 no.4
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• pp.103-114
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• 2021

A conceptual model can be used to manage complexity in both the design and implementation phases of the system development life cycle. Such a model requires a firm grasp of the abstract principles on which a system is based, as well as an understanding of the high-level nature of the representation of entities and processes. In this context, models can have distinct architectural characteristics. This paper discusses model multiplicity (e.g., unified modeling language [UML]), model singularity (e.g., object-process methodology [OPM], thinging machine [TM]), and a heterogeneous model that involves multiplicity and singularity. The basic idea of model multiplicity is that it is not possible to present all views in a single representation, so a number of models are used, with each model representing a different view. The model singularity approach uses only a single unified model that assimilates its subsystems into one system. This paper is concerned with current approaches, especially in software engineering texts, where multimodal UML is introduced as the general-purpose modeling language (i.e., UML is modeling). In such a situation, we suggest raising the issue of multiplicity versus singularity in modeling. This would foster a basic appreciation of the UML advantages and difficulties that may be faced during modeling, especially in the educational setting. Furthermore, we advocate the claim that a multiplicity of views does not necessitate a multiplicity of models. The model singularity approach can represent multiple views (static, behavior) without resorting to a collection of multiple models with various notations. We present an example of such a model where the static representation is developed first. Then, the dynamic view and behavioral representations are built by incorporating a decomposition strategy interleaved with the notion of time.

• Bulletin of the Korean Mathematical Society
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• v.37 no.4
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• pp.697-710
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• 2000

We investigate the existence of solutions of the nonlinear heat equation under Dirichlet boundary conditions on $\Omega$ and periodic condition on the variable t, $Lu-D_tu$+g(u)=f(x, t). We also investigate a relation between multiplicity of solutions and the source terms of the equation.

• Journal of the Korean Mathematical Society
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• v.42 no.2
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• pp.191-202
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• 2005

A condition on forcing term insuring the multiplicity of Dirichlet-periodic solutions of nonlinear dissipative hyperbolic equations is discussed. The nonlinear term is assumed to have coercive growth.