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  • Title/Summary/Keyword: weak regularity

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MEASURE-VALUED SOLUTIONS FOR STOCHASTIC DIFFERENTIAL EQUATIONS ON HILBERT SPACES DRIVEN BY LÉVY MEASURE AND THEIR OPTIMAL CONTROL

  • Nasir U. Ahmed
    • Communications of the Korean Mathematical Society
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    • v.39 no.4
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    • pp.1035-1057
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    • 2024
  • In this paper we consider a general class of nonlinear stochastic differential equations on Hilbert spaces determined by nonstandard infinitesimal generators (drift, diffusion, jump-kernel) and driven by Lévy process (measure). The infinitesimal generators are assumed to be only continuous and bounded on bounded sets. Under such relaxed assumptions, these equations do not have solutions in the usual sense (classical, strong, mild and weak). We prove existence of measure-valued solutions and consider several control problems (including control of the range of vector measures) and prove existence of partially observed optimal feedback controls. This paper is an extension of our previous studies on similar problems for deterministic as well as stochastic differential equations driven by cylindrical Brownian motion.

REGULARITY AND MULTIPLICITY OF SOLUTIONS FOR A NONLOCAL PROBLEM WITH CRITICAL SOBOLEV-HARDY NONLINEARITIES

  • Alotaibi, Sarah Rsheed Mohamed;Saoudi, Kamel
    • Journal of the Korean Mathematical Society
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    • v.57 no.3
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    • pp.747-775
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    • 2020
  • In this work we investigate the nonlocal elliptic equation with critical Hardy-Sobolev exponents as follows, (P)\;\{(-{\Delta}_p)^su={\lambda}{\mid}u{\mid}^{q-2}u+{\frac{{\mid}u{\mid}^{p{^*_s}(t)-2}u}{{\mid}x{\mid}^t}}{\hspace{10}}in\;{\Omega},\\u=0{\hspace{217}}in\;{\mathbb{R}}^N{\backslash}{\Omega}, where Ω ⊂ ℝN is an open bounded domain with Lipschitz boundary, 0 < s < 1, λ > 0 is a parameter, 0 < t < sp < N, 1 < q < p < ps where ps=NpNsp, ps(t)=p(Nt)Nsp, are the fractional critical Sobolev and Hardy-Sobolev exponents respectively. The fractional p-laplacian (-∆p)su with s ∈ (0, 1) is the nonlinear nonlocal operator defined on smooth functions by (Δp)su(x)=2limϵ0RNBϵu(x)u(y)p2(u(x)u(y))xyN+psdy, x ∈ ℝN. The main goal of this work is to show how the usual variational methods and some analysis techniques can be extended to deal with nonlocal problems involving Sobolev and Hardy nonlinearities. We also prove that for some α ∈ (0, 1), the weak solution to the problem (P) is in C1,α(ˉΩ).

A Multimedia Data Prefetching Based on 2 Dimensional Block Structure (이차원 블록 구조에 근거한 선인출 기법)

  • Kim, Seok-Ju
    • Journal of Korea Multimedia Society
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    • v.7 no.8
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    • pp.1086-1096
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    • 2004
  • In case of a multimedia application which deals with streaming data, in terms of cache management, cache loses its efficiency due to weak temporal locality of the data. This means that when data have been brought into cache, much of the data are supposed to be replaced without being accessed again during its service. However, there is a good chance that such multimedia data has a commanding locality in it. In this paper, to take advantage of the memory reference regularity which typically innates even in the multimedia data showing up its weak temporal locality, a method is suggested. The suggested method with the feature of dynamic regular-stride reference prefetching can identify for 2-dimensional array format(block pattern). The suggested method is named as block-reference-prediction-technique (BRPT) since it identifies a block pattern and place an address to be prefetched by the regulation of the block format. BRPT proved to be reassuring to reduce memory reference time significantly for applications having abundant block patterns although new rule has complicated the prefetching system even further.

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WELL-POSEDNESS AND ASYMPTOTIC BEHAVIOR OF PARTLY DISSIPATIVE REACTION DIFFUSION SYSTEMS WITH MEMORY

  • Vu Trong Luong;Nguyen Duong Toan
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.1
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    • pp.161-193
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    • 2024
  • In this paper, we consider the asymptotic behavior of solutions for the partly dissipative reaction diffusion systems of the FitzHugh-Nagumo type with hereditary memory and a very large class of nonlinearities, which have no restriction on the upper growth of the nonlinearity. We first prove the existence and uniqueness of weak solutions to the initial boundary value problem for the above-mentioned model. Next, we investigate the existence of a uniform attractor of this problem, where the time-dependent forcing term h ∈ L2b(ℝ; H-1(ℝN)) is the only translation bounded instead of translation compact. Finally, we prove the regularity of the uniform attractor A, i.e., A is a bounded subset of H2(ℝN) × H1(ℝN) × L2µ(ℝ+, H2(ℝN)). The results in this paper will extend and improve some previously obtained results, which have not been studied before in the case of non-autonomous, exponential growth nonlinearity and contain memory kernels.

Experimental Design of S box and G function strong with attacks in SEED-type cipher (SEED 형식 암호에서 공격에 강한 S 박스와 G 함수의 실험적 설계)

  • 박창수;송홍복;조경연
    • Journal of the Korea Institute of Information and Communication Engineering
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    • v.8 no.1
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    • pp.123-136
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    • 2004
  • In this paper, complexity and regularity of polynomial multiplication over GF(2n) are defined by using Hamming weight of rows and columns of the matrix ever GF(2) which represents polynomial multiplication. It is shown experimentally that in order to construct the block cipher robust against differential cryptanalysis, polynomial multiplication of substitution layer and the permutation layer should have high complexity and high regularity. With result of the experiment, a way of constituting S box and G function is suggested in the block cipher whose structure is similar to SEED, which is KOREA standard of 128-bit block cipher. S box can be formed with a nonlinear function and an affine transform. Nonlinear function must be strong with differential attack and linear attack, and it consists of an inverse number over GF(28) which has neither a fixed pout, whose input and output are the same except 0 and 1, nor an opposite fixed number, whose output is one`s complement of the input. Affine transform can be constituted so that the input/output correlation can be the lowest and there can be no fixed point or opposite fixed point. G function undergoes linear transform with 4 S-box outputs using the matrix of 4×4 over GF(28). The components in the matrix of linear transformation have high complexity and high regularity. Furthermore, G function can be constituted so that MDS(Maximum Distance Separable) code can be formed, SAC(Strict Avalanche Criterion) can be met, and there can be no weak input where a fixed point an opposite fixed point, and output can be two`s complement of input. The primitive polynomials of nonlinear function affine transform and linear transformation are different each other. The S box and G function suggested in this paper can be used as a constituent of the block cipher with high security, in that they are strong with differential attack and linear attack with no weak input and they are excellent at diffusion.

A Study on the Condition of Location According to the Formed Time in the Clan Village (동족(同族)마을의 설촌(設村)시기에서 나타난 입지(立地) 특성에 관한 연구)

  • Park, Myung-Duk;Park, Eon-Kon
    • Journal of architectural history
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    • v.1 no.1 s.1
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    • pp.68-87
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    • 1992
  • This study is the conditions of location according to formed the times in the clan village. The results of this study are as follows ; 1. in the 15th century, the characterestics of the village established residencial place where mountain stream flowed surrounded by the mountain and deep in the mountains with superior quality land. That's because Sa-dae-bu put equal importance on beautiful scenery and practical benefit for living. Stream House provided economical foundation for Sa-dae-bu to be able to keep confucial manners by putting limit their economic status to small medium sized land owner. Topographical condition such as valley or hollow separated from the exterior maintained unification of consanguineous village in self sufficient farming society and held on to independent territory against external to be able to stay away from turbulent days so that they formed residential area of Sa-dae-bu clan. And the valley where flowed clean water was considered as the connection of continuous place where distinctiveness of form in each curve and and factor of calm and dynamic scenery of the clean stream. Scholars in the middle of Chosun Dynasty located in the utopia as place for confucious retirement to study, a place for refinement by combination with the nature or as a way of spacial practice based on Confucious view of nature. 2. in the 16th-l7th century, Most of existing consanguineous villages adopt deep in the mountains for refuge. at that place, upward rank was established by settlement of the ancestor who entered in the village first, the principal was placed in the center of the village and since descendants became numerous, it was serialized as the space of descendants. So, it was arranged in the order of social rank. Most of the villages showed development step by step started from precaution by apperance of the mountain to the lower part. It's because the topography of valley around the village worked as the natural hedge against external force and genealogy of the clan, regularity of social status, order of entrance into the village were reflected into residencial destribution. Also, order of the rank coincided with the one of aspects on geomancy. Genealogical rank within the village represented spacial rank. Houses of descendants and branch families were placed lower than the principal which showed worship to the principal. 3. In 18th century after, as the village was settled nearby cultivated land considering economical loss caused by long distance between residencial area and cultivated land, direction of sect followed by development of village expanded from the front part of the village to the rear part. The principal that was poped out to the front presented frontage over exterior. Therefore, residencial area of branch families expanded to the rear starting from the principal. This represented a slice of social structure at that time. after 18th century, spirit was percieved superior over material, After then, development of cultivation and expantion of land created difference of economic strength within one village. In order to maintain and show off the status of Yang-ban, economic power of indigenous land owner became fundamental, so, sense to worship and to keep the principal became weak eventually. Taking advantage of that situation, residencial area of branch family expanded to the rear part of the principal which showed dual disposition conflicted with each other. However, these clan rules were destroyed and new rules were created after 18th century because of the situation and consciousness at that time.

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