• Journal of the Korean Mathematical Society
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• v.37 no.6
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• pp.915-927
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• 2000

This paper is concerned with degenerate Volterra equations Mu(t) + ∫(sub)0(sup)t k(t-s) Lu(s)ds = f(t) in Banach spaces both in the hyperbolic case, and the parabolic one. The key assumption is played by the representation of the underlying space X as a direct sum X = N(T) + R(T), where T is the bounded linear operator T = ML(sup)-1. Hyperbolicity means that the part T of T in R(T) is an abstract potential operator, i.e., -T(sup)-1 generates a C(sub)0-semigroup, and parabolicity means that -T(sup)-1 generates an analytic semigroup. A maximal regularity result is obtained for parabolic equations. We will also investigate the cases where the kernel k($.$) is degenerated or singular at t=0 using the results of Pruss[8] on analytic resolvents. Finally, we consider the case where $\lambda$ is a pole for ($\lambda$L + M)(sup)-1.

• Journal of the Korean Mathematical Society
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• v.48 no.2
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• pp.267-288
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• 2011

In this paper, we construct eight infinite families of ternary linear codes associated with double cosets with respect to certain maximal parabolic subgroup of the special orthogonal group $SO^-$(2n, q). Here q is a power of three. Then we obtain four infinite families of recursive formulas for power moments of Kloosterman sums with square arguments and four infinite families of recursive formulas for even power moments of those in terms of the frequencies of weights in the codes. This is done via Pless power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of "Gauss sums" for the orthogonal groups $O^-$(2n, q).

• Journal of the Korean Mathematical Society
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• v.57 no.3
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• pp.585-602
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• 2020

In this paper, we construct four infinite families of binary linear codes associated with double cosets with respect to a certain maximal parabolic subgroup of the orthogonal group O⁺(2n, 2^r). And we obtain two infinite families of recursive formulas for the power moments of Kloosterman sums and those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes. This is done via Pless' power moment identity and by utilizing the explicit expressions of exponential sums over those double cosets related to the evaluations of "Gauss sums" for the orthogonal groups O⁺(2n, 2^r).

• Journal of the Korean Mathematical Society
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• v.46 no.6
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• pp.1193-1206
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• 2009

C. Guenther studied the stability of DeTurck flows by using maximal regularity theory and center manifolds, but these arguments can not solve the stability of Ricci flows because the Ricci flow equation is not strictly parabolic. Recognizing this deficiency, the present paper considers and obtains the stability of Ricci flows, and of quasi-Ricci flows in view of some Killing conditions.

• The Pure and Applied Mathematics
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• v.22 no.1
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• pp.75-90
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• 2015

In this paper the existence of global solutions of the parabolic cross-diffusion systems with cooperative reactions is obtained under certain conditions. The uniform boundedness of $W_{1,2}$ norms of the local maximal solution is obtained by using interpolation inequalities and comparison results on differential inequalities.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.871-894
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• 1997

For a lifted nontrivial additive character $\lambda'$ and a multiplicative character $\chi$ of the finite field with $q^2$ elements, the 'Gauss' sums $\Sigma\lambda'$(tr $\omega$) over $\omega$ $\in$ SU(2n + 1, $q^2$) and $\Sigma\chi$(det $\omega$)$\lambda'$(tr $\omega$) over $\omega$ $\in$ U(2n + 1, $q^2$) are considered. We show that the first sum is a polynomial in q with coefficients involving certain new exponential sums and that the second one is a polynomial in q with coefficients involving powers of the usual twisted Kloosterman sums and the average (over all multiplicative characters of order dividing q-1) of the usual Gauss sums. As a consequence we can determine certain 'generalized Kloosterman sum over nonsingular Hermitian matrices' which were previously determined by J. H. Hodges only in the case that one of the two arguments is zero.

• Bulletin of the Korean Mathematical Society
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• v.49 no.4
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• pp.705-714
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• 2012

In this paper, we study nonlinear equation arising in MEMS modeling electrostatic actuation. We will prove the local and global existence of solutions of the generalized parabolic MEMS equation. We present that there exists a constant ${\lambda}^*$ such that the associated stationary problem has a solution for any ${\lambda}$ < ${\lambda}^*$ and no solution for any ${\lambda}$ > ${\lambda}^*$. We show that when ${\lambda}$ < ${\lambda}^*$ the global solution converges to its unique maximal steady-state as $t{\rightarrow}{\infty}$. We also obtain the condition for the existence of a touchdown time $T{\leq}{\infty}$ for the dynamical solution. Furthermore, there exists $p_0$ > 1, as a function of $p$, the pull-in voltage ${\lambda}^*(p)$ is strictly decreasing with respect to 1 < $p$ < $p_0$, and increasing with respect to $p$ > $p_0$.

• Structural Engineering and Mechanics
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• v.63 no.1
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• pp.55-64
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• 2017

The paper concerns concrete carbonation, the phenomena that occurs in every type of climate, especially in urban-industrial areas. In European Standards, including Eurocode (EC) for concrete structures the demanded durability of construction located in the conditions of the carbonation threat is mainly assured by the selection of suitable thickness of reinforcement cover. According to EC0 and EC2, the thickness of the cover in the particular class of exposure depends on the structural class/category and concrete compressive strength class which is determined by cement content and water-cement ratio (thus the quantitative composition) but it is not differentiated for various cements, nor additives (i.e., qualitative composition), nor technological types of concrete. As a consequence the selected thickness of concrete cover is in fact a far estimation - sometimes too exaggerated (too safe or too risky). The paper presents the elaborated "self-terminated carbonation model" that includes abovementioned factors and enables to indicate the maximal possible depth of carbonation. This is possible because presented model is a hyperbolic function of carbonation depth in time (the other models published in the literature use the parabolic function that theoretically assume the infinite increase of carbonation depth value). The paper discusses the presented model in comparison to other models published in the literature, moreover it contains the algorithm of concrete cover design with use of the model as well as an example of calculation of the cover thickness.

• Progress in Medical Physics
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• v.18 no.2
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• pp.73-80
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• 2007

Fruitful findings have been produced from five out of sixty cells which were obtained from each 63 individual Aplisia caught at the Jeju coast. Spiking patterns of three out of five cells, such as relaxation oscillator, bursting within a short time of the inter-burst interval, chaotic bursting, period doubling sequences, bursting with long trains of action potentials separated by short silent periods, regular repeated beating or elliptic bursting, and silent states had been changed in order as the temperature was lowered to $10^{\circ}C\;from\;32^{\circ}C$. In the intervals of every about 40 minutes repeated ups and downs of temperature produced similar firing patterns at the allowable temperature ranges. The other two cells showed difference from these. The amplitudes of the action potentials of the two cells will not be highly decreased in 24 hours. Average spike frequencies, the inter-burst interval, peak to peak spike amplitude of action potentials, minimum potential values are compared and analyzed by using the computer programme. The spike frequencies according to temperature show the distribution of bell type, with maximal spike frequencies at intermediate temperatures and minimal ones at either end. The most common pattern consist of high spike frequency during failing and low one during rising temperatures.