• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.221-227
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• 2014

Our goal is to relax a sufficient condition for the exponential almost sure stability of a certain class of stochastic differential equations. Compared to the existing theory, we prove the almost sure stability, replacing Lipschitz continuity and linear growth conditions by the existence of a strong solution of the underlying stochastic differential equation. This result is extendable for the regime-switching system. An explicit example is provided for the illustration purpose.

• Journal of the Korean Mathematical Society
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• v.49 no.3
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• pp.515-536
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• 2012

This work focuses on the general decay stability of nonlinear stochastic differential equations with unbounded delay. A Razumikhin-type theorem is first established to obtain the moment stability but without almost sure stability. Then an improved edition is presented to derive not only the moment stability but also the almost sure stability, while existing Razumikhin-type theorems aim at only the moment stability. By virtue of the $M$-matrix techniques, we further develop the aforementioned Razumikhin-type theorems to be easily implementable. Two examples are given for illustration.

• Journal of the Korean Mathematical Society
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• v.61 no.2
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• pp.227-253
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• 2024

This paper studies the existence of weak solutions and the stability of stationary solutions to stochastic 3D globally modified Navier-Stokes equations with unbounded delays in the phase space BCL_-∞(H). We first prove the existence and uniqueness of weak solutions by using the classical technique of Galerkin approximations. Then we study stability properties of stationary solutions by using several approach methods. In the case of proportional delays, some sufficient conditions ensuring the polynomial stability in both mean square and almost sure senses will be provided.

• Structural Engineering and Mechanics
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• v.52 no.3
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• pp.525-540
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• 2014

The Lyapunov exponent and moment Lyapunov exponents of Hill's equation with frequency and damping coefficient fluctuated by correlated wideband random processes are studied in this paper. The method of stochastic averaging, both the first-order and the second-order, is applied. The averaged $It\hat{o}$ differential equation governing the pth norm is established and the pth moment Lyapunov exponents and Lyapunov exponent are then obtained. This method is applied to the study of the almost-sure and the moment stability of the stationary solution of the thin simply supported beam subjected to time-varying axial compressions and damping which are small intensity correlated stochastic excitations. The validity of the approximate results is checked by the numerical Monte Carlo simulation method for this stochastic system.

• Journal of Nutrition and Health
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• v.3 no.3
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• pp.129-135
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• 1970

From the previous studies, F-P-4 formula was found to be comparable to full fat dry milk in its nutritive value and feeding performance. However, an attempt was made in order to make sure whether or not any possibility might exist, by which further improvement of nutritive quality and simultaneous reduction of product costs may be achieved. Using F-P-4 as a control, modifications were made in new formulas, F-P-5, F-P-6 and F-P-7 by reducing FPC, eliminating yeast from the mixture, and by enriching with methionine as needed. In particular, F-P-7 is completely free of FPC, hydrogenated oil and yeast. Yet, levels of total protein and fat were kept equal to those of F-P-4 in all formulas. An animal feeding test for all formulas using 10 female rats per group for 8 weeks and an infant feeding trial for F-P-5 and F-P-6 with 5 of each female infants under age of one for one month were conducted along with F-P-4 as a control. Almost the same results were obtained with F-P-4, 5 and 6, but F-P-7 showed the lowest body weight gain. FER of F-P-5 and 6 was 0.20 as was with F-P-4, while that of F-P-7 was 0.16. Acceptability to infants was excellent; growth, appearance and biochemical data were normal. As an example F-P-4 packed in 0.04mm polyethylene bags was used for storage study at $25^{\circ}C$ and relative humidity of $65{\sim}85%$ for 8 months. Although viable bacterial counts and vitamin C contents were reduced, peroxide and TBA values were increased gradually during such storage. Since there are also significant changes in color and organoleptic quality, the expected shelf life under the given conditions is considered to be about 2 months and thus further works are needed both on the product and packaging in order to improve the storage stability. Either elimination of yeast form F-P-4, that is F-P-5, or partial replacement of FPC with methionine, that is F-P-6 may well reduce material costs about 10%. Considering blending process of ingredients, F-P-5 is thus found to be the best formula developed. While F-P-7 free of FPC is inferior in its nutritive quality than that of others, but significantly superior than of rice. Furthermore, the material cost of the product can be reduced about 20% from that of F-P-4. And thus this vegetable blend is considered to be useful as a low cost supplementary food mixture for growing children.