• The Mathematical Education
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• v.15 no.1
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• pp.18-21
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• 1976

Lagrange proved that any positive integer is the sum of at most four squares. We consider a elliptic function f$_{\alpha}$(v│$\tau$) of periods 1. $\tau$ derived from $\theta$-functions. From the important number-theoretical interpretation (equation omitted) we obtain $A_4$(n) the number of representations entations of n as a sum of 4-squares.m of 4-squares.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.161-173
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• 2013

This paper is concerned with the space $\mathcal{K}_{w^*}(X^*,Y)$ of $weak^*$ to weak continuous compact operators from the dual space $X^*$ of a Banach space X to a Banach space Y. We show that if $X^*$ or $Y^*$ has the Radon-Nikod$\acute{y}$m property, $\mathcal{C}$ is a convex subset of $\mathcal{K}_{w^*}(X^*,Y)$ with $0{\in}\mathcal{C}$ and T is a bounded linear operator from $X^*$ into Y, then $T{\in}\bar{\mathcal{C}}^{{\tau}_{\mathcal{c}}}$ if and only if $T{\in}\bar{\{S{\in}\mathcal{C}:{\parallel}S{\parallel}{\leq}{\parallel}T{\parallel}\}}^{{\tau}_{\mathcal{c}}}$, where ${\tau}_{\mathcal{c}}$ is the topology of uniform convergence on each compact subset of X, moreover, if $T{\in}\mathcal{K}_{w^*}(X^*, Y)$, here $\mathcal{C}$ need not to contain 0, then $T{\in}\bar{\mathcal{C}}^{{\tau}_{\mathcal{c}}}$ if and only if $T{\in}\bar{\mathcal{C}}$ in the topology of the operator norm. Some properties of $\mathcal{K}_{w^*}(X^*,Y)$ are presented.

• Korean Journal of Animal Reproduction
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• v.23 no.4
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• pp.365-369
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• 1999

Interferon tau (IFN$\tau$), the pregnancy recognition signal in ruminants, suppresses transcription of the estrogen receptor (ER) gene in the endometrial luminal (LE) and superficial glandular epithelium (sGE) to prevent oxytocin receptor (OTR) expression and pulsatile release of luteolytic prostaglandin $F_{2{\alpha}}$ (PGF), Interferon regulatory factors one (IRF-l) and two (IRF-2) are transcription factors induced by IFN$\tau$ that activate and silence gene expression, respectively. Available results suggest that IFN$\tau$ acts directly on LE and sGE during pregnancy to induce sequentially IRF-l and then IRF-2 gene expression to silence transcription of ER and OTR genes, block the luteolytic mechanism to maintenance a functional corpus luteum (CL) and, signal maternal recognition of pregnancy. The theory for maternal recognition of pregnancy in pigs is that the uterine endometrium of cyclic gilts secretes PGF in an endocrine direction, toward the uterine vasculature for transport to the CL to exert its luteolytic effect. However, in pregnant pigs, estrogens secreted by the conceptuses are responsible, perhaps in concert with effects of prolactin and calcium, for exocrine secretion of PGF into the uterine lumen where it is sequestered to exert biological effects and / or be metabolized to prevent luteolysis.

• Tribology and Lubricants
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• v.15 no.1
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• pp.8-16
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• 1999

The electrorheological (ER) behavior of suspensions in silicone oil of phosphoric ester cellulose powder (average particle size : 18$\pm$1 ${\mu}{\textrm}{m}$) was investigated on the elevated temperature up to 10$0^{\circ}C$. For development of anhydrous ER fluids using at wide temperature range, it should be researched to how the effect of temperature on the ER activities. As a first step, the anhydrous ER suspensions mixing with the phosphoric ester cellulose particles which were made from the phosphoric ester reaction of cellulose were measured. As increasing the temperature, not only the analysis of electrical properties such as dielectric constant current density and electrical conductivity but also the rheological properties of ER fluids were studied. From the experimental results, the temperature had a large influence to the ER properties of anhydrous ER fluids. The current density, conductivity and elecoorheological effect ($\tau$$_{A}$$\tau$$_{0}$) of phosphoric ester cellulose ER fluids were proportional to the temperature with power law. And the shear stress of them was closely related with the square of dielectric constant mismatch parameter ($\beta$$^2$) under constant shear rate and electric field.d.

• Tribology and Lubricants
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• v.14 no.2
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• pp.1-9
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• 1998

The electrorheological (ER) behavior of suspensions in silicone oil of phosphated cellulose particles (average particle size 17.77 ${\mu}{\textrm}{m}$) was investigated at room temperature with electric fields up to 2.5 KV/mm. In this paper, for development of anhydrous ER suspensions using at wide temperature range, we would like to know fundamental understandings on the ER activity. As a first step, the anhydrous ER suspensions dispersed the phosphated cellulose particles were measured, and not only the electrical characteristics such as dielectric constant, current density and electrical conductivity but also the rheological properties on strength of electric field and quantity of dispersed phase were studied. From the experimental results, the anhydrous ER suspensions dispersed phosphated cellulose particles showed a stable current density and very high performance of ER effect $(\tau/\tau_0=1030)$ on the 2.5 KV/mm and the dynamic yield stress $(\tau_y)$ was in exponential proportion to the strength of electric fields.

• Journal of Biomedical Engineering Research
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• v.13 no.1
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• pp.55-64
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• 1992

AB -type block copolymers of poly($\tau$-benzyl L-glutamate) (PBLG, a segment) and polyoxypropylene (POP, Bsegment) were synthesized and they were reacted with polyoxypropylene his(6-aminohexyl) (POE) to enhance the hydrophilicity of the poly- mers. The degree of swelling of the POE -crosslinked block copolymers (CPB -2) was higher than that of the block dopolymers (PB). The amount of 5-fluorouracil re- leased from CPB-2 block copolymers was greater than that from the PB block copol- ymers when they were incubated in the same period. These results indicate that the amount of 5-fluorouracil released from the matrices is proportional to the hydrophilicity of the polymers. Observing polymer surfaces by scanning electron mi- croscope, pores (diameter ; $0.5-1.5{\mu}$m) were appeared on the surface of CPB-2

• East Asian mathematical journal
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• v.32 no.3
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• pp.399-412
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• 2016

In this paper, we study the viscoelastic Kirchhoff type equation with the following nonlinear source and time-varying delay $$u_{tt}-M(x,t,{\parallel}{\nabla}u(t){\parallel}^2){\Delta}u+{\int_{0}^{t}}h(t-{\tau})div[a(x){\nabla}u({\tau})]d{\tau}\\+{\parallel}u{\parallel}^{\gamma}u+{\mu}_1u_t(x,t)+{\mu}_2u_t(x,t-s(t))=0.$$ Under the smallness condition with respect to Kirchhoff coefficient and the relaxation function and other assumptions, we prove the uniform decay rate of the Kirchhoff type energy.

• Proceedings of the KIEE Conference
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• 2007.07a
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• pp.977-978
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• 2007

The phase angle which is mainly depend on time constant $\tau$ of the calculable 10 A and 0.1 ${\Omega}ac/dc$ resistor has been analysed. The low values of resistors are usually inductive and time constant $\tau$ very large compare to high values of resistors. The numerical analysis has been shown that the time constant $\tau$ becomes zero when introduce the compensation capacitance geometrically to the resistive coil with proper dimension. As a result a very low phase angle can be achieved within the realizable dimensions.

• Journal of the Korean Mathematical Society
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• v.41 no.1
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• pp.95-105
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• 2004

In this paper we will consider the weighted composition operators W = $uC_{\tau}$ between $L^{p}$$(X,\sum,\mu$) spaces and Orlicz spaces $L^{\phi}$$(X,\sum,\mu$) generated by measurable and non-singular transformations $\tau$ from X into itself and measurable functions u on X. We characterize the functions u and transformations $\tau$ that induce weighted composition operators between $L^{p}$ -spaces by using some properties of conditional expectation operator, pair (u,${\gamma}$) and the measure space $(X,\sum,\mu$). Also, some other properties of these types of operators will be investigated.

• Kyungpook Mathematical Journal
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• v.47 no.1
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• pp.1-20
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• 2007

In this paper, oscillation and nonoscillation criteria are established for nonlinear neutral delay differential equations with several positive and negative coefficients $$[x(t)-R(t)x(t-r)]^{\prime}+\sum_{i=1}^{m}Pi(t)H_i(x(t-{\tau}_i))-\sum_{j=1}^{n}Q_j(t)H_j(x(t-{\sigma}_j))=0$$. Our criteria improve and extend many results known in the literature. In addition we prove that under appropriate hypotheses, if every solution of the associated linear equation with constant coefficients, $$y^{\prime}(t)+\sum_{i=1}^{m}(p_i-\sum_{k{\in}J_i}qk)y(t-{\tau}_i)=0$$, oscillates, then every solution of the nonlinear equation also oscillates.