• Honam Mathematical Journal
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• v.1 no.1
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• pp.1-9
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• 1979

Positive Local Coordmate($(x^1,x^2,{\cdots}x^n)$)을 갖는 Oriented Manifold M을 생각한다. M이 Compact Carrier를 갖는 경우, M위의 n차(次) Differential Form ${\phi}^{(n)}$의 적분(積分)을 $${\int}{\phi}^{(n)}=\sum_{\alpha}{\int}_{-{\infty}}^{\infty}{\cdots}{\int}_{-{\infty}}^{\infty}f_{\alpha}{\phi}^{(n)}dx^1{\cdots}dx^n$$로 정의(定義)하며 (정의(定義) 7), M위의 p 차(次)의 Differential form $\beta^{(p)}$와 Differential simplex $S^{(p)}=(S^{(p)},\;{\pi},\;{\varepsilon})$에 대하여 $S^{(p)}$위의 $\beta^{(p)}$의 적분(積分)을 $${\int}_{^{(p)}S}{\beta}^{(p)}={\int}_{S^{(p)}}{\varepsilon}{\pi}^*{\beta}^{(p)}={\int}_{E^p}f{\cdot}{\varepsilon}{\cdot}{\pi}^*{\beta}^{(p)}$$로 정의(定義)한다 (정의(定義) 9). 단(但) $\bar{S}^{(p)}$는 $S^{(p)}=(p_0{\cdot}p_1{\cdots}p_p)$에 의(依)하여 Spanning 되는 $E^p$의 Subspace이고 f는 $\bar{S}^{(p)}$의 특성함수(特性函數)이다. 이때 (n-1)차(次)의 Differential Form ${\beta}^{(n-1)}$이 Compart인 Carrier를 가지면 ${\int}d{\beta}^{(n-1)}=0$이 됨을 고찰(考察)하며(정리(定理 8), (p-1)차(次) Differential Form ${\beta}^{(p-1)$과 p차(次) Differential Chain $C^{(p)}$에 관(關)하여 $${\int}_{C^{(p)}}d{\beta}^{(p-1)}={\int}_{{\partial}C^{(p)}}{\beta}^{(p-1)}$$이 성립(成立)함을 구명(究明)하려 한다(정리(定理) 10).

• Archives of Pharmacal Research
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• v.27 no.3
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• pp.357-360
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• 2004

Four crystal forms of ketorolac have been obtained by recrystallization in organic solvents under variable conditions. Different ketorolac polymorphs and pseudo polymorph were characterized by X-ray powder diffraction crystallography (XRD), Differential scanning calorimetry (DSC) and thermogravimetric analysis (TGA). In the dissolution studies in water at $37{\pm}0.5^{\circ}C$ four crystal forms showed different patterns. The solubility of Form I were the highest. The solubility decreased in rank order: Form I> Form II > Form III > Form IV. Form land Form III were shown to have a good physical stability at room temperature for 60 days. However, Form II is converted to Form III and Form IV is converted to Form I after 60 days storage. Therefore, these observations indicate that crystalline polymorphism for ketorolac is readily inter-convertible and the relationship may have to taken into consideration in the formulation of the drug.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.4
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• pp.289-294
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• 2018

In this work, a recently developed semi-analytic technique, so called the residual power series method, is generalized to process higher-dimensional linear and nonlinear partial differential equations. The solutions obtained takes a form of an infinite power series which can, in turn, be expressed in a closed exact form. The results reveal that the proposed generalization is very effective, convenient and simple. This is achieved by handling the (m+1)-dimensional Burgers equation.

• Kyungpook Mathematical Journal
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• v.28 no.2
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• pp.185-196
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• 1988

We study existence of polynomial integrating factors and solutions F(x, y)=c of first order nonlinear differential equations. We characterize the homogeneous case, and give algorithms for finding existence of and a basis for polynomial solutions of linear difference and differential equations and rational solutions or linear differential equations with polynomial coefficients. We relate singularities to nature of the solution. Solution of differential equations in closed form to some degree might be called more an art than a science: The investigator can try a number of methods and for a number of classes of equations these methods always work. In particular integrating factors are tricky to find. An analogous but simpler situation exists for integrating inclosed form, where for instance there exists a criterion for when an exponential integral can be found in closed form. In this paper we make a beginning in several directions on these problems, for 2 variable ordinary differential equations. The case of exact differentials reduces immediately to quadrature. The next step is perhaps that of a polynomial integrating factor, our main study. Here we are able to provide necessary conditions based on related homogeneous equations which probably suffice to decide existence in most cases. As part of our investigations we provide complete algorithms for existence of and finding a basis for polynomial solutions of linear differential and difference equations with polynomial coefficients, also rational solutions for such differential equations. Our goal would be a method for decidability of whether any differential equation Mdx+Mdy=0 with polynomial M, N has algebraic solutions(or an undecidability proof). We reduce the question of all solutions algebraic to singularities but have not yet found a definite procedure to find their type. We begin with general results on the set of all polynomial solutions and integrating factors. Consider a differential equation Mdx+Ndy where M, N are nonreal polynomials in x, y with no common factor. When does there exist an integrating factor u which is (i) polynomial (ii) rational? In case (i) the solution F(x, y)=c will be a polynomial. We assume all functions here are complex analytic polynomial in some open set.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1145-1166
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• 2022

In this paper, for a transcendental meromorphic function f and α ∈ ℂ, we have exhaustively studied the nature and form of solutions of a new type of non-linear differential equation of the following form which has never been investigated earlier: $$f^n+{\alpha}f^{n-2}f^{\prime}+P_d(z,f)={\sum\limits_{i=1}^{k}}{p_i(z)e^{{\alpha}_i(z)},$$ where P_d(z, f) is a differential polynomial of f, p_i's and α_i's are non-vanishing rational functions and non-constant polynomials, respectively. When α = 0, we have pointed out a major lacuna in a recent result of Xue [17] and rectifying the result, presented the corrected form of the same equation at a large extent. In addition, our main result is also an improvement of a recent result of Chen-Lian [2] by rectifying a gap in the proof of the theorem of the same paper. The case α ≠ 0 has also been manipulated to determine the form of the solutions. We also illustrate a handful number of examples for showing the accuracy of our results.

• Archives of Pharmacal Research
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• v.7 no.1
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• pp.47-52
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• 1984

Four polymorphic forms (I, II, III and IV) of hydrochlorothiazide have been characterized on the basis of x-ray diffractometry and differential thermal analysis. Form I was obtained by crystallization from N, N-dimethylformamide and Form II was crystallized from hot methanol. Form III was precipitated from sodium hydroxide aqueous solution by treatment with hydrochloric acid and Form IV was crystallized from 50% methanol. The metastable form I was a most stable form among four polymorphs, which was stable more than ten months at room temperature. The thermodynamic parameters such as heat of solution, enthalpy, entropy, free energy difference and transition temperature were determined by the measurement of intrinsic dissolution rate. The transition temperature and the heat of transition between the metastable Form I an Form II were determined to be $299.15^{\circ}$K and 5.03 Kcal/mole, respectively and free energy difference ($\delta$ F) was 302. 13 cal/mole. Diuretic action of these four polymorphic forms was also evaluated by monitoring the difference in urinary excretion of sodium, potassium and magnesium in rats.

• Journal of Pharmaceutical Investigation
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• v.20 no.1
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• pp.35-42
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• 1990

A new form of sulfabenzamide was characterized using X-ray diffraction patterns and differential scanning calorimetry. Solubility studies demonstrated that, of the sulfabenzamide polymorphs, the new form was more soluble than form I. Compression of the new form at compression force of 1000 $kg/cm{^2}$ didn't induce polymorphic change in the crystal. Similar patterns were also produced through grinding. The effects of some diluents on the polymorphic transformation from the new form into form I by grinding and compression were also studied. Three diluents, $Avicel^{\circledR}$, lactose and starch showed no influence on the polymorphic transformation. The new form seemed to be more suitable for the pharmaceutical preparation.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1471-1479
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• 2013

In this paper, we study the non-existence of finite order entire solutions of nonlinear differential-difference of the form $$f^n+Q(z,f)=h$$, where $n{\geq}2$ is an integer, $Q(z,f)$ is a differential-difference polynomial in $f$ with polynomial coefficients, and $h$ is a meromorphic function of order ${\leq}1$.

• Korean Journal of Mathematics
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• v.25 no.1
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• pp.45-60
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• 2017

This paper focuses on estimation of an non-integral quadratic function (NIQF) and integral quadratic function (IQF) of a random signal in dynamic system described by a linear stochastic differential equation. The quadratic form of an unobservable signal indicates useful information of a signal for control. The optimal (in mean square sense) and suboptimal estimates of NIQF and IQF represent a function of the Kalman estimate and its error covariance. The proposed estimation algorithms have a closed-form estimation procedure. The obtained estimates are studied in detail, including derivation of the exact formulas and differential equations for mean square errors. The results we demonstrate on practical example of a power of signal, and comparison analysis between optimal and suboptimal estimators is presented.

• Archives of Pharmacal Research
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• v.23 no.4
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• pp.381-384
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• 2000

It is well recognized that physicochemical properties of drugs are affected by the type of polymorphic crystalline form of drugs. Clarithromycin is known to exist in at least three polymorphic crystalline forms. Since conventional means to obtain the most thermodynamically stable form (Form II) for the antibiotics is known to be associated with a low purity of the stable form, we developed a novel method to improve the purity of the crystalline form by a modification of the preparation process. The new method involved a simple recrystallization of clarithromycin in solvents having 5-12 carbon atoms (e.g., hexane and heptane) or ethers with 4-10 carbon atoms (e.g., isopropyl ether) and, thus, less likely to be associated with the problem in purity of resulting crystal. Differential scanning calorimetry and powder X-ray diffraction were used to compare the crystalline form of the resultant powder with Form IIcrystal prepared by the conventional method. The crystal prepared by the new method was identical to Form IIcrystal of the conventional method as evidenced by the lack of the exothermic peak near 11$0^{\circ}C$ in differential calorimetry scan, indicating that Form IIcrystal could be readily prepared by the new process. Therefore, these data indicated that the improvement in the purity of the Form IIcrystal for clarithromycin as well as a significant cost reduction is likely by the novel method.