• 한국학교수학회논문집
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• 제10권2호
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• pp.187-206
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• 2007

본 연구에서는 중학교 3학년 수학학습부진아가 함수 분야에서 나타나는 어려움과 이에 대한 지도방안에 대해 알아보고자 하는 목적으로 좀 더 심층적인 학습 부진 요인을 알아보기 위해 설문지, 인터뷰, 녹음과 관찰 등을 통한 사례연구를 하였다. 그 결과 수학학습부진아는 일반 학생들이 겪고 있는 어려움과 유사한 어려움을 겪고 있었다. 다음으로 수학학습부진아는 일반 학생들이 겪고 있는 어려움과 차이를 보이는 어려움으로 문제를 이해하는 데 어려움, 선수 학습 결손으로 인한 어려움, 형식화하여 답을 찾으려고 하는 어려움, 대수 기호의 구분을 정확히 하지 못함으로써 겪는 어려움 그리고 두 점을 지나는 직선의 기울기를 구하는데 있어 어려움을 겪는 것으로 나타났다.

• 대한수학회지
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• 제53권2호
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• pp.363-379
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• 2016

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [Integral Transforms Spec. Funct. 23 (2012), 659-683] and the second Appell function [Appl. Math. Comput. 219 (2013), 8332-8337] by means of the incomplete Pochhammer symbols $({\lambda};{\kappa})_{\nu}$ and $[{\lambda};{\kappa}]_{\nu}$, we introduce here the family of the incomplete generalized ${\tau}$-hypergeometric functions $2{\gamma}_1^{\tau}(z)$ and $2{\Gamma}_1^{\tau}(z)$. The main object of this paper is to study these extensions and investigate their several properties including, for example, their integral representations, derivative formulas, Euler-Beta transform and associated with certain fractional calculus operators. Further, we introduce and investigate the family of incomplete second ${\tau}$-Appell hypergeometric functions ${\Gamma}_2^{{\tau}_1,{\tau}_2}$ and ${\gamma}_2^{{\tau}_1,{\tau}_2}$ of two variables. Relevant connections of certain special cases of the main results presented here with some known identities are also pointed out.

• 대한수학회보
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• 제53권1호
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• pp.91-102
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• 2016

A chief factor H/K of G is called F-central in G provided $(H/K){\rtimes}(G/C_G(H/K)){\in}{\mathfrak{F}}$. A normal subgroup N of G is said to be ${\pi}{\mathfrak{F}}$-hypercentral in G if either N = 1 or $N{\neq}1$ and every chief factor of G below N of order divisible by at least one prime in ${\pi}$ is $\mathfrak{F}$-central in G. The symbol $Z_{{\pi}{\mathfrak{F}}}(G)$ denotes the ${\pi}{\mathfrak{F}}$-hypercentre of G, that is, the product of all the normal ${\pi}{\mathfrak{F}}$-hypercentral subgroups of G. We say that a subgroup H of G is ${\pi}{\mathfrak{F}}$-embedded in G if there exists a normal subgroup T of G such that HT is s-quasinormal in G and $(H{\cap}T)H_G/H_G{\leq}Z_{{\pi}{\mathfrak{F}}}(G/H_G)$, where $H_G$ is the maximal normal subgroup of G contained in H. In this paper, we use the ${\pi}{\mathfrak{F}}$-embedded subgroups to determine the structures of finite groups. In particular, we give some new characterizations of p-nilpotency and supersolvability of a group.

• 대한수학회지
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• 제48권6호
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• pp.1249-1268
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• 2011

For imaginary quadratic number fields F = $\mathbb{Q}(\sqrt{{\varepsilon}p_1{\ldots}p_{t-1}})$, where ${\varepsilon}{\in}${-1,-2} and distinct primes $p_i{\equiv}1$ mod 4, we give condition of 8-ranks of class groups C(F) of F equal to 1 or 2 provided that 4-ranks of C(F) are at most equal to 2. Especially for F = $\mathbb{Q}(\sqrt{{\varepsilon}p_1p_2)$, we compute densities of 8-ranks of C(F) equal to 1 or 2 in all such imaginary quadratic fields F. The results are stated in terms of congruence relation of $p_i$ modulo $2^n$, the quartic residue symbol $(\frac{p_1}{p_2})4$ and binary quadratic forms such as $p_2^{h+(2_{p_1})/4}=x^2-2p_1y^2$, where $h+(2p_1)$ is the narrow class number of $\mathbb{Q}(\sqrt{2p_1})$. The results are also very useful for numerical computations.

• 한국수학교육학회지시리즈A:수학교육
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• 제56권1호
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• pp.81-100
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• 2017

The concept of variable is a big idea to develop algebraic thinking. Variable has multiple meanings such as the unknown, a tool for generalization, and the relationship between varying quantities. In this study we analyzed in what ways the meanings of variable were presented in the current elementary mathematics textbooks and workbooks. The results showed that the most frequent meaning of variable was 'the unknown', 'a tool for generalization', and 'the relationship between varying quantities' in order. A close look at the results revealed that the same symbol was often used in representing different values of variable as the unknown. In taking variable as a tool for generalization, questions to provoke generalization were sometimes included not in the textbooks but in the teachers' manuals. The main focus in dealing with variable as the relationship between varying quantities was on finding out the dependent values compared to the independent ones. Building on these results, this study is expected to suggest implications for how to deal with variable concept in elementary mathematics instructional materials.

• 대한수학회지
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• 제8권1호
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• pp.25-30
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• 1971

Put ($$^*$$) $$G[x,y]={\sum}\limits^{p+q=n}_{p,q=0}[-n]_{p+q}c_{p,q}x^py^q$$, where $[{\lambda}]_m$ is the Pocbhammer symbol and the $c_{p,q}$ are arbitrary constants. Making use of the specialized forms of some of his earlier results (see [8] and [9] the author derives here bilateral generating functions of the type ($$^{**}$$) $${\sum}\limits^{\infty}_{n=0}{\frac{[\lambda]_n}{n!}}_2F_1[\array{{\rho}-n,\;{\alpha};\\{\lambda}+{\rho};}x]\;G[y,z]t^n$$ where ${\alpha}$, ${\rho}$ and ${\lambda}$ are arbitrary complex numbers. In particular, it is shown that when G[y, z] is a double hypergeometric polynomial, the right-band member of ($^{**}$) belongs to a class of general triple hypergeometric functions introduced by the author [7]. An interesting special case of ($^{**}$) when ${\rho}=-m,\;m$ being a nonnegative integer, yields a class of bilateral generating functions for the Jacobi polynomials $\{P_n{^{{\alpha},{\beta}}}(x)\}$ in the form ($$^{***}$$) $${\sum\limits^{\infty}_{n=0}}\(\array{m+n\\n}\)P{^{({\alpha}-n,{\beta}-n)}_{m+n}(x)\;G[y,z]{\frac{t^n}{n!}}$$, which provides a unification of several known results. Further extensions of ($^{**}$) and ($^{***}$) with G[y, z] replaced by an analogous multiple sum $H\[y_1,{\cdots},y_m\]$ are also discussed.

• 대한수학회논문집
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• 제18권1호
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• pp.117-126
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• 2003

Let {$X_{nk}$\mid$1\;{\leq}\;k\;{\leq}\;n,\;n\;{\geq}\;1$} be an array of row negatively associated (NA) random variables which satisfy $P($\mid$X_{nk}$\mid$\;>\;x)\;{\leq}\;P($\mid$X$\mid$\;>\;x)$. For weighed sums ${{\Sigma}_{k=1}}^{Tn}\;a_kX_{nk}$ indexed by random variables {$T_n$\mid$n\;{\geq}$1$}, we establish a general weak law of large numbers (WLLN) of the form $({{\Sigma}_{k=1}}^{Tn}\;a_kX_{nk}\;-\;v_{[nk]})\;/b_{[an]}$ under some suitable conditions, where $\{a_n$\mid$n\;\geq\;1\},\; \{b_n$\mid$n\;\geq\;1\}$ are sequences of constants with $a_n\;>\;0,\;0\;<\;b_n\;\rightarrow \;\infty,\;n\;{\geq}\;1$, and {$v_{an}$\mid$n\;{\geq}\;1$} is an array of random variables, and the symbol [x] denotes the greatest integer in x.

• 충청수학회지
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• 제24권2호
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• pp.187-204
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• 2011

B. C. Carlson [Some extensions of Lardner's relations between $_0F_3$ and Bessel functions, SIAM J. Math. Anal. 1(2) (1970), 232-242] presented several useful relations between Bessel and generalized hypergeometric functions that generalize some earlier results. Here, by simply splitting Srivastava's hypergeometric function $H_B$ into eight parts, we show how some useful and generalized relations between Srivastava's hypergeometric functions $H_B$ and $F^{(3)}$ can be obtained. These main results are shown to be specialized to yield certain relations between functions $_0F_1$, $_1F_1$, $_0F_3$, ${\Psi}_2$, and their products including different combinations with different values of parameters and signs of variables. We also consider some other interesting relations between the Humbert ${\Psi}_2$ function and $Kamp\acute{e}$ de $F\acute{e}riet$ function, and between the product of exponential and Bessel functions with $Kamp\acute{e}$ de $F\acute{e}riet$ functions.

• Kyungpook Mathematical Journal
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• 제59권4호
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• pp.617-629
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• 2019

A term is called linear if each variable which occurs in the term, occurs only once. A hypersubstitution is said to be linear if it maps any operation symbol to a linear term of the same arity. Linear hypersubstitutions have some importance in Theoretical Computer Science since they preserve recognizability [7]. We show that the collection of all linear hypersubstitutions forms a monoid. Linear hypersubstitutions are used to define linear hyperidentities. The set of all linear term operations of a given algebra forms with respect to certain superposition operations a function algebra. Hypersubstitutions define endomorphisms on this function algebra.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제19권1호
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• pp.43-57
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• 2012

By simply splitting the hypergeometric Saran function $F_E$ into eight parts, we show how some useful and generalized relations between $F_E$ and Srivas- tava's hypergeometric function $F^{(3)}$ can be obtained. These main results are shown to be specialized to yield certain relations between functions $_0F_1$, $_1F_1$, $_0F_3$, ${\Psi}_2$, and their products including different combinations with different values of parameters and signs of variables.