• Journal of Educational Research in Mathematics
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• v.13 no.1
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• pp.13-28
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• 2003

Practice of proof is considered in, the view of language and meta-mathematics, recognizing the role of proof that is the means of communication and development of mathematical understanding. Linguistic components in proof texts are symbol, verbal language and visual text, and contain the implicit knowledge in the meta-mathematics view. This study investigates the functions of linguistic elements according to Halliday's functional grammar and the rhetoric skills in proof texts in math textbook, teacher's note, and student's written text. We need to inquire into the aspects of language for mathematics learning process and the understanding and use of students' language.

• Kyungpook Mathematical Journal
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• v.11 no.1
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• pp.93-99
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• 1971

In this paper, the author has established the formulae for product of two generalized hypergeometric polynomials by defining the polynomial in the form $$F_n(x)=x^{({\delta}-1)n}{_{p+{\delta}}F_q}\[\array{{\Delta}({\delta},-n),&a_1,&{\cdots}{\cdots},&a_p\\&b_1,&{\cdots}{\cdots},&b_q};\;{\lambda}x^c\]$$, where the symbol ${\Delta}({\delta},-n)$ represents the set of ${\delta}$-parameters: $${\frac{-n}{\delta}},\;{\frac{-n+1}{\delta}},\;{\cdots}{\cdots},\;{\frac{-n+{\delta}-1}{\delta}}$$ and ${\delta}$, n are positive integers. A number of known as well as new results have been also obtained with proper choice of parameters.

• Kyungpook Mathematical Journal
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• v.51 no.2
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• pp.139-143
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• 2011

In this paper we consider mappings ${\sigma}$ which map the binary operation symbol f to the term ${\sigma}$(f) which do not necessarily preserve the arities. We call these mappings generalized hypersubstitutions. Any generalized hypersubstitution ${\sigma}$ can be extended to a mapping $\hat{\sigma}$ on the set of all terms of type ${\tau}$ = (2). We de ne a binary operation on the set $Hyp_G$(2) of all generalized hypersubstitutions of type ${\tau}$ = (2) by using this extension The set $Hyp_G$(2) together with the identity generalized hypersubstitution ${\sigma}_{id}$ which maps f to the term f($x_1,x_2$) forms a monoid. We determine all regular elements of this monoid.

• Kyungpook Mathematical Journal
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• v.52 no.2
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• pp.109-136
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• 2012

We aim at presenting certain unexpected functional relations among various hypergeometric functions of one or several variables (for example, see the identities in Corollary 5) by making use of Carlson's method employed in his work (Some extensions of Lardner's relations between $_0F_3$ and Bessel functions, SIAM J. Math. Anal. 1(2)(1970), 232-242).

• Kyungpook Mathematical Journal
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• v.45 no.3
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• pp.395-404
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• 2005

Let α be a complex number with 𝕽α > 0. Let the functions f and g be analytic in the unit disc E = {z : |z| < 1} and normalized by the conditions f(0) = g(0) = 0, f'(0) = g'(0) = 1. In the present article, we study the differential subordinations of the forms $${\alpha}{\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}+{\frac{zf^{\prime}(z)}{f(z)}}{\prec}{\alpha}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}}+{\frac{zg^{\prime}(z)}{g(z)}},\;z{\in}E,$$ and $${\frac{z^2f^{{\prime}{\prime}}(z)}{f(z)}}{\prec}{\frac{z^2g^{{\prime}{\prime}}(z)}{g(z)}},\;z{\in}E.$$ As consequences, we obtain a number of sufficient conditions for star likeness of analytic maps in the unit disc. Here, the symbol ' ${\prec}$ ' stands for subordination

• Kyungpook Mathematical Journal
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• v.63 no.1
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• pp.45-60
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• 2023

In this paper we suggest an enhanced Geraghty-type contractive mapping for examining the existence properties of classical nonlinear operators with or without prior degenerates. The nonlinear operators are proved to exist with the imposition of the Geraghty-type condition in a non-empty closed subset of complete metric spaces. To showcase some efficacies of the Geraghty-type condition, convergent rate and stability are deduced. The results are used to study some asymptotic properties of perturbed integral and hypergeometric operators. The results also extend and generalize some existing Geraghty-type conditions.

• Communications of Mathematical Education
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• v.24 no.2
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• pp.475-502
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• 2010

In this paper, the ability to use mathematical representations in solving math problem was analyzed according to student assessment levels using 113 first-year high school students, and the characteristics of their representation usage according to student assessment levels were also examined. For this purpose, problems were presented that could be solved using various mathematical representations, and the students were asked to solve them using a maximum of three different methods. Also, based on the comparative analysis results of a paper evaluation, six students were selected and interviewed, and the reasons for their representation usage differences were analyzed according to their student assessment levels. The results of the analysis show that over 50% of high ranking students used two or more representations in all questions to solve problems, but with middle ranking students, there were deviations depending on the difficulty of the questions. Low ranking students failed to use representation in diverse ways when solving problems. As for characteristics of symbol usage, high ranking students preferred using formulas and used mathematical representations efficiently while solving problems. In contrast, middle and low ranking students mostly used tables or pictures. Even when using the same representations, high ranking students' representations were expressed in a more structurally refined manner than those by middle and low ranking students.

• Journal of Communications and Networks
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• v.12 no.5
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• pp.433-441
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• 2010

Multihop transmission is a promising technique that helps in achieving broader coverage (excellent network connectivity) and preventing the impairment of wireless channels. This paper proposes a cluster-based multihop wireless network that makes use of the advantages of multihop relaying, i.e., path loss gain, and partial relay selection in each hop, i.e., spatial diversity. In this partial relay selection, the node with the maximum instantaneous channel gain will serve as the sender for the next hop. With the proposed protocol, the transmit power and spectral efficiency can be improved over those in the case of direct transmission and conventional multihop transmission. Moreover, at a high signal-to-noise ratio (SNR), the performance of the system with at least two nodes in each cluster is dependent only on the last hop and not on any of the intermediate hops. For a practically feasible decode-and-forward relay strategy, a compact expression for the probability density function of the end-to-end SNR at the destination is derived. This expression is then used to derive closed-form expressions for the outage probability, average symbol error rate, and average bit error rate for M-ary square quadrature amplitude modulation as well as to determine the spectral efficiency of the system. In addition, the probability of SNR gain over direct transmission is investigated for different environments. The mathematical analysis is verified by various simulation results for demonstrating the accuracy of the theoretical approach.

• The Journal of the Korea institute of electronic communication sciences
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• v.12 no.6
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• pp.1027-1034
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• 2017

In this paper, we propose a new function embedding method that can measure mathematical projections of probability amplitude, probability, average expectation and matrix elements of stationary-state unit matrix at all control operation points of quantum gates. The function embedding method in this paper is to embed orthogonal normalization condition of probability amplitude for each control operating point into a binary scalar operator by using Dirac symbol and Kronecker delta symbol. Such a function embedding method is a very effective means of controlling the arithmetic power function of a unitary gate in a unitary transformation which expresses a quantum gate function as a tensor product of a single quantum. We present the results of evolutionary operation and projective measurement when we apply the proposed function embedding method to the ternary 2-qutrit cNOT gate and compare it with the existing methods.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.35 no.3C
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• pp.314-319
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• 2010

The chirp spread spectrum (CSS) technique that spreads the data signal over a wider frequency band via chirp signal for transmission has attracted much attention in the field of wireless communications due to capability to resist multipath fading signals. However, there has been little mathematical analysis for the performance of CSS-based communication systems in the multipath fading environments. In this paper, we study the influence of the multipath channel on the direct modulation (DM) scheme with M-ary phase shift keying (MPSK). When a chirp signal is transmitted on the Rayleigh fading channel and affected by additive white Gaussian noise (AWGN), we derive the theoretical performance. From numerical results, it is confirmed that the analytic symbol error rate (SER) agrees closely with the empirical SER.