• The Journal of Korean Institute of Communications and Information Sciences
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• v.12 no.3
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• pp.285-291
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• 1987

Seismic waves are attenuated by losses of energy as they propagate through the earth. One way to model this numerically is to make the velocity a complex number, the real part giving the phase velocity and the imaginary part the attenuation. This models wave propagation in a medium for which the logarithmic decrement is independent of frequency(attenuation coefficient is proportional to frequncy). The aim is to modify forward and inverse numerical modeling so that attenuation can be specified as a function of position.

• Honam Mathematical Journal
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• v.31 no.4
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• pp.463-478
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• 2009

Let k be an imaginary quadratic field, ℌ the complex upper half plane, and let ${\tau}{\in}k{\cap}$ℌ, q = exp(${\pi}i{\tau}$). And let n, t be positive integers with $1{\leq}t{\leq}n-1$. Then $q^{{\frac{n}{12}}-{\frac{t}{2}}+{\frac{t^2}{2n}}}{\prod}^{\infty}_{m=1}(1-q^{nm-t})(1-q^{nm-(n-t)})$ is an algebraic number [10]. As a generalization of this result, we find several infinite series and products giving algebraic numbers using Ramanujan's $_{1{\psi}1}$ summation. These are also related to Rogers-Ramanujan continued fractions.

• Journal of Ocean Engineering and Technology
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• v.8 no.2
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• pp.80-92
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• 1994

The dynamic response characteristics of a floating ocean city are examined for presenting the basic data for the design of huge offshore structures supported by a large number of floating bodies in waves. The numerical approach which is accurate in linear system is based on combination of a three dimensional source distribution method, wave interaction theory and the finite element method of using the space frame element. The hydrodynamic interactions among the floating bodies are taken into account in their exact form within the context of linear potential theory in the motion and structural analysis. The method is applicable to an arbitrary number of three dimensional bodies having any individual body geometries and geometrical arrangement with the restriction that the circumscribed, bottom-mounted. Imaginary vertical cylinder for each body does not contain any part of the other body. The validity of this procedure was verified by comparing with experimental results obtained in the literature.

• Bulletin of the Korean Mathematical Society
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• v.56 no.4
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• pp.815-827
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• 2019

It is well known that the denominator of the Dedekind sum s(c, d) divides 2 gcd(d, 3)d and that no smaller denominator independent of c can be expected. In contrast, here we prove that we usually get a smaller denominator in S(H, d), the sum of the s(c, d)'s over all the c's in a subgroup H of order n > 1 in the multiplicative group $(\mathbb{Z}/d\mathbb{Z})^*$. First, we prove that for p > 3 a prime, the sum 2S(H, p) is a rational integer of the same parity as (p-1)/2. We give an application of this result to upper bounds on relative class numbers of imaginary abelian number fields of prime conductor. Finally, we give a general result on the denominator of S(H, d) for non necessarily prime d's. We show that its denominator is a divisor of some explicit divisor of 2d gcd(d, 3).

• Journal of the Korean Mathematical Society
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• v.60 no.5
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• pp.1087-1107
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• 2023

Let E be an elliptic curve defined over ℚ of conductor N, c the Manin constant of E, and m the product of Tamagawa numbers of E at prime divisors of N. Let K be an imaginary quadratic field where all prime divisors of N split in K, P_K the Heegner point in E(K), and III(E/K) the Shafarevich-Tate group of E over K. Let 2u_K be the number of roots of unity contained in K. Gross and Zagier conjectured that if P_K has infinite order in E(K), then the integer c · m · u_K · |III(E/K)| $\frac{1}{2}$ is divisible by |E(ℚ)_tor|. In this paper, we prove that this conjecture is true if E(ℚ)_tor ≅ ℤ/2ℤ or ℤ/4ℤ except for two explicit families of curves. Further, we show these exceptions can be removed under Stein-Watkins conjecture.

• School Mathematics
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• v.10 no.3
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• pp.357-374
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• 2008

The complex number system is supposed to introduce first chapter in the first grade of high school. When number system is expanded to complex numbers, the main aim is to understand preservation of algebraic structure with regard to the flow of curriculum and textbook. This research reviewed overall alternative articulation and treatment of textbooks from a logical viewpoint. Two research questions are developed below. First, in the structure of the current curriculum, when we consider student's 'level', how are the alternative articulation and treatment of textbooks in complex unit on a logical point of view? Second, What are more logical alternative articulation and treatment? What alternative articulation and treatment are suitable for a running goal? and what are the improvement which is definitive?

• The Journal of the Korea institute of electronic communication sciences
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• v.12 no.6
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• pp.1181-1188
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• 2017

The unit and fundamental units of number fields are important to number field sieves testing primality of more than 400 digits integers and number field seive factoring the number in RSA cryptosystem, and multiplication of ideals and counting class number of the number field in imaginary quadratic cryptosystem. To minimize the time and space in H/W implementation of cryptosystems using fundamental units, in this paper, we introduce the Dirichlet's unit Theorem and propose our process of generating the fundamental units of the number field. And then we present the algorithm generating our fundamental units of the number field to minimize the time and space in H/W implementation and implementation results using the algorithm over the number field.

• The Mathematical Education
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• v.51 no.1
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• pp.1-19
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• 2012

In this study, contents of 'the 2007 revised curriculum handbook' and 16 kinds of mathematics textbooks were analyzed first. The purpose of this study is to examine the understanding state of students at general high schools by making questionnaires to survey the understanding state on contents of chapter of complex number based on above analysis. Results of research can be summarized as follows. First, the content of chapter of complex number in textbook was not logically organized. In the introduction of imaginary number unit, two kinds of marks were presented without any reason and it has led to two kinds of notation of negative square root. There was no explanation of difference between delimiter symbol and operator symbol at all. The concepts were presented as definition without logical explanations. Second, students who learned with textbook in which problems were pointed out above did not have concept of complex number for granted, and recognized it as expansion of operation of set of real numbers. It meant that they were confused of operation of complex numbers and did not form the image about number system itself of complex number. Implications from this study can be obtained as follows. First, as we came over to the 7th curriculum, the contents of chapter of complex number were too abbreviated to have the logical configuration of chapter in order to remove the burden for learning. Therefore, the quantitative expansion and logical configuration fit to the level for high school students corresponding to the formal operating stage are required for correct configuration of contents of chapter. Second, teachers realize the importance of chapter of complex number and reconstruct the contents of chapter to let students think conceptually and logically.

• Journal of the Korean Institute of Telematics and Electronics C
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• v.34C no.2
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• pp.29-37
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• 1997

A new algorithm and parallel architecture for high-speed complex number multiplication is presented, and a prototype chip based on the proposed approach is designed. By employing redundant binary (RB) arithmetic, an N-bit complex number multiplication is simplified to two RB multiplications (i.e., an addition of N RB partial products), which are responsible for real and imaginary parts, respectively. Also, and efficient RB encoding scheme proposed in this paper enables to generate RB partial products without additional hardware and delay overheads compared with binary partial product generation. The proposed approach leads to a highly parallel architecture with regularity and modularity. As a results, it results in much simpler realization and higher performance than the classical method based on real multipliers and adders. As a test vehicle, a prototype 8-b complex number multiplier core has been fabricated using $0.8\mu\textrm{m}$ CMOS technology. It contains 11,500 transistors on the area of about $1.05 \times 1.34 textrm{mm}^2$. The functional and speed test results show that it can safely operate with 200 MHz clock at $V_{DD}=2.5 V$, and consumes about 90mW.

• Journal for History of Mathematics
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• v.25 no.4
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• pp.69-82
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• 2012

The objective of this paper is to study De Morgan's perspective on teaching and learning complex numbers. De Morgan's didactical approaches reflect the process of development of his thoughts about algebra from universal arithmetic, symbolic algebra to meaning algebra. De Morgan develop his perspective on algebra by justifying and explaining complex numbers. This implies that teaching and learning complex numbers is a catalyst for mathematical development of De Morgan.