• Kyungpook Mathematical Journal
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• v.13 no.2
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• pp.235-240
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• 1973

A k-dimensional vector bundle is a bundle ${\xi}=(E,P,B,F^k)$ with fibre $F^k$ satisfying the local triviality, where F is the field of real numbers R or complex numbers C ([1], [2] and [3]). Let $Vect_k(X)$ be the set consisting of all isomorphism classes of k-dimensional vector bundles over the topological space X. Then $Vect_F(X)=\{Vect_k(X)\}_{k=0,1,{\cdots}}$ is a semigroup with Whitney sum (${\S}1$). For a pair (X, A) of topological spaces, a difference isomorphism over (X, A) is a vector bundle morphism ([2], [3]) ${\alpha}:{\xi}_0{\rightarrow}{\xi}_1$ such that the restriction ${\alpha}:{\xi}_0{\mid}A{\longrightarrow}{\xi}_1{\mid}A$ is an isomorphism. Let $S_k(X,A)$ be the set of all difference isomorphism classes over (X, A) of k-dimensional vector bundles over X with fibre $F^k$. Then $S_F(X,A)=\{S_k(X,A)\}_{k=0,1,{\cdots}}$, is a semigroup with Whitney Sum (${\S}2$). In this paper, we shall prove a relation between $Vect_F(X)$ and $S_F(X,A)$ under some conditions (Theorem 2, which is the main theorem of this paper). We shall use the following theorem in the paper. THEOREM 1. Let ${\xi}=(E,P,B)$ be a locally trivial bundle with fibre F, where (B, A) is a relative CW-complex. Then all cross sections S of ${\xi}{\mid}A$ prolong to a cross section $S^*$ of ${\xi}$ under either of the following hypothesis: (H1) The space F is (m-1)-connected for each $m{\leq}dim$ B. (H2) There is a relative CW-complex (Y, X) such that $B=Y{\times}I$ and $A=(X{\times}I)$ ${\cap}(Y{\times}O)$, where I=[0, 1]. (For proof see p.21 [2]).

• Journal of Broadcast Engineering
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• v.25 no.4
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• pp.587-597
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• 2020

In this paper, we analyze the normalization that occurs when processing digital hologram and propose an optimized quantization method. In JPEG Pleno, which standardizes the compression of holograms, full complex holograms are defined as complex numbers with 32-bit or 64-bit precision, and the range of values varies greatly depending on the method of hologram generation and object type. Such data with high precision and wide dynamic range are converted to fixed-point or integer numbers with lower precision for signal processing and compression. In addition, in order to reconstruct the hologram to the SLM (spatial light modulator), it is approximated with a precision of a value that can be expressed by the pixels of the SLM. This process can be refereed as a normalization process using quantization. In this paper, we introduce a method for normalizing high precision and wide range hologram using quantization technique and propose an optimized method.

• Wind and Structures
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• v.21 no.1
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• pp.25-40
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• 2015

Oscillating motions of flexible cylinders are associated to some extent with the aerodynamic response of plants. Tandem motions of reeds with flexible stems in a colony are experimentally investigated using an array of flexible cylinders made of polydimethylsiloxane (PDMS). Consecutive images of flexible cylinders subjected to oncoming wind are recorded with a high-speed camera. To quantify oscillating motions, the average bending angle and displacement of flexible cylinders are evaluated using point-tracking method and spectral analysis. The tandem motions of flexible cylinders are closely related to the flow characteristics around the cylinders. Thus, the dynamic motions of a tandem arrangement of flexible cylinders are investigated with varying numbers of cylinders arranged in-line, numbers of cylinders in a group (behaving like a single body), and Reynolds numbers (Re). When the number of cylinders in a group increases, the damping effect caused by the support of downstream cylinders is pronounced. These results would be provide useful information on the tandem-arranged design of complex structures and energy harvesting devices.

• Journal of the Korean Mathematical Society
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• v.44 no.1
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• pp.55-107
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• 2007

Let k be an imaginary quadratic field, h the complex upper half plane, and let $\tau{\in}h{\cap}k,\;q=e^{{\pi}i\tau}$. In this article, we obtain algebraic numbers from the 130 identities of Rogers-Ramanujan continued fractions investigated in [28] and [29] by using Berndt's idea ([3]). Using this, we get special transcendental numbers. For example, $\frac{q^{1/8}}{1}+\frac{-q}{1+q}+\frac{-q^2}{1+q^2}+\cdots$ ([1]) is transcendental.

• Journal for History of Mathematics
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• v.26 no.4
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• pp.259-275
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• 2013

It has been proposed since modern times that we need to consult the history of mathematics in teaching mathematics, and some modifications of this principle were made recently by Lakatos, Freudenthal, and Brousseau. It may be necessary to have a direction which we consult when modifying the history of mathematics for students. In this article, we analyse the elements of the cognitive interest in Hamilton's discovery of the quaternions and in the history of discovery of imaginary numbers, and we investigate the effects of these elements on attention of the students of nowadays. These works may give a direction to the historic-genetic principle in teaching mathematics.

• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.977-998
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• 2003

Let k be an imaginary quadratic field, η the complex upper half plane, and let $\tau$ $\in$ η $textsc{k}$, p = $e^{{\pi}i{\tau}}$. In this article, using the infinite product formulas for g2 and g3, we prove that values of certain infinite products are transcendental whenever $\tau$ are imaginary quadratic. And we derive analogous results of Berndt-Chan-Zhang ([4]). Also we find the values of (equation omitted) when we know j($\tau$). And we construct an elliptic curve E : $y^2$ = $x^3$ ＋ 3 $x^2$ ＋ {3-(j/256)}x ＋ 1 with j = j($\tau$) $\neq$ 0 and P = (equation omitted) $\in$ E.

• The Transactions of the Korean Institute of Electrical Engineers
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• v.39 no.4
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• pp.341-348
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• 1990

This paper describes an algorithm for evaluating the Loss of Load Probability (LOLP) and calculating the production cost for all the generators in the system using Fast Hartley Transform (FHT). It also suggests the deconvolution procedure which is necessary for the generation expansion planning. The FHT is as fast as or faster than the Fast Fourier Transform (FFT) and serves for all the uses such as spectral, digital processing, and convolution to which the FFT is normally applied. The transformed function using FFT has complex numbers. However, the transformed function using FHT has real numbers and the convolution become quite simple. This method has been applied for the IEEE reliability test system and practical size model system. The test results show the effectiveness of the proposed method.

• Journal of computational fluids engineering
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• v.18 no.3
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• pp.79-86
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• 2013

Preconditioned compressible Navier-Stokes equations are solved for almost incompressible flows. Unstructured meshes are utilized for spatial discretization of complex flow domain. Effectiveness of the current preconditioning algorithm, with respect to various Reynolds numbers and Mach numbers, is demonstrated by the solution of canonical problems for incompressible flows, e.g. driven cavity flows.

• Journal of the Korea Institute of Building Construction
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• v.8 no.3
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• pp.119-125
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• 2008

The volume of material and number of workers mobilized has been on the rise in line with the domestic projects getting higher, larger and complex currently. Particularly for the project in downtown, delivering the resources in timely manner is very crucial in carrying out the overall project as scheduled. Inappropriate lifting plan often causes inefficiency over the entire project, resulting in increase in schedule and cost. Despite of such importance of lifting plan for architectural work, lifting plans at the most of domestic projects, except a few cases for large scale high-rise buildings, have been heavily dependent on engineer's personal experience and intuition alone. To deal with such problems, the study was intended to develop and suggest a systematic and objective process for determining the lift, categorizing the lifting equipment into the two types, one for workers and another for material.

• Journal of Korean Society of Industrial and Systems Engineering
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• v.20 no.43
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• pp.321-331
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• 1997

Measurement of attributes is often highly subjective and imprecise, yet most MADM methods lack provisions for handling imprecise data. Frequently, decision makers must establish a ranking within a finite set of alternatives with respect to multiple attributes which have varying degrees of importance. The problem is more complex if the evaluations of alternatives according to each attribute are not expressed in precise numbers, but rather in fuzzy numbers. Analysis must allow for lack of precision and partial truth. The advantages of a fuzzy approach for MADM are that a decision maker can obtain efficient solutions all at once without trial and error, and that this approach provides better support for judging the interactive improvement of solutions in comparison with o decision making method. The algorithm used in this study is based on the concepts of vague set theory. Linguistic variables and vague values are used to facilitate a decision maker's subjective assessment about attribute weightings and the appropriateness of alternative versus selection attributes in order to obtain final scores which are called vague appropriateness indices. A numerical example is presented to show the practical applicability of this approach.