• Title/Summary/Keyword: special polynomials

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ANALYSIS OF THE 90/150 CA GENERATED BY LINEAR RULE BLOCKS

  • CHO, SUNG-JIN;KIM, HAN-DOO;CHOI, UN-SOOK;KIM, JIN-GYOUNG;KANG, SUNG-WON
    • Journal of applied mathematics & informatics
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    • v.37 no.1_2
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    • pp.23-35
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    • 2019
  • Self-reciprocal polynomials are important because it is possible to specify only half of the coefficients. The special case of the self-reciprocal polynomial, the maximum weight polynomial, is particularly important. In this paper, we analyze even cell 90/150 cellular automata with linear rule blocks of the form < $a_1,{\cdots},a_n,d_1,d_2,b_n,{\cdots},b_1$ >. Also we show that there is no 90/150 CA of the form < $U_n{\mid}R_2{\mid}U^*_n$ > or < $\bar{U_n}{\mid}R_2{\mid}\bar{U^*_n}$ > whose characteristic polynomial is $f_{2n+2}(x)=x^{2n+2}+{\cdots}+x+1$ where $R_2$ =< $d_1,d_2$ > and $U_n$ =< $0,{\cdots},0$ >, and $\bar{U_n}$ =< $1,{\cdots},1$ >.

Characteristic Polynomial of 90 UCA and Synthesis of CA using Transition Rule Blocks (90 UCA의 특성다항식과 전이규칙 블록을 이용한 CA 합성법)

  • Choi, Un-Sook;Cho, Sung-Jin
    • The Journal of the Korea institute of electronic communication sciences
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    • v.13 no.3
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    • pp.593-600
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    • 2018
  • Cellular automata (CA) have been applied to effective cryptographic system design. CA is superior in randomness to LFSR due to the fact that its state is updated simultaneously by local interaction. To apply these CAs to the cryptosystem, a study has been performed how to synthesize CA corresponding to given polynomials. In this paper, we analyze the recurrence relations of the characteristic polynomial of the 90 UCA and the characteristic polynomial of the 90/150 CA whose transition rule is <$00{\cdots}001$>. And we synthesize the 90/150 CA corresponding to the trinomials $x^{2^n}+x+1(n{\geq}2)$ satisfying f(x)=f(x+1) using the 90 UCA transition rule blocks and the special transition rule block. We also analyze the properties of the irreducible factors of trinomials $x^{2^n}+x+1$ and propose a 90/150 CA synthesis algorithm corresponding to $x^{2^n}+x^{2^m}+1(n{\geq}2,n-m{\geq}2)$.

New prototypes of target transfer functions for time domain specification (시간영역 설계명세를 위한 목표전달함수의 새로운 표준형)

  • Kim, Sin-Gu;Kim, Yeong-Cheol
    • Journal of Institute of Control, Robotics and Systems
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    • v.5 no.8
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    • pp.889-897
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    • 1999
  • This paper deals with a problem searching a target transfer function to meet the time-domain specifications for feedback system with given plant transfer function. For the Type I system, we first define three forms of transient response to unit step input, which are named by F, M, S-type. These are charaacterized as follows ; F-type has fast initial response and slow approach to the steady sate after reaching at 90% of the steady state value, S-type has slow initial response but fast approach to the steady state, and M-type is denoted by highly smooth response between F-type and S-type. Three prototypes corresponding to each form are proposed, time. For the order $n{\geq}4$, after determining admissible root structures of target characteristic polynomials empirically and expressing such polynomial coefficients by using special parameters ${\gamma}_i$ and $\epsilon$, the optimal prototypes that minimize the integral of the squared of the modified errors(ISME) have been obtained. Since the step responses of these prototypes have almost same wave forms irrespective to the order, the desired settling time or the rise time can be converted into the equibalent time constant $\tau$ and thus it is easy to obtain a target transfer function. It is shown through a design example that the present prototype is very useful for meeting the time-domain specifications and has been compared with different methods with a viewpoint of pertinence.

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Investigation of the behavior of a crack between two half-planes of functionally graded materials by using the Schmidt method

  • Zhou, Zhen-Gong;Wang, Biao;Wu, Lin-Zhi
    • Structural Engineering and Mechanics
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    • v.19 no.4
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    • pp.425-440
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    • 2005
  • In this paper, the behavior of a crack between two half-planes of functionally graded materials subjected to arbitrary tractions is resolved using a somewhat different approach, named the Schmidt method. To make the analysis tractable, it is assumed that the Poisson's ratios of the mediums are constants and the shear modulus vary exponentially with coordinate parallel to the crack. By use of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. This process is quite different from those adopted in previous works. Numerical examples are provided to show the effect of the crack length and the parameters describing the functionally graded materials upon the stress intensity factor of the crack. It can be shown that the results of the present paper are the same as ones of the same problem that was solved by the singular integral equation method. As a special case, when the material properties are not continuous through the crack line, an approximate solution of the interface crack problem is also given under the assumption that the effect of the crack surface interference very near the crack tips is negligible. It is found that the stress singularities of the present interface crack solution are the same as ones of the ordinary crack in homogenous materials.

Ring of Four Almonds and the Omar Khayyam's Triangle in Islamic Art Design (이슬람 예술 디자인에서 회전하는 알몬드와 오마르 하얌의 삼각형)

  • Park, Jeanam;Park, Mingu
    • Journal for History of Mathematics
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    • v.32 no.4
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    • pp.159-173
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    • 2019
  • In this paper, we examine the brief history of the ring of four almonds regarding Mesopotamian mathematics, and present reasons why the Omar Khayyam's triangle, a special right triangle in a ring of four almonds, was essential for artisans due to its unique pattern. We presume that the ring of four almonds originated from a point symmetry figure given two concentric squares used in the proto-Sumerian Jemdet Nasr period (approximately 3000 B.C.) and a square halfway between two given concentric squares used during the time of the Old Akkadian period (2340-2200 B.C.) and the Old Babylonian age (2000-1600 B.C.). Artisans tried to create a new intricate pattern as almonds and 6-pointed stars by subdividing right triangles in the pattern of the popular altered Old Akkadian square band at the time. Therefore, artisans needed the Omar Khayyam's triangle, whose hypotenuse equals the sum of the short side and the perpendicular to the hypotenuse. We presume that artisans asked mathematicians how to construct the Omar Khayyam's triangle at a meeting between artisans and mathematicians in Isfahan. The construction of Omar Khayyam's triangle requires solving an irreducible cubic polynomial. Omar Khayyam was the first to classify equations of integer polynomials of degree up to three and then proceeded to solve all types of cubic equations by means of intersections of conic sections. Omar Khayyam's triangle gave practical meaning to the type of cubic equation $x^3+bx=cx^2+a$. The work of Omar Khayyam was completed by Descartes in the 17th century.