Characteristic Ratio Symmetric Polynomials and Their Root Characteristics

Young Chol Kim 제어·로봇·시스템학회 2021 International Journal of Control, Automation, and Vol.19 No.5

For a real polynomial p(s)=a_ns^n+...+a_1s+a_0, its characteristic ratios are defined by \alpha_i :=a_i^2/a_{i-1}a_{i+1} for i =1, 2, ..., n-1, and the generalized time constant is defined by \tau := a_1/a_0. In contrast, every coefficient of the polynomial p(s) can be represented in terms of \alpha_i and \tau. We present a novel family of polynomials named characteristic ratio symmetric (CRS), where a polynomial p(s) is said to be CRS if \alpha_i = \alpha_{n-i} for 1 <= i <= (n-1) with any \tau. This paper deals with the relationships between the roots and {\alpha_i, t} of a CRS polynomial. It is shown that some of the roots of the CRS polynomial are on the circle of a specific radius \omega_c while the rest appear in fourtuples {\lambda_i, \omega_c^2 / \lambda_i, \lambda_i^*, \omega_c^2 / \lambda_i^*}. For CRS polynomials of the fifth or lower order, we derive that the damping ratio and natural frequency of every root of these polynomials can be uniquely represented in terms of only {\alpha_1, \alpha_2, \tau} or {\alpha_1, \tau} for less than third order. It is also shown that a special polynomial named K-polynomial is a CRS polynomial and the damping of an nth-order K-polynomial can be adjusted by just choosing a single parameter \alpha_1.

A SURVEY ON SOME OLD AND NEW IDENTITIES ASSOCIATED WITH LAPLACE DISTRIBUTION AND BERNOULLI NUMBERS

ZEHRA SELIN ASKAN,IREM KUCUKOGLU,Yilmaz Simsek 장전수학회 2020 Advanced Studies in Contemporary Mathematics Vol.30 No.4

The purpose of this paper is to give some survey on old and new identities related to the characteristic function, the Laplace distribution and special numbers and polynomials with comparative results and observations. Additionally, we give some computation formulas for the higher-order moments of some kinds of random variables with the Laplace distribution in terms of the Bernoulli numbers of the first kind, the Euler numbers of the second kind and Riemann zeta function by using the techniques of generating functions and characteristic function of the aforementioned random variables. Finally, with the aid of the Hankel determinants formed by the moments corresponding to the weight function that reveals the orthogonality feature of the orthogonal polynomials, we give futher remarks and observations on not only orthogonality properties of some orthogonal polynomials such as the Hermite polynomials, but also construction methods of the three-term recurrence relations for the orthogonal polynomials.

CHARACTERISTIC POLYNOMIALS OF SOME WEIGHTED GRAPH BUNDLES AND ITS APPLICATION TO LINKS

Sohn, Moo-Young,Lee, Ja-Eun 國立 昌原大學校 基礎科學硏究所 1994 基礎科學硏究所論文集 Vol.6 No.-

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weighted K_(2) (??_(2)) bundles over a weighted graph Γ_(ω) can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs are Γ As an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

CHARACTERISTIC POLYNOMIALS OF SOME WEIGHTED GRAPH BUNDLES AND ITS APPLICATION TO LNKS

SOHN, MOO YOUNG,LEE, JAEUN 경북대학교 위상수학 기하학연구센터 1995 硏究論文集 Vol.1 No.-

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weighted K_(2) (??_(2))- bundles over a weighted graph Γ_(ω) can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs are Γ As an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

CHARACTERISTIC POLYNOMIALS OF SOME WEIGHTED GRAPH BUNDLES AND ITS APPLICATION TO LINKS

LEE,JAEUN,SOHN,MOO YOUNG 國立昌原大學校 基礎科學硏究所 1994 基礎科學硏究所論文集 Vol.6 No.-

In this paper, we introduce weighted graph bundles and study their characteristic polynomial. In particular, we show that the characteristic polynomial of a weighted K₂(K₂)-bundles over a weighted graph Γω can be expressed as a product of characteristic polynomials two weighted graphs whose underlying graphs are Γ As an application, we compute the signature of a link whose corresponding weighted graph is a double covering of that of a given link.

Relation between the Irreducible Polynomials that Generates the Same Binary Sequence Over Odd Characteristic Field

Ali, Md. Arshad,Kodera, Yuta,Park, Taehwan,Kusaka, Takuya,Nogmi, Yasuyuki,Kim, Howon The Korea Institute of Information and Commucation 2018 Journal of information and communication convergen Vol.16 No.3

A pseudo-random sequence generated by using a primitive polynomial, trace function, and Legendre symbol has been researched in our previous work. Our previous sequence has some interesting features such as period, autocorrelation, and linear complexity. A pseudo-random sequence widely used in cryptography. However, from the aspect of the practical use in cryptographic systems sequence needs to generate swiftly. Our previous sequence generated by utilizing a primitive polynomial, however, finding a primitive polynomial requires high calculating cost when the degree or the characteristic is large. It’s a shortcoming of our previous work. The main contribution of this work is to find some relation between the generated sequence and irreducible polynomials. The purpose of this relationship is to generate the same sequence without utilizing a primitive polynomial. From the experimental observation, it is found that there are (p - 1)/2 kinds of polynomial, which generates the same sequence. In addition, some of these polynomials are non-primitive polynomial. In this paper, these relationships between the sequence and the polynomials are shown by some examples. Furthermore, these relationships are proven theoretically also.

Every polynomial over a field containing F_16 is a strict sum of four cubes and one expression A^2+A

Luis H. Gallardo 대한수학회 2009 대한수학회보 Vol.46 No.5

Let q be a power of 16. Every polynomial P 2 F_q[t] is a strict sum P = A^2 + A + B^3 + C^3 + D^3 + E^3. The values of A,B,C,D,E are effectively obtained from the coefficients of P. The proof uses the new result that every polynomial Q ∈ F_q[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: Q = F^2 + F + tG^2. This improves for such q’ and such Q’ a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials F, G,H for the strict representation Q = F^2+F +GH. Observe that the latter representation may be considered as an analogue in characteristic 2 of the strict representation of a polynomial Q by three squares in odd characteristic. Let q be a power of 16. Every polynomial P 2 F_q[t] is a strict sum P = A^2 + A + B^3 + C^3 + D^3 + E^3. The values of A,B,C,D,E are effectively obtained from the coefficients of P. The proof uses the new result that every polynomial Q ∈ F_q[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: Q = F^2 + F + tG^2. This improves for such q’ and such Q’ a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials F, G,H for the strict representation Q = F^2+F +GH. Observe that the latter representation may be considered as an analogue in characteristic 2 of the strict representation of a polynomial Q by three squares in odd characteristic.

Characteristic polynomial of a generalized complete product of matrices

Hwang, S.G.,Park, J.W. North Holland [etc.] 2011 Linear algebra and its applications Vol.434 No.5

For a simple graph G, let G@? denote the complement of G relative to the complete graph and let P<SUB>G</SUB>(x)=det(xI-A(G)) where A(G) denotes the adjacency matrix of G. The complete product G@?H of two simple graphs G and H is the graph obtained from G and H by joining every vertex of G to every vertex of H. In [2]P<SUB>G@?H</SUB>(x) is represented in terms of P<SUB>G</SUB>, P<SUB>G@?</SUB>, P<SUB>H</SUB> and P<SUB>H@?</SUB>. In this paper we extend the notion of complete product of simple graphs to that of generalized complete product of matrices and obtain their characteristic polynomials.

90 UCA의 특성다항식과 전이규칙 블록을 이용한 CA 합성법

최언숙(Un Sook Choi),조성진(Sung Jin Cho) 한국전자통신학회 2018 한국전자통신학회 논문지 Vol.13 No.3

효과적인 암호시스템 설계에 셀룰라 오토마타(이하 CA)가 적용되고 있다. CA는 국소적 상호작용에 의해 상태가 동시에 업데이트되는 성질이 있어서 LFSR보다 랜덤성이 우수하다. 이런 CA를 암호 시스템에 적용하기 위해 주어진 다항식에 대응하는 CA를 합성하는 방법에 대한 연구가 진행되었다. 본 논문에서는 90 UCA의 특성다항식과 전이규칙이 <00 … 001>인 90/150 CA의 특성다항식의 점화관계를 분석한다. 또한 f(x)=f(x+1)을 만족하는 삼항다항식 x²ⁿ+x+1에 대응하는 90/150 CA를 90 UCA 전이규칙 블록과 특별한 전이규칙 블록을 이용하여 합성한다. 또한 x²ⁿ+x+1의 기약인수에 관한 성질을 분석한 후 x²ⁿ+x²ᵐ+1(n≥2, n-m≥2)에 대응하는 90/150 CA 합성 알고리즘을 제안한다. 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 … 001>. And we synthesize the 90/150 CA corresponding to the trinomials x²ⁿ+x+1(n≥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 x²ⁿ+x+1 and propose a 90/150 CA synthesis algorithm corresponding to x²ⁿ+x²ᵐ+1(n≥2, n-m≥2).

EVERY POLYNOMIAL OVER A FIELD CONTAINING 𝔽₁₆ IS A STRICT SUM OF FOUR CUBES AND ONE EXPRESSION A² + A

Gallardo, Luis H. Korean Mathematical Society 2009 대한수학회보 Vol.46 No.5

Let q be a power of 16. Every polynomial $P\in\mathbb{F}_q$[t] is a strict sum $P=A^2+A+B^3+C^3+D^3+E^3$. The values of A,B,C,D,E are effectively obtained from the coefficients of P. The proof uses the new result that every polynomial $Q\in\mathbb{F}_q$[t], satisfying the necessary condition that the constant term Q(0) has zero trace, has a strict and effective representation as: $Q=F^2+F+tG^2$. This improves for such q's and such Q's a result of Gallardo, Rahavandrainy, and Vaserstein that requires three polynomials F,G,H for the strict representation $Q=F^2$+F+GH. Observe that the latter representation may be considered as an analogue in characteristic 2 of the strict representation of a polynomial Q by three squares in odd characteristic.