Diophantine equations and Fermat's last theorem for multivariate(skew-)polynomials

Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.

Solving Some Problems and Elimination in Systems of Polynomial Equations

In a factorial ring, we can define the p.g.c.d. of two elements (defined to the nearest unit) and the notion of prime elements between them. More generally, Bezout’s identity characterizes two prime elements in a main ring. A ring that satisfies the property of the theorem is called a Bezout ring. We have given some geometry theorems that can be proved algebraically, although the methods of geometry and, in particular, of projective geometry are by far the most beautiful. Most geometric problems actually involve polynomial equations and can be translated into the language of polynomial ideals. We have given a few examples of a different nature without pretending to make a general theory.

Linear Functional Equations and Twisted Polynomials

A certain variety of non-switched polynomials provides a uni-figure representation for a wide range of linear functional equations. This is properly adapted for the calculations. We reinterpret from this point of view a number of algorithms.

A proof of maximum contention-free property of interleavers for Turbo codes using permutation polynomials over integer rings

It is well known that interleavers play a critical role in Turbo coding/decoding schemes, and contention-free interleaver design has become a serious problem in the paraUelization of Turbo decoding, which is indispensable to meet the demands for high throughput and low latency in next generation mobile communication systems. This paper unveils the fact that interleavers based on permutation polynomials modulo N are contention-free for every window size W, a factor of the intedeaver length N, which, also called maximum contention-free interleavers.

The Some Properties of Skew Polynomial Rings

This paper mainly studies some properties of skew polynomial ring related to Morita invariance, Armendariz and (quasi)-Baer. First, we show that skew polynomial ring has no Morita invariance by the counterexample. Then we prove a necessary condition that skew polynomial ring constitutes Armendariz ring. We lastly investigate that condition of skew polynomial ring is a (quasi)-Baer ring, and verify that the conditions is necessary, but not sufficient by example and counterexample.

ON ABEL-GONTSCHAROFF-GOULD’S POLYNOMIALS

In this paper a connective study of Gould's annihilation coefficients and Abel-Gontscharoff polynomials is presented. It is shown that Gould's annihilation coefficients and Abel-Gontscharoff polynomials are actually equivalent to each other under certain linear substitutions for the variables. Moreover, a pair of related expansion formulas involving Gontscharoff s remainder and a new form of it are demonstrated, and also illustrated with several examples.

Monomial Derivations without Darboux Polynomials

In this paper, it is proved that a monomial derivation d of k[x, y, z] has no Darboux polynomials if and only if d is a strict derivation with a trivial ring of constants, and we give the specific conditions when it has no Darboux polynomials.

Legendre Polynomial and Nonlinear Oscillating Point-Like Charged Particle

In this article we explore the kinematics of a point-like charged particle placed within the interior plane of a charged ring. Analytically we formulate the electric field of the ring along a representative diagonal. Graph of the field as a function of the distance from the center of the ring assists foreseeing oscillating movement of the charged particle. We formulate the equation of motion;this is a nonlinear differential equation. Applying Computer Algebra System (CAS), specifically Mathematica [1] we solve the equation numerically. Utilizing the solution we quantify the kinematic quantities of interest including oscillations period. Although the equation of motion is nonlinear its period is regulated. For better understanding we take an advantage of Mathematica animation features animating the nonlinear oscillations.

THE DISCUSSION ON ALGEBRAIC PROPERTIESOF POLYNOMIAL MATRICES

By using the results about polynomial matrix in the system and control theory. this paper gives some discussions about the algebraic properties of polynomial,matrices. The obtained main results include that the,ring of rr x n polynomial matrices is a principal ideal and principal one-sided ideal ring.

*-诣零McCoy环

研究了具有对合映射*-诣零McCoy环的性质,给出了一批*-诣零McCoy环例子,并讨论了其扩张和*-斜多项式环的*-诣零McCoy性,证明了(1)设*-环R满足nil(R[x])=nil(R)[x],则环R是*-诣零McCoy环当且仅当环R[x]是*-诣零McCoy环;(2)设R[x;*]是*-斜多项式环,如果R是*-可逆环,则R[x;*]是*-诣零McCoy环。

函数域中与相交多项式相关的计数估计

设F_(q)[t]为q个元的有限域F_(q)上的多项式环,设h∈F_(q)[t][x]为一个相交多项式,A■F_(q)[t]为由一些次数小于N的多项式构成的集合.利用密度增量估计构造4元组序列,当A的密度适当大时,得到了方程x-y=z解数的估计,其中:x、y∈A,z∈h(F_(q)[t])\{0}.

关于Seksenbaev-Robinson定理

剩余有限群也被称为是可以有限逼近的群,其特性常常由它的有限商群的性质决定.Seksenbaev定理断言:若对无限多个素数p,多重循环群G都是剩余有限p-群,则G是有限生成的无挠幂零群.Robinson把该定理推广为:设G是有限秩的可解群,若对无限多个素数p,G都是剩余有限p-群,则G是有限秩的无挠幂零群.这是无限可解群里的两个经典结果.本文证明了有关无限可解群的两个剩余有限性定理.本文的结果完善了Seksenbaev-Robinson定理.

Ramp Scheme Based on CRT for Polynomial Ring over Finite Field

Chinese Reminder Theorem(CRT)for integers has been widely used to construct secret sharing schemes for different scenarios,but these schemes have lower information rates than that of Lagrange interpolation-based schemes.In ASIACRYPT 2018,Ning,et al.constructed a perfect(r,n)-threshold scheme based on CRT for polynomial ring over finite field,and the corresponding information rate is one which is the greatest case for a(r,n)-threshold scheme.However,for many practical purposes,the information rate of Ning,et al.scheme is low and perfect security is too much security.In this work,the authors generalize the Ning,et al.(r,n)-threshold scheme to a(t,r,n)-ramp scheme based on CRT for polynomial ring over finite field,which attains the greatest information rate(r−t)for a(t,r,n)-ramp scheme.Moreover,for any given 2≤r_(1)<r_(2)≤n,the ramp scheme can be used to construct a(r_(1),n)-threshold scheme that is threshold changeable to(r′,n)-threshold scheme for all r′∈{r_(1)+1,r_(1)+2,···,r_(2)}.The threshold changeable secret sharing(TCSS)scheme has a greater information rate than other existing TCSS schemes of this type.

Extensions of McCoy Rings

A ring R is said to be right McCoy if the equation f（x）g（x） = 0, where y（x） and g（x） are nonzero polynomials of R[x], implies that there exists nonzero s E R such that f（x）s = 0. It is proven that no proper （triangular） matrix ring is one-sided McCoy. It is shown that for many polynomial extensions, a ring R is right Mccoy if and only if the polynomial extension over R is right Mccoy.

Linearly McCoy Rings and Their Generalizations

A ring R is called linearly McCoy if whenever linear polynomials f（x）, g（x） e R[x]／{0） satisfy f（x）g（x） ： O, then there exist nonzero elements r, s ∈ R such that f（x）r ： sg（x） =0. For a ring endomorphism α, we introduced the notion of α-skew linearly McCoy rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x]. A number of properties of this generalization are established and extension properties of α-skew linearly McCoy rings are given.

On the Cozero-Divisor Graphs of Commutative Rings

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by , is a graph with vertices in , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in are adjacent if and only if and . In this paper, we investigate some combinatorial properties of the cozero-divisor graphs and such as connectivity, diameter, girth, clique numbers and planarity. We also study the cozero-divisor graphs of the direct products of two arbitrary commutative rings.

Annihilator on Co-commutators with Derivations on Lie Ideals in Prime Rings

Let R be a prime ring of characteristic different from 2, d and 9 two derivations of R at least one of which is nonzero, L a non-central Lie ideal of R, and a ∈ R. We prove that if a（d（u）u - ug（u）） = 0 for any u ∈ L, then either a = O, or R is an sa-ring, d（x） = [p, x], and g（x） = -d（x） for some p in the Martindale quotient ring of R.

Parametric Stabilization of the Ring and Linear Neural Network with Two Delays

This paper is devoted to the problem of stabilizing a Hopfield-type neural network with bi-directional ring architecture and two delays. The delay-independent and delay-dependent stability conditions are explicitly presented by the method of the characteristic roots and the skill of mathematical analysis. Moreover, if a link between the adjacent two neurons is cut, the ring neural network turns to a linear one, and the stability results are also established. Furthermore, a comparative analysis for the ring and linear network shows that the stability domain is enlarged after the breaking.

Analytic solutions of the double ring-shaped Coulomb potential in quantum mechanics

The exact solutions of the Schr6dinger equation with the double ring-shaped Coulomb potential are presented, including the bound states, continuous states on the ＂k/2π scale＂, and the calculation formula of the phase shifts. The polar angular wave functions are expressed by constructing the so-called super-universal associated Legendre polynomials. Some special cases are discussed in detail.

大环径比O形复合金属密封件制备工艺及静力学性能研究

针对工业领域中对于大口径耐高温的金属密封件的迫切需求,开发一种将勾连交叉缠绕而成的螺旋网状金属橡胶嵌入有一定开口槽的不锈钢细环中的大环径比O形复合金属密封件,探究大环径比O形复合金属密封件的制备工艺,包含金属橡胶弹性内芯以及不锈钢包覆层两部分的制备工艺。研究4种不同孔隙度的金属橡胶密封件在准静态常温下的力学性能,分析载荷、孔隙度对金属橡胶密封件力学性能的影响。研究结果表明,复合金属橡胶密封件在高承载下具有良好的密封特性,当载荷增大后,密封件的力学性能更加趋于稳定;且随着孔隙度的减小,承载能力越来越大。基于静力学性能试验,建立含有不锈钢包覆的金属橡胶密封件的多项式拟合函数,发现六阶多项式能够较好地对试验数据进行拟合,由此可计算分析金属橡胶密封件的静刚度与位移之间的变化规律,从而可以在工程上得到更好的应用。