SELF-DUAL PERMUTATION CODES OVER FORMAL POWER SERIES RINGS AND FINITE PRINCIPAL IDEAL RINGS

In this paper, we study self-dual permutation codes over formal power series rings and finite principal ideal rings. We first give some results on the torsion codes associated with the linear codes over formal power series rings. These results allow for obtaining some conditions for non-existence of self-dual permutation codes over formal power series rings. Finally, we describe self-dual permutation codes over finite principal ideal rings by examining permutation codes over their component chain rings.

A Note on z-Ideals and z°-Ideals of the Formal Power Series Rings and Polynomial Rings in an Infinite Set of Indeterminates

Let A be a ring.In this paper we generalize some results introduced by Aliabad and Mohamadian.We give a relation bet ween the z-ideals of A and t hose of the formal power series rings in an infinite set of indetermiiiates over A.Consider A[[Xa]]3 and its subrings A[[X_(A)]]_(1),A[[X_(A)]]_(2),and A[[X_(A)]]_(α),where a is an infinite cardinal number.In fact,a z-ideal of the rings defined above is of the form I+(X_(A))i,where i=1,2,3 or an infinite cardinal number and I is a z-ideal of A.In addition,we prove that the same condition given by Aliabad and Mohamadian can be used to get a relation between the minimal prime ideals of the ring of the formal power series in an infinite set of indeterminates and those of the ring of coefficients.As a natural result,we get a relation between the z°-ideals of the formal power series ring in an infinite set of indeterminates and those of the ring of coefficients.

Two Kinds of Series Involving the Reciprocals of Binomial Coefficients

By applying the theory of formal power series,the author obtains the closed forms for two kinds of infinite series involving the reciprocals of binomial coefficients,and the author gets another closed form for the infinite series Σr≥m tn＋r/（n＋rr）.

Analysis of Formal and Analytic Solutions for Singularities of the Vector Fractional Differential Equations

In this article, we study on the existence of solution for a singularities of a system of nonlinear fractional differential equations （FDE）. We construct a formal power series solution for our considering FDE and prove convergence of formal so- lutions under conditions. -We use the Caputo fractional differential operator and the nonlinearity depends on the fractional derivative of an unknown function.

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.

ON THE CHARACTERIZATION OF CYCLIC CODES OVER TWO CLASSES OF RINGS

Let R be a finite chain ring with maximal ideal （7） and residue field F,and letγ be of nilpotency index t. To every code C of length n over R, a tower of codes C = （C ： γ0） C_ （C： 7） C ... C_ （C： γ2） C_ .-. C_ （C：γ^t-1） can be associated with C, where for any r C R, （C ： r） = {e C Rn I re E C}. Using generator elements of the projection of such a tower of codes to the residue field F, we characterize cyclic codes over R. This characterization turns the condition for codes over R to be cyclic into one for codes over the residue field F. Furthermore, we obtain a characterization of cyclic codes over the formal power series ring of a finite chain ring.

ALGEBRAIC OPERATION OF SPECIAL MATRICES RELATED TO METHOD OF LEAST SQUARES

The following situation in using the method of least squares to solve problems often occurs.After m experiments completed and a solution of least squares obtained,the ( m+1 ) th experiment is made further in order to improve the results.A method of algebraic operation of special matrices involved in the problem is given in this paper for obtaining a new solution for the m +1 experiments based upon the old solution for the primary m experiments. This method is valid for more general matrices.

Nonlinear Inverse Relations of the Bell Polynomials via the Lagrange Inversion Formula (Ⅱ)

In this paper,by means of the classical Lagrange inversion formula,the authors establish a general nonlinear inverse relation as the solution to the problem proposed in the paper[J.Wang,Nonlinear inverse relations for the Bell polynomials via the Lagrange inversion formula,J.Integer Seq.,Vol.22(2019),Article 19.3.8].As applications of this inverse relation,the authors not only find a short proof of another nonlinear inverse relation due to Birmajer,et al.(2012),but also set up a few convolution identities concerning the Mina polynomials.