scholarly journals Cogredient Standard Forms of Symmetric Matrices over Galois Rings of Odd Characteristic

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yonglin Cao
Keyword(s):  
Prime Number ◽  
Explicit Formula ◽  
Galois Ring ◽  
Symmetric Matrices ◽  
Galois Rings ◽  
Positive Integers

Let R=GR(ps,psm) be a Galois ring of characteristic ps and cardinality psm, where s and m are positive integers and p is an odd prime number. Two kinds of cogredient standard forms of symmetric matrices over R are given, and an explicit formula to count the number of all distinct cogredient classes of symmetric matrices over R is obtained.

COORDINATE SETS OF GENERALIZED GALOIS RINGS

2004 ◽  
Vol 03 (01) ◽  
pp. 31-48 ◽  
Author(s):  
S. GONZÁLEZ ◽  
C. MARTÍNEZ ◽  
I. F. RÚA ◽  
V. T. MARKOV ◽  
A. A. NECHAEV
Keyword(s):  
Prime Number ◽  
Closed Subset ◽  
Galois Ring ◽  
Galois Rings ◽  
Characteristic P ◽  
Nonassociative Ring

A Generalized Galois Ring (GGR) S is a finite nonassociative ring with identity of characteristic pn, for some prime number p, such that its top-factor [Formula: see text] is a semifield. It is well-known that if S is an associative Galois Ring (GR), then it contains a multiplicatively closed subset isomorphic to [Formula: see text], the so-called Teichmüller Coordinate Set (TCS). In this paper we show that the existence of a TCS characterizes GR in the class of all GGR S such that the multiplicative loop [Formula: see text] is right (or left) primitive.

scholarly journals Normal Bases on Galois Ring Extensions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 702
Author(s):  
Aixian Zhang ◽  
Keqin Feng
Keyword(s):  
Signal Processing ◽  
Finite Field ◽  
Block Cipher ◽  
Field Extension ◽  
Galois Ring ◽  
Normal Basis ◽  
Galois Rings ◽  
Ring Extension ◽  
Galois Fields ◽  
Normal Bases

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

scholarly journals The restricted Burnside problem for Moufang loops

2021 ◽  
pp. 1-11
Author(s):  
ALEXANDER GRISHKOV ◽  
LIUDMILA SABININA ◽  
EFIM ZELMANOV
Keyword(s):  
Prime Number ◽  
Burnside Problem ◽  
Moufang Loops ◽  
Restricted Burnside Problem ◽  
Positive Integers

Abstract We prove that for positive integers $m \geq 1, n \geq 1$ and a prime number $p \neq 2,3$ there are finitely many finite m-generated Moufang loops of exponent $p^n$ .

INDEX-DEPENDENT DIVISORS OF COEFFICIENTS OF MODULAR FORMS

2013 ◽  
Vol 09 (07) ◽  
pp. 1841-1853 ◽  
Author(s):  
B. K. MORIYA ◽  
C. J. SMYTH
Keyword(s):  
Prime Number ◽  
Modular Forms ◽  
Fourier Coefficients ◽  
Survey Methods ◽  
Integer Sequences ◽  
Positive Integers

We evaluate [Formula: see text] for a certain family of integer sequences, which include the Fourier coefficients of some modular forms. In particular, we compute [Formula: see text] for all positive integers n for Ramanujan's τ-function. As a consequence, we obtain many congruences — for instance that τ(1000m) is always divisible by 64000. We also determine, for a given prime number p, the set of n for which τ(pn-1) is divisible by n. Further, we give a description of the set {n ∈ ℕ : n divides τ(n)}. We also survey methods for computing τ(n). Finally, we find the least n for which τ(n) is prime, complementing a result of D. H. Lehmer, who found the least n for which |τ(n)| is prime.

Symbol-Pair Distances of Repeated-Root Negacyclic Codes of Length 2s over Galois Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 581-600
Author(s):  
Hai Q. Dinh ◽  
Hualu Liu ◽  
Roengchai Tansuchat ◽  
Thang M. Vo
Keyword(s):  
Finite Field ◽  
Galois Ring ◽  
Maximum Distance ◽  
Galois Rings ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Negacyclic Codes ◽  
Singleton Bound ◽  
Maximum Distance Separable ◽  
Special Case

Negacyclic codes of length [Formula: see text] over the Galois ring [Formula: see text] are linearly ordered under set-theoretic inclusion, i.e., they are the ideals [Formula: see text], [Formula: see text], of the chain ring [Formula: see text]. This structure is used to obtain the symbol-pair distances of all such negacyclic codes. Among others, for the special case when the alphabet is the finite field [Formula: see text] (i.e., [Formula: see text]), the symbol-pair distance distribution of constacyclic codes over [Formula: see text] verifies the Singleton bound for such symbol-pair codes, and provides all maximum distance separable symbol-pair constacyclic codes of length [Formula: see text] over [Formula: see text].

scholarly journals Integers free of small prime factors in arithmetic progressions

2000 ◽  
Vol 157 ◽  
pp. 103-127 ◽  
Author(s):  
Ti Zuo Xuan
Keyword(s):  
Prime Number ◽  
Short Range ◽  
Error Term ◽  
Real Zero ◽  
Prime Number Theorem ◽  
Arithmetic Progressions ◽  
Real Character ◽  
Dickman Function ◽  
Prime Factors ◽  
Positive Integers

For real x ≥ y ≥ 2 and positive integers a, q, let Φ(x, y; a, q) denote the number of positive integers ≤ x, free of prime factors ≤ y and satisfying n ≡ a (mod q). By the fundamental lemma of sieve, it follows that for (a,q) = 1, Φ(x,y;a,q) = φ(q)-1, Φ(x, y){1 + O(exp(-u(log u- log2 3u- 2))) + (u = log x log y) holds uniformly in a wider ranges of x, y and q.Let χ be any character to the modulus q, and L(s, χ) be the corresponding L-function. Let be a (‘exceptional’) real character to the modulus q for which L(s, ) have a (‘exceptional’) real zero satisfying > 1 - c0/log q. In the paper, we prove that in a slightly short range of q the above first error term can be replaced by where ρ(u) is Dickman function, and ρ′(u) = dρ(u)/du.The result is an analogue of the prime number theorem for arithmetic progressions. From the result can deduce that the above first error term can be omitted, if suppose that 1 < q < (log q)A.

scholarly journals Asymptotic counting theorems for primitive juggling patterns

2019 ◽  
Vol 15 (05) ◽  
pp. 1037-1050
Author(s):  
Erik R. Tou
Keyword(s):  
Prime Number ◽  
Generating Functions ◽  
Arbitrary Number ◽  
Mathematical Theory ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Theoretical Framework ◽  
Positive Integers ◽  
Infinite Set ◽  
Fixed Period

The mathematics of juggling emerged after the development of siteswap notation in the 1980s. Consequently, much work was done to establish a mathematical theory that describes and enumerates the patterns that a juggler can (or would want to) execute. More recently, mathematicians have provided a broader picture of juggling sequences as an infinite set possessing properties similar to the set of positive integers. This theoretical framework moves beyond the physical possibilities of juggling and instead seeks more general mathematical results, such as an enumeration of juggling patterns with a fixed period and arbitrary number of balls. One problem unresolved until now is the enumeration of primitive juggling sequences, those fundamental juggling patterns that are analogous to the set of prime numbers. By applying analytic techniques to previously-known generating functions, we give asymptotic counting theorems for primitive juggling sequences, much as the prime number theorem gives asymptotic counts for the prime positive integers.

scholarly journals 2-Adic valuations of Stirling numbers of the first kind

2019 ◽  
Vol 15 (09) ◽  
pp. 1827-1855 ◽  
Author(s):  
Min Qiu ◽  
Shaofang Hong
Keyword(s):  
Explicit Formula ◽  
Symmetric Functions ◽  
Stirling Numbers ◽  
Elementary Symmetric Functions ◽  
Disjoint Cycles ◽  
Adic Valuation ◽  
Positive Integers

Let [Formula: see text] and [Formula: see text] be positive integers. We denote by [Formula: see text] the 2-adic valuation of [Formula: see text]. The Stirling numbers of the first kind, denoted by [Formula: see text], count the number of permutations of [Formula: see text] elements with [Formula: see text] disjoint cycles. In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, and Adelberg made some progress on the study of the [Formula: see text]-adic valuations of [Formula: see text]. In this paper, by introducing the concept of [Formula: see text]th Stirling numbers of the first kind and providing a detailed 2-adic analysis, we show an explicit formula on the 2-adic valuation of [Formula: see text]. We also prove that [Formula: see text] holds for all integers [Formula: see text] between 1 and [Formula: see text]. As a corollary, we show that [Formula: see text] if [Formula: see text] is odd and [Formula: see text]. This confirms partially a conjecture of Lengyel raised in 2015. Furthermore, we show that if [Formula: see text], then [Formula: see text] and [Formula: see text], where [Formula: see text] stands for the [Formula: see text]th elementary symmetric functions of [Formula: see text]. The latter one supports the conjecture of Leonetti and Sanna suggested in 2017.

scholarly journals Further Results on a Curious Arithmetic Function

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Long Chen ◽  
Kaimin Cheng ◽  
Tingting Wang
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Function ◽  
Rational Numbers ◽  
Arithmetic Properties ◽  
Adic Valuation ◽  
Positive Integers

Let p be an odd prime number and n be a positive integer. Let vpn, N∗, and Q+ denote the p-adic valuation of the integer n, the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn for any positive integer n. We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the p-adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime.

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