Search

In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above 2. As an application, we show that for each isogeny factor of the Jacobian of the p -th Fermat curve where 2 is a quadratic residue modulo p , there are infinitely many twists whose analytic rank is zero. Also, for a certain hyperelliptic curve over the 11-th cyclotomic field whose Jacobian has complex multiplication, there are infinitely many twists whose analytic rank is zero. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, Q be the set of quadratic residues modulo p, and N be the set of quadratic non-residues modulo p. In this paper, we observe that the problem of finding the number of solutions of ∑i=1kx i2ηi = ζ over 픽p, where (ηi, p−1 2 ) = 1 is equivalent to the problem of counting the multiplicity (see Definition 1) of Q, N, or zero in the set of sums of k quadratic residues, which are not necessarily distinct, depending on whether ζ is a quadratic residue, a non-residue, or zero, respectively. Thus, we count the multiplicity of Q, N, or zero in the set of sums of k quadratic residues using a graphical approach. In addition, we compute the number of times the sum of k squares and s non-squares over 픽p yields a square, non-square, or zero. Furthermore, we obtain an explicit formula for the sum of k distinct quadratic residues (non-residues) modulo p, where 2 ≤ k ≤ 5. The results obtained in this study are useful in the factorization of the polynomial xp − 1 over 픽q, where q is a quadratic residue modulo p, and are also helpful in developing time-efficient computer programs. [ABSTRACT FROM AUTHOR]

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime p. The sum over the distinct residues can sometimes be computed independent of the prime p ; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the value set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree. [ABSTRACT FROM AUTHOR]

Modulo a prime number, we define semi-primitive roots as the square of primitive roots. We present a method for calculating primitive roots from quadratic residues, including semi-primitive roots. We then present progressions that generate primitive and semi-primitive roots, and deduce an algorithm to obtain the full set of primitive roots without any gcd calculation. Next, we present a method for determining irreducible quadratic forms with arbitrarily large conjectured asymptotic density of primes (after Shanks, [1][2]). To this end, we propose an algorithm for calculating the square root modulo p, based on the Tonelli-Shanks algorithm [3]. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with the identity 1R, and let R* be the multiplicative group of units in R. An element a ∈ R* is called an exceptional unit if there exists a b ∈ R* such that a + b = 1R. We set R** to be the set of all exceptional units in R. In this paper, we consider the residue-class ring Zn. For any positive integers n, s, and c ∈ Zn, let Ns(n, c) :=#{(x1, ..., xs) ∈ (Z** n ) s : x1 + ... + xs ≡ c (mod n)} . In 2016, Sander (J.Number Theory 159 (2016)) got a formula for N2(n, c). Later on, Yang and Zhao (Monatsh. Math. 182 (2017)) extended Sander’s theorem to finite terms by using exponential sum theory. In this paper, using matrix theory, we present an explicit formula for Ns(n, c). This extends and improves earlier results. [ABSTRACT FROM AUTHOR]

Paley graphs form a nice link between the distribution of quadratic residues and graph theory. These graphs possess remarkable properties which make them useful in several branches of mathematics. Classically, for each prime number p we can construct the corresponding Paley graph using quadratic and non-quadratic residues modulo p. Therefore, Paley graphs are naturally associated with the Legendre symbol at p which is a quadratic Dirichlet character of conductor p. In this article, we introduce the generalized Paley graphs. These are graphs that are associated with a general quadratic Dirichlet character. We will then provide some of their basic properties. In particular, we describe their spectrum explicitly. We then use those generalized Paley graphs to construct some new families of Ramanujan graphs. Finally, using special values of L-functions, we provide an effective upper bound for their Cheeger number. As a by-product of our approach, we settle a question raised in [Cramer et al.: The isoperimetric and Kazhdan constants associated to a Paley graph, Involve 9 (2016), 293–306] about the size of this upper bound. [ABSTRACT FROM AUTHOR]

Copyright of Arab Journal for Scientific Publishing is the property of Research & Development of Human Recourses Center (REMAH) and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The book "Number Theory Through the Eyes of Sophie Germain: An Inquiry Course" by David Pengelley explores the correspondence between Sophie Germain and renowned mathematicians Joseph-Louis Lagrange, Adrien-Marie Legendre, and Carl Friedrich Gauss as they attempted to find a proof for Fermat's Last Theorem. Although Germain's plan was ultimately unsuccessful, it represented significant progress on the problem and provided insights into the theorem. The book takes an inquiry-based approach, using primary source materials to teach undergraduate students elementary number theory. It offers a unique and engaging perspective on the subject, allowing students to discover and prove results themselves. [Extracted from the article]

Let S = { a 1 , a 2 , ⋯ , a n } be a finite set of non-zero integers. In [5], Karthick Babu and Anirban Mukhopadhyay calculated the explicit structure of the Galois group of multi-quadratic field Q (a 1 , a 2 , ⋯ , a n) over Q . For a positive integer d ⩾ 3 , ζ d denotes the primitive d-th root of unity. In this paper, we calculate the explicit structure of the Galois group of Q (a 1 , a 2 , ⋯ , a n , ζ d) over Q in terms of its action on ζ d and a i for 1 ⩽ i ⩽ n . [ABSTRACT FROM AUTHOR]

We present new constructions of binary quantum codes from quaternary linear Hermitian self-dual codes. Our main ingredients for these constructions are nearly self-orthogonal cyclic or duadic codes over F 4 . An infinite family of 0-dimensional binary quantum codes is provided. We give minimum distance lower bounds for our quantum codes in terms of the minimum distance of their ingredient linear codes. We also present new results on the minimum distance of linear cyclic codes using their fixed subcodes. Finally, we list many new record-breaking quantum codes obtained from our constructions. [ABSTRACT FROM AUTHOR]

The main goal of this research was to design and study the best structure, location, and shape of acoustic diffusers to be fitted on the ceilings of multipurpose auditoriums. Their absorbing properties can enhance the acoustics when installed on high ceilings, and behind suspended reflecting panels, by mitigating or nullifying specular reflections that could overcome the panels and, thus, avoiding time delay gaps exceeding 30–40 ms compared with the direct sound. For this purpose, a typical medium-sized room, with inclined floors, a stage, and 20 rows of seats, was considered. The allocation and height of the considered diffusers were based on the Schroeder quadratic residue sequence, and they were modeled as rectangles, wedges, cylinders, and Y-shaped elements. A standardized speech source spectrum was analyzed for up to five different receiver locations. In this way, the attenuation parameter as a function of frequency was evaluated and compared between the candidate diffusers in order to identify the best absorber. The simulations were undertaken with a software tool previously validated by the authors called PARDOS, which incorporates an innovative formulation based on the uniform theory of diffraction (UTD) to analyze multiple diffractions and reflections of acoustic waves. The results show that the new Y-shaped diffusers proposed, tuned for the hearing frequency band from 250 Hz up to 10,000 Hz, attained the best acoustic performance in terms of absorption. [ABSTRACT FROM AUTHOR]

In this work, we investigate essential definitions, defining G as a simple graph with vertices in Z n and subgraphs Γ u and Γ q as unit residue and quadratic residue graphs modulo n , respectively. The investigation extends to the degree of G , Γ u , and Γ q , illuminating the properties of these subgraphs in the context of quadratic congruences. [ABSTRACT FROM AUTHOR]

Copyright of Environmental & Social Management Journal / Revista de Gestão Social e Ambiental is the property of Environmental & Social Management Journal and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper, we introduce secure groups as a cryptographic scheme representing finite groups together with a range of operations, including the group operation, inversion, random sampling, and encoding/decoding maps. We construct secure groups from oblivious group representations combined with cryptographic protocols, implementing the operations securely. We present both generic and specific constructions, in the latter case specifically for number-theoretic groups commonly used in cryptography. These include Schnorr groups (with quadratic residues as a special case), Weierstrass and Edwards elliptic curve groups, and class groups of imaginary quadratic number fields. For concreteness, we develop our protocols in the setting of secure multiparty computation based on Shamir secret sharing over a finite field, abstracted away by formulating our solutions in terms of an arithmetic black box for secure finite field arithmetic or for secure integer arithmetic. Secure finite field arithmetic suffices for many groups, including Schnorr groups and elliptic curve groups. For class groups, we need secure integer arithmetic to implement Shanks' classical algorithms for the composition of binary quadratic forms, which we will combine with our adaptation of a particular form reduction algorithm due to Agarwal and Frandsen. As a main result of independent interest, we also present an efficient protocol for the secure computation of the extended greatest common divisor. The protocol is based on Bernstein and Yang's constant-time 2-adic algorithm, which we adapt to work purely over the integers. This yields a much better approach for multiparty computation but raises a new concern about the growth of the Bézout coefficients. By a careful analysis, we are able to prove that the Bézout coefficients in our protocol will never exceed 3 max (a , b) in absolute value for inputs a and b. We have integrated secure groups in the Python package MPyC and have implemented threshold ElGamal and threshold DSA in terms of secure groups. We also mention how our results support verifiable multiparty computation, allowing parties to jointly create a publicly verifiable proof of correctness for the results accompanying the results of a secure computation. [ABSTRACT FROM AUTHOR]

Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697–728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263–284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear with a frequency which is a perfect t th power. In addition, the partitions that we study here have smallest part greater than or equal to s for some given natural number s. Our inequalities hold after a certain bound, which for given t is a polynomial in s, a major improvement over the previously known bound in the case t = 1 . To prove these inequalities, our methods involve constructing injective maps between the relevant sets of partitions. The construction of these maps crucially involves concepts from analysis and calculus, such as explicit maps used to prove countability of N t , and Jensen's inequality for convex functions, and then merge them with techniques from number theory such as Frobenius numbers, congruence classes, binary numbers and quadratic residues. We also show a connection of our results to colored partitions. Finally, we pose an open problem which seems to be related to power residues and the almost universality of diagonal ternary quadratic forms. [ABSTRACT FROM AUTHOR]

Private Simultaneous Messages (PSM) model is a minimal model for secure multiparty computation. Feige, Kilian, and Naor (STOC 1994) and Ishai (Cryptology and Information Security Series 2013) constructed PSM protocols based on quadratic residues. In this paper, we define QR-PSM protocols as a generalization of these protocols. A QR-PSM protocol is a PSM protocol whose decoding function outputs the quadratic residuosity modulo p of what is computed from messages. We design a QR-PSM protocol for any symmetric function f : { 0 , 1 } n → { 0 , 1 } of communication complexity O (n 2) . As far as we know, it is the most efficient PSM protocol for symmetric functions since the previously known best PSM protocol was of O (n 2 log n) (Beimel et al., CRYPTO 2014). We also study the sizes of the underlying finite fields F p in the protocols since the communication complexity of a QR-PSM protocol is proportional to the bit length of the prime p. We show that there is a prime p ≤ (1 + o (1)) N 2 2 2 N - 2 such that any length-N pattern of quadratic (non)residues appears modulo p (and hence it can be used for general QR-PSM protocols), which improves the Peralta's known result (Mathematics of Computation 1992) by a constant factor (1 + 2) 2 . [ABSTRACT FROM AUTHOR]

In this paper, we discuss the adjacency matrices of finite undirected simple graphs over a finite prime field F p. We apply symmetric (row and column) elementary transformations to the adjacency matrix over F p in order to get a direct sum decomposition by other adjacency matrices. In this paper, we give a complete description of the direct sum decomposition of the adjacency matrix of any graph over F p for any odd prime p. Our key tool is quadratic residues of F p. [ABSTRACT FROM AUTHOR]

Following V. I. Arnold, we define the stochasticity parameter |$S(U)$| of a subset |$U$| of |${\mathbb {Z}}/M{\mathbb {Z}}$| to be the sum of squares of the consecutive distances between elements of |$U$|⁠. In this paper, we study the stochasticity parameter of the set |$R_{M}$| of quadratic residues modulo |$M$|⁠. We present a method that allows to find the asymptotics of |$S(R_{M})$| for a set of |$M$| of positive density. In particular, we obtain the following two corollaries. Denote by |$s(k)=s(k,{\mathbb {Z}}/M{\mathbb {Z}})$| the average value of |$S(U)$| over all subsets |$U\subseteq {\mathbb {Z}}/M{\mathbb {Z}}$| of size |$k$|⁠ , which can be thought of as the stochasticity parameter of a random set of size |$k$|⁠. Let |${\mathfrak {S}}(R_{M})=S(R_{M})/s(|R_{M}|)$|⁠. We show that (a) |$\varliminf _{M\to \infty }{\mathfrak {S}}(R_{M})<1<\varlimsup _{M\to \infty }{\mathfrak {S}}(R_{M})$|⁠ ; (b) the set |$\{ M\in {\mathbb {N}}: {\mathfrak {S}}(R_{M})<1 \}$| has positive lower density. [ABSTRACT FROM AUTHOR]

An original CSIKE post-quantum cryptography algorithm is proposed as a modification of the CSIDH algorithm but with one public key instead of two. The conditions for its implementation on two classes of non-cyclic Edwards curves are substantiated. The properties of quadratic and twisted supersingular Edwards curves that form pairs of quadratic twist of order p + 1 ≡ 0 mod 8 over a prime field Fp are considered. A modification of the CSIDH algorithm and the CSIKE algorithm are presented, which are generated on the isogenies of these curves instead of the traditional arithmetic of curves in the Montgomery form. For isogenies of degrees 3, 5, and 7, the parameters of isogenic chains of non-cyclic supersingular Edwards curves are calculated and tabulated for p = 839. The implementation of the key encapsulation scheme with its encryption by Alice with Bob's public key is considered. A new randomized CSIKE algorithm with random equiprobable selection of the curve from two classes at each step of the isogeny chain is proposed. The probability of a successful side-channel attack in a randomized algorithm is estimated. It is proposed to abandon the calculation of the isogenic function ϕ(R) of a random point R, which significantly speeds up the algorithm. [ABSTRACT FROM AUTHOR]

Let P(t) ± i = t 2k±itm be a non square polynomial and Q(t) ± i = 4k 2 t4k−2+i 2m² t2m−2± 4imkt2k+m−2 − 4t 2k ∓ 4itm − 1 be a polynomial, such that k ≥ 2m and i ∈ {1, 2}. In this paper, we consider the number of integer solutions of Diophantine equation E: x 2 − P(t) ± i y 2 − 2P′ (t) ± i x + 4P(t) ± i y + Q(t) ± i = 0. We extend a previous results given by A. Tekcan and A. Chandoul et al. We also derive some recurrence relations on the integer solutions of a Pell equation. [ABSTRACT FROM AUTHOR]

In the present paper, we give proofs of the existence of a 3‐design in the extended ternary quadratic residue code of length 14 and the extended quaternary quadratic residue code of length 18. [ABSTRACT FROM AUTHOR]

This paper shows electromagnetic field uniformity in an electromagnetic compatibility (EMC) chamber that is used as a test facility for measuring electromagnetic interference and radiated immunity of 6G communication systems. While not defined yet, 6G radio frequency will work in the wavelength ranges above 95 GHz. With this reason, this paper designed a schroeder-type quadratic residue diffuser for 95 GHz to generate a uniform electromagnetic field in the EMC chamber and studied the field uniformity characteristics in it. To analyze the distribution of electromagnetic fields inside the EMC chamber, finite-difference time-domain (FDTD) numerical analysis method is used. The simulation results show that the EMC chamber with this diffuser satisfies the requirements of international standards and has improved the field uniformity in the chamber. [ABSTRACT FROM AUTHOR]

We compute Hirano's formula for the mod 2 arithmetic Dijkgraaf–Witten invariant Z k for the ring of integers of the quadratic field k = Q ( p 1 ⋯ p r ) , where p i 's are distinct prime numbers with p i ≡ 1 (mod 4) , and give a simple formula for Z k in terms of the graph obtained from quadratic residues among p 1 , ... , p r . Our result answers the question posed by Ken Ono. We also give a density formula for mod 2 arithmetic Dijkgraaf–Witten invariants. [ABSTRACT FROM AUTHOR]

For a modular elliptic curve A = X 0 (4 9) and its quadratic twists A (M) , we give equivalent conditions such that the 2 -Selmer group S 2 (A (M) / ℚ) is minimal, namely, it is of order 2. One of these conditions is described by the L -value L (A (M) / ℚ , 1). The other conditions are described by quadratic and biquadratic residue symbol, so explicit and computable (and one can compute the density of M). Also we prove the full Birch–Swinnerton-Dyer conjecture when the equivalent conditions are satisfied. This generalizes a result by J. Coates, Y. Li, Y. Tian and S. Zhai. [ABSTRACT FROM AUTHOR]

For all q\ge 2 and for all invertible residue classes a modulo q, there exists a natural number that is congruent to a modulo q and that is the product of exactly three primes, all of which are below (10^{15}q)^{5/2}. [ABSTRACT FROM AUTHOR]

The study on graphs emerging from different algebraic structures such as groups, rings, fields, vector spaces, etc. is a prominent area of research in mathematics, as algebra and graph theory are two mathematical fields that focus on creating and analysing structures. There are numerous studies linking algebraic structures and graphs, which began with the introduction of Cayley graphs of groups. Several algebraic graphs have been defined on rings, a fast-growing area in the literature. In this article, we systematically review the literature on some variants of Cayley graphs that are defined on rings and highlight the properties and characteristics of such graphs, to showcase the research in this area. [ABSTRACT FROM AUTHOR]

A Latin square of order n $n$ is an n×n $n\times n$ matrix of n $n$ symbols, such that each symbol occurs exactly once in each row and column. For an odd prime power q $q$ let Fq ${{\mathbb{F}}}_{q}$ denote the finite field of order q $q$. A quadratic Latin square is a Latin square L[a,b] ${\rm{ {\mathcal L} }}[a,b]$ defined by (L[a,b])i,j=i+a(j−i)ifj−iis a quadratic residue inFq,i+b(j−i)otherwise, ${({\mathscr{L}}[a,b])}_{i,j}=\left\{\begin{array}{cc}i+a(j-i) & \,\text{if}\,\,j-i\,\,\text{is a quadratic residue in}\,\,{{\mathbb{F}}}_{q},\\ i+b(j-i) & \,\text{otherwise},\end{array}\right.$ for some {a,b}⊆Fq $\{a,b\}\subseteq {{\mathbb{F}}}_{q}$ such that ab $ab$ and (a−1)(b−1) $(a-1)(b-1)$ are quadratic residues in Fq ${{\mathbb{F}}}_{q}$. Quadratic Latin squares have previously been used to construct perfect 1‐factorisations, mutually orthogonal Latin squares and atomic Latin squares. We first characterise quadratic Latin squares which are devoid of 2×2 $2\times 2$ Latin subsquares. Let G $G$ be a graph and F ${\rm{ {\mathcal F} }}$ a 1‐factorisation of G $G$. If the union of every pair of 1‐factors in F ${\rm{ {\mathcal F} }}$ induces a Hamiltonian cycle in G $G$ then F ${\rm{ {\mathcal F} }}$ is called perfect, and if there is no pair of 1‐factors in F ${\rm{ {\mathcal F} }}$ which induce a Hamiltonian cycle in G $G$ then F ${\rm{ {\mathcal F} }}$ is called antiperfect. We use quadratic Latin squares to construct new examples of antiperfect 1‐factorisations of complete graphs and complete bipartite graphs. We also demonstrate that for each odd prime p $p$, there are only finitely many orders q $q$, which are powers of p $p$, such that quadratic Latin squares of order q $q$ could be used to construct perfect 1‐factorisations of complete graphs or complete bipartite graphs. [ABSTRACT FROM AUTHOR]

Let GF(l) be the Galois field with l = pm elements where p is a prime number and integer m ≥ 1. Here, we present three constructions for linear codes over GF(l) (depending on the parity of l) by using the quadratic residue approach and obtain some sufficient conditions for these codes to be LCD with respect to the Euclidean and Hermitian inner products, respectively. Furthermore, several examples of codes, including optimal and near to optimal codes, are provided to support our study. [ABSTRACT FROM AUTHOR]

In this paper, we consider the simultaneous Pell equations and where is a positive integer and is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution in positive integers. When a solution exists, assuming the positive solutions of the Pell equation are and with odd, then the only solution of the system is given by or or or . In this paper, we consider the simultaneous Pell equations and where is a positive integer and is squarefree and has at most three prime divisors. We obtain the necessary and sufficient conditions that the above simultaneous Pell equations have positive integer solutions by using only the elementary methods of factorization, congruence, the quadratic residue and fundamental properties of Lucas sequence and the associated Lucas sequence. Moreover, we prove that these simultaneous Pell equations have at most one solution in positive integers. When a solution exists, assuming the positive solutions of the Pell equation are and with odd, then the only solution of the system is given by or or or. [ABSTRACT FROM AUTHOR]

Copyright of Basic Sciences Journal of Textile Universities / Fangzhi Gaoxiao Jichu Kexue Xuebao is the property of Basic Sciences Journal of Textile Universities and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This work studies Diophantine equations of the form AX -BY = Z². Specifically, we determine the nonnegative integer solutions (pM, a, b, c) of the exponential Diophantine equation (pM)a - (pM +1)b = c² and its more generalized form (pM)a-(pM +1)b = c2n, where pM is a Mersenne prime number. Moreover, we also deal with the Diophantine equation (pM)a - (qM + 1)b = c², where pM and qM are both Mersenne primes. We solve these equations with the aid of elementary methods in number theory like the factoring technique and the modular arithmetic method. We also utilize Mihailescu's Theo- rem, the concepts of quadratic residue and Legendre symbol, and some properties of Mersenne primes for our Diophantine analysis. Results show that both (pM)a - (pM + 1)b = c² and (pM)a - (pM + 1)b = c2n have trivial solutions which only exist when a = 0 and b = 0, while (pM)a - (qM + 1)b = c² has two positive integer solutions. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. We study the distribution of consecutive quadratic residues modulo p in non-homogeneous Beatty sequences ℬ α , β = { ⌊ α n + β ⌋ : n = 1 , 2 , ... } , where α , β ∈ ℝ and α is irrational. Furthermore, we study the pseudorandom properties of sequences formed by the Legendre symbol and Beatty sequences. In addition, we give upper bounds for certain character sums over ℬ α , β . [ABSTRACT FROM AUTHOR]

Let p be an odd prime number and A : = ℤ / p 2 ℤ. Assume that either (i) p = 3 or (ii) p ≥ 5 and p is congruent to either 5 or 1 1 modulo 1 2 (respectively, assume that p ≥ 5 and p is congruent to either 1 or 7 modulo 1 2). Then there exist exactly five (respectively, exactly seven) isomorphism classes of rings R for which there exists a tower A ⊂ ℬ ⊂ R of ramified (integral minimal) ring extensions such that ℬ is the only ring properly contained between A and R. We produce a set of isomorphism class representatives of such R and, for each such R , we show that, up to isomorphism, the corresponding ring ℬ is the idealization A (+) p . One consequence, for each integer n > 1 whose prime-power factorization n = ∏ p i e i (with pairwise distinct prime numbers p i and positive integers e i ) satisfies e i ≤ 4 for all i , is a classification, up to isomorphism, of the rings that have cardinality n and exactly two proper (unital) subrings. [ABSTRACT FROM AUTHOR]

Copyright of Basic Sciences Journal of Textile Universities / Fangzhi Gaoxiao Jichu Kexue Xuebao is the property of Basic Sciences Journal of Textile Universities and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

A professor and her students were working through a routine problem on isomorphisms that required finding the multiplicative inverse of 43 modulo 50. The professor, however, simply asked for "the inverse" of 43 modulo 50 and got a surprising response. One student quickly answered that the inverse of 43 modulo 50 was 7. Thinking that the student was responding with the additive inverse, the professor restated her question and asked for the multiplicative inverse. The student checked her work and said that 7 is the multiplicative inverse of 43 mod 50. The observation that the additive and multiplicative inverse of an element could be the same led to a research project into the questions of when, why, and how frequently this phenomenon occurs. Answering these questions led to tools such asthe Chinese remainder theorem, the fundamental theorem of cyclic groups, and quadratic residues. [ABSTRACT FROM AUTHOR]

For any > 0 , there exists q 0 () such for any q ≥ q 0 () and any invertible residue class a modulo  q , there exists a natural number that is congruent to a modulo  q and that is the product of exactly three primes, all of which are below q 3 2 + . If we restrict our attention to odd moduli q that do not have prime factors congruent to 1 mod 4, we can find such primes below q 1 1 8 + . If we further restrict our set of moduli to prime q that are such that (q − 1 , 4 ⋅ 7 ⋅ 1 1 ⋅ 1 7 ⋅ 2 3 ⋅ 2 9) = 2 , we can find such primes below q 6 5 + . Finally, for any > 0 , there exists q 0 () such that when q ≥ q 0 () , there exists a natural number that is congruent to a modulo  q and that is the product of exactly four primes, all of which are below q (log q) 6 . [ABSTRACT FROM AUTHOR]

Assuming the Generalized Riemann Hypothesis, it is known that the smallest quadratic non-residue modulo a prime p is less than or equal to (\log p)^2. Our aim in this paper is to establish the distribution of quadratic non-residues in even smaller intervals of size (\log p)^A with A >1, for almost all primes p. [ABSTRACT FROM AUTHOR]

Nowadays, Radio frequency identification (RFID) plays an important role in many real-life applications for remotely identifying objects. RFID system works over a wireless communication environment. Because of this, the RFID system is not secure against various attacks such as user private data leakage, location tracking, and replay attack. To overcome these security flaws, we propose a quadratic residue-based authentication scheme for the RFID system. The scheme uses square root properties of quadratic residue to prevent existing possible attacks. Formal and informal security analysis of the proposed scheme shows that the proposed scheme resists various attacks. In addition, we use BAN logic and Scyther tool to simulate the scheme. The simulation results show that the proposed scheme withstands all possible attacks. Performance evaluation illustrates that the proposed scheme is efficient under a resource-constraints environment. Moreover, the proposed scheme does not store the private information of RFID tags in its database and identifies a tag with constant-time complexity. [ABSTRACT FROM AUTHOR]

Let S be a nonempty finite subset of ℕ and : S → { − 1 , 1 } be an arbitrary map (choice of signs for S). We will say that  S  has residue pattern   modulo  p  if  s p = (s)   ∀ s ∈ S , where ⋅ p is the Legendre symbol mod  p. For a given nonempty finite subset S of ℕ with a choice of signs and a real number 1 ≤ c < 1 2 1 1 , we obtain an asymptotic formula for the number of primes p of the form p = ⌊ n c ⌋ such that S has residue pattern modulo p which also satisfies p ≡ f  (mod  d) where f , d are integers with 1 ≤ f ≤ d and (f , d) = 1. For an irrational α > 1 and a real β ≥ 0 , we also obtain an asymptotic formula for the same but primes p of the form p = ⌊ α n + β ⌋ under certain assumption on α. [ABSTRACT FROM AUTHOR]

The conventional soft decision decoding (SDD) methods require various hard decision decoders (HDDs) based on different codes or re-manipulate the generator matrix by the complicated Gaussian elimination technique according to the bit reliability. This paper presents a general multi-class neural network (NN)-based decoder for the short linear block codes, where no HDD and Gaussian elimination are required once the NN is constructed. This network architecture performs multi-classification to select the messages with high occurrence probabilities and chooses the best codeword on a maximum likelihood basis. Simulation results show that the developed approach outperforms the existing deep neural network (DNN)-based decoders in terms of decoding time and bit error rate (BER). The error-correcting performance is also superior to the conventional Chase-II algorithm and is close to the ordered statistics decoding (OSD) in most cases. For Bose–Chaudhuri–Hocquenghem (BCH) codes, the SNR is improved by 1dB to 4dB as the BER is 10−4. For the (23, 12) quadratic residue (QR) code, the SNR is improved by 2dB when the BER is 10−3. The developed NN-based decoder is quite general and applicable to various short linear block codes with good BER performance. [ABSTRACT FROM AUTHOR]

The explosive growth of connected devices demands high quality and reliability in data transmission and storage. Error correction codes (ECCs) contribute to this in ways that are not very apparent to the end user, yet indispensable and effective at the most basic level of transmission. This paper presents an investigation of the performance and analysis of two decoders that are based on hash techniques and syndrome calculation over a Rayleigh channel. These decoders under study consist of two main features: a reduced complexity compared to other competitors and good error correction performance over an additive white gaussian noise (AWGN) channel. When applied to decode some linear block codes such as Bose, Ray-Chaudhuri, and Hocquenghem (BCH) and quadratic residue (QR) codes over a Rayleigh channel, the experiment and comparison results of these decoders have shown their efficiency in terms of guaranteed performance measured in bit error rate (BER). For example, the coding gain obtained by syndrome decoding and hash techniques (SDHT) when it is applied to decode BCH (31, 11, 11) equals 34.5 dB, i.e., a reduction rate of 75% compared to the case where the exchange is carried out without coding and decoding process. [ABSTRACT FROM AUTHOR]

Throughout, we will work only with lattice triangles in HT ht , with the lattice HT ht . Let I C i be a simple polygon in HT ht with vertices in HT ht . Let HT ht be a lattice triangle in HT ht . Let HT ht be two lattice polygons. [Extracted from the article]

Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive ℓ -tuples of quadratic residues, studying corresponding algebraic curves and their Jacobians, which happen to be products of Jacobians of hyperelliptic curves. In the second part we state the last unpublished result of Lydia Goncharova on squares such that their differences are also squares, reformulate it in terms of algebraic geometry of a K3 surface, and prove it. The core of this theorem is an unexpected relation between the number of points on the K3 surface and that on a CM elliptic curve. [ABSTRACT FROM AUTHOR]

We study the Vapnik–Chervonenkis (VC) dimension of the set of quadratic residues (i.e. squares) in finite fields, F q , when considered as a subset of the additive group. We conjecture that as q → ∞ , the squares have the maximum possible VC-dimension, viz. (1 + o (1)) log 2 ⁡ q. We prove, using the Weil bound for multiplicative character sums, that the VC-dimension is ⩾ (1 2 + o (1)) log 2 ⁡ q. We also provide numerical evidence for our conjectures. The results generalize to multiplicative subgroups Γ ⊆ F q × of bounded index. [ABSTRACT FROM AUTHOR]

The article presents the discussion on maximum residue levels for azoxystrobin, prosulfocarb, sedaxane and valifenalate in or on certain products. Topics include handling after entry of consignments of certain animals, germinal products and products of animal origin; and name entered in the register of protected designations of origin and protected geographical indications (Pitina (PGI)).

Fekete polynomials associate with each prime number p a polynomial with coefficients −1 or 1 except the constant term, which is 0. These coefficients reflect the distribution of quadratic residues modulo p. These polynomials were already considered in the 19th century in relation to the studies of Dirichlet L -functions. In our paper, we introduce two closely related polynomials. We then express their special values at several integers in terms of certain class numbers and generalized Bernoulli numbers. Additionally, we study the splitting fields and the Galois group of these polynomials. In particular, we propose two conjectures on the structure of these Galois groups. We also provide some computational evidence toward the validity of these conjectures. [ABSTRACT FROM AUTHOR]

Let p ≥ 5 be a prime. In 1801, Gauss proved that the sum of distinct quadratic residues modulo p is congruent to 0 modulo p. A study by Stetson in 1904 showed that the sum of distinct triangular residues modulo p is congruent to 1=16 modulo p. Both of these results were extended in 2017 by Gross, Harrington, and Minott, who studied the sum of distinct quadratic polynomial residues modulo p. In this article, we determine the sum of distinct cubic polynomial residues modulo p and prove a conjecture of Gross, Harrington, and Minott. We further consider the sum of distinct residues modulo p for polynomials of higher degree. [ABSTRACT FROM AUTHOR]