Residue Class Research Articles

Mod-$\phi $ convergence: Approximation of discrete measures and harmonic analysis on the torus

In this paper, we relate the framework of mod-$\phi$ convergence to the construction of approximation schemes for lattice-distributed random variables. The point of view taken here is that of Fourier analysis in the Wiener algebra, allowing the computation of asymptotic equivalents in the local, Kolmogorov and total variation distances. By using signed measures instead of probability measures, we are able to construct better approximations of discrete lattice distributions than the standard Poisson approximation. This theory applies to various examples arising from combinatorics and number theory: number of cycles in (possibly coloured) permutations, number of prime divisors (possibly within different residue classes) of a random integer, number of irreducible factors of a random polynomial, etc. One advantage of the approach developed in this paper is that it allows us to deal with approximations in higher dimensions as well. In this setting, we can explicitly see the influence of the correlations between the components of the random vectors in our asymptotic formulas.

The smallest invariant factor of the multiplicative group

Let [Formula: see text] denote the least invariant factor in the invariant factor decomposition of the multiplicative group [Formula: see text]. We give an asymptotic formula, with order of magnitude [Formula: see text], for the counting function of those integers [Formula: see text] for which [Formula: see text]. We also give an asymptotic formula, for any even [Formula: see text], for the counting function of those integers [Formula: see text] for which [Formula: see text]. These results require a version of the Selberg–Delange method whose dependence on certain parameters is made explicit, which we provide in Appendix A. As an application, we give an asymptotic formula for the counting function of those integers [Formula: see text] all of whose prime factors lie in an arbitrary fixed set of reduced residue classes, with implicit constants uniform over all moduli and sets of residue classes.

Temporal and Spatial Occurrence of NSAIDs in Taihu Lake and Relevant Risk Assessment

Non-steroidal anti-inflammatory drugs (NSAIDs) are a class of drug residues with a high frequency of detection in Taihu Lake. However, little information is available about the occurrence of typical NSAID mixtures in Taihu Lake as a whole across the four seasons. Therefore, for each season, the concentrations of five typical NSAIDs including diclofenac, ibuprofen, indomethacin, naproxen, and ketoprofen were monitored in the water of Taihu Lake by the method of high-performance liquid chromatography-mass spectrometry (HPLC-MS/MS) at 19 transects covering the entire lake. The temporal and spatial occurrence of NSAID mixtures in the water of Taihu Lake and their correlation with environmental factors were analyzed, and the mixture risk quotient (MRQ) model was also used to assess the ecological risk of the NSAID mixtures. The research results are as follows:① The concentrations of NSAIDs in the northern, western, and eastern waters of Taihu Lake are at a higher level compared to those in the central waters. Ketoprofen is the main contributor to the contamination of NSAID mixtures in all regions of Taihu Lake. ② The concentrations of NSAIDs in Taihu Lake are higher in summer (15.9-134.3 ng·L-1) and autumn (16.4-144.6 ng·L-1) but lower in spring (25.3-72.5 ng·L-1) and winter (14.6-57.4 ng·L-1), being significantly correlated with water conductivity and pH, respectively. ③ The MRQ model evaluation reveals that there are nine sections in Taihu Lake showing a high ecological risk (MRQ>1) from NSAID mixtures throughout the year. The ecological risk of the NSAID mixtures at a medium or high level (MRQ>0.1) lasts for a long time spanning the spring, summer, and autumn seasons, of which the risk is greatest in autumn. Overall, the pollution caused by the NSAID mixtures in the water of Taihu Lake should not be ignored, and especially great attention should be paid to the pollution in autumn.

On the Probability That Two Random Integers Are Coprime

We show that there is a nonempty class of finitely additive probabilities on $\mathbb{N}^{2}$ such that for each member of the class, each set with limiting relative frequency $p$ has probability $p$. Hence, in that context the probability that two random integers are coprime is $6/\pi ^{2}$. We also show that two other interpretations of “random integer,” namely residue classes and shift invariance, support any number in $[0,6/\pi ^{2}]$ for that probability. Finally, we specify a countably additive probability space that also supports $6/\pi ^{2}$.

Об одном алгоритмическом решении задачи восстановления остатка числа в системе остаточных классов

При выполнении операций расширения диапазона представления чисел, деления, определения переполнения, масштабирования, контроля ошибок вычислений возникает задача восстановления остатка числа по данному модулю на основании остатков этого числа по остальным модулям системы. Табличное выполнение операции восстановления остатка числа реализуется с помощью базового алгоритма. Метод решения основан на определении остатка по данному модулю на основании полученных остатков по остальным модулям системы. Такое определение выполняется последовательным вычитанием констант из полученных остатков и суммированием этих констант к результататам, которые формируются по данному модулю. При этом константы на каждой итерации выбираются в зависимости от значения остатка в анализируемом разряде. При несомненном достоинстве метода сохраняются требования к быстродействию выполнения операции восстановления остатка числа. Целью исследования является аналитическое рассмотрение подхода к ускоренной реализации базовой операции восстановления остатка числа по данному модулю на основании остатков этого числа по остальным модулям системы. Одна из реализаций алгоритма состоит в одновременном его выполнении по базовому варианту для искомого числа и числа, обратного искомому. При этом искомый остаток определяется по значению остатка того из чисел, для которого первым получается результат поиска. Приведены варианты реализации алгоритма с переходами от представления числа в прямом коде к представлению этого числа в обратном коде и от представления числа в обратном коде к его представлению в прямом коде. Рассмотренный алгоритм реализации в системе остаточних классов базовой немодульной операции восстановления значения остатка числа по данному модулю на основании значений остатков этого числа по остальным модулям системы обеспечивает получение искомого результата. На основе предложенных подходов достигается ускоренная реализация базовой операции восстановления остатка числа по данному модулю. Представляется целесообразным применить предложенные подходы в качестве перспективных направлений исследований этой операции в системе остаточных классов.

IMPLEMENTATION OF THE ARITHMETIC ADDITION OPERATION IN THE SYSTEM OF RESIDUAL CLASSES

Предметом статті є розробка методу реалізації арифметичної операції додавання чисел, які представлені у системі залишкових класів (СЗК). Даний метод ґрунтується на основі використання принципу кільцевого зсуву (ПКЗ). Метою статті є зменшення часу реалізації арифметичної операції додавання чисел, які представлені у СЗК. Задачі : провести аналіз і виявити недоліки існуючих систем числення, які використовуються при побудові комп'ютерних систем і компонентів, дослідити можливі шляхи усунення виявлених недоліків, зробити оцінку часу реалізації арифметичної операції додавання на основі використання розробленого методу. Методи дослідження : методи аналізу та синтезу комп'ютерних систем, теорія чисел, теорія кодування у СЗК. Отримано наступні результати . Показано, що основним принципом реалізації арифметичних операцій у позиційних системах числення, є використання двійкових суматорів. При цьому використання таких систем числення обумовлює наявність міжрозрядних зв'язків, що в свою чергу збільшує час реалізації арифметичних операцій, а також зумовлює виникнення помилок. Встановлено, що використання непозиційної системи числення у залишкових класах дозволяє усунути зазначені недоліки, за рахунок використання основних властивостей СЗК. Особливість розробленого методу полягає у тому, що результат операції додавання чисел може бути знайдений послідовними циклічними зсувами бітів вмісту блоків даних по відповідним модулям СЗК. Використання ПКЗ дозволяє виключити вплив межрозрядних зв'язків між залишками чисел, що оброблюються, це дозволяє зменшити час реалізації операції додавання двох чисел у СЗК. Висновки . Результати проведених досліджень показали, що використання СЗК дозволяє усунути існуючі недоліки позиційних систем числення, а також підвищує швидкодію реалізації арифметичних операцій.

Discussion on the Structure of the Reduced Residue Class Additive Group in Cyclic Group

Discussion on the Structure of the Reduced Residue Class Additive Group in Cyclic Group

Distribution of the Number of Square-Free Divisors in Residue Classes

The notion of weekly uniform distribution of arithmetical functions was introduced by W. Narckiewicz. For many functions in various papers, the problem of existence of such distribution was solved, and asymptotic formulas for frequency of hits of function values in residue classes coprime with a module were established. These asymptotic formulas contained, as a rule, only principal part of the asymptotics. A similar problem is solved in the present paper for the function which gives the number of square-free divisors of a natural number, but the asymptotic formula is the asymptotic series.

CONTROL AND CORRECTION OF DATA ERRORS IN A RESIDUE CLASS

The subject of the research in the article is the methods of control and correction of single errors of integer data are presented in the residue class (RC), which allows to increase the efficiency of using RC when building computer systems and components. The purpose of the article is to develop a method for correcting single data errors in RC. Tasks: to study the code structures presented in RC to determine the possibility of control and correction of data errors; to investigate the effect of RC properties on data control and correction operations; to develop a method for correcting single data errors in RC. Research methods: methods of analysis and synthesis of computer systems, number theory, coding theory in RC. The following results are obtained. An analysis of the correcting capabilities of codes in RC showed the high efficiency of using non-positional code structures, which is due to the presence of primary and secondary redundancy in such structures. The article presents a method for correcting one-time data errors in RC. Examples of detecting and correcting data errors in RC code are given, which confirms the theoretical results obtained. Conclusions. Studies have shown that the use of codes in RC makes it possible to build an effective system for monitoring and correcting data errors with the introduction of minimal code redundancy. That is, when certain conditions are met, the introduction of one control base allows not only monitoring, but also correction of single data errors

Inclusive prime number races

Let $\pi(x;q,a)$ denote the number of primes up to $x$ that are congruent to $a$ modulo $q$. A prime number race, for fixed modulus $q$ and residue classes $a_1, \ldots, a_r$, investigates the system of inequalities $\pi(x;q,a_1) > \pi(x;q,a_2) > \cdots > \pi(x;q,a_r)$. The study of prime number races was initiated by Chebyshev and further studied by many others, including Littlewood, Shanks-Renyi, Knapowski-Turan, and Kaczorowski. We expect that this system of inequalities should have arbitrarily large solutions $x$, and moreover we expect the same to be true no matter how we permute the residue classes $a_j$; if this is the case, and if the logarithmic density of the set of such $x$ exists and is positive, the prime number race is called inclusive. In breakthrough research, Rubinstein and Sarnak proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet $L$-functions. We show that the same conclusion can be reached assuming the generalized Riemann hypothesis and a substantially weaker linear independence hypothesis. In fact, we can assume that almost all of the zeros may be involved in $\mathbb{Q}$-linear relations; and we can also conclude more strongly that the associated limiting distribution has mass everywhere. This work makes use of a number of ideas from probability, the explicit formula from number theory, and the Kronecker-Weyl equidistribution theorem.

On the average number of cyclic subgroups of the groups {mathbb {Z}}_{n_1} times {mathbb {Z}}_{n_2}times {mathbb {Z}}_{n_3} with n_1,n_2,n_3le x

Let {{mathbb {Z}}}_{n} be the additive group of residue classes modulo n. Let c(n_1,n_2,n_3) denote the number of cyclic subgroups of the group {{mathbb {Z}}}_{n_1}times {{mathbb {Z}}}_{n_2}times {{mathbb {Z}}}_{n_3}, where n_1, n_2 and n_3 are arbitrary positive integers. In this paper we obtain an asymptotic formula for the sum sum _{n_1,n_2,n_3le _x} c(n_1,n_2,n_3).

Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

AbstractWe generalize current known distribution results on Shanks–Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let $\unicode[STIX]{x1D70B}(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$. For a fixed integer $q$ and distinct invertible congruence classes $a_{0},a_{1},\ldots ,a_{D}$, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real $x$ for which the inequalities $\unicode[STIX]{x1D70B}(x;q,a_{0})>\unicode[STIX]{x1D70B}(x;q,a_{1})>\cdots >\unicode[STIX]{x1D70B}(x;q,a_{D})$ are simultaneously satisfied admits a logarithmic density.

On relative projectivity of lattices in Auslander-Reiten components for group rings

Let G be a finite group, Q a p-subgroup of G and P a Sylow p-subgroup of Let be a complete discrete valuation ring of characteristic zero with residue class field of characteristic p > 0. Suppose that the group ring is of infinite representation type and is sufficiently large. Let be a stable Auslander-Reiten component containing a Scott -lattice S(Q) with vertex Q. Then all the -lattices in have P as their vertices. Also, we show that there exists an indecomposable -lattice L with vertex P such that a kG-module is Q-projective.Communicated by J. Zhang

A Reduced Collatz Dynamics Maps to a Residue Class, and Its Count of x/2 over the Count of 3∗x+1 Is Larger than ln3/ln2

We propose reduced Collatz conjecture and prove that it is equivalent to Collatz conjecture but more primitive due to reduced dynamics. We study reduced dynamics (that consists of occurred computations from any starting integer to the first integer less than it) because it is the component of original dynamics (from any starting integer to 1). Reduced dynamics is denoted as a sequence of “I” that represents (3∗x+1)/2 and “O” that represents x/2. Here, 3∗x+1 and x/2 are combined together because 3∗x+1 is always even and thus followed by x/2. We discover and prove two key properties on reduced dynamics: (1) Reduced dynamics is invertible. That is, given reduced dynamics, a residue class that presents such reduced dynamics can be computed directly by our derived formula. (2) Reduced dynamics can be constructed algorithmically, instead of by computing concrete 3∗x+1 and x/2 step by step. We discover the sufficient and necessary condition that guarantees a sequence consisting of “I” and “O” to be a reduced dynamics. Counting from the beginning of a sequence, if and only if the count of x/2 over the count of 3∗x+1 is larger than ln3/ln2, reduced dynamics will be obtained (i.e., current integer will be less than starting integer).

On smooth square‐free numbers in arithmetic progressions

A. Booker and C. Pomerance (2017) have shown that any residue class modulo a prime $p\ge 11$ can be represented by a positive $p$-smooth square-free integer $s = p^{O(\log p)}$ with all prime factors up to $p$ and conjectured that in fact one can find such $s$ with $s = p^{O(1)}$. Using bounds on double Kloosterman sums due to M. Z. Garaev (2010) we prove this conjecture in a stronger form $s \le p^{3/2 + o(1)}$ and also consider more general versions of this question replacing $p$-smoothness of $s$ by the stronger condition of $p^{\alpha}$-smoothness. Using bounds on multiplicative character sums and a sieve method, we also show that we can represent all residue classes by a positive square-free integer $s\le p^{2+o(1)}$ which is $p^{1/(4e^{ /2})+o(1)}$-smooth. Additionally, we obtain stronger results for almost all primes $p$.

Anzahl theorems of matrices over the residue class ring modulo psqt and their applications

Let h=psqt be its decomposition into a product of powers of two distinct primes, and Zh be the residue class ring modulo h. In this paper, we determine the Smith normal forms of matrices, compute the number of the orbits of m×n matrices under the action of the group GLm(Zh)×GLn(Zh) and the length of each orbit over Zh. As applications, we determine the clique number, the independence number and the chromatic number of the bilinear forms graph, construct a family of association schemes by using 1×n matrices and compute the intersection numbers and the character table over Zh for s=t=1.

Temperament and performance of Nellore bulls classified for residual feed intake in a feedlot system

This study aimed to evaluate the performance in feedlot and temperament of Nellore bulls classified by residual feed intake. The residual feed intake was calculated as the difference between the observed and predicted dry matter intake. Bulls classified as low residual feed intake had lower dry matter intake (kg day-1) and dry matter intake (g kg-1 d-1) of body weight, and were more efficient in feed conversion ratio than those classified as medium and high. The average daily gain didn't differ among residual feed intake classes and was 1.69 kg day-1, 1.82 kg day-1 and 1.71 kg day-1 for bulls classified as low, medium, or high, respectively. The residual feed intake was positively associated with dry matter intake, feed conversion ratio and subcutaneous fat thickness. The subcutaneous fat thickness was lower in bulls classified as low residual feed intake than in those with medium and high. No differences were observed in flight speed and reactivity score among residual feed intake classes. Overall, we concluded that bulls classified as low residual feed intake consumed less dry matter than high, with no differences in average daily gain, temperamentand had better feed efficiency, albeit their subcutaneous fat thickness was lower.

Beurling numbers whose number of prime factors lies in a specified residue class

We find asymptotics for $S_{K,c}(x)$, the number of positive integers below $x$ whose number of prime factors is $c \; \mathrm{mod}\; K$. We study this question in the context of Beurling integers.

Physical Layer Security for Multiple-Input Multiple-Output Systems by Alternating Orthogonal Space-Time Block Codes

Algorithms are provided to build a large set of unique complex-valued orthogonal space-time block codes (STBCs) from a known or standard STBC. A physical layer security (PLS) scheme is proposed to take advantage of this set by alternating the STBC in use over a multiple-input single-output (MISO) or multiple-input multiple-output (MIMO) communications link between base station (BS) and user equipment (UE). A practical procedure is proposed and demonstrated to build individual STBCs from the set without use of a lookup table. The sufficient statistic is given and proven to allow for maximal ratio combining (MRC) by the intended receiver for all STBCs in the set. An algorithm is offered for the UE to update the MRC matrix in use as the STBC alternates. Definitions are provided for cryptograms, key residues (KRs), key residue classes (KRCs), and message and cryptogram residue classes pertaining to STBC PLS schemes. These definitions are used to present analysis of the information-theoretic security of the proposed PLS scheme to include message and key equivocation. Theoretical expected bit error rate (BER) for a passive eavesdropper is proven and plotted along with Monte Carlo simulations for confirmation. Discussion of different attack models is provided. Cost and attack complexity are compared between the proposed scheme and two related techniques from the literature.

On the Set of Primitive Triples of Natural Numbers Satisfying the Diophantine Equation of Pythagor

The main task considered in the article is to find the condition primitive integer solutions of the Diophantine Pithagorean equation x²+y²=z² It is known that for this purpose it is enough to find primive solution of x, y such that x is even and y is odd. In this paper, in particular, we proved that the z of a primitive solution is a Prime number of the form 4k+1. It is prove in this paper that any right triangle with integer side lengths has a hypotenuse equal to a Prime of the form 4k+1and we show with the help of the descent axiom how to find primitive solutions of x and y in this case. We divide the search for primitive solutions (x, y, z) of right triangles into two cases: 1) the hypotenuse of such triangles is a Prime number of the form 4k+1 and 2) the hypotenuse of such triangles is a composite number. In section 3 we use formulas known to the ancient Hindus to find primitive solutions of Pithagorean equations in cases where m and n<m are mutually Prime numbers of different parity, and this happens in cases where, for example, n=4, and m is a Prime number ending in 3 or 7. In such cases, the hypotenuse z=m²+n² is an compaund number ending in 5. To find primes ending in 3 and 7, we refer the reader to our paper, which presents algorithms for constructing all primes and twin primes. The proposed paper also presents a generalization of Euclid's fundamental result on the infinity of the set of Primes, namely, it is shown that all twin primes are in residue classes (1, 3), (2, 4), (4, 1), and there are infinity many such twins.