Residue Class Research Articles

Three classes of partitioned difference families and their optimal constant composition codes

<p style='text-indent:20px;'>Cyclotomy, firstly introduced by Gauss, is an important topic in Mathematics since it has a number of applications in number theory, combinatorics, coding theory and cryptography. Depending on <inline-formula><tex-math id="M2">\begin{document}$ v $\end{document}</tex-math></inline-formula> prime or composite, cyclotomy on a residue class ring <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb{Z}}_{v} $\end{document}</tex-math></inline-formula> can be divided into classical cyclotomy or generalized cyclotomy. Inspired by a foregoing work of Zeng et al. [<xref ref-type="bibr" rid="b40">40</xref>], we introduce a generalized cyclotomy of order <inline-formula><tex-math id="M4">\begin{document}$ e $\end{document}</tex-math></inline-formula> on the ring <inline-formula><tex-math id="M5">\begin{document}$ {\rm GF}(q_1)\times {\rm GF}(q_2)\times \cdots \times {\rm GF}(q_k) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ q_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q_j $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M8">\begin{document}$ i\neq j $\end{document}</tex-math></inline-formula>) may not be co-prime, which includes classical cyclotomy as a special case. Here, <inline-formula><tex-math id="M9">\begin{document}$ q_1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ q_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ \cdots $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ q_k $\end{document}</tex-math></inline-formula> are powers of primes with an integer <inline-formula><tex-math id="M13">\begin{document}$ e|(q_i-1) $\end{document}</tex-math></inline-formula> for any <inline-formula><tex-math id="M14">\begin{document}$ 1\leq i\leq k $\end{document}</tex-math></inline-formula>. Then we obtain some basic properties of the corresponding generalized cyclotomic numbers. Furthermore, we propose three classes of partitioned difference families by means of the generalized cyclotomy above and <inline-formula><tex-math id="M15">\begin{document}$ d $\end{document}</tex-math></inline-formula>-form functions with difference balanced property. Afterwards, three families of optimal constant composition codes from these partitioned difference families are obtained, and their parameters are also summarized.</p>

The inverse sieve problem for algebraic varieties over global fields

Let K be a global field and let Z be a geometrically irreducible algebraic variety defined over K . We show that if a big set S\subseteq Z of rational points of bounded height occupies few residue classes modulo \mathfrak{p} for many prime ideals \mathfrak{p} , then a positive proportion of S must lie in the zero set of a polynomial of low degree that does not vanish at Z . This generalizes a result of Walsh who studied the case when S\subseteq \{0,\ldots ,N\}^{d} .

The first digit of the discriminant of Eisenstein polynomials as an invariant of totally ramified extensions of p-adic fields

Let K be an extension of the p-adic numbers with uniformizer π. Let φ and ψ be Eisenstein polynomials over K of degree n that generate isomorphic extensions. We show that if the cardinality of the residue class field of K divides n(n−1), then v(disc(φ)− disc(ψ))>v(disc(φ)). This makes the first (nonzero) digit of the π-adic expansion of disc(φ) an invariant of the extension generated by φ. Furthermore we find that noncyclic extensions of degree p of the field of p-adic numbers are uniquely determined by this invariant.

A complete classification of de mersenne´s primes and its iomplications for computing

A study of Mersenne’s primes is carried out using the multiplicative group modulo 360 and a complete classification is obtained by its residual classes. This allows the search for Mersenne’s primes to be classified into four subgroups mutually exclusive (disjoint) and contributes to the ordered selection of exponents to be computationally tested. According to this idea, Mersenne’s trapeze is presented with the purpose of giving a geometric representation for this classification. Finally, from one of the theorems presented and proven for primes in modulo 360, a conjecture is established that could be solved by computing.

Rapid detection of pesticide residues in Chinese herbal medicines by molecularly imprinted membrane electrospray ionization mass spectrometry

Chinese herbal medicines (CHMs) play an increasingly important role in the field of medicine and affects public health in the world. Although more and more strict has been employed to ensure the quality and safety of CHMs, pesticide residues in CHMs remain a serious issue and are the bottleneck for the global development of CHMs. In this work, we applied molecularly imprinted membrane electrospray mass spectrometry (MIM-ESI MS) for rapid detecting 4 classes of pesticide residues in CHMs, including organophosphorus (OPP), carbamates, pyrethroids and neonicotinoids in CHMs. Compared with our previous ambient ionization method MESI, MIM-ESI is capable of achieving a ∼50-fold increase in the detection limit of conventional analytical methods owing to the specificity recognition and unique enrichment of MIM. The optimal experimental conditions were determined, and the method was further validated for its sensitivity and specificity. Our data showed that MIM-ESI MS is applicable for the direct quantitation of pesticide residues in CHMs. This detection technology may help to ensure the quality of CHMs in the future.

Histopathological examination of surgical breast cancer specimens after neoadjuvant chemotherapy using digital radiography

The article provides a literature overview on significance, pathologic assessment of residual disease problems, and digital radiography (DR) potential in breast cancer (BC) after neoadjuvant therapy (NAT). Within the framework of the paper, the authors carry out an analysis of the Russian and English-language publications from PubMed, Google Scholar, ClinicalTrials.gov, eLibrary, and Cyberleninka. The comparison of the Russian clinical guidelines for BC diagnosis and the European and American guidelines revealed a lack of information on DR usage in the morphological assessment. The review showed the international experience in DR usage and demonstrated the relevance of the solution of morphological assessment problems in BC regression degree after NAT due to necessary clinical trial protocol standardization and increased predictive residual tumor class significance. The DR facilitated the morphological identification of metal markers implanted into the tumor bed, microcalcifications, altered foci, and improved tumor bed visibility, which is important for further objective status assessment of the resection margins and residual cancer burden class. The authors consider it necessary to conduct a study to optimize the residual tumor assessment using DR.

Long gaps in sieved sets

For each prime p , let I_p \subset \mathbb{Z}/p\mathbb{Z} denote a collection of residue classes modulo p such that the cardinalities |I_p| are bounded and about 1 on average. We show that for sufficiently large x , the sifted set \{n \in \mathbb{Z}: n (\mathrm {mod}\: p) \not \in I_p \: \mathrm {for \: all} p \leq x\} contains gaps of size x (\log x)^{\delta} depends only on the densitiy of primes for which I_p \neq \emptyset . This improves on the "trivial'' bound of \gg x . As a consequence, for any non-constant polynomial f: \mathbb{Z} \to \mathbb{Z} with positive leading coefficient, the set \{n \leq X: f(n) \; \mathrm {composite}\} contains an interval of consecutive integers of length \ge (\log X) (\log\log X)^{\delta} for sufficiently large X , where \delta > 0 depends only on the degree of f .

On products of primes and square-free integers in arithmetic progressions

We obtain an asymptotic formula for the number of ways to represent every reduced residue class as a product of a prime and square-free integer. This may be considered as a relaxed version of a conjecture of Erd\"os, Odlyzko, and S\'ark\"ozy.

On the Sums Running over Reduced Residue Classes Evaluated at Polynomial Arguments

For a given polynomial G we study the sums φm(n) := ∑′km and φG(n) = ∑′G(k) where m ≥ 0 is a fixed integer and ∑′ runs through all integers k with 1 ≤ k ≤ n and gcd(k, n) = 1. Although, for m ≥ 1 the function φm is not multiplicative, analogue to the Euler function we obtain expressions for φm(n) and φG(n). Also, we estimate the averages ∑n≤x φm(n) and ∑n≤xφG(n), as more as, the alternative averages ∑n≤x(−1)n−1φm(n) and ∑n≤x(−1)n−1φG(n).

Erratum: Limiting properties of the distribution of primes in an arbitrarily large number of residue classes

AbstractAs pointed out by Alexandre Bailleul, the paper mentioned in the title contains a mistake in Theorem 2.2. The hypothesis on the linear relation of the almost periods is not sufficient. In this note, we fix the problem and its minor consequences on other results in the same paper.

Method for increasing the dynamic range of digital images based on modular arithmetic

The article proposes a new method of synthesis of digital images from several digital images to increase the dynamic range of intensity. The method is based on modular arithmetic. The increased range is divided into several residual classes. The full value of the intensity is calculated as the solution of the system of comparisons of residues for given modules. An analysis is made of the stability of the proposed method to the nonlinearity of the transfer characteristic and digital image recording devices under various types of interference. To this end, the maximum range for changing the brightness of a digital image is limited. By selecting the magnitude of the modules and the maximum brightness value, the distance between the diagonals in the decision table is selected so that this distance is greater than the level of distortion and (or) interference. Experimental validation of the method has been performed.

Scaling algorithm designed to reduce circuit costs for modular digital filters

Currently, there is a tendency to increase the speed and accuracy of data processing in computer networks (CN). It is possible to meet these requirements through the use of new technologies in CN telecommunication devices that rely on residue class codes (RNS). In these codes, input values are presented as sets of remainder moduli of the RNS base selected. As a result, arithmetic operations are performed in parallel, which provides an increased speed of signal processing. Therefore, RNS codes are used in high-speed modular digital filters (MDF). To advance the accuracy of digital signal processing in MDF, the number of RNS bases is increased, which leads to greater circuit costs. It is possible to remedy this shortcoming by scaling an MDF response. Hence, it is crucial to develop a scaling algorithm to reduce circuit costs for MDF implementation.

METHOD FOR CONTROLLING DATA PRESENTED IN THE DEDUCTION CLASS

The article proposes a method for increasing the reliability of control of data presented in the class of residues. The procedures for obtaining a positional attribute of a non-positional code and forming a one-time code that is the basis of the proposed method for controlling data in the deduction class are considered. A method for controlling data in the deduction class is given and described. An example of implementing the control method for a specific class of deductions is considered. Since the developed method of operational control of data in the class of deductions and devices for its implementation has a very low reliability of control, a method for increasing the reliability of control based on the wellknown method of operational control of information in the class of deductions is proposed.The results of calculations and comparative analysis of the reliability of control of data in class of residues showed that with an increase in the bit grid of processed data, the efficiency of non-positional coding in the class of residues increases significantly.

Recovery of thorium and rare earths from leachate of ion-absorbed rare earth radioactive residues with N1923 and Cyanex® 572

Ion-absorbed rare earth ores radioactive residues (IREORR) are a class of waste residue from the production of rare earth elements (REEs). Because of its radioactive dose, IREORR are usually stored in waste warehouses. IREORR are difficult to be disposed of. However, it contains relatively high concentrations of REEs, which can be considered as a valuable secondary resource. In this paper, a novel process is developed for the separation of thorium (Th) and recovery of REEs from IREORR hydrochloric acid leachate with primary amine N1923 and Cyanex® 572, respectively. The effects of sulfate concentration, extractant concentration and pH on N1923 extraction in hydrochloric acid solution were investigated in detail. The results show that the extraction capacity of N1923 can be improved by adding sulfate to the solution and increasing the concentration of N1923. Acidity has little effect on the extraction of Th when pH is higher than 1. As for the stripping, REEs are more easily stripped from loaded organic phase than Th, and nitric acid is a better stripping agent than hydrochloric acid. Combined with the extraction of Cyanex® 572 for REEs, a fractional extraction experiment for separating Th and enriching of REEs was performed. The yield of Th is higher than 99.9% and the concentration of REEs is enriched to 183.84 g/L.

THE TASK OF CHOOSING A RELIABLE PATH FOR MESSAGE TRANSMISSION IN A COMPUTER NETWORK

The article provides a calculation and comparative analysis of the reliability and productivity of computer systems in a positional binary number system and in a non-positional number system in residual classes (residual number system – RNS), for calculations and comparative we consider practical task. The main goal is to solve the task of choosing a reliable path for message transmission in a computer network. Calculation and comparative evaluation of the reliability and performance of the computer system in the RNS and the existing in the positional binary number system computer system APO-221 of the product 15E1235 (automatic message switching center - ASC) when solving the basic task of the ASC – the task of choosing the transmission path of a formalized message (path selection algorithm (PSA))

There are Infinitely Many Fibonacci Primes

We invent a novel algorithm and solve the Fibonacci prime conjecture by an interaction between proof and algorithm. From the entire set of natural numbers successively deleting the residue class 0 mod a prime, we retain this prime and possibly delete another one prime retained, then we invent a recursive sieve method, a modulo algorithm on finite sets of natural numbers, for indices of Fibonacci primes. The sifting process mechanically yields a sequence of sets of natural numbers, which converges to the index set of all Fibonacci primes. The corresponding cardinal sequence is strictly increasing. The algorithm reveals a structure of particular order topology of the index set of all Fibonacci primes, then we readily prove that the index set of all Fibonacci primes is an infinite set based on the existing theory of the structure. Some mysteries of primes are hidden in second order arithmetics.

PIPELINE MULTIPLIER OF POLYNOMIALS MODULO WITH ANALYSIS OF HIGH-ORDER BITS OF THE MULTIPLIER

Among public-key cryptosystems, cryptosystems built on the basis of a polynomial system of residual classes are special. Because in these systems, arithmetic operations are performed at high speed. There are many algorithms for encrypting and decrypting data presented in the form of polynomials. The paper considers data encryption based on the multiplication of polynomials modulo irreducible polynomials. In such a multiplier, the binary image of a multiply polynomial can serve as a fragment of encrypted text. The binary image of the multiplier polynomial is the secret key and the binary representation of the irreducible polynomial is the module. Existing sequential polynomial multipliers and single-cycle matrix polynomial multipliers modulo do not provide the speed required by the encryption block. The paper considers the possibility of multiplying polynomials modulo on a Pipeline in which architectural techniques are laid in order to increase computing performance. In the conclusion of the work, the time gain of the multiplication modulo is shown by the example of the multiplication of five triples of polynomials. Verilog language was used to describe the scheme of the Pipeline multiplier. Used FPGA Artix-7 from Xilinx companies. The developed Pipeline multiplier can be used for cryptosystems based on a polynomial system of residual classes, which can be implemented in hardware or software.

Алгоритм преобразования представлений чисел в системах остаточных классов

Алгоритм преобразования представлений чисел в системах остаточных классов

UNEXPECTED AVERAGE VALUES OF GENERALIZED VON MANGOLDT FUNCTIONS IN RESIDUE CLASSES

In order to study integers with few prime factors, the average of $\unicode[STIX]{x1D6EC}_{k}=\unicode[STIX]{x1D707}\ast \log ^{k}$ has been a central object of research. One of the more important cases, $k=2$ , was considered by Selberg [‘An elementary proof of the prime-number theorem’, Ann. of Math. (2)50 (1949), 305–313]. For $k\geq 2$ , it was studied by Bombieri [‘The asymptotic sieve’, Rend. Accad. Naz. XL (5)1(2) (1975/76), 243–269; (1977)] and later by Friedlander and Iwaniec [‘On Bombieri’s asymptotic sieve’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4)5(4) (1978), 719–756], as an application of the asymptotic sieve. Let $\unicode[STIX]{x1D6EC}_{j,k}:=\unicode[STIX]{x1D707}_{j}\ast \log ^{k}$ , where $\unicode[STIX]{x1D707}_{j}$ denotes the Liouville function for $(j+1)$ -free integers, and $0$ otherwise. In this paper we evaluate the average value of $\unicode[STIX]{x1D6EC}_{j,k}$ in a residue class $n\equiv a\text{ mod }q$ , $(a,q)=1$ , uniformly on $q$ . When $j\geq 2$ , we find that the average value in a residue class differs by a constant factor from the expected value. Moreover, an explicit formula of Weil type for $\unicode[STIX]{x1D6EC}_{k}(n)$ involving the zeros of the Riemann zeta function is derived for an arbitrary compactly supported ${\mathcal{C}}^{2}$ function.

On the addition of values of a quadratic polynomial at units modulo n

Let $$\mathbb{Z}_{n}$$ be the ring of residue classes modulo n, $$\mathbb{Z}_{n}^{*}$$ be its unit group, and let $$f(z)=az^{2}+bz$$ be an integral quadratic polynomial with $$b\neq0$$ and $${\rm gcd}(a,b)=1$$ . In this paper, for any integer c and positive integer n we give a formula for the number of solutions of the congruence equation $$f(x)+f(y)\equiv c({\rm mod}\, n)$$ with x, y units $$({\rm mod}\, n)$$ . This partly solves a problem posed by Yang and Tang in [11].