Residue Class Research Articles

An exact test for exponentiality against renewal increasing mean residual life class

In this paper, we develop an exact test for testing exponentiality against renewal increasing mean residual life class. The critical values are presented for different sample sizes. Asymptotic properties of the test statistic are also studied. Some numerical results are presented for evaluating the performance of the proposed test. The test procedure is illustrated by using two real data sets. The proposed method is then modified to incorporate right censored observations and the efficiency loss due to censoring is evaluated using Pitman's asymptotic efficacy value.

Classification of Crop Residue Cover in High-Resolution RGB Images Using Machine Learning

HighlightsA machine learning framework estimated residue cover in RGB images taken at three resolutions from 88 locations.The best results primarily used texture features, the RFE-SVM feature selection method, and the SVM classifier.Accounting for shadows and plants plus modifying and optimizing the texture features may improve performance.An automated system developed using machine learning is a viable strategy to estimate residue cover from RGB images obtained with handheld or UAV platforms.Abstract. Maintaining plant residue on the soil surface contributes to sustainable cultivation of arable land. Applying machine learning methods to RGB images of residue could overcome the subjectivity of manual methods. The objectives of this study were to use supervised machine learning while identifying the best feature selection method, the best classifier, and the most effective image feature types for classifying residue levels in RGB imagery. Imagery was collected from 88 locations in 40 row-crop fields in five Missouri counties between early May and late June in 2018 and 2019 using a tripod-mounted camera (0.014 cm pixel-1 ground sampling distance, GSD) and an unmanned aerial vehicle (UAV, 0.05 and 0.14 GSD). At each field location, 50 contiguous 0.3 × 0.2 m region of interest (ROI) images were extracted from the imagery, resulting in a dataset of 4,400 ROI images at each GSD. Residue percentages for ground truth were estimated using a bullseye grid method (n = 100 points) based on the 0.014 GSD images. Representative color, texture, and shape features were extracted and evaluated using four feature selection methods and two classifiers. Recursive feature elimination using support vector machine (RFE-SVM) was the best feature selection method, and the SVM classifier performed best for classifying the amount of residue as a three-class problem. The best features for this application were associated with texture, with local binary pattern (LBP) features being the most prevalent for all three GSDs. Shape features were irrelevant. The three residue classes were correctly identified with 88%, 84%, and 81% 10-fold cross-validation scores for the 2018 training data and 81%, 69%, and 65% accuracy for the 2019 testing data in decreasing resolution order. Converting image-wise data (0.014 GSD) to location residue estimates using a Bayesian model showed good agreement with the location-based ground truth (r2 = 0.90). This initial assessment documents the use of RGB images to match other methods of estimating residue, with potential to replace or be used as a quality control for line-transect assessments. Keywords: Feature selection, Soil erosion, Support vector machine, Texture features, Unmanned aerial vehicle.

Joint distribution in residue classes of polynomial-like multiplicative functions

Under fairly general conditions, we show that families of integer-valued polynomial-like multiplicative functions are uniformly distributed in coprime residue classes modulo $p$, where $p$ is a growing prime (or nearly prime) modulus. This can be seen as

Residual classes based mathematical model of the computer system's reliability

It is also known, and it is practically shown that the use of a unposition number system in residual classes (RNS) allows you to drastically increase the speed of the computer system (CS). There are various approaches and difficulties regarding to the assessment of the reliability of the CS in RNS. So, the existing options of mathematical models do not always satisfy a comprehensive assessment of the reliability of the CS in RNS. The variant of mathematical model of reliability of CS, that functions in RNS, is considered in the article, based on the use of a mathematical model of structural sliding backup in the case of a loaded mode of operation of the backup elements, which is used in the positional binary number system (PBNS). A calculation and comparative security of the trooped computing structure (TCS) analysis was carried out in PBNS with an ideal majority element and the CS in RNS with an ideal reliability machine. The results of analysis showed the following. At the initial stage of the functioning of computational systems, reliability, by probability of trouble-free operation, the CS in RNS with two control bases above the reliability of the trooked positional computing structure, widely used in the PDS and with a smaller addition of an additional equipment insertion. It supposes the effective use of RNS for the increase of reliability of CS and computing devices on the initial stage of their functioning. For example, for the increase of reliability the practical use of unposition code structures is possible in RNS in the side digital calculable complexes of ballistic rockets and in the computing devices of pilotless aircrafts of brief action.

Mersenne Numbers in the Bases of Systems of Residual Classes when Transmitting Data in Serial Communication Channels

In this paper, we consider the system of residual classes for data representation in communication channels. In this case, it is possible to perform parallel data processing without taking into account digit transfers. However, the most of communication channels are serial; therefore, we also consider the possibility of serial transmission of parallel data. We show that the use of the system of residual classes of Mersenne numbers as a base does not lead to decreasing the data transmission rate. We estimate the amount of information contained in the positions of the numbers of a positional code.

Improvement of the algorithm for calculating the interval number in the code of the residual class system

Improvement of the algorithm for calculating the interval number in the code of the residual class system

Pesticide Residues in Fresh Fruit and Vegetables from Farm to Fork in the Kampala Metropolitan Area, Uganda.

This study assessed concentrations of pesticide residues in fruits and vegetables from farm-to-fork in Kampala Metropolitan Area, Uganda. A total of 160 samples of fruit and vegetables collected from farms, markets, streets, restaurants and homes were analysed using liquid chromatography–tandem mass spectrometry; and Gas Chromatograph–Mass Spectrometer for dithiocarbamates. Multiple pesticide residues were detected in majority of the samples (95.6%). The proportions of the most frequently detected pesticides residue classes were organophosphates (91.3%), carbamates (67.5%), pyrethroids (60.0%) dithiocarbamates (48.1%) and neonicotinoids (42.5%). Among organophosphates, propotamophos, acephate, fonofos, monocrotophos and dichlorvos were the most detected active ingredients; aminocarb, methomyl and pirimicarb were the commonly detected carbamates; while imidacloprid, a neonicotinoid and lambda-cyhalothrin, pyrethroid were also highly detected. Twenty-seven pesticide were tested at all stages, of which the concentrations either decreased or increased along the chain. Multiple pesticide residues occurred in commonly consumed fruit and vegetables with decreasing or increasing concentrations from farm-to-fork.

Integer Programming Formulations For The Frobenius Problem

The Frobenius number of a set of relatively prime positive integers α1,α2,…,αn such that α1< α2< …< αn, is the largest integer that can not be written as a nonnegative integer linear combination of the given set. Finding the Frobenius number is known as the Frobenius problem, which is also named as the coin exchange problem or the postage stamp problem. This problem is closely related with the equality constrained integer knapsack problem. It is known that this problem is NP-hard. Extensive research has been conducted for finding the Frobenius number of a given set of positive integers. An exact formula exists for the case n=2 and various formulas have been derived for all special cases of n = 3. Many algorithms have been proposed for n≥4. As far as we are aware, there does not exist any integer programming approach for this problem which is the main motivation of this paper. We present four integer linear programming formulations about the Frobenius number of a given set of positive integers. Our first formulation is used to check if a given positive integer is the Frobenius number of a given set of positive integers. The second formulation aims at finding the Frobenius number directly. The third formulation involves the residue classes with respect to the least member of the given set of positive integers, where a residue table is computed comprising all values modulo that least member, and the Frobenius number is obtained from there. Based on the same approach underlying the third formulation, we propose our fourth formulation which produces the Frobenius number directly. We demonstrate how to use our formulations with several examples. For illustrative purposes, some computa-tional analysis is also presented.

Grafting Hydrophobic Amino Acids Critical for Inhibition of Protein-Protein Interactions on a Cell-Penetrating Peptide Scaffold.

Stapled peptides are a promising class of conformationally restricted peptides for modulating protein-protein interactions (PPIs). However, the low membrane permeability of these peptides is an obstacle to their therapeutic applications. It is common that only a few hydrophobic amino acid residues are mandatory for stapled peptides to bind to their target proteins. Hoping to create a novel class of membrane-permeable PPI inhibitors, the phenylalanine, tryptophan, and leucine residues that play a critical role in inhibiting the p53-HDM2 interaction were grafted into the framework of CADY2─a cell-penetrating peptide (CPP) having a helical propensity. Two analogues (CADY-3FWL and CADY-10FWL) induced apoptotic cell death but lacked the intended HDM2 interaction. Pull-down experiments followed by proteomic analysis led to the elucidation of nesprin-2 as a candidate binding target. Nesprin-2 is considered to play a role in the nuclear translocation of β-catenin upon activation of the Wnt signaling pathway, which leads to the expression of antiapoptosis proteins and cell survival. Cells treated with the two analogues showed decreased nuclear localization of β-catenin and reduced mRNA expression of related antiapoptotic proteins. These data suggest inhibition of β-catenin nuclear translocation as a possible mode of action of the described cell-penetrating stapled peptides.

Proof of the Bessenrodt\u2013Ono Inequality by Induction

In 2016 Bessenrodt–Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalizations by several authors have been given; on partitions with rank in a given residue class by Hou–Jagadeesan and Males, on k-regular partitions by Beckwith–Bessenrodt, on k-colored partitions by Chern–Fu–Tang, and Heim–Neuhauser on their polynomization, and Dawsey–Masri on the Andrews spt-function. The proofs depend on non-trivial asymptotic formulas related to the circle method on one side, or a sophisticated combinatorial proof invented by Alanazi–Gagola–Munagi. We offer in this paper a new proof of the Bessenrodt–Ono inequality, which is built on a well-known recursion formula for partition numbers. We extend the proof to the result by Chern–Fu–Tang and its polynomization. Finally, we also obtain a new result.

Neoadjuvant Chemotherapy and Immunotherapy in Luminal B-like Breast Cancer: Results of the Phase II GIADA Trial.

The role of immunotherapy in hormone receptor (HR)-positive, HER2-negative breast cancer is underexplored. The neoadjuvant phase II GIADA trial (NCT04659551, EUDRACT 2016-004665-10) enrolled stage II-IIIA premenopausal patients with Luminal B (LumB)-like breast cancer (HR-positive/HER2-negative, Ki67 ≥ 20%, and/or histologic grade 3). Patients received: three 21-day cycles of epirubicin/cyclophosphamide followed by eight 14-day cycles of nivolumab, triptorelin started concomitantly to chemotherapy, and exemestane started concomitantly to nivolumab. Primary endpoint was pathologic complete response (pCR; ypT0/is, ypN0). A pCR was achieved by 7/43 patients [16.3%; 95% confidence interval (CI), 7.4-34.9]; the rate of residual cancer burden class 0-I was 25.6%. pCR rate was significantly higher for patients with PAM50 Basal breast cancer (4/8, 50%) as compared with other subtypes (LumA 9.1%; LumB 8.3%; P = 0.017). Tumor-infiltrating lymphocytes (TIL), immune-related gene-expression signatures, and specific immune cell subpopulations by multiplex immunofluorescence were significantly associated with pCR. A combined score of Basal subtype and TILs had an AUC of 0.95 (95% CI, 0.89-1.00) for pCR prediction. According to multiplex immunofluorescence, a switch to a more immune-activated tumor microenvironment occurred following exposure to anthracyclines. Most common grade ≥3 treatment-related adverse events (AE) during nivolumab were γ-glutamyltransferase (16.7%), alanine aminotransferase (16.7%), and aspartate aminotransferase (9.5%) increase. Most common immune-related AEs were endocrinopathies (all grades 1-2; including adrenal insufficiency, n = 1). Luminal B-like breast cancers with a Basal molecular subtype and/or a state of immune activation may respond to sequential anthracyclines and anti-PD-1. Our data generate hypotheses that, if validated, could guide immunotherapy development in this context.

On Schneider\u2019s Continued Fraction Map on a Complete Non-Archimedean Field

Let {mathcal {M}} denote the maximal ideal of the ring of integers of a non-Archimedean field K with residue class field k whose invertible elements, we denote k^{times }, and a uniformizer we denote pi . In this paper, we consider the map T_{v}: {mathcal {M}} rightarrow {mathcal {M}} defined by Tv(x)=πv(x)x-b(x),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} T_v(x) = \\frac{\\pi ^{v(x)}}{x} - b(x), \\end{aligned}$$\\end{document}where b(x) denotes the equivalence class to which frac{pi ^{v(x)}}{x} belongs in k^{times }. We show that T_v preserves Haar measure mu on the compact abelian topological group {mathcal {M}}. Let {mathcal {B}} denote the Haar sigma -algebra on {mathcal {M}}. We show the natural extension of the dynamical system ({mathcal {M}}, {mathcal {B}}, mu , T_v) is Bernoulli and has entropy frac{#( k)}{#( k^{times })}log (#( k)). The first of these two properties is used to study the average behaviour of the convergents arising from T_v. Here for a finite set A its cardinality has been denoted by # (A). In the case K = {mathbb {Q}}_p, i.e. the field of p-adic numbers, the map T_v reduces to the well-studied continued fraction map due to Schneider.

Polynomial functions on rings of dual numbers over residue class rings of the integers

AbstractThe ring of dual numbers over a ringRisR[α] =R[x]/(x2), whereαdenotesx+ (x2). For any finite commutative ringR, we characterize null polynomials and permutation polynomials onR[α] in terms of the functions induced by their coordinate polynomials (f1,f2∈R[x], wheref=f1+αf2) and their formal derivatives onR.We derive explicit formulas for the number of polynomial functions and the number of polynomial permutations on ℤpn[α] forn≤p(pprime).

Сравнение чисел и анализ переполнения в модулярной арифметике

The method of modular arithmetic consists in operating not with a number, but with its remainders after division by some integers. In the modular number system or the number system in the residual classes, a multi-bit integer in the positional number system is represented as a sequence of several positional numbers. These numbers are the remainders (residues) of dividing the original number into some modules that are mutually prime integers. The advantage of the modular representation is that it is very simple to perform addition, subtraction and multiplication operations. In parallel execution of operations, the use of modular arithmetic can significantly reduce the computation time. However, there are drawbacks to modular representation that limit its use. These include a slow conversion of numbers from modular to positional representation; the complexity of comparing numbers in modular representation; the difficulty in performing the division operation; and the difficulty of determining the presence of an overflow. The use of modular arithmetic is justified if there are fast algorithms for calculating a number from a set of remainders. This article describes a fast algorithm for converting numbers from modular representation to positional representation based on a geometric approach. The review is carried out for the case of a comparison system with two modules. It is also shown that as a result of increasing numbers in positional calculus, they successively change in a spiral on the surface of a two-dimensional torus. Based on this approach, a fast algorithm for comparing numbers and an algorithm for detecting an overflow during addition and multiplication of numbers in modular representation were developed. Consideration for the multidimensional case is possible when analyzing a multidimensional torus and studying the behavior of the turns on its surface.

On Totally Split Primes in High-Degree Torsion Fields of Elliptic Curves

Abstract Analogously to primes in arithmetic progressions to large moduli, we can study primes that are totally split in extensions of ${\mathbb {Q}}$ of high degree. Motivated by a question of Kowalski we focus on the extensions ${\mathbb {Q}}(E[d])$ obtained by adjoining the coordinates of $d$-torsion points of a non-CM elliptic curve $E/{\mathbb {Q}}$. We show that for almost all integers $d$ there exists a non-CM elliptic curve $E/{\mathbb {Q}}$ and a prime $p<|\text {Gal}({\mathbb {Q}}(E[d])/{\mathbb {Q}})|= d^{4-o(1)}$, which is totally split in ${\mathbb {Q}}(E[d])$. Note that such a prime $p$ is not accounted for by the expected main term in the Chebotarev Density Theorem. Furthermore, we prove that for almost all $d$ that factorize suitably there exists a non-CM elliptic curve $E/{\mathbb {Q}}$ and a prime $p$ with $p^{0.2694} < d$, which is totally split in ${\mathbb {Q}}(E[d])$. To show this we use work of Kowalski to relate the question to the distribution of primes in certain residue classes modulo $d^2$. Hence, the barrier $p < d^4$ is related to the limit in the classical Bombieri–Vinogradov Theorem. To break past this we make use of the assumption that $d$ factorizes conveniently, similarly as in the works on primes in arithmetic progression to large moduli by Bombieri, Friedlander, Fouvry, and Iwaniec, and in the more recent works of Zhang, Polymath, and the author. In contrast to these works we do not require any of the deep exponential sum bounds (i.e., sums of Kloosterman sums or Weil/Deligne bound). Instead, we only require the classical large sieve for multiplicative characters and we apply Harman’s sieve method to obtain a combinatorial decomposition for primes.

Analytic and Combinatorial Number Theory

Класс рядов Дирихле, соответствующих периодическим арифметическим функциям $f$, включает в себя дзета-функцию Римана и $L$-функции характеров Дирихле. В работе изучается распределение значений таких рядов Дирихле $L(s;f)$ и их аналитических продолжений в окрестности критической прямой (которая является осью симметрии соответствующего функционального уравнения типа Римана). В частности, для заданного комплексного числа $a\neq 0$ и четных или нечетных периодических функций $f$ находится количество $a$-значений $\Delta $-множителя функционального уравнения, доказывается существование среднего для значений рядов Дирихле $L(s;f)$ в этих точках, показывается, что ординаты этих $a$-значений равномерно распределены по модулю $1$, и этот результат применяется для доказательства дискретной теоремы универсальности.

Dirichlet Series with Periodic Coefficients and Their Value-Distribution near the Critical Line

The class of Dirichlet series associated with a periodic arithmetical function $$f$$ includes the Riemann zeta-function as well as Dirichlet $$L$$ -functions to residue class characters. We study the value-distribution of these Dirichlet series $$L(s;f)$$ and their analytic continuation in the neighbourhood of the critical line (which is the axis of symmetry of the related Riemann-type functional equation). In particular, for a fixed complex number $$a\neq 0$$ , we find for an even or odd periodic $$f$$ the number of $$a$$ -points of the $$\Delta$$ -factor of the functional equation, prove the existence of the mean of the values of $$L(s;f)$$ taken at these points, show that the ordinates of these $$a$$ -points are uniformly distributed modulo one and apply this to show a discrete universality theorem.

Classifying Residues in Mechanically Stable and Unstable Substructures Based on a Protein Sequence: The Case Study of the DnaK Hsp70 Chaperone.

Artificial proteins can be constructed from stable substructures, whose stability is encoded in their protein sequence. Identifying stable protein substructures experimentally is the only available option at the moment because no suitable method exists to extract this information from a protein sequence. In previous research, we examined the mechanics of E. coli Hsp70 and found four mechanically stable (S class) and three unstable substructures (U class). Of the total 603 residues in the folded domains of Hsp70, 234 residues belong to one of four mechanically stable substructures, and 369 residues belong to one of three unstable substructures. Here our goal is to develop a machine learning model to categorize Hsp70 residues using sequence information. We applied three supervised methods: logistic regression (LR), random forest, and support vector machine. The LR method showed the highest accuracy, 0.925, to predict the correct class of a particular residue only when context-dependent physico-chemical features were included. The cross-validation of the LR model yielded a prediction accuracy of 0.879 and revealed that most of the misclassified residues lie at the borders between substructures. We foresee machine learning models being used to identify stable substructures as candidates for building blocks to engineer new proteins.

On a density property of the residual order of $a \pmod{pq}$

We consider a distribution property of the residual order (the multiplicative order) of the residue class $a \pmod{pq}$. It is known that the residual order fluctuates irregularly and increases quite rapidly. We are interested in how the residual orders $a \pmod{pq}$ distribute modulo 4 when we fix $a$ and let $p$ and $q$ vary. In this paper we consider the set $S(x) = \{(p, q); p, q \ \text{are distinct primes,} \ pq \leq x \}$, and calculate the natural density of the set $\{(p, q) \in S(x); \ \text{the residual order of} \ a \pmod{pq} \equiv l \pmod{4}\}$. We show that, under a simple assumption on $a$, these densities are $\{5/9,\, 1/18,\, 1/3,\, 1/18 \}$ for $l= \{0, 1, 2, 3 \}$, respectively. For $l = 1, 3$ we need Generalized Riemann Hypothesis.

Efficient Communication Protocols for Non DHT-based Pyramid Tree P2P Architecture

In this paper, we have considered a recently reported 2-layer non-DHT-based structured P2P network. Residue Class based on modular arithmetic has been used to realize the overlay topology. At the heart of the architecture (layer-1), there exists a tree like structure, known as pyramid tree. It is not a conventional tree. A node i in this tree represents the cluster-head of a cluster of peers which are interested in a particular resource of type Ri (i.e. peers with a common interest). The cluster-head is the first among these peers to join the system. Root of the tree is assumed to be at level 1. Such a tree is a complete one if at each level j, there are j number of nodes. It is an incomplete one if only at its leaf level, say k, there are less than k number of nodes. Layer 2 consists of the different clusters. The network has some unique structural properties, e.g. each cluster has a diameter of only 1 overlay hop and the diameter of the network is just (2+2d); d being the number of levels of the layer-1 pyramid tree and d depends only on the number of distinct resources. Therefore, the diameter of the network is independent of the number of peers in the whole network. In the present work, we have used some such properties to design low latency intra and inter cluster data lookup protocols. Our choice of considering non-DHT and interest-based overlay networks is justified by the following facts: 1) intra-cluster data lookup protocol has constant complexity and complexity of inter-cluster data lookup is O(d) if tree traversal is used and 2) search latency is independent of the total number of peers present in the overlay network unlike any structured DHT-based network (as a matter fact unlike any existing P2P network, structured or unstructured). Experimental results as well show superiority of the proposed protocols to some noted structured networks from the viewpoints of search latency and complexity involved in it. In addition, we have presented in detail the process of handling churns and proposed a simple yet very effective technique related to cluster partitioning, which, in turn, helps in reducing the number of messages required to be exchanged to handle churns.