Residue Class Research Articles

On the 𝑝-adic zeros of the Tribonacci sequence

Let ( T n ) n ∈ Z (T_n)_{n\in {\mathbb Z}} be the Tribonacci sequence and for a prime p p and an integer m m let ν p ( m ) \nu _p(m) be the exponent of p p in the factorization of m m . For p = 2 p=2 Marques and Lengyel found some formulas relating ν p ( T n ) \nu _p(T_n) with ν p ( f ( n ) ) \nu _p(f(n)) where f ( n ) f(n) is some linear function of n n (which might be constant) according to the residue class of n n modulo 32 32 and asked if similar formulas exist for other primes p p . In this paper, we give an algorithm which tests whether for a given prime p p such formulas exist or not. When they exist, our algorithm computes these formulas. Some numerical results are presented.

Sign changes in statistics for plane partitions

Recent work of Cesana, Craig and the third author shows that the trace of plane partitions is asymptotically equidistributed in residue classes mod b. Applying a technique of the first two authors and Garnowski, we prove asymptotic formulas for the secondary terms in this equidistribution, which are controlled by certain complex numbers generated by a twisted MacMahon-type product. We further carry out a similar analysis for a statistic related to plane overpartitions.

Categorization of hotspots into three types – weak, moderate and strong to distinguish protein–protein versus protein–peptide interactions

Protein–protein and protein–peptide interactions (PPI and PPepI) belong to a similar category of interactions, yet seemingly subtle differences exist among them. To characterize differences between protein–protein (PP) and protein–peptide (PPep) interactions, we have focussed on two important classes of residues-hotspot and anchor residues. Using implicit solvation-based free energy calculations, a very large-scale alanine scanning has been performed on benchmark datasets, consisting of over 5700 interface residues. The differences in the two categories are more pronounced, if the data were divided into three distinct types, namely – weak hotspots (having binding free energy loss upon Ala mutation, ΔΔG, ∼2–10 kcal/mol), moderate hotspots (ΔΔG, ∼10–20 kcal/mol) and strong hotspots (ΔΔG ≥ ∼20 kcal/mol). The analysis suggests that for PPI, weak hotspots are predominantly populated by polar and hydrophobic residues. The distribution shifts towards charged and polar residues for moderate hotspot and charged residues (principally Arg) are overwhelmingly present in the strong hotspot. On the other hand, in the PPepI dataset, the distribution shifts from predominantly hydrophobic and polar (in the weak type) to almost similar preference for polar, hydrophobic and charged residues (in moderate type) and finally the charged residue (Arg) and Trp are mostly occupied in the strong type. The preferred anchor residues in both categories are Arg, Tyr and Leu, possessing bulky side chain and which also strike a delicate balance between side chain flexibility and rigidity. The present knowledge should aid in effective design of biologics, by augmentation or disruption of PPIs with peptides or peptidomimetics. Communicated by Ramaswamy H. Sarma

Polynomial equations for matrices over integers modulo a prime power and the cokernel of a random matrix

Given a prime p and a positive integer k, let Mn(Z/pkZ) be the ring of n×n matrices over Z/pkZ. We consider the number of solutions X∈Mn(Z/pkZ) to the polynomial equation P(X)=0, where P(t) is a monic polynomial in (Z/pkZ)[t] whose reduction modulo p is square-free over the finite field Fp of p elements. Noting that P(X)=0 if and only if cok(P(X))≃(Z/pkZ)n, we give a conjectural generalization of counting solutions to P(X)=0 as the distribution of the cokernel cok(P(X)) of P(X) up to isomorphisms, where X is a uniform random matrix in Mn(Z/pkZ). This distribution involves an explicit formula when we fix the residue class of X modulo p. We prove this conjecture for the special case when the image of P(t) in Fp[t] modulo p is irreducible. We explain how the distribution we obtain is closely related to the Cohen-Lenstra distribution in number theory. Our proof involves algebraic and combinatorial arguments in linear algebra over Z/pkZ and builds upon a previous work of Cheong and Kaplan.

Hermitian matrices of roots of unity and their characteristic polynomials

We investigate spectral conditions on Hermitian matrices of roots of unity. Our main results are conjecturally sharp upper bounds on the number of residue classes of the characteristic polynomial of such matrices modulo ideals generated by powers of (1−ζ), where ζ is a root of unity. We also prove a generalisation of a classical result of Harary and Schwenk about a relation for traces of powers of a graph-adjacency matrix, which is a crucial ingredient for the proofs of our main results.

ON THE DENSITY OF SUMSETS, II

AbstractArithmetic quasidensities are a large family of real-valued set functions partially defined on the power set of $\mathbb {N}$ , including the asymptotic density, the Banach density and the analytic density. Let $B \subseteq \mathbb {N}$ be a nonempty set covering $o(n!)$ residue classes modulo $n!$ as $n\to \infty $ (for example, the primes or the perfect powers). We show that, for each $\alpha \in [0,1]$ , there is a set $A\subseteq \mathbb {N}$ such that, for every arithmetic quasidensity $\mu $ , both A and the sumset $A+B$ are in the domain of $\mu $ and, in addition, $\mu (A + B) = \alpha $ . The proof relies on the properties of a little known density first considered by Buck [‘The measure theoretic approach to density’, Amer. J. Math.68 (1946), 560–580].

Diverging prognostic effects of CD155 and CD73 expressions in locally advanced triple-negative breast cancer.

Immune checkpoint inhibition, combined with novel biomarkers, may provide alternative pathways for treating chemotherapy-resistant triple-negative breast cancer (TNBC). This study investigates the expression of new immune checkpoint receptors, including CD155 and CD73, which play a role in T and natural killer (NK) cell activities, in patients with residual TNBC after neoadjuvant chemotherapy (NAC). The expression of biomarkers was immunohistochemically examined by staining archival tissue from surgical specimens (n = 53) using specific monoclonal antibodies for PD-L1, CD155, and CD73. Of those, 59.2% (29/49) were found to be positive (>1%) for PD-L1 on the tumour and tumour-infiltrating lymphocytes (TILs), while CD155 (30/53, 56.6%) and CD73 (24/53, 45.3%) were detected on tumours. Tumour expressions of CD155 and CD73 significantly correlated with PD-L1 expression on the tumour (p = 0.004 for CD155, p = 0.001 for CD73). Patients with CD155 positivity ≥10% were more likely to have a poor chemotherapy response, as evidenced by higher MDACC Residual Cancer Burden Index scores and Class II/III than those without CD155 expression (100% vs 82.6%, p = 0.03). At a median follow-up time of 80 months (range, 24-239), patients with high CD73 expression showed improved 10-year disease-free survival (DFS) and disease-specific survival (DSS) rates compared to those with low CD73 expression. In contrast, patients with CD155 (≥10%) expression exhibited a decreasing trend in 10-year DFS and DSS compared to cases with lower expression, although statistical significance was not reached. However, patients with coexpression of CD155 (≥10%) and low CD73 were significantly more likely to have decreased 10-year DFS and DSS rates compared to others (p = 0.005). These results demonstrate high expression of CD73 and CD155 in patients with residual tumours following NAC. CD155 expression was associated with a poor response to NAC and poor prognosis in this chemotherapy-resistant TNBC cohort, supporting the use of additional immune checkpoint receptor inhibitor therapy. Interestingly, the interaction between CD155 and CD73 at lower levels resulted in a worse outcome than either marker alone, which calls for further investigation in future studies.

Characterizing absolutely irreducible integer-valued polynomials over discrete valuation domains

Rings of integer-valued polynomials are known to be atomic, non-factorial rings furnishing examples for both irreducible elements for which all powers factor uniquely (absolutely irreducibles) and irreducible elements where some power has a factorization different from the trivial one.In this paper, we study irreducible polynomials F∈Int(R) where R is a discrete valuation domain with finite residue field and show that it is possible to explicitly determine a number S∈N that reduces the absolute irreducibility of F to the unique factorization of FS.To this end, we establish a connection between the factors of powers of F and the kernel of a certain linear map that we associate to F. This connection yields a characterization of absolute irreducibility in terms of this so-called fixed divisor kernel. Given a non-trivial element v of this kernel, we explicitly construct non-trivial factorizations of Fk, provided that k≥L, where L depends on F as well as the choice of v. We further show that this bound cannot be improved in general. Additionally, we provide other (larger) lower bounds for k, one of which only depends on the valuation of the denominator of F and the size of the residue class field of R.

The method of statistical processing of data of RGB pixels of color images based on the system of residual classes

The method of statistical processing of data of RGB pixels of color images based on the system of residual classes

Deep Local Analysis deconstructs protein-protein interfaces and accurately estimates binding affinity changes upon mutation.

The spectacular recent advances in protein and protein complex structure prediction hold promise for reconstructing interactomes at large-scale and residue resolution. Beyond determining the 3D arrangement of interacting partners, modeling approaches should be able to unravel the impact of sequence variations on the strength of the association. In this work, we report on Deep Local Analysis, a novel and efficient deep learning framework that relies on a strikingly simple deconstruction of protein interfaces into small locally oriented residue-centered cubes and on 3D convolutions recognizing patterns within cubes. Merely based on the two cubes associated with the wild-type and the mutant residues, DLA accurately estimates the binding affinity change for the associated complexes. It achieves a Pearson correlation coefficient of 0.735 on about 400 mutations on unseen complexes. Its generalization capability on blind datasets of complexes is higher than the state-of-the-art methods. We show that taking into account the evolutionary constraints on residues contributes to predictions. We also discuss the influence of conformational variability on performance. Beyond the predictive power on the effects of mutations, DLA is a general framework for transferring the knowledge gained from the available non-redundant set of complex protein structures to various tasks. For instance, given a single partially masked cube, it recovers the identity and physicochemical class of the central residue. Given an ensemble of cubes representing an interface, it predicts the function of the complex. Source code and models are available at http://gitlab.lcqb.upmc.fr/DLA/DLA.git.

METHOD OF ULTRASONIC PHASE TWO-COORDINATE INFORMATION REGISTRATION FOR AUTOMATED NON-DESTRUCTIVE TESTING SYSTEMS

The article is devoted to the issue of improving the coordinate registration of information in means of automated non-destructive testing. Existing systems of non-contact registration of spatial position sensors of non-destructive testing are mostly focused on the use of signal fields of different physical nature, for example, acoustic or optical, which are created around the position of these sensors and amplitude methods of signal processing. These methods do not provide the necessary accuracy of determining the coordinates in the case of using small-aperture sensors of non-destructive testing.
 The basis of the proposed technical solution is the trilateration method. The distance between the emitter of ultrasonic vibration, which is structurally connected to the sensor, and the two receivers is determined by the phase method. This method provides increased accuracy in determining the coordinates of the sensor. Eliminating the ambiguity of phase measurements is based on the combination of the multi-frequency method and the numerical system of residual classes. This makes it possible to minimize the probability of incorrect resolution of the ambiguity of the results of phase distance measurements and to ensure high accuracy of coordinate determination (with an absolute error of the order of mm). The idea of the method and the main calculation formulas that make it possible to determine the modules of the numerical system of residual classes and the values of the operating frequencies of ultrasonic oscillations are presented. Matching the discretes for determining the phase shifts of signals with the modules of the numerical system of residual classes at different operating frequencies makes it possible to reduce the resolution of the ambiguity of the results of phase measurements to the task of restoring integers in the positional counting system from their representation by residuals in the numerical system of residual classes. The mechanism of occurrence of gross errors in the recovery of whole numbers is considered. An algorithm for eliminating this error is proposed. In general, the correctness of the algorithm for determining the sensor coordinates is confirmed by the simulation results. The considered method of two-coordinate information registration can be used both in the development of new automated means of non-destructive testing, and for the improvement of existing ones, which involve a manual method of scanning the surface of testing objects.

IMPROVING THE QUALITY OF THE FUNCTIONING OF TELECOMMUNICATION SYSTEMS BASED ON THE INTEGRATION OF NEURAL NETWORKS

The article proposes the integration of neural networks as a parallel element base in telecommunications systems. In this case, their main advantage, the ability to self-study or adapt to external conditions, will be used. For telecommunication systems in conditions of interference, this ability will significantly increase their noise immunity, reliability, operability, etc. The article considers an example of integration of a neural network into a discrete matched signal filter. It is noted that the use of parallel mathematics in signal processing, such as a system of residual classes, will lead to the maximum effect of increasing the quality parameters of such elements of telecommunications.
 Purpose – introduction to neural networks; application of neural networks in digital filtering; possible directions of integration of neural network technologies.
 Method or methodology of the work: harmonic analysis, Fourier transform, correlation analysis were used in the article.
 Results: The introduction of mass parallelism in telecommunications becomes feasible when integrating neural networks into telecommunication elements. The efficiency becomes maximum when using simultaneously parallel mathematics of the system of residual classes
 Practical implications: the use of parallel mathematics in signal processing, such as a system of residual classes, will lead to the maximum effect of increasing the quality parameters of such elements of telecommunications.

Multiplicative functions in short arithmetic progressions

AbstractWe study for bounded multiplicative functions sums of the form establishing that their variance over residue classes is small as soon as , for almost all moduli , with a nearly power‐saving exceptional set of . This improves and generalizes previous results of Hooley on Barban–Davenport–Halberstam type theorems for such , and moreover our exceptional set is essentially optimal unless one is able to make progress on certain well‐known conjectures. We are nevertheless able to prove stronger bounds for the number of the exceptional moduli in the cases where is restricted to be either smooth or prime, and conditionally on GRH we show that our variance estimate is valid for every . These results are special cases of a “hybrid result” that we establish that works for sums of over almost all short intervals and arithmetic progressions simultaneously, thus generalizing the Matomäki–Radziwiłł theorem on multiplicative functions in short intervals. We also consider the maximal deviation of over all residue classes in the square root range , and show that it is small for “smooth‐supported” , again apart from a nearly power‐saving set of exceptional , thus providing a smaller exceptional set than what follows from Bombieri–Vinogradov type theorems. As an application of our methods, we consider Linnik‐type problems for products of exactly three primes, and in particular prove a ternary approximation to a conjecture of Erdős on representing every element of the multiplicative group as the product of two primes less than .

Large prime gaps and probabilistic models

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval [1,x]. Our results are stated in terms of the extremal bounds in the interval sieve problem. The same methods also allow us to rigorously relate the validity of the Hardy-Littlewood conjectures for an arbitrary set (such as the actual primes) to lower bounds for the largest gaps within that set.

THE STRUCTURE OF THE COMPUTER SYSTEM IN THE RESIDUAL CLASSES

The subject of the article is the formulation and solution of the inverse problem of optimal redundancy in the system of residual classes (RNS) based on the use of the dynamic programming method. The solution of this problem makes it possible to improve the reliability of the operation of computer systems and components (CSC) in the RNS. The purpose of the article is to increase the reliability of the functioning of CSC, which are built on the basis of the use of RNS, without reducing the speed of calculations, as well as to calculate and compare the reliability, in terms of the probability of failure-free operation, of CSC in RNS and a tripled computing system that operates in a positional binary number system (PNS). Tasks: to analyze the influence of the number system used on the reliability of the CSC, taking into account the primary and secondary redundancy; to synthesize a computing system in RNS for a l-byte bit grid based on the use of the passive fault tolerance method (constant structural redundancy); formulate and solve the inverse problem of optimal redundancy in RNS based on the use of the dynamic programming method; to check the correctness of the results obtained, calculate the conditional amount of computer system equipment in the residual classes; evaluate the efficiency of using RNS to improve reliability when building a redundant CSC in relation to a redundant CSC in the PNS. Research methods: methods of analysis and synthesis of computer systems, number theory, coding theory in RNS, reliability theory. The following results are obtained. The paper shows that the use of PNS as a number system does not allow a radical increase in the performance and reliability of CSC. In this regard, the article developed the concept of using RNS as a number system for constructing a CSC. Based on this, the inverse problem of optimal redundancy in RNS is formulated and solved. Conclusions. As shown by the results of calculations and comparative analysis, the use of RNS provides a higher reliability of the CSC than the majority three-channel computing system in the PNS. The obtained research results can be used for the synthesis of fault-tolerant computer structures in RNS.

A digital signature scheme based on problem of solving polynomial congruence in residue class modulo n

Many digital signature schemes have been proposed based on different mathematical problems. Some of them are based on factoring into primes, discrete logarithm problem, elliptic curve discrete logarithm problem, lattice problem, multivariate quadratic problem etc. In this work a signature scheme is proposed which security is based on problem of solving polynomial congruence modulo some positive integer. In proposed scheme degree of polynomial is not fixed, so degree can be changed to increase security. It is also efficient because in all the phases modulo addition and multiplication are used and no need to calculate higher exponent

Linear and circular single‐change covering designs revisited

AbstractA single‐change covering design (SCCD) is a ‐set and an ordered list of blocks of size where every pair from must occur in at least one block. Each pair of consecutive blocks differs by exactly one element. This is a linear single‐change covering design, or more simply, a single‐change covering design. A single‐change covering design is circular when the first and last blocks also differ by one element. A single‐change covering design is minimum if no other smaller design can be constructed for a given . In this paper, we use a new recursive construction to solve the existence of circular SCCD() for all and three residue classes of circular SCCD() modulo 16. We solve the existence of three residue classes of SCCD modulo 16. We prove the existence of circular SCCD, for all , using difference methods.

An algebraic construction technique for codes over Hurwitz integers

Let $\alpha$ be a prime Hurwitz integer. $\mathcal{H}_{\alpha}$, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of $\mathcal{H},$ which is the set of all Hurwitz integers. In this study, we present an algebraic construction technique, which is a modulo function formed depending on two modulo operations, for codes over Hurwitz integers. We consider left congruent modulo $\alpha,$ and the domain of related modulo function is $\mathbb{Z}_{N(\alpha)},$ which is residual class ring of ordinary integers with $N(\alpha)$ elements. Therefore, we obtain the residue class rings of Hurwitz integers with $N(\alpha)$ size. In addition, we present some results for mathematical notations used in two modulo functions, and for the algebraic construction technique formed depending upon two modulo functions. Moreover, we presented graphs obtained by graph layout methods, such as spring, high-dimensional, and spiral embedding, for the set of the residual class obtained with respect to the related modulo function in the rings of Hurwitz integers.

Quasi-quadratic modules in valuation rings and valued fields

We define quasi-quadratic modules in a commutative ring generalizing the notion of quadratic modules. The main result is a structure theorem of quasi-quadratic modules in a subring A of a 2-henselian valued field $$(K,\text{val})$$ whose residue class field F is of characteristic $$\neq 2$$ . We further assume that the valuation ring B is contained in A. Set $$H=\rm{val}(A ^{\times} )$$ and $$G_{\geq e}=\{g \in G \mid g \geq e\}$$ . The notation $$\mathfrak X_R$$ denotes the set of all the quasi-quadratic modules in a commutative ring~ R. Our structure theorem asserts that there exists a one-to-one correspondence between $$\mathfrak X_A$$ and a subset $$\mathcal T_F^{ H \cup G_{\geq e}}$$ of $$\prod_{g \in H \cup G_{\geq e}}\mathfrak X_F$$ . We explicitly construct the map $$\Theta\colon \mathfrak X_A \rightarrow \mathcal T_F^{ H \cup G_{\geq e}}$$ and its inverse. We also give explicit expressions of $$\Theta(\mathcal M \cap \mathcal N)$$ and $$\Theta(\mathcal M+\mathcal N)$$ for $$\mathcal M, \mathcal N \in \mathfrak X_A$$ . In addition, we briefly investigate the case in which the field F is of characteristic two in the appendix as well.

Transformation in the system of residual number classes from one system of modules to another

The purpose of the study. The purpose of the study is an analytical consideration of the system of residual classes for the implementation of the operation of converting numbers from one system of residual classes to another. Method. System analysis, number theory, and the Chinese remainder theorem are tools of the research methodology. The method uses the representation of the number both by its remainders and in the polyadic code. The methodology is based on determining the positional characteristics for this module on the basis of the received positional characteristics for the remaining modules of the original system, with the subsequent construction on their basis of the residuals for the modules of the sought system. The proposed method is algorithmically simple. The results. The theoretical substantiation of the considered approach to obtain an effective solution of the non-modular transformation operation in the system of residual classes for the transition from the representation of a number by one system of modules to its representation by another system of modules is performed. It is shown that the proposed method provides the desired result. On the basis of the proposed approach, an increase in the performance of the conversion operation is achieved. Scientific novelty. The theoretical justification of the presented approach to the solution of the non-modular transformation operation in the system of residual classes for the transition from the representation of a number in one system of modules to its representation in another system of modules is proposed. It seems appropriate to apply the proposed approach as a prospective direction of research of complex operations in the system of residual classes. Practical significance. It follows from the importance of theoretical conclusions and obtained research results.It is based on the fact that there is a reasonably simple and effective approach to solving the problem of performing a non-modular conversion operation in the system of residual classes for the transition from the representation of a number in one system of modules to its representation in another system of modules. The considered solutions have high speed and can be effective in the development of modular computing structures for promising information technologies.