Residue Class Research Articles

Method of Data Control in the Residue Classes

Method of Data Control in the Residue Classes

Random integral matrices and the Cohen-Lenstra heuristics

We prove that given any $\epsilon>0$, random integral $n\times n$ matrices with independent entries that lie in any residue class modulo a prime with probability at most $1-\epsilon$ have cokernels asymptotically (as $n\rightarrow\infty$) distributed as in the distribution on finite abelian groups that Cohen and Lenstra conjecture to be the distribution for class groups of imaginary quadratic fields. This shows the Cohen-Lenstra distribution is universal for finite abelian groups given by generators and random relations---that the distribution of quotients does not depend on the way in which we choose (sufficiently nice) relations. This is a refinement of a result on the distribution of ranks of random matrices with independent entries in ${\Bbb Z}/ p{\Bbb Z}$. This is interesting especially in light of the fact that these class groups are naturally cokernels of square matrices. We also prove the analogue for $n\times (n+u)$ matrices.

Signed Product Cordial of the Sum and Union of Two Fourth Power of Paths and Cycles

A simple graph is said to be signed product cordial if it admits ±1 labeling that satisfies certain conditions. Our aim in this paper is to contribute some new results on signed product cordial labeling and present necessary and sufficient conditions for signed product cordial of the sum and union of two fourth power of paths. We also study the signed product cordiality of the sum and union of fourth power cycles The residue classes modulo 4 are accustomed to find suitable labelings for each class to achieve our task. We have shown that the union and the join of any two fourth power of paths are always signed product cordial. Howover, the join and union of fourth power of cycles are only signed codial with some expectional situations.

Grothendieck–Messing deformation theory for varieties of K3 type

Let R be an artinian local ring with perfect residue class field k. We associate to certain 2-displays over the small ring of Witt vectors W(R) a crystal on SpecR. Let X be a scheme of K3 type over SpecR. We define a perfect bilinear form on the second crystalline cohomology group X which generalizes the Beauville–Bogomolov form for hyper-Kahler varieties over ℂ. We use this form to prove a lifting criterion of Grothendieck–Messing type for schemes of K3 type. The crystalline cohomology Hcrys2(X∕W(R)) is endowed with the structure of a 2-display such that the Beauville–Bogomolov form becomes a bilinear form in the sense of displays. If X is ordinary, the infinitesimal deformations of X correspond bijectively to infinitesimal deformations of the 2-display of X with its Beauville–Bogomolov form. For ordinary K3 surfaces X∕R we prove that the slope spectral sequence of the de Rham–Witt complex degenerates and that Hcrys2(X∕W(R)) has a canonical Hodge–Witt decomposition.

The formulation and solution of the task of the optimum reservation in the system of residual classes

The formulation and solution of the task of the optimum reservation in the system of residual classes

Data control method, which presented by code of non-positioning system of deduction class calculation

The paper proposes a new method of monitoring data presented in non-positional residue class system. For code in residue class system, test bases are included in the general code structure of data containing a set of information bases. In this case, the balances that represent operations for informational and control grounds simultaneously and independently participate in the process of information processing. The result of the information processing can be monitored either step by step or at the end of all calculations, since the error that occurred in any residue, does not apply (does not “multiply”) to the remaining residues. The control method proposed based on the principle of comparison, further creates prerequisites for developing effective methods for diagnosing and correcting errors in deduction class. The disadvantage of the proposed method is the relatively low control efficiency. This circumstance makes it necessary to increase the efficiency of data processing system control in deduction class by reducing the execution time of the above operations by developing and using, for example, methods and means for implementing the positional features of the non-positional deduction class code.

Methods for Comparing Numbers in Non-Positional Notation of Residual Classes

Methods for Comparing Numbers in Non-Positional Notation of Residual Classes

MODULES OF THE RESIDUAL CLASSESʼ SYSTEM AS AN INFORMATION ACCURACY TOOL

An analysis of the means and principles of electronic intelligence, as well as technologies, that allow access to information, shows that in modern conditions, the need for increasing the effectiveness of countering them is growing. That is why the urgent task is the search or synthesis of signal structures that provide the necessary level of resistance of information messages to the possibility of unauthorized access. To this end, the article explores the principles of signal generation based on timer and position coding. These principles are very different. In positional coding, the length of individual signal segments in the code structure can be equal , and so on. Thus, the distance between the moments of modulation is a multiple of the length and the exact number of code words . When using timer signal constructions, the code word will consist of several information sections that must correspond to the condition , where are integers. Therefore, cannot be less , but contains information about the number. The article shows the number of implementations of timer signal structures and power of the position code, entropy calculated indicating the probability of errors in various parameters. Also considered is a modular time consolidation method and recommendations are given for its use to increase the secrecy of information by generating uncertainty during transmission during encryption. In addition, it is proved that the use of modules of the system of residual classes will increase the efficiency of secrecy of information transmitted by a signal constructed on the basis of timer signal structures.

Performance, carcass traits, and non-carcass components of feedlot finished lambs from different residual feed intake classes

This study aimed to assess the performance, carcass traits, and non-carcass components of feedlot finished Texel crossbred lambs from different residual feed intake classes (RFI). Forty-seven uncastrated male Texel crossbred lambs (¾ Texel + ¼ Pantaneira) tested here were 4-month-old with an initial weight of 29.9 ± 5.5 kg. The lambs were confined for 70 days for individual dry matter intake (DMI) and average daily gain (ADG) assessment. The diet containing corn silage was provided as roughage, while the concentrate consisted of corn grain, soybean meal, urea, and mineral mixture with a 40:60 roughage to concentrate ratio and 76.34% total digestible nutrients (TDN). After confinement for 70 days, the lambs were slaughtered to assess carcass traits and non-carcass components. Based on the RFI, lambs were divided into three classes according to the standard deviation (sd): Positive RFI (inefficient, 0.5 above the mean), Negative RFI (efficient, 0.5 below the mean), and Medium RFI (intermediate). Classes with Negative (efficient) and Positive RFI (inefficient) showed no differences in ADG (0.321 vs 0.306 kg; P > 0.05). Dry matter intake (g d-1), and percentage of body weight (BW) differed significantly between the RFI classes (P < 0.05). Compared to lambs in the inefficient class, those in the efficient class (Negative RFI) showed a 9% reduction in DMI and had the same ADG. The lambs in either Positive or Negative RFI classes showed no weight differences between non-carcass components (P > 0.05). The RFI classes showed no differences in shrunk body weight, hot carcass weight, hot dressing, and weight of commercial cuts (P > 0.05). They showed no differences in neck, shoulder + shank, loin, rack, flank steak, rack cap off, and leg weights (P > 0.05). Our results indicate that improvement in feed efficiency, as a function of the RFI index, does not compromise performance and carcass traits of Texel crossbred lambs.

Potential of Physalis peruviana calyces as a low-cost valuable resource of phytoprostanes and phenolic compounds.

In Colombia, agro-industrial residues represent an enormous economic and environmental problem, which could be reduced if different techniques for the addition of value to such residues were implemented by this industrial sector. One of the fruits with the highest export rates is Physalis peruviana (goldenberry); however, this fruit is generally marketed without its calyx, generating a large amount of residues. To develop a strategy to add value to these residues, it is essential to know their chemical composition. In the present work, phytoprostanes (PhytoPs)-new active oxylipins-have been detected for the first time in Physalis peruviana calyces by ultra-high performance liquid chromatography triple quadrupole tandem mass spectrometry (UHPLC-QqQ-MS/MS), F1t -phytoprostanes and D1t -phytoprostanes being the predominant and minor classes, respectively. In addition, we were able to characterize the phenolic compounds profile of this matrix using LC-IT-DAD-MS/MS, describing six phenolic derivatives for the first time therein. This study increases our knowledge of the chemical composition of the calyces of this fruit and thereby supports the recycling of this class of residue. Consequently, goldenberry calyces could be used as phytotherapeutic, nutraceutic, or cosmetic ingredients for the development of diverse natural products. © 2018 Society of Chemical Industry.

On short products of primes in arithmetic progressions

We give several families of reasonably small integers $k, \ell \ge 1$ and real positive $\alpha, \beta \le 1$, such that the products $p_1\ldots p_k s$, where $p_1, \ldots, p_k \le m^\alpha$ are primes and $s \le m^\beta$ is a product of at most $\ell$ primes, represent all reduced residue classes modulo $m$. This is a relaxed version of the still open question of P. Erdos, A. M. Odlyzko and A. Sarkozy (1987), that corresponds to $k = \ell =1$ (that is, to products of two primes). In particular, we improve recent results of A. Walker (2016).

On Distribution of Elements of Subgroups in Arithmetic Progressions Modulo a Prime

Let $$\mathbb{F}_p$$ be the field of residue classes modulo a large prime number p. We prove that if $$\mathcal{G}$$ is a subgroup of the multiplicative group $$\mathbb{F}_p^*$$ and if $$\mathcal{I} \subset \mathbb{F}_p$$ is an arithmetic progression, then $$|\mathcal{G} \cap \mathcal{I}| = (1 + o(1))|\mathcal{G}|\mathcal{I}|/p + R$$ , where $$|R| < (|\mathcal{I}|^{1/2} + |\mathcal{G}|^{1/2} + |\mathcal{I}|^{1/2}|\mathcal{G}|^{3/8}p^{-1/8})p^{o(1)}$$ . We use this bound to show that the number of solutions to the congruence xn ≡ λ (mod p), x ∈ ℕ, L < x < L + p/n, is at most p1/3−1/390+o(1) uniformly over positive integers n, λ and L. The proofs are based on results and arguments of Cilleruelo and the author (2014), Murphy, Rudnev, Shkredov and Shteinikov (2017) and Bourgain, Konyagin, Shparlinski and the author (2013).

Encrypting Numbers through Square Grid Graphs

Cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. Encryption is essential for ensured and trusted delivery of sensitive information whereas the decryption is the process of decoding the data which has been encrypted in to a secret format. An authorized user can only decrypt data since decryption requires a secret key or password. There are several methods to encrypt the numbers. In this paper, a combinatorial technique to encrypt and decrypt numbers through labeled strong face of a square grid graph using residue class of Z3 has been investigated.

Determination and Quantitation of Residual Solvents in Natural Food Ingredients Using a Static Headspace Gas Chromatography with Flame Ionization Detection and Mass Spectrometric Detection Method

The determination of residual solvents is critical for quality control in food production and manufacturing processes. A gas chromatographic and mass spectrometric detection (MSD) method was developed and validated to complement the static headspace flame ionization detection (FID) technique for identification, confirmation, and quantitation of solvent residues in botanicals, using Coffeeberry extract and pomegranate powder as test samples. Relative standard deviations (RSDs) of less than 12% were obtained for residual solvents in water and dimethyl sulfoxide. Coffeeberry extract supplemented with10 μg/g of Residual Solvents Class 3 Mix showed MSD and FID recoveries mainly of 91 to 121% and 77 to 110%, respectively, while those samples supplemented with 100 μg/g of Residual Solvents Class 3 Mix displayed MSD and FID recoveries of 105 to 123% and 87 to 112%, respectively. For pomegranate samples supplemented with 10 μg/g of Residual Solvents Class 3 Mix, MSD and FID recoveries of 95 to 124% and 72 to 151% were observed, respectively. Those samples supplemented with 100 μg/g of Residual Solvents Class 3 Mix exhibited MSD and FID recoveries of 109 to 135% and 97 to 127%, respectively. Thus, the developed procedure was demonstrated to be suitable for the analysis of residual solvents in natural food ingredients and is expected to find numerous industrial applications (e.g., process control and impurity detection in final products).

Prime Residue Class of Uniform Charges on the Integers

There is a probability charge on the power set of the integers that gives probability 1 / p to every residue class modulo a prime p. There exists such a charge that gives probability w to the set of prime numbers iff $$w \in [0,1/2]$$. Similarly, there is such a charge that gives probability x to a residue class modulo c, where c is composite, iff $$x \in [0,1/y]$$, where y is the largest prime factor of c.

Explicit equivalence of quadratic forms over [formula omitted

We propose a randomized polynomial time algorithm for computing non-trivial zeros of quadratic forms in 4 or more variables over Fq(t), where Fq is a finite field of odd characteristic. The algorithm is based on a suitable splitting of the form into two forms and finding a common value they both represent. We make use of an effective formula for the number of fixed degree irreducible polynomials in a given residue class. We apply our algorithms for computing a Witt decomposition of a quadratic form, for computing an explicit isometry between quadratic forms and finding zero divisors in quaternion algebras over quadratic extensions of Fq(t).

Orthogonal gauge fixing of first order gravity

We consider the first order connection formulation of 4D general relativity in the "orthogonal" gauge. We show how the partial gauge fixing of the phase space canonical coordinates leads to the appearance of second class constraints in the theory. We employ the gauge unfixing procedure in order to successfully complete the Dirac treatment of the system. While equivalent to the inversion of the Dirac matrix, the gauge unfixing allows us to work directly with the reduced phase space and the ordinary Poisson bracket. At the same time, we explicitly derive the new set of residual first class constraints preserving the partial gauge fixing, which are linear combinations of the original constraints, and these turn out to contain nonlinear terms. While providing an explicit example of how to consistently recast general relativity in a given partial gauge, the main motivation of this classical analysis is the application of the Quantum Reduced Loop Gravity program to a Schwarzschild black hole geometry.

Destackification with restricted root operations

We exhibit a modified version of Bergh’s functorial destackification algorithm which employs mth root operations only for positive integers m not divisible by a given prime p when applied to stacks whose geometric stabilizer group schemes do not contain pth roots of unity. When p is the characteristic of a given base field, this has the consequence that it is possible to destackify a tame Deligne–Mumford stack over the base field staying entirely within the realm of tame Deligne–Mumford stacks, rather than requiring more general tame Artin stacks. The modifications are based on two observations. First, destackification is possible for stacks with 3-torsion diagonalizable stabilizers, as for 2-torsion stabilizers, with just ordinary blow-ups, and not the more general stacky blow-ups that enlarge stabilizer groups. Second, modulo a given prime greater than or equal to 5, every nonzero residue class can be expressed as a product of smaller primes different from p.

A note on primes in certain residue classes

Given positive integers [Formula: see text], we prove that the set of primes [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density relative to the set of all primes which is at least [Formula: see text], where [Formula: see text] is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density which is at least [Formula: see text].

On the spaces generated by vectors with coordinates in incomplete residue systems

We characterize one-dimensional subspaces of three-dimensional space over a finite field of odd prime order, containing a vector whose coordinates are constrained to half of the nonzero residue classes.