Integer Research Articles

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Annotation The properties of the atomic nucleus were studied, in particular the weight of quarks and the energy of gluons were calculated. The nuclear potential and quark mass were calculated from the kinetic energy and momentum. This is determined using integers on which the mass of quarks and the energy of gluons depend. The mass of quarks is approximately known, and a dependence on integer numbers of the quark mass is selected that is close to the experimental value. This allows us to more accurately determine the mass of quarks that satisfy the kinetic energy of the quarks and their momentum. The energy of the atomic nucleus and the energy of the atom are determined using STR formulas at imaginary speed.

A delineation of new classes of exponential dispersion models supported on the set of nonnegative integers

A delineation of new classes of exponential dispersion models supported on the set of nonnegative integers

Stable skyrmion bundles at room temperature and zero magnetic field in a chiral magnet

Topological spin textures are characterized by magnetic topological charges, Q, which govern their electromagnetic properties. Recent studies have achieved skyrmion bundles with arbitrary integer values of Q, opening possibilities for exploring topological spintronics based on Q. However, the realization of stable skyrmion bundles in chiral magnets at room temperature and zero magnetic field — the prerequisite for realistic device applications — has remained elusive. Here, through the combination of pulsed currents and reversed magnetic fields, we experimentally achieve skyrmion bundles with different integer Q values — reaching a maximum of 24 at above room temperature and zero magnetic field — in the chiral magnet Co8Zn10Mn2. We demonstrate the field-driven annihilation of high-Q bundles and present a phase diagram as a function of temperature and field. Our experimental findings are consistently corroborated by micromagnetic simulations, which reveal the nature of the skyrmion bundle as that of skyrmion tubes encircled by a fractional Hopfion.

A Short Proof that Rational Algebraic Integers are Integers

Summary We present a short proof that a rational algebraic integer is an integer.

Chaotic fuzzy iterated function systems

It is observed that IFSs are defined based on the concept that the iterates take only an integer number of times. This work studies the dynamics of functions, where a function can iterate r times for every r ∈ R. Utilizing concepts from fuzzy sets, r-times iterates of a function are defined for r ∈ R. The study demonstrates that the chaotic property can be generalized to this new iterative concept. The chaotic behavior of a function is then extended using this iterative concept.

Additive and geometric transversality of fractal sets in the integers

AbstractBy juxtaposing ideas from fractal geometry and dynamical systems, Furstenberg proposed a series of conjectures in the late 1960's that explore the relationship between digit expansions with respect to multiplicatively independent bases. In this work, we introduce and study — in the discrete context of the integers — analogs of some of the notions and results surrounding Furstenberg's work. In particular, we define a new class of fractal sets of integers that parallels the notion of ‐invariant sets on the 1‐torus and investigate the additive and geometric independence between two such fractal sets when they are structured with respect to multiplicatively independent bases. Our main results in this direction parallel the works of Furstenberg, Hochman–Shmerkin, Shmerkin, Wu, and Lindenstrauss–Meiri–Peres and include: a classification of all subsets of the positive integers that are simultaneously ‐ and ‐invariant; integer analogs of two of Furstenberg's transversality conjectures pertaining to the dimensions of the intersection and the sumset of ‐ and ‐invariant sets and when and are multiplicatively independent; and a description of the dimension of iterated sumsets for any ‐invariant set . We achieve these results by combining ideas from fractal geometry and ergodic theory to build a bridge between the continuous and discrete regimes. For the transversality results, we rely heavily on quantitative bounds on the ‐dimensions of projections of restricted digit Cantor measures obtained recently by Shmerkin. We end by outlining a number of open questions and directions regarding fractal subsets of the integers.

Expressive power of ReLU and step networks under floating-point operations

The study of the expressive power of neural networks has investigated the fundamental limits of neural networks. Most existing results assume real-valued inputs and parameters as well as exact operations during the evaluation of neural networks. However, neural networks are typically executed on computers that can only represent a tiny subset of the reals and apply inexact operations, i.e., most existing results do not apply to neural networks used in practice. In this work, we analyze the expressive power of neural networks under a more realistic setup: when we use floating-point numbers and operations as in practice. Our first set of results assumes floating-point operations where the significand of a float is represented by finite bits but its exponent can take any integer value. Under this setup, we show that neural networks using a binary threshold unit or ReLU can memorize any finite input/output pairs and can approximate any continuous function within an arbitrary error. In particular, the number of parameters in our constructions for universal approximation and memorization coincides with that in classical results assuming exact mathematical operations. We also show similar results on memorization and universal approximation when floating-point operations use finite bits for both significand and exponent; these results are applicable to many popular floating-point formats such as those defined in the IEEE 754 standard (e.g., 32-bit single-precision format) and bfloat16.

A Discrete Version of the Half-Logistic Distribution Based on the Mimicking of the Probability Density Function

We introduce a count distribution obtained as a discrete analogue of the continuous half-logistic distribution. It is derived by assigning to each non-negative integer value a probability proportional to the corresponding value of the density function of the parent model. The main features of this new distribution, in particular related to its shape, moments, and reliability properties, are described. Parameter estimation, which can be carried out resorting to different methods including maximum likelihood, is discussed, and a numerical comparison of their performances, based on Monte Carlo simulations, is presented. The applicability of the proposed distribution is proved on two real datasets, which have been already fitted by other well-established count distributions. In order to increase the flexibility of this counting model, a generalization is finally suggested, which is obtained by adding a shape parameter to the continuous one-parameter half-logistic and then applying the same discretization technique, based on the mimicking of the density function.

Practical multi-party private set intersection cardinality and intersection-sum protocols under arbitrary collusion1

Private set intersection cardinality (PSI-CA) and private intersection-sum with cardinality (PSI-CA-sum) are two primitives that enable data owners to learn the intersection cardinality of their data sets, with the difference that PSI-CA-sum additionally outputs the sum of the associated integer values of all the data that belongs to the intersection (i.e., intersection-sum). However, to the best of our knowledge, all existing multi-party PSI-CA (MPSI-CA) protocols are either limited by high computational cost or face security challenges under arbitrary collusion. As for multi-party PSI-CA-sum (MPSI-CA-sum), there is even no formalization for this notion at present, not to mention secure constructions for it. In this paper, we first present an efficient MPSI-CA protocol with two non-colluding parties. This protocol significantly decreases the number of parties involved in expensive interactive procedures, leading to a significant enhancement in runtime efficiency. Our numeric results demonstrate that the running time of this protocol is merely one-quarter of the time required by our proposed MPSI-CA protocol that is secure against arbitrary collusion. Therefore, in scenarios where performance is a priority, this protocol stands out as an excellent choice. Second, we successfully construct the first MPSI-CA protocol that achieves simultaneous practicality and security against arbitrary collusion. Additionally, we also conduct implementation to verify its practicality (while the previous results under arbitrary collusion only present theoretical analysis of performance, lacking real implementation). Numeric results show that by shifting the costly operations to an offline phase, the online computation can be completed in just 12.805 seconds, even in the dishonest majority setting, where 15 parties each hold a set of size 2 16 . Third, we formalize the concept of MPSI-CA-sum and present the first realization that ensures simultaneous practicality and security against arbitrary collusion. The computational complexity of this protocol is roughly twice that of our MPSI-CA protocol. Besides the main results, we introduce the concepts and efficient constructions of two novel building blocks: multi-party secret-shared shuffle and multi-party oblivious zero-sum check, which may be of independent interest.

An Inclusive Distance Irregularity Strength of n-ary Tree

An inclusive distance vertex irregular labelling of a simple graph G is a function of the vertex set of to positive integer set such that the sum of its vertex label and the labels of all vertices adjacent to the vertex are distinct. The minimum of maximum label of the vertices is said to be inclusive distance irregularity strength of G, denoted by dis(G). The purpose of this research is showing that dis(T_{n,2})= (n^2+2)/2 where T_{n,2} is a complete n-ary tree to level two.

Differentially Private Consensus for Second-Order Multiagent Systems With Quantized Communication.

This article considers the differentially private consensus problem of discrete-time second-order multiagent systems with partially measurable states and limited communication channel capacity, where only the integer-value information of agents can be transmitted. To reduce the potential risk of state information disclosure in digital communication, a differentially private consensus algorithm via dynamic encoding-decoding is proposed for the second-order multiagent system to make agents achieve mean-square consensus by transmitting quantized integer values with privacy protection. To deal with the uncertainty of the quantizer saturation, the statistical analysis is given for the boundedness of the input of quantizers. It is shown that the expectation of the minimum memory capacity of quantizers is 2 bits. Finally, some simulation results are given to visualize our conclusions.

Exploring the Relationship of Self-Compassion and Gratitude: A Multidimensional Analysis Among Sports Professionals

The aim of this research is to investigate the intricate connection between self-compassion and gratitude among athletes using a multidimensional approach. By delving into the dimensions of both self-compassion and gratitude, this study aims to uncover how these two psychological constructs interact within the context of sports. Through comprehensive analysis, we seek to understand how self-compassion and gratitude manifest in athletes' lives and how they may influence each other. This exploration is vital for enhancing our understanding of the psychological dynamics within sports environments, potentially offering insights into strategies for promoting well-being and performance among athletes. A descriptive research design was employed, with correlational and independent t-test analyses conducted to explore relationships and gender-based differences. Pre-therapy data collection involved assessing participants' baseline levels of negative traits, such as fear, anxiety, depression, and worry, using validated scales, questionnaires, and interviews. Data simulation involved generating random integer values within specified ranges to initialise pre-therapy scores for fear, anxiety, depression, and worry, thereby creating a diverse participant group.Results reveal no significant relationship between gratitude and self-compassion, nor any gender-based differences. However, cognitive-behavioural therapy significantly reduced fear, anxiety, depression, and worry among participants. The p-values of 0.0000 for all measures indicate highly significant differences between pre-therapy and post-therapy scores, affirming the therapy's effectiveness. This highlights the importance of cognitive-behavioural interventions in addressing psychological well-being. While this study provides insights into gratitude and self-compassion among volleyball players, further research is needed to explore interventions enhancing these psychological factors.

Trapping of deformable active particles by a periodic background potential.

The dynamic behaviors, specifically trapping and sorting, of active particles interacting with periodic substrates have garnered significant attention. This study investigates numerically the trapping of soft, deformable particles on a periodic potential substrate, which can be experimentally verified through optical tweezers. The research demonstrates that multiple factors, including the relative size of traps, self-propelled velocity, shape parameters, ratio of particles to traps, and translational diffusion, can influence the trapping effect. Within certain parameter boundaries, it is shown that all particles can be consistently trapped. The research reveals that stable trapping typically occurs at median values of the relative trap size. An increase in the self-propelled velocity, the shape parameter, and the translational diffusion coefficient tends to facilitate the escapement of the particles from the traps. It is noteworthy that particles with larger shape parameters can escape even when the restoring force exceeds the self-propelled force. In addition, as the ratio of particles to traps grows, the fraction of trapped particles steadily reduces. Notably, rigid particles are consistently divided and trapped by traps closely approximating an integer multiple of the particles' area, up until the ratio reaches the aforesaid integer value. These findings can potentially enhance the understanding of the interactive effects between active deformable particles and periodic substrates. Moreover, this work suggests a different experimental approach to sort active particles based on rigidity disparities.

Programming Language Selection for The Development of Deep Learning Library

Recently, deep learning has become very successful in various applications, leading to an increasing need for software tools to keep up with the rapid pace of innovation in deep learning research. As a result, we suggested the development of a software library related to deep learning that would be useful for researchers and practitioners in academia and industry for their research endeavors. The programming language is the core of deep learning library development, so this paper describes the selection stage to find the most suitable programming language for developing a deep learning library based on two criteria, including coverage on many projects and the ability to handle high-dimensional array processing. We addressed the comparison of programming languages with two approaches. First, we looked for the most demanding programming languages for AI Jobs by conducting a data-driven approach against the data gathered from several Job-Hunting Platforms. Then, we found the findings that imply Python, C++, and Java as the top three. After that, we compared the three most widely used programming languages by calculating interval time to three different programs that contain an array of exploitation processes. Based on the result of the experiments that were executed in the computer terminal, Java outperformed Python and C++ in two of the three experiments conducted with 5,4047 milliseconds faster than C++ and 231,1639 milliseconds faster than Python to run quick sort algorithm for arrays that contain 100.000 integer values.

Super (a,d)-P_2⨀P_m-Antimagic Total Labeling of Corona Product of Two Paths

Graph labeling involves mapping the elements of a graph (edges and vertices) to a set of positive integers. The concept of an anti-magic super outer labeling (a,d)-H pertains to assigning labels to the vertices and edges of a graph using natural numbers {1,2,3,...,p+q}. The weights of the outer labels H form an arithmetic sequence {a,a+d,a+2d,...,a+(k-1)d}, where 'a' represents the first term, 'd' is the common difference, and 'k' denotes the total number of outer labels, with the smallest label assigned to a vertex. This study investigates the super (a,d)-P_2⨀P_m-antimagic total labeling of the corona product P_n⨀P_m, where n and m are both greater than or equal to 3. We define the labeling functions for vertices and edges based on the partitioning of labels into three subsets. Using k-balanced and (k,δ)-anti balanced multisets, we demonstrate that for m being odd, P_n⨀P_m is super (9m^2 n+4mn+m-n+3,1)-P_2 ⨀▒P_(m ) -antimagic, and for m being even, P_n⨀P_m is super (9m^2 n+4mn+m-2n+5,3)-P_2 ⨀▒P_(m ) -antimagic. The labeling scheme is illustrated through examples. For the case when m is odd, an antimagic total labeling of P_3 ⨀▒P_3 forms a super (282,1)- P_2 ⨀▒P_(3 ) -antimagic labeling. In the case of even m, an antimagic total labeling of P_3 ⨀▒P_(4 ) results in a super (483,3)- P_2 ⨀▒P_(4 ) -antimagic labeling. Both of these examples provide insights into the antimagic properties of corona products.

On the Tower Factorization of Integers

Under the fundamental theorem of arithmetic, any integer n > 1 can be uniquely written as a product of prime powers pa ; factoring each exponent a as a product of prime powers qb , and so on, one will obtain what is called the tower factorization of n. Here, given an integer n > 1, we study its height h(n), that is, the number of “floors” in its tower factorization. In particular, given a fixed integer k ≥ 1 , we provide a formula for the density of the set of integers n with h(n) = k. This allows us to estimate the number of floors that a positive integer will have on average. We also show that there exist arbitrarily long sequences of consecutive integers with arbitrarily large heights.

Testability of positive integers

Given a set [Formula: see text] of some positive integers, a positive integer [Formula: see text] is [Formula: see text]-testable if there are two disjoint subsets [Formula: see text] and [Formula: see text] of [Formula: see text] such that [Formula: see text]. The testability of the set [Formula: see text] of positive integers not exceeding [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that there is a set [Formula: see text] with [Formula: see text] positive integers satisfying that each [Formula: see text] is [Formula: see text]-testable. In this short paper, we prove that [Formula: see text] and propose an algorithm to output the representation of an arbitrary positive integer by the form [Formula: see text], where [Formula: see text] and [Formula: see text] are two disjoint sets of some non-negative integers not exceeding [Formula: see text].

Digraphs of power maps over finite nilpotent groups

Given a finite group G and an integer t, let G(φt/G) be the directed graph with vertex set G and directed edges g→gt,g∈G. We prove that nilpotent groups are precisely the class of groups yielding a special symmetry on the graphs G(φt/G) that has appeared several times in the literature (considering functions over various algebraic structures). We then explore the question on when two finite nilpotent groups yield the same digraphs as t runs over Z: this defines an equivalence relation ∼φ. We completely describe the set of integers M in which the relation ∼φ is equivalent to the isomorphism of groups when one restricts to the set of nilpotent groups of order M. Our proofs rely on arguments from number theory and combinatorics.

Weak Mean Random Attractor of Reversible Selkov Lattice Systems Driven By Locally Lipschitz Lévy Noises

This paper is concerned with weak pullback mean random attractor of reversible Selkov lattice systems defined on the entire integer set \(\mathbb{Z}\) driven by locally Lipschitz Lévy noises. Firstly, we formulate the stochastic lattice equations to an abstract system defined in the non-concrete space \(\ell^2\times\ell^2\) of square-summable sequences. Secondly, we establish the global well-posedness of the systems with locally Lipschitz diffusion terms. Under certain conditions, we show that the long-time dynamics can be captured by a weakly compact and weakly attracting mean random attractor in the Bochner space \(L^2(\Omega,\ell^2\times\ell^2)\). To overcome the difficulty caused by the drift and diffusion terms, we adopt a stopping time technique to prove the convergence of solutions in probability. The mean random dynamical systems theory proposed by Wang (J. Differ. Equ., 31:2177-2204, 2019) is used to deal with the difficulty caused by the nonlinear noise.

Lattice Ordered G􀀀Semirings

Objectives: The main objective of this paper is to derive some of the results of lattice ordered semirings, distributive lattice, lattice ideals and morphisms. Methods: To establish the results, we use some conditions like commutativity, simple, multiplicative idempotent, additively idempotent, and finally, use the concept of lattice ideal in semirings. Findings: First we give some examples of lattice ordered semirings and then study some results regarding lattices, distributive lattices, commutative lattice ordered semirings and finally lattice ideals and morphisms. The unique feature of this study is that the concept of gamma is new for the study of lattices. Novelty: We consider a condition (c.f. Theorem 4.1.5) for an additively idempotent semiring due to which it becomes a distributive lattice ordered semiring. Again, in general, the sum of ideals of a semiring need not be ideal. Indeed, and are ideals of is a set of non-negative integers. Clearly, (say) is not a ideal, because , but . However, this condition does not hold in the case of a lattice ordered semiring. AMS Mathematics subject classification (2020): 16Y60. Keywords: Lattices, additive idempotent, multiplicative Γ-idempotent, k-ideal, lattice ideal, Γ-morphism