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Number Theory - Science topic

The study of properties of integers and prime numbers.
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Publications related to Number Theory (10,000)
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Authentication plays a critical role in the security of quantum key distribution (QKD) protocols. We propose using Polynomial Hash and its variants for authentication of variable length messages in QKD protocols. Since universal hashing is used not only for authentication in QKD but also in other steps in QKD like error correction and privacy ampli...
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We prove that the reciprocal sum Sk(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_k(x)$$\end{document} of the least common multiple of k≥3\documentclass[12pt]{min...
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In advancing upon the Temporal Mechanics zero-dimensional number theory, here is presented the description of the zero-dimensional number theory as scaled with the charge of the electron and the speed of light. The two key equations for timespace as the Fibonacci equation for time and Euler's equation for space are then given physical relevance by...
Preprint
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In this paper, we analyze several equations concerning Stable Solutions for Boundary Reactions and "Inflation after Planck". We obtain new possible mathematical connections with some sectors of Number Theory and String Theory.
Preprint
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In this paper (Part III), we analyze some equations concerning the mathematics of String Theory and Broken Supersymmetry.We describe the new possible connections with some sectors of Number Theory and MRB Constant
Preprint
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We present and discuss the many results obtained concerning a famous limit theorem, the local limit theorem, which has many interfaces, with Number Theory notably, and for which, in spite of considerable efforts, the question concerning conditions of validity of the local limit theorem, has up to now no satisfactory solution. These results mostly c...
Article
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Pseudorandom sequences, sometimes shortened as sequences, have played a key role in the applications of digital communications, cryptography and computer science. This research field is an example of scientific research directly born from the real world applications. Specifically, the research on sequences stems from the application of the sequence...
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In recent work, Miezaki introduced the notion of a sphericalT-design in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^2$$\end{document}, where T is a po...
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In this article, we prove that a general version of Alladi’s formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom A or Axiom A#\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\...
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In this paper, we study the problem on the sum of four rational numbers with a fixed product. We will show that when d in an odd positive integer, the equation xy+dyz+zw+dwx=4dn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgr...
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This design-based study addresses the issue of how to digitally support students’ problem-solving by providing heuristics, in the absence of the teacher. The problem is that, so far, digital tutoring systems lack the ability to diagnose students’ needs in open problem situations. Our approach is based on students’ ability to self-diagnose and find...
Article
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Setting out from König & Smith [ Acta Cryst. (2019), B 75 , 788–802; Acta Cryst. (2021), B 77 , 861], we present an analytic description of nominal wurtzite-structure nanowire (NWire) cross sections, focusing on the underlying geometric–crystallographic description and on the associated number theory. For NWires with diameter d Wire [ i ], we predi...
Article
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We provide examples of multiplicative functions f supported on the k-free integers such that at primes f(p)=±1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(p)=\pm 1...
Preprint
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This article deals with two questions. 1. The ideas related to topological quantum computation suggests that it might make sense to replace quantum states with representations of the Galois group or even the coefficient space of state space with a quantum analog of a number field with tensor product and direct sum replacing the multiplication and s...
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Let $\chi$ be a primitive character modulo a prime $q$, and let $\delta > 0$. It has previously been observed that if $\chi$ has large order $d \geq d_0(\delta)$ then $\chi(n) \neq 1$ for some $n \leq q^{\delta}$, in analogy with Vinogradov's conjecture on quadratic non-residues. We give a new and simple proof of this fact. We show, furthermore, th...
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In this paper, we study generalized Kobayashi-Stieltjes type operators and obtain some of their relations with other known operators for example , Kobayashi's integral operator, Borel-Leroy operators and others. The results obtained by these operators are in terms of Voigt functions, Hurwitz zeta functions and hypergeometric functions which are use...
Preprint
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The objective of this paper is to present some open problems about the n-cyclic refined neutrosophic rings. These questions concern the n-cyclic refined neutrosophic rings, n-cyclic refined neutrosophic number theory, and n-cyclic refined neutrosophic analysis. Main Discussion Open Problem 1: Find an algorithm to solve a linear n-cyclic refined neu...
Conference Paper
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Providing high-order thinking skills is one of the main goals in the 21st-century learning concept which aims to form toughness and high-quality thinking skills in analyzing, evaluating, and looking for alternative solutions to the problems at hand. One of the efforts that can be made to improve teacher competence in the 21st-century learning conce...
Conference Paper
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Carl F. Gauss once stated that "number theory is the queen of mathematics." However, C. Goldbach's conjecture queried 280 years ago has set a persistent challenge to the exploration of the foundations of mathematics in general and number theory in particular in order to explain the ultimate essences of human abstract reasoning and inference. This i...
Article
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This is a review of Mederos, B.; Pérez-Cabrera, I.; Takane, M.; Tapia Sánchez, G.; Zavala, B. A method to construct all the paving matroids over a finite set. (English) £ ¢ ¡ Zbl 07557831 Paving matroids were defined in [J. Hartmanis, Can. J. Math. 11, 97-106 (1959; Zbl 0089.37002)] through the concept of d-partitions in number theory. Paving mat...
Article
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In this paper, we provide an overall perspective on the teaching and learning of discrete mathematics. Our aim is to highlight what research has been conducted in this area and to connect it to existing research ideas for future work. We begin by characterizing discrete mathematics and its role in the school curriculum, highlighting themes, topics,...
Article
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This work presents a formal proof of Goldbach conjecture based on a novel theory of Mirror-Prime Decomposition (MPD) for arbitrary even integers. A new concept of mirror primes is introduced as a set of pairs of primes that are symmetrically adjacent to any pivotal even number n_e on both sides in finite distance k bounded by 1 <= k <= (n_e/2) − 2....
Preprint
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We discuss a phenomenon where Optimal Transport leads to a remarkable amount of combinatorial regularity. Consider infinite sequences $(x_k)_{k=1}^{\infty}$ in $[0,1]$ constructed in a greedy manner: given $x_1, \dots, x_n$, the new point $x_{n+1}$ is chosen so as to minimize the Wasserstein distance $W_2$ between the empirical measure of the $n+1$...
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Introduction: A twin prime is a prime number that is either 2 less or 2 more than another prime number (e.g., the twin prime pair (3, 5)). The question of whether there exists infinitely many twin primes has been one of the great open questions in number theory for many years. The objective of this work was to investigate / characterize the statist...
Book
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This book brings together the impact of Prof. John Horton Conway, the playful and legendary mathematician's wide range of contributions in science which includes research areas—Game of Life in cellular automata, theory of finite groups, knot theory, number theory, combinatorial game theory, and coding theory. It contains transcripts where some emin...
Preprint
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Examined here is a proposed zero-dimensional number theory as the process of labelling zero-dimensional space as a point and zero-dimensional time as a moment as the different mathematical values of 0 and 1 respectively. By such it can be shown how zero-dimensional time in being mathematically labelled as a unit can form relationships between zero-...
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This is an expository article that concerns the various related notions of algebraic idele class characters, the Groessencharaktere of Hecke, and cohomological automorphic representations of GL(1), all under the general title of algebraic Hecke characters. The first part of the article systematically lays the foundations of algebraic Hecke characte...
Article
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The bare rudiments of the principle of mathematical induction as a method of proof date back to ancient times. In the contemporary university milieu, the demonstrative scheme is taught as part of a course in discrete mathematics, set theory, number theory, graph theory, group theory, game theory, linear algebra, logic, and combinatorics. In theoret...
Presentation
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Presentation at Number Theory Conference in Honour of Kalman Gyory, Janos Pintz and Andras Sarkozy, Debrecen, 4.-8.7.2022.
Preprint
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Let $(u_n)_{n \ge 0}$ be a nondegenerate Lucas sequence and $g_u(n)$ be the arithmetic function defined by $\gcd(n, u_n).$ Recent studies have investigated the distributional characteristics of $g_u$. Numerous results have been proven based on the two extreme values $1$ and $n$ of $g_{u}(n)$. Sanna investigated the average behaviour of $g_{u}$ and...
Book
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This monograph presents the proofs of 4 important conjectures in the field of number theory: -The Beal's conjecture. -The Riemann Hypothesis. -The c < R 1.63 conjecture. -The abc conjecture is true. We give in detail all the proofs. Résumé. Cette monographie présente les preuves de 4 conjectures importantes dans le domaine de la théorie des nomb...
Preprint
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Decoupling is a recent development in Fourier analysis, which has applications in harmonic analysis, PDE, and number theory. We survey some applications of decoupling and some of the ideas in the proof. This survey is aimed at a general mathematical audience. It is based on my 2022 ICM talk.
Article
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Motivated by an interdisciplinary question of whether a resource-storing mechanism favors “the wealthy do fight” or “the wealthy do not fight,” we establish a new model based on spatial prisoner's dilemma (SPD) game where a time-accumulating payoff is allowed, and the probability of game participation depending on wealthiness is introduced. Althoug...
Article
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In the paper, the authors collect, discuss, and find out several connections, equivalences, closed-form formulas, and combinatorial identities concerning partial Bell polynomials, falling factorials, rising factorials, extended binomial coefficients, and the Stirling numbers of the first and second kinds. These results are new, interesting, importa...
Article
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The objective of this paper is to present 40 open problems about the n-cyclic refined neutrosophic rings. These questions concern the n-cyclic refined neutrosophic rings, n-cyclic refined neutrosophic number theory, and n-cyclic refined neutrosophic analysis. They will represent the future of the study of neutrosophic n-cyclic refined rings and the...
Preprint
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Given a finite abelian group $G$ and a subset $J\subset G$ with $0\in J$, let $D_{G}(J,N)$ be the maximum size of $A\subset G^{N}$ such that the difference set $A-A$ and $J^{N}$ have no non-trivial intersection. Recently, this extremal problem has been widely studied for different groups $G$ and subsets $J$. In this paper, we generalize and improve...
Research
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This Special Issue, "Advances in Linear Recurrence System", welcomes submissions from a broad interdisciplinary area. Typical interdisciplinary uses of recurrence relations are to describe the kinetics of physical, chemical, and biological processes. In biology, some of the best-known difference equations originated from the attempt to model popula...
Article
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In this paper, we have presented a brief review of arithmetic functions that appear in introductory number theory. First, we have discussed Modius function and Euler's totient function then we have provided a brief review on Mnagoldt function. Furthermore, theorems relating several arithmetic functions are also derived.
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In this paper, we address the case of a particular class of function referred to as the rational equivariant functions. We investigate which elliptic zeta functions arising from integrals of power of ℘, where ℘ is the Weierstrass ℘-function attached to a rank two lattice of C, yield rational equivariant functions. Our concern in this survey is to p...
Article
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This work expects to solve the problem that the traditional corporate bond model for default risk assessment has low accuracy and poor data collection and storage robustness. Firstly, this work uses the mature Kealhofer, McQuown, and Vasicek (KMV) model to evaluate the default risk of corporate bonds. Secondly, Blockchain (BC) technology's Informat...
Article
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This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin's generic absoluteness results for second order number theory and the theory of universally Baire set...
Poster
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The objective of ICEPAM-2022 is to provide an international platform for scientists, researchers and educators to present and discuss the most recent innovations, trends and applications related to various branches of Mathematics like Algebra, Analysis, Number Theory, Numerical Analysis, Optimization etc. The aim of the conference is to provide a k...
Article
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Mobile medicine plays a significant role in optimizing medical resource allocation, improving medical efficiency, etc. Identifying and analyzing user concern elements from active online reviews can help to improve service quality and enhance product competitiveness in a targeted manner. Based on the latent Dirichlet allocation (LDA) topic model, th...
Preprint
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Algebraic K-theory has applications in a broad range of mathematical subjects, from number theory to functional analysis. It is also fiendishly hard to calculate. Presently there are two main inroads: motivic and cyclic homology. I've been asked to present an overview of the applications of topological cyclic homology to algebraic K-theory "from a...
Preprint
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The most important open question in the theory of Hadamard matrices is that of existence (the Hadamard conjecture). The generalization of Sylvester's construction proves that if H_n and H_m are Hadamard matrices of orders n and m, respectively, then H_n x H_m given by the Kronecker product is a Hadamard matrix of order nm. This result is used to pr...
Article
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In this paper, we demonstrate the Collatz conjecture using the mathematical complete induction method. We show that this conjecture is satisfied for the first values of natural numbers, and in analyzing the sequence generated by odd numbers, we can deduce a formula for the general term of the Collatz sequence for any odd natural number n after seve...
Article
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Balancing the project’s time, cost, and quality involves deciding on different implementation methods for each project activity. Time and cost are minimized simultaneously, and the quality of the final product or service is maximized. Indeed, the use of search methods to determine the optimal execution methods of each activity requires the evaluati...
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We characterize the bipartite graphs that minimize the (first-degree based) entropy, among all bipartite graphs of given size, or given size and (upper bound on the) order. The extremal graphs turn out to be complete bipartite graphs, or nearly complete bipartite. Here we make use of an equivalent representation of bipartite graphs by means of Youn...
Article
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We investigate a special sequence of random variables A(N) defined by an exponential power series with independent standard complex Gaussians (X(k))k≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidema...
Article
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We study a process of generating random positive integer weight sequences \(\{ W_n \}\) where the gaps between the weights \(\{ X_n = W_n - W_{n-1} \}\) are i.i.d. positive integer-valued random variables. The main result of the paper is that if the gap distribution has a moment generating function with large enough radius of convergence, then the...
Article
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In this paper we give a full description of the inequalities that can occur between overpartition ranks modulo $$ c\ge 2. $$ c ≥ 2 . If $$ \overline{N}(a,c,n) $$ N ¯ ( a , c , n ) denotes the number of overpartitions of n with rank congruent to a modulo c , we prove that for any $$ c\ge 7 $$ c ≥ 7 and $$ 0\le a<b\le \left\lfloor \frac{c}{2}\right\r...
Article
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The partition function p[1cℓd](n) can be defined using the generating function ∑n=0∞p[1cℓd](n)qn=∏n=1∞1(1-qn)c(1-qℓn)d.In Mestrige (Res Number Theory 6(1), Paper No. 5, 2020), we proved an infinite family of congruences for this partition function for ℓ=11. In this paper, we extend the ideas that we have used in Mestrige (2020) to prove infinite fa...
Article
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COVID-19 has shown that quarantine (or self-isolation) may be the only available tool against an unknown infectious disease if neither an effective vaccine nor anti-viral medication is available. Motivated by the fact that a considerable number of people were not compliant with the request for self-quarantine made by public authorities, this study...
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Over the years, the study of Bailey transform, Bailey lemma, Bailey pair, their variants and their applications are the major subjects of interest. Of course, it is due to the efficiency of the Bailey transform and lemma in producing many ordinary and q-hypergeometric identities, multiple series summation and transformation formulas, and the Rogers...
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When vertical columns of water are connected, the water reaches the same level in each column. This is traditionally explained using properties and laws of fluid pressure. In an unusual approach here, we ignore the idea of pressure, and instead bring together mathematical results from calculus, statistics, number theory and logic to derive the same...
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Guztwiller's Trace Formula is central to the semiclassical theory of quantum energy levels and spectral statistics in classically chaotic systems. Motivated by recent developments in Random Matrix Theory and Number Theory, we elucidate a hierarchical structure in the way periodic orbits contribute to the Trace Formula that has implications for the...
Article
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We prove a global local rigidity result for character varieties of 3-manifolds into $\rm{SL}_2$. Given a 3-manifold with toric boundary $M$ satisfying some technical hypotheses, we prove that all but a finite number of its Dehn fillings $M_{p/q}$ are globally locally rigid in the following sense: every irreducible representation $\rho:\pi_1(M_{p/q}...
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In this article the connection of quantum gravitation, as it is understood in the TGD framework, with topological quantum computation (TQC) is considered. I sketched the first TGD based vision about DNA as a TQCer for about 13 years ago. In particular, a model of the system consisting of DNA and nuclear/cell membrane system acting as a TQCer was di...
Conference Paper
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The main motivation of the conference is to foster collaboration among researchers from South Eastern Europe who work on algebra, algebraic geometry, number theory, cryptography, computational algebra, and related areas. The conference will be combined with online talks and face to face talk to include as large audience and participation as possibl...
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In this article we solve one of the oldest and celebrated problems in number theory, namely the existence or nonexistence of odd perfect numbers. We know there be no number of this type having less than 100 digits. A number is said to be perfect if it is the sum of its proper divisors. Euclid in his The Elements ninth book gives a formula for all e...
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In this paper (part III), we analyze various Equations and Lagrangians. We describe the new possible mathematical connections with π, ζ(2), 4096, 1729, MRB Constant, some sectors of Number Theory and String Theory
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In this paper (part II), we analyze various Equations and Lagrangians. We describe the new possible mathematical connections with π, ζ(2), 4096, 1729, MRB Constant, some sectors of Number Theory and String Theory
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Recently, Rattan and the first author (Ann. Comb. 25 (2021) 697-728) proved a conjectured inequality of Berkovich and Uncu (Ann. Comb. 23 (2019) 263-284) concerning partitions with an impermissible part. In this article, we generalize this inequality upon considering t impermissible parts. We compare these with partitions whose certain parts appear...
Article
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Background Despite the great concern triggered by the environmental crisis worldwide, the loss of temporal key functions and processes involved in biodiversity maintenance has received little attention. Species are restricted in their life cycles by environmental variables because of their physiological and behavioral properties; thus, the timing a...
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We introduce the adjoint homological Selmer module for an SL$_2$-representation of a knot group, which may be seen as a knot theoretic analogue of the dual adjoint Selmer module for a Galois representation. We then show finitely generated torsion-ness of our adjoint Selmer module, which are widely known as conjectures in number theory, and give som...
Article
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Digital signatures are unsuitable for specific applications that are sensitive on a personal or commercial level because they are universally verifiable. Jakobsson et al. proposed the Designated Verifier Signature (DVS) system, which only allows the intended verifier to validate a message’s signature. It prohibits the disclosure of a conviction to...
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In this paper, we analyze various Lagrangians. We describe the new possible mathematical connections with π, ζ(2), 4096, 1729 and MRB Constant, some sectors of Number Theory and String Theory 11 M.Nardelli studied at
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Applications of Rasiowa-Sikorski Lemma in arithmetic (I). The twin primes conjecture is true in the standard model of Peano arithmetic Abstract. The paper is concerned with the old conjecture that there are infinitely many twin primes. In the paper we show that this conjecture is true, that is, it is true in the standard model of arithmetic. The pr...
Article
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Adopting the Mahler measure from number theory, we introduce it to toric quiver gauge theories, and study some of its salient features and physical implications. We propose that the Mahler measure is a universal measure for the quiver, encoding its dynamics with the monotonic behaviour along a so-called Mahler flow including two special points at i...
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In statistical mechanics one packages the possible energies of a system into a partition function. In number theory, and elsewhere in mathematics, one packages the spectrum of a phenomenon, say the prime numbers, into a $\zeta$-function or more generally into an L-function. These packaging functions have symmetries and properties not at all apparen...
Book
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In order to place this book in a modern and dynamic context, we have introduced where possible and throughout the text, a number of algorithms and examples worked with the Mathematica® programming language, to efficiently solve some problems. of number theory. This program has a Wolfram Cloud online interface where it is possible to run the command...
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Linear algebra, Number Theory, Series, Arithmetic Series, Education, High school.
Article
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Статья содержит научную биографию Иоганна Альбрехта Эйлера (1734--1800), историю рукописи «История геометрии», её публикацию и комментарии. И.А. Эйлер, старший сын Леонарда Эйлера, родился в Санкт-Петербурге, юность и молодость провёл с отцом в Берлине, где служил инспектором Берлинской обсерватории; в 32-летнем возрасте вернулся с семьёй в Петербу...
Article
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In this part of a series of two papers, we extend the theorems discussed in Part I for infinite series. We then use these theorems to develop distinct novel results involving the Hurwitz zeta function, Riemann zeta function, polylogarithms and Fibonacci numbers. In continuation, we obtain some zeros of the newly developed zeta functions and explain...
Article
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In the paper, the author presents two expansions, complete monotonicity, minimality, some determinantal inequalities, and product inequalities of the Catalan numbers and other sequences involving the Catalan numbers in combinatorics and number theory and verifies the equivalence of two integral representations for the Catalan numbers.
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In this paper, we first give a simple combinatorial proof of Tepper's identity. Then, as a by product of this interesting identity we present another proof of the well-known Wilson's identity in number theory. Finally, we obtain a generalization of Tepper's identity for any polynomial with real coefficients.
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This work is a continuation of the author's series of works on the use of artificial intelligence systems for research in the field of number theory (higher arithmetic) and statistics. The paper solves the problem of studying the dependence of the level of consistency of natural numbers, as systems of prime factors, on the magnitude of the number,...
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This work is a continuation of the author's series of works on the use of artificial intelligence systems for research in the field of number theory (higher arithmetic) and statistics. The paper solves the problem of studying the dependence of the level of consistency of natural numbers, as systems of prime factors, on the magnitude of the number,...
Preprint
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In this paper, we analyze various equations concerning the MRB Constant and QFTs-Coleman-de Luccia Instantons. We obtain new possible mathematical connections with some sectors of Number Theory and String Theory
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The algorithm of Shor for prime factorization is a hybrid algorithm consisting of a quantum part and a classical part. The main focus of the classical part is a continued fraction analysis. The presentation of this is often short, pointing to text books on number theory. In this contribution, we present the relevant results and proofs from the theo...
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Motivated by the Langlands program in representation theory, number theory and geometry, the theory of representations of a reductive $p$-adic group over a coefficient ring different from the field of complex numbers has been widely developped during the last two decades. This article provides a survey of basic results obtained in the 21st century.
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In this paper (Part III), we analyze some equations concerning the mathematics of String Theory and Broken Supersymmetry. We describe the new possible connections with some sectors of Number Theory and MRB Constant
Preprint
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In this paper, we analyze some equations concerning the mathematics of String Theory. New possible connections with some sectors of Number Theory and MRB Constant 11 M.Nardelli studied at