Odd Positive Integer Research Articles

Algorithms for representing positive odd integers as the sum of arithmetic progressions

This paper delves into the historical and recent developments in this area of mathematical inquiry, tracing the evolution from Wheatstone’s representation of powers of an integer as sums of arithmetic progressions to extensions of Sylvester’s Theorem (Sylvester and Franklin, [14]). Sylvester’s Theorem, a result that determines the representability of positive integers as sums of consecutive integers, has been the foundation for numerous extensions, including the representation of integers as sums of specific arithmetic progressions and powers of such progressions. The recent works of Ho et al. [3] and Ho et al. [4] have further expanded on Sylvester’s Theorem, offering a procedural approach to compute the representability of positive integers in the context of arithmetic progressions. In this paper, efficient algorithms to compute the number of ways to represent an odd positive integer as sums of powers of arithmetic progressions are presented.

A generalization of a theorem of Brass and Schmeisser

Let n be an odd positive integer. It was proved by Brass and Schmeisser that for every quadrature Q=α1f(x1)+⋯+αmf(xm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Q}=\\alpha _1f(x_1)+\\dots +\\alpha _mf(x_m)$$\\end{document}(with positive weights) of order at least n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} and for every n-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-$$\\end{document}convex function f, the value of Q on f lies between the values of Gauss-Legendre and Gauss-Lobatto quadratures of order n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} calculated for the same function f. We generalize this result in two directions, replacing Q by an integral with respect to a given measure and allowing the number n to any positive integer (for even n Gauss-Radau quadratures replace Gauss-Legendre and Gauss-Lobatto ones).

On integers of the form p + 2a2 + 2b2

In this paper, it is proved that there is a positive proportion of positive odd integers which can be uniquely represented as [Formula: see text], where [Formula: see text] is a prime and [Formula: see text] are two positive integers with [Formula: see text].

Analysis of Collatz Conjecture Rules

Abstract A proof of the Collatz Conjecture is presented. Changing the perspective of the problem from looking at the pattern of the positive integers to looking at the conjecture rules made the proof possible. The conjecture rule for even numbers organizes all positive integers into unique sets and the rule for odd numbers interconnects the unique sets into dendritic pathways to “1.” Infinite loops, other than the minor 4-2-1 loop after reaching “1,” and values continually increasing to infinity are shown to be mathematically impossible. The proof predicted a general equation that shows all positive integers reach a final value of “1," and calculates the values and locations of the odd positive integers during the iterations for each tested positive integer. JEL classification numbers: CO2; C65. Keywords: Collatz Conjecture, Proof, Rules, Dendritic pathway.

Dynamics of some compact structures and moment of inertia in f(ℛ,𝒯 ) gravity

The primary objective of this study is to conduct a thorough examination of relativistic structures within the framework of [Formula: see text] gravity, where [Formula: see text] is the Ricci scalar and [Formula: see text] is the trace of the energy–momentum tensor. We investigate the emerging characteristics of compact stars, focusing on the [Formula: see text] model, where [Formula: see text] is the coupling constant. In the initial sections of the paper, we concentrate on the examination of exact interior solutions involving perfect fluid. With this aim, we introduce the metric coefficient [Formula: see text], where [Formula: see text] and [Formula: see text] are arbitrary constants to be computed from the matching conditions and [Formula: see text] is a positive odd integer. It is worth noticing that we select a group of 10 significant compact stars from recent studies, namely, Vela [Formula: see text], EXO [Formula: see text], [Formula: see text], SMC[Formula: see text], Her[Formula: see text], PSR [Formula: see text], PSR[Formula: see text], Cen[Formula: see text], [Formula: see text] and LMC[Formula: see text]. To investigate the viability of our model, we conduct various checks, including the analysis of energy density and pressure components, equilibrium, and energy conditions, stability analysis and the adiabatic index. An important part of this work provides the slowly rotating property of stars, accompanied by the derivation of the moment of inertia. The relationship of mass, energy density, radius and moment of inertia for stellar objects are observed. Our findings demonstrate that obtained results are physically acceptable and consistent with our considered model.

Oscillatory Properties of Second Order Half-Linear Delay Difference Equations

This study explores, some necessary and sufficient conditions that are established for oscillatory properties of second order half-linear delay difference equations of the form Δ(p(ξ)(Δx(ξ))^r )+q(ξ)x^s (σ(ξ))=0, for ξ≥ξ_0 . Under the assumption ∑_(t=ξ_0)^(ξ-1) 1/(p^(1/r) (t))=∞. Two cases are considered for r<s and r>s, where r and s are the quotients of two positive odd integers. The effectiveness and applicability of the result are illustrated through few examples.

Gray Images of Constacyclic Codes over a Non-chain Extension of Z 4

Let [Formula: see text] be the ring of integers modulo 4. We study the [Formula: see text]-constacyclic and [Formula: see text]-cyclic codes over the non-chain ring [Formula: see text] for a unit [Formula: see text] in [Formula: see text]. We define several Gray maps and find that the respective Gray images of a quasi-cyclic code over [Formula: see text] are cyclic, quasi-cyclic or permutation equivalent to this code. For an odd positive integer [Formula: see text], we determine the generator polynomials of cyclic and [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text]. Further, we prove that a [Formula: see text]-cyclic code of length [Formula: see text] is a [Formula: see text]-constacyclic code if [Formula: see text] is odd, and a [Formula: see text]-quasi-twisted code if [Formula: see text] is even. A few examples are also incorporated, in which two parameters are new and one is best known to date.

On Hopf algebras whose coradical is a cocentral abelian cleft extension

This paper is a first step toward the full description of a family of Hopf algebras whose coradical is isomorphic to a semisimple Hopf algebra Kn , n an odd positive integer, obtained by a cocentral abelian cleft extension. We describe the simple Yetter-Drinfeld modules, compute the fusion rules and determine the finite-dimensional Nichols algebras for some of them. In particular, we give the description of the finite-dimensional Nichols algebras over simple modules over K 3. This includes a family of 12-dimensional Nichols algebras { B ξ } depending on 3rd roots of unity. Here, B 1 is isomorphic to the well-known Fomin-Kirillov algebra, and B ξ ≃ B ξ 2 as graded algebras but B 1 is not isomorphic to B ξ as algebra for ξ ≠ 1 . As a byproduct we obtain new Hopf algebras of dimension 216.

On integers of the form [formula omitted

Let r1,…,rt be positive integers and let R2(r1,…,rt) be the set of positive odd integers that can be represented as p+2k1r1+⋯+2ktrt, where p is a prime and k1,…,kt are positive integers. It is easy to see that if r1−1+⋯+rt−1<1, then the set R2(r1,…,rt) has asymptotic density zero. In this paper, we prove that if r1−1+⋯+rt−1≥1, then the set R2(r1,…,rt) has a positive lower asymptotic density. Several conjectures and open problems are posed for further research.

On the hulls of cyclic codes of oddly even length over [formula omitted

In this paper, we study hulls of cyclic codes of length 2n (i.e., oddly even length) over the ring Z4 of integers modulo 4 by viewing these codes as ideals of the quotient ring Z4[x]/<x2n−1>, where n is an odd positive integer. We express the generator polynomials of the hull of each cyclic code C of length 2n over Z4 in terms of the generator polynomials of the code C, and we note that the 2-dimension of the hull of the code C is an integer Θ satisfying 0≤Θ≤2n. We also obtain an enumeration formula for cyclic codes of length 2n over Z4 with hulls of a given 2-dimension. Besides this, we study the average 2-dimension, denoted by E(2n), of the hulls of cyclic codes of length 2n over Z4 and establish a formula for E(2n) that handles well. With the help of this formula, we deduce that E(2n)=5n6 when n∈N2, and we study the growth rate of E(2n) with respect to the length 2n when n∉N2, where N2 denotes the set of all positive integers ω such that w divides 2i+1 for some positive integer i. Further, when n∉N2, we derive lower and upper bounds on E(2n) and show that these bounds are attained at n=7. We also illustrate these results with some examples. As an application of these results, we construct some entanglement-assisted quantum error-correcting codes (EAQECCs) over Z4 with specific parameters from cyclic codes of oddly even length over Z4.

The 2-Class Group of Certain Families of Imaginary Triquadratic Fields

ABSTRACT Our goal in this paper is to compute the 2-rank of the class group of the fields K = ℚ ( − 2 , q , d ) \[\mathbb{K}=\mathbb{Q}\left( \sqrt{-2},\sqrt{q},\sqrt{d} \right)\] for any odd positive squarefree integer d and any prime integer q ≡ 5 (mod 8). Furthermore, we give the list of these fields whose 2-class groups are trivial, cyclic, of type (2, 2), (2, 4) or (2, 2, 2).

Event-Based Fuzzy Adaptive Consensus Tracking for Stochastic High-Order Nonlinear Multiagent Networks With Specified-Time Convergence

In this paper, a specified-time event-triggered fuzzy adaptive control algorithm is developed to solve the consensus tracking control problem for stochastic high-order nonlinear multi-agent networks, which is intrinsically challenging due to the existence of stochasticity and high-order (positive odd integers greater than one) terms. More precisely, a novel specified-time performance function (STPF) is incorporated into time-varying high-order tan-type barrier Lyapunov function to guarantee that the tracking errors remain under time-varying constraints within specified time. Combining fuzzy logic systems with the adding on power integrator technique, an adaptive approximation policy is introduced to handle the system uncertainties. Moreover, a new switching threshold event-triggered mechanism is devised to determine the control signals updating instants, which reduces the transmission and computation burden, while resizing the triggering threshold in real-time. The Zeno phenomenon is excluded by guaranteeing that the triggering intervals is lower bounded by a positive constant. Two simulation examples are provided to demonstrate the effectiveness of the designed algorithm.

The inverse Mellin transform via analytic continuation

We present a method to calculate the x-space expressions of massless or massive operator matrix elements in QCD and QED containing local composite operator insertions, depending on the discrete Mellin index N, directly, without computing the Mellin-space expressions in explicit form analytically. Here N belongs either to the even or odd positive integers. The method is based on the resummation of the operators into effective propagators and relies on an analytic continuation between two continuous variables. We apply it to iterated integrals as well as to the more general case of iterated non-iterative integrals, generalizing the former ones. The x-space expressions are needed to derive the small-x behaviour of the respective quantities, which usually cannot be accessed in N-space. We illustrate the method for different (iterated) alphabets, including non-iterative 2F1 and elliptic structures, as examples. These structures occur in different massless and massive three-loop calculations. Likewise the method applies even to the analytic closed form solutions of more general cases of differential equations which do not factorize into first-order factors.

Decentralized adaptive finite‐time fault‐tolerant control for a class of high‐power interconnected nonlinear systems via graph theory

SummaryThis article investigates the decentralized adaptive finite‐time fault‐tolerant control (FTC) for unknown strong interconnected nonlinear systems with high‐powers and actuator faults. The powers are the ratio of positive odd rational integers and may be different, which makes the presented scheme different from existing work on interconnected nonlinear systems. The considered interconnections are governed by some boundary functions that grow nonlinearly in the states of all subsystems. First, the difficulty brought by high powers to controller design is overcome by introducing the adding one power integrator method. Then, based on backstepping method and FTC theory, a decentralized adaptive algorithm is proposed without approximate structure (neural network, fuzzy logic, etc.), which adaptively compensates the impact of actuator faults. Besides, with the aid of algebraic graph theory, the system is stable under practical finite‐time concept. Finally, the effectiveness of the theoretical findings are validated by conducting two simulation examples.

Characterizing an odd [1, b]-factor on the distance signless Laplacian spectral radius

Let G be a connected graph of even order n. An odd [1, b]-factor of G is a spanning subgraph F of G such that dF(v) ∈ {1, 3, 5, ⋯, b} for any v ∈ V(G), where b is positive odd integer. The distance matrix Ɗ(G) of G is a symmetric real matrix with (i, j)-entry being the distance between the vertices vi and vj. The distance signless Laplacian matrix Q(G) of G is defined by Q(G), where Tr(G) is the diagonal matrix of the vertex transmissions in G. The largest eigenvalue η1(G) of Q(G) is called the distance signless Laplacian spectral radius of G. In this paper, we verify sharp upper bounds on the distance signless Laplacian spectral radius to guarantee the existence of an odd [1, b]-factor in a graph; we provide some graphs to show that the bounds are optimal.

Neural-adaptive specified-time constrained consensus tracking control of high-order nonlinear multi-agent systems with unknown control directions and actuator faults

In this paper, a specified-time neural-adaptive control algorithm is developed to solve the consensus tracking control problem of high-order nonlinear multi-agent systems with full-state constraints, unknown control directions, and actuator faults, which is intrinsically challenging due to the existence of high-order (positive odd integers greater than one) terms. More precisely, a novel specified-time performance function (STPF) is skillfully incorporated into the time-varying high-order log-type barrier Lyapunov function (BLF) to guarantee that the tracking errors remain under time-varying constraints within specified time. To handle the mixed unknown control directions, the hybrid Nussbaum function is innovatively employed in the distributed high-order nonlinear multi-agent scenario. By combining radial basis function neural networks (RBFNNs) with the adding-one-power-integrator technique, an adaptive approximation policy is introduced to derive the neural adaptive controllers. Moreover, actuator faults are also considered in this work. The variable-separable lemma is exploited to extract the fault signals in a “linear-like” manner. Comparative simulations are provided to demonstrate the effectiveness of the designed control framework.

Congruence relations for r-colored partitions

Let ℓ≥5 be prime. For the partition function p(n) and 5≤ℓ≤31, Atkin found a number of examples of primes Q≥5 such that there exist congruences of the form p(ℓQ3n+β)≡0(modℓ). Recently, Ahlgren, Allen, and Tang proved that there are infinitely many such congruences for every ℓ. In this paper, for a wide range of c∈Fℓ, we prove congruences of the form p(ℓQ3n+β0)≡c⋅p(ℓQn+β1)(modℓ) for infinitely many primes Q. For a positive integer r, let pr(n) be the r-colored partition function. Our methods yield similar congruences for pr(n). In particular, if r is an odd positive integer for which ℓ>5r+19 and 2r+2≢2±1(modℓ), then we show that there are infinitely many congruences of the form pr(ℓQ3n+β)≡0(modℓ). Our methods involve the theory of modular Galois representations.

Statistical Analysis of Descending Open Cycles of Collatz Function

Collatz dynamic systems present a statistical space that can be studied rigorously. In a previous study, the author presented Collatz space in a unique dynamic numerical mode by tabulating a sequential correlation pattern of division by 2 of Collatz function’s even numbers until the numbers became odd with a consecutive occurrence, following an attribute of a 50:50 probability of division by 2 once (ascending behavior) as opposed to division by 2 more than once (descending behavior). In this paper, we describe the path of the Collatz function as sequences comprised of groups of the function’s iterates (open cycles) that end up with the first odd integer that is less than the starting odd integer. The descending behavior of the open cycles is attributed to a deterministic factor as observation of the cycles’ sequences shows. We do statistical analysis on 4 large samples of open cycles and orbits to 1. We define R(n) as the cycles’ deterministic variable defined as the ratio of division by 2 once to division by 2 more than once. We use statistical analysis to study the randomness of the orbits of the cycles’ starting odd positive integers as well as orbits to 1 up to 1,002,097,149.

Even partition functions and $2$-adic analysis

Let $\mathcal A$ denote a set of positive integers, and let $p(\mathcal A,n)$ denote the associated partition function. Let $\beta $ be an odd positive integer, and let $P(z)$ be a polynomial in $ \mathbb F_2[z]$ of order $\beta $ such that $P(0)=1$. J.-L

Study on the oscillation of solution to second-order impulsive systems

&lt;abstract&gt;&lt;p&gt;In the present article, we set the if and only if conditions for the solutions of the class of neutral impulsive delay second-order differential equations. We consider two cases when it is non-increasing and non-decreasing for quotient of two positive odd integers. Our main tool is the Lebesgue's dominated convergence theorem. Examples illustrating the applicability of the results are also given, and state an open problem.&lt;/p&gt;&lt;/abstract&gt;