Infinite Series Research Articles

Some infinite series summations involving linear recurrence relations of order 2 and 3

This paper extends known results of second and third order recursive sequences through extensive formulations of properties of the roots of their characteristic equations, some are old but most are new. They are applied to novel studies of $\sum_{n=0}^{\infty}{\frac{a_{mn}}{10^{n+1}}, \ m=1,2,3}$, including their convergence criteria, and applied to many standard sequences, as particular cases of a generic $\left\{a_n\right\}$. The detailed development of the algebra of the pertinent theorems, and their associated lemmas and corollaries, should open up new vistas for interested number theorists with the concluding results on series values.

Line arrangements with many triple points

AbstractIn this paper, we construct an infinite series of line arrangements in characteristic two, each featuring only triple intersection points. This finding challenges the existing conjecture that suggests the existence of only a finite number of such arrangements, regardless of the characteristic. Leveraging the theory of matroids and employing computer algebra software, we rigorously examine the existence and non-existence across various characteristics of line arrangements with up to 19 lines maximizing the number of triple intersection points.

Existence of a Periodic and Seasonal INAR Process

A spectral criterion involving the model parameters is given for the existence and uniqueness of a periodically correlated and seasonal non‐negative integer‐valued autoregressive process. The structure of the mean and covariance functions of the periodically stationary distribution of the model is derived using its implicit state‐space representation. Two infinite series representations for the process, the moving average, and the immigrant generation, are established. Based on the latter representation, a novel and parallelizable simulation method is proposed to generate the process.

Fourier-Matsubara series expansion for imaginary-time correlation functions.

A Fourier-Matsubara series expansion is derived for imaginary-time correlation functions that constitutes the imaginary-time generalization of the infinite Matsubara series for equal-time correlation functions. The expansion is consistent with all known exact properties of imaginary-time correlation functions and opens up new avenues for the utilization of quantum Monte Carlo simulation data. Moreover, the expansion drastically simplifies the computation of imaginary-time density-density correlation functions with the finite temperature version of the self-consistent dielectric formalism. Its existence underscores the utility of imaginary-time as a complementary domain for many-body physics.

Nonlocal Massive Gravity from Einstein Gravity.

We present a top-down construction of a three-dimensional nonlocal theory of massive gravity. This "nonlocal massive gravity" (NLMG) is obtained as the gravitational theory induced by Einstein gravity on a brane inserted in anti-de Sitter space modified by an overall minus sign. The theory involves an infinite series of increasingly complicated higher-derivative corrections to the Einstein-Hilbert action, with the quadratic term coinciding with new massive gravity. We obtain an analytic formula for the quadratic action of NLMG and show that its linearized spectrum consists of an infinite tower of positive-energy massive spin-2 modes. We compute the Newtonian potential and show that the introduction of the infinite series of terms makes it behave as ∼1/r at short distances, as opposed to the logarithmic behavior encountered when the series is truncated at any finite order. We use this and input from brane-world holography to argue that the theory may contain asymptotically flat black hole solutions.

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

Modular relations involving generalized digamma functions

Generalized digamma functions ψk(x), studied by Ramanujan, Deninger, Dilcher, Kanemitsu, Ishibashi etc., appear as the Laurent series coefficients of the Hurwitz zeta function. In this paper, a modular relation of the form Fk(α)=Fk(1/α) containing infinite series of ψk(x), or, equivalently, between the generalized Stieltjes constants γk(x), is obtained for any k∈N. When k=0, it reduces to a famous transformation given on page 220 of Ramanujan's Lost Notebook. For k=1, an integral containing Riemann's Ξ-function, and corresponding to the aforementioned modular relation, is also obtained along with its asymptotic expansions as α→0 and α→∞. Carlitz-type and Guinand-type finite modular relations involving ψj(m)(x),0≤j≤k,m∈N∪{0}, are also derived, thereby extending previous results on the digamma function ψ(x). The extension of Guinand's result for ψj(m)(x),m≥2, involves an interesting combinatorial sum h(r) over integer partitions of 2r into exactly r parts. This sum plays a crucial role in an inversion formula needed for this extension. This formula has connection with the inversion formula for the inverse of a triangular Toeplitz matrix. The modular relation for ψj′(x) is subtle and requires delicate analysis.

An investigation on the torsional vibration of a FG strain gradient nanotube

AbstractIn this work, an attention is paid to the prediction of torsional vibration frequencies of functionally graded porous nanotubes based on the Lam strain gradient elasticity theory. The nanotubes are formed of functionally graded porous nanomaterials that vary in the radial direction. This study also aims to obtain the analytical solution of the strain gradient model presented by Lam for torsional vibration response, in a simple manner, for different rigid or restrained boundary conditions. The torsion angle of a functionally graded nanotube is defined by an infinite Fourier series. Then, the Stokes’ transformation is applied to force the boundary conditions to the desired state. An eigenvalue problem is established with the help of the two systems of equations obtained. This eigenvalue problem, which includes deformable springs at both ends of the nanotube, appears as a general analytical solution that can find torsional vibration frequencies. It is shown that the vibrational responses can be significantly influenced by the through‐radius gradings of material, material length scale parameters and deformable springs of the functionally graded nanotubes and consequently can be predicted by giving proper values to torsional spring parameters.

Gauss’ Second Theorem for F12(1/2)-Series and Novel Harmonic Series Identities

Two summation theorems concerning the F12(1/2)-series due to Gauss and Bailey will be examined by employing the “coefficient extraction method”. Forty infinite series concerning harmonic numbers and binomial/multinomial coefficients will be evaluated in closed form, including eight conjectured ones made by Z.-W. Sun. The presented comprehensive coverage for the harmonic series of convergence rate “1/2” may serve as a reference source for readers.

WITHDRAWN: Accurate and efficient analytical simulation of free vibration for embedded nonlocal CNTRC beams with general boundary conditions

This study aims to evaluate the free vibrational response of restrained nanobeam enriched by nanocomposites based on an exact Fourier series approach. In order to capture the small size effects on the dynamical response, Eringen’s differential form of nonlocal elasticity is used which employs the one size (nonlocal) parameter. Within the framework of Rayleigh and Bernoulli-Euler beam theories to include the effect of nonlocality and by employing Fourier sine series together with Stokes’ transformation, a system of linear equations is obtained then solved by using the coefficient matrix. The combined effects of elastic boundary conditions, elastic medium, dispersion patterns and nonlocal effects are examined by solving this eigen-value problem constructed with Fourier infinite series. Free vibration frequencies are calculated for carbon nanotube reinforced nanobeam under different rigid or restrained boundary conditions, including an elastic medium Winkler-Pasternak type. A comprehensive parametric study is performed, with the particular focus on the influences of distribution pattern, elastic boundary and medium parameter on the free vibrational response of the composite nanobeam with different graded distributions. It is concluded that the addition of a small amount of carbon nanotube material can reinforce the stiffness of the nanobeam, and its free vibration performance is significantly affected by the distribution patterns, elastic medium and boundary conditions.

On the Universal Presence of Mathematics for Incompletely Predictable Problems in Rigorous Proofs for Riemann Hypothesis, Modified Polignac’s and Twin Prime Conjectures

We validly ignore even prime number 2. Based on all arbitrarily large number of even Prime gaps 2, 4, 6, 8, 10...; the complete set and its derived subsets of Odd Primes fully comply with the Prime number theorem for Arithmetic Progressions, and our derived Generic Squeeze theorem and Theorem of Divergent-to-Convergent series conversion for Prime numbers. With these conditions being satisfied by all Odd Primes, we argue Polignac's and Twin prime conjectures are proven to be true when they are usefully treated as Incompletely Predictable Problems. In so doing [and with Riemann hypothesis being a special case], this action also support the generalized Riemann hypothesis formulated for Dirichlet L-function. By broadly applying Hodge conjecture, Grothendieck period conjecture and Pi-Circle conjecture to Dirichlet eta function (proxy function for Riemann zeta function), Riemann hypothesis is separately proven to be true when it is usefully treated as Incompletely Predictable Problem. Crucial connections exist between these Incompletely Predictable Problems and Quantum field theory whereby eligible (sub-)functions and (sub-)algorithms are treated as infinite series.

Two Classes of Infinite Series through Contour Integration

Two Classes of Infinite Series through Contour Integration

The Recurrence and Transience of Random Walks

This article reviews the research history and application fields of random walks and abstracts random walks into a time-homogeneous Markov chain to study their recurrent and transient properties. For one-dimensional and two-dimensional random walks, the likelihood of returning in steps and the probability of returning for the first time in steps of each state are first introduced, along with their relationship. Then the Stirling's formula is given, which is utilized to estimate the probability of returning in n steps, and the convergence and divergence of infinite series is used to prove that random walk in one dimension is recurrent. Random walk in two dimension has similar properties, which is a bit more complicated by using the relation between polynomials. Higher dimensional random walks need to consider special states first, and then generalize them to other states. Finally, this paper concluded that one-dimensional and two-dimensional random walks are recurrent, and random walks with dimensions higher than two are transient.

Analyzing Lower Limb Dynamics in Human Gait Using Average Value-Based Technique

The motivation of this study is to develop effective and economical assistive technologies for people with physical disabilities. The novelty in this manuscript is the application of the average value-based technique to accurately represent the involved biomechanics of the lower limb joints during the human gait cycle. This mathematical formulation of lower limb joints’ biomechanics forms the first objective for modeling and final exoskeleton prototype development. To account for modeling the characteristics of human locomotion, the nth-order linear differential equation with constant coefficients is considered with appropriate modification. The physical characteristics of an individual are represented by the constant coefficients (P0, P1, P2, and P3) of the modified infinite series, which are obtained by processing experimental data collected using an optical technique. The differential terms of the infinite series are replaced by difference terms (δbavg, δ2bavg, and δ3bavg) since the data were captured as a set of digital values. The work presented here is based on the experimental results of individuals suitably categorized according to their physical nature like age and other physical structure. The optically monitored positional values of the lower limb joints of the individual subjects while they are completing the gait cycles are used for getting values of different terms of the model. The data collected through the conduct of experiments are used for finding the values of the terms of the differential equation. The model is effectively validated through experimental results. It was determined that the representation’s accuracy fell within the ±5% acceptable tolerance limit. The model is prepared for healthy as well as disabled persons, through which the disability is quantified. The resulting model can be used to develop assistive devices for people with physical disabilities. This results in the rehabilitation process and thereby helps the reintegration into society, subsequently allowing them to lead a normal life.

SOLUTION OF FUZZY VOLTERRA INTEGRAL EQUATIONS USING COMPOSITE METHOD

With the rapid development of of fuzzy Integral equations, the problem of finding its solution becomes increasingly important for knowledge propagation and putting researchers wisdom to work. A recent development trend of fuzzy integral equations is modeling many problems in the field of science and technology. If the unknown function in the considered equation has a solution in terms of infinite series expansion, this proposed method becomes more accurate to find the exact solution. According on the parametric form of a fuzzy number, a fuzzy integral equation converts to two systems of integral equations in the crisp case, and then proposed method is achieved to obtain the exact fuzzy solutions of fuzzy Volterra integral equations. To validate the effectiveness of proposed method, some examples of considered equations are solved and the results show that it achieved the best performance.

Option pricing under jump diffusion model

We provide an European option pricing formula written in the form of an infinite series of Black–Scholes-type terms under double Lévy jumps model, where both the interest rate and underlying price are driven by Lévy processes. The series solution converges with a radius of convergence, and it is complemented by some numerical experiments to demonstrate its speed of convergence.

Physical meaning of conditionally convergent series: the calculation of the Madelung constant

Conditionally convergent series are infinite series whose result depends on the order of the sum. One of the most famous examples of conditionally convergent series of interest in Physics is the calculation of Madelung’s constant α in ionic crystals. The appearance of this type of series is quite disturbing to students and often causes misunderstandings. In this work we analyze the physical meaning of the conditional convergence from a pedagogical point of view. The problem is posed using a toy model of ionic crystal in which the lattice sums can be calculated explicitly for various forms of expansion of the crystal about a central core. It is seen directly how the Coulomb series does not converge to α when there are charge accumulations on the surfaces. Therefore, it becomes clear what the appropriate strategy should be when choosing the order of summation to arrive at the correct value of α.

On the stability analysis of a restrained functionally graded nanobeam in an elastic matrix with neutral axis effects

Abstract In this work, a general eigenvalue solution of an arbitrarily constrained nonlocal strain gradient nanobeam made of functionally graded material is presented for the first time for the stability response by the effect of the Winkler foundation. Elastic springs at the ends of the nanobeam are considered in the formulation, which have not been considered in most studies. In order to analyze deformable boundary conditions, linear equation systems are derived in terms of infinite power series by using the Fourier sine series together with the Stokes’ transform. The higher-order force boundary conditions are used to obtain a coefficient matrix including different end conditions, power-law index, elastic medium, and small-scale parameters. A general eigenvalue problem of technical interest, associated with nonlocal strain gradient theory, is mathematically evaluated and presented in detail. Parametric results are obtained to investigate the effects of material length scale parameter, Winkler stiffness, power-law index, nonlocal parameter, and elastic springs at the ends. In addition, the effects of the other higher-order elasticity theories simplified from nonlocal strain gradient theory are also investigated and some benchmark results are presented.

Arakelov class groups of random number fields

AbstractThe main purpose of the paper is to formulate a probabilistic model for Arakelov class groups in families of number fields, offering a correction to the Cohen–Lenstra–Martinet heuristic on ideal class groups. To that end, we show that Chinburg’s $$\Omega (3)$$ Ω ( 3 ) conjecture implies tight restrictions on the Galois module structure of oriented Arakelov class groups. As a consequence, we construct a new infinite series of counterexamples to the Cohen–Lenstra–Martinet heuristic, which have the novel feature that their Galois groups are non-abelian.

A Note on Summability of Infinite Series

The purpose of the present paper is to get the necessary and sufficient conditions for absolute matrix summability of infinite series.