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Kilmer and Zheng (2021) recently introduced a generalized version of the alternating harmonic series. In this paper, we introduce a new generalization of the alternating harmonic series. A special case of our generalization converges to the Kilmer-Zheng series. Then we investigate several interesting and useful properties of this generalized, such as a summation formula related to the Hurwitz -Lerch Zeta function, a duplication formula, an integral representation, derivatives, and the recurrence relationship. Some important special cases of the main results are also discussed. [ABSTRACT FROM AUTHOR]

In the present paper, we focused on finding an initial basic feasible solution for transportation problems. The new method is compared to four different proposed mean methods namely, Proposed Arithmetic Mean (PAM), Proposed Harmonic Mean (PHM), Proposed Geometric Mean (PGM) and Proposed Quadratic Mean (PQM) methods along with existing traditional methods like North West Corner Method (NWCM), Least Cost Method (LCM) and Vogel Approximation Method (VAM). Also, the deviation percentage of the outcomes has been figured out in contrast to distinct methods. The approach thus developed, involves elementary arithmetic and statistics, which could be of keen interest even for beginners and is beneficial to decision makers dealing with the problem of supply chain as well as logistics. Suitable examples are given for examining optimality and to validate the results obtained. [ABSTRACT FROM AUTHOR]

The focus of this research is the examination of the theoretical and practical knowledge regarding the convergence and divergence of the series. In line with the research focus, it is aimed to reveal the approaches of pre-service elementary mathematics teachers on the convergence and divergence of the series with the help of the real-life problem in the context of theory and application by using the concept of harmonic series. The research carried out the qualitative research method is designed according to the case study. The data of the research have consisted of two written questions and one problem and four interview questions which were formed by the researchers and their validity and reliability were ensured. The study concluded that pre-service teachers have theoretical knowledge on the convergence of series and harmonic series. Also, their perception of series, harmonic series, convergence and divergence concepts changed in the application process and they performed different approaches in problem-solving. [ABSTRACT FROM AUTHOR]

In this paper, we extend tools developed in [26] to study harmonic numbers sums. We use these objects to compute Euler-related sums and to explore their connections with classical Euler sums. We establish some new relations between Euler sums and related sums. The relations obtained allow us to find some nice closed-form representations of nonlinear Euler sums through zeta values and linear sums. [ABSTRACT FROM AUTHOR]

This article describes the discovery and the subsequent proof of a hypothesis concerning harmonic series. The whole situation happened directly in the course of the process of a maths lesson. The formulation of the hypothesis was supported by the computer. The hypothesis concerns an interesting connection between harmonic series and the Euler's number e. Let Hn denote the n'th partial sum of the harmonic series. Let's notice the sums H1, H4, H11, …, where the partial sum reaches 1, 2, 3, … for the first time. Let's mark the relevant indexes 1, 4, 11, … as P1, P2, P3, …. So Pn is the index of such a partial sum for which the following is true: Hpn-1

For any positive decreasing to zero sequence an such that Φ an diverges we consider the related series Φ knan and Φ jnan. Here, kn and jn are real sequences such that kn ∈ {0, 1} and jn ∈ {-1, 1}. We study their convergence and characterize it in terms of the density of 1's in the sequences kn and jn. We extend our results to series Φ mnan, with mn ∈ {-1, 0, 1} and apply them to study some associated random series. [ABSTRACT FROM AUTHOR]

For ∈ 2 N and p ∈ R the nth partial sum of the alternating p-series, known also as alternating generalized harmonic number of order p... [ABSTRACT FROM AUTHOR]

We consider subsums of the harmonic series, and determine conditions for their convergence. We apply these conditions to determine convergence for a family of series that generalizes Kempner's series. [ABSTRACT FROM AUTHOR]

We present an easily defined countable family of permutations of the natural numbers for which explicit rearrangements (i.e., the sums induced by the permutations) can be computed. The digamma function proves to be the key tool for the computations found here for the alternating harmonic series. The permutations φ under consideration are simple in the sense that φ○φ is the identity function. We show that the countable set of rearrangements obtained from the permutations considered below is dense in the reals. [ABSTRACT FROM PUBLISHER]

We show that the module of Stark units associated to a sign-normalized rank one Drinfeld module can be obtained from Anderson's equivariant A-harmonic series. We apply this to obtain a class formula à la Taelman and to prove a several variable log-algebraicity theorem, generalizing Anderson's log-algebraicity theorem. We also give another proof of Anderson's log-algebraicity theorem using shtukas and obtain various results concerning the module of Stark units for Drinfeld modules of arbitrary rank. [ABSTRACT FROM AUTHOR]

The aim of this note is to establish new spectral bounds for the harmonic matrix. [ABSTRACT FROM AUTHOR]

We evaluate the three-loop massive vacuum bubble diagrams in terms of polylogarithms up to weight six. We also construct the basis of irrational constants being harmonic polylgarithms of arguments e . [ABSTRACT FROM AUTHOR]

It is shown that the deterministic infinite trigonometric products with parameters $$ p\in (0,1]\ \& \ s>\frac{1}{2}$$ , and variable $$t\in \mathbb {R}$$ , are inverse Fourier transforms of the probability distributions for certain random series $$\Omega _{p}^\zeta (s)$$ taking values in the real $$\omega $$ line; i.e. the $${\text{ Cl }_{p;s}^{}}(t)$$ are characteristic functions of the $$\Omega _{p}^\zeta (s)$$ . The special case $$p=1=s$$ yields the familiar random harmonic series, while in general $$\Omega _{p}^\zeta (s)$$ is a 'random Riemann- $$\zeta $$ function,' a notion which will be explained and illustrated-and connected to the Riemann hypothesis. It will be shown that $$\Omega _{p}^\zeta (s)$$ is a very regular random variable, having a probability density function (PDF) on the $$\omega $$ line which is a Schwartz function. More precisely, an elementary proof is given that there exists some $$K_{p;s}^{}>0$$ , and a function $$F_{p;s}^{}(|t|)$$ bounded by $$|F_{p;s}^{}(|t|)|\!\le \! \exp \big (K_{p;s}^{} |t|^{1/(s+1)})$$ , and $$C_{p;s}^{}\!:=\!-\frac{1}{s}\int _0^\infty \ln |{1-p+p\cos \xi }|\frac{1}{\xi ^{1+1/s}}\mathrm{{d}}\xi $$ , such that the regularity of $$\Omega _{p}^\zeta (s)$$ follows. Incidentally, this theorem confirms a surmise by Benoit Cloitre, that $$\ln {\text{ Cl }_{{{1}/{3}};2}^{}}(t) \sim -C\sqrt{t}\; \left( t\rightarrow \infty \right) $$ for some $$C>0$$ . Graphical evidence suggests that $${\text{ Cl }_{{{1}/{3}};2}^{}}(t)$$ is an empirically unpredictable (chaotic) function of t. This is reflected in the rich structure of the pertinent PDF (the Fourier transform of $${\text{ Cl }_{{{1}/{3}};2}^{}}$$ ), and illustrated by random sampling of the Riemann- $$\zeta $$ walks, whose branching rules allow the build-up of fractal-like structures. [ABSTRACT FROM AUTHOR]

It is well known that the sound produced by string instruments has a well defined pitch. Essentially, this is due to the fact that all the resonance frequencies of the string have integer ratio with the smallest eigenfrequency. However, it is enough to use Ashbaugh–Benguria bound for the ratio of the smallest two eigenfrequencies to conclude that it is impossible to build a drum with a uniform density membrane satisfying harmonic relations on the eigenfrequencies. On the other hand, it is known since the antiquity, that a drum can produce an almost harmonic sound by using different densities, for example adding a plaster to the membrane. This idea is applied in the construction of some Indian drums like the tabla or the mridangam. In this work we propose a density and shape optimization problem of finding a composite membrane that satisfy approximate harmonic relations of some eigenfrequencies. The problem is solved by a domain decomposition technique applied to the Method of Fundamental Solutions and Hadamard shape derivatives for the optimization of inner and outer boundaries. This method allows to present new configurations of membranes, for example a two-density membrane for which the first 21 eigenfrequencies have approximate five harmonic relations or a three-density membrane for which the first 45 eigenfrequencies have eight harmonic relations, both involving some multiple eigenfrequencies. [ABSTRACT FROM AUTHOR]

The dynamic behavior of a rectangular crack in a three-dimensional (3D) orthotropic elastic medium is investigated under a harmonic stress wave based on the non-local theory. The two-dimensional (2D) Fourier transform is applied, and the mixed-boundary value problems are converted into three pairs of dual integral equations with the unknown variables being the displacement jumps across the crack surfaces. The effects of the geometric shape of the rectangular crack, the circular frequency of the incident waves, and the lattice parameter of the orthotropic elastic medium on the dynamic stress field near the crack edges are analyzed. The present solution exhibits no stress singularity at the rectangular crack edges, and the dynamic stress field near the rectangular crack edges is finite. [ABSTRACT FROM AUTHOR]

We try and describe here the general scheme of homogeneous fibre bundles and the harmonicity equations for their sections and how it applies to geometric structures arising from a reduction of the structure group. Among G-structures, we particularly pay attention to almost complex and almost contact structures and the relevant notions of harmonic sections and harmonic maps. We then review some of the more salient properties and results in these two instances. In the last section, we detail the proof of harmonicity for the class nearly cosymplectic. [ABSTRACT FROM AUTHOR]

The Moho surface can be determined according to isostatic theories, and among them, the recentVening Meinesz-Moritz (VMM) theory of isostasy has been successfully applied for this purpose. In this paper, this method is studied from a theoretical prospective and its connection to the Airy-Heiskanen (AH) and Vening Meinesz original theories are presented. Jeffrey's inverse solution to isostasy is developed according to the recent developments of the VMM method and both are compared in similar situations. It is shown that they are generalizations of the AH model in a global and continuous domain. In the VMM spherical harmonic solution for Moho depth, the mean Moho depth contributes only to the zero-degree term of the series, while in Jeffrey's solution it contributes to all frequencies. In addition, the VMM spherical harmonic series is improved further so that the mean Moho can contribute to all frequencies of the solution. This modification makes the VMM global solution superior to the Jeffrey one, but in a global scale, the difference between both solutions is less than 3 km. Both solutions are asymptotically convergent and we present two methods to obtain smooth solutions for Moho from them. [ABSTRACT FROM AUTHOR]

The well-known limit comparison test is only applicable for series with nonnegative terms. Thus, it can be used only for proving or disproving the absolute convergence of a series. In this note we formulate and justify a modified version of the limit comparison test for general series. The test can be used to prove the convergence of conditionally convergent series. [ABSTRACT FROM AUTHOR]

An accurate image-charge method (ICM) is developed for ionic interactions outside a core-shell structured dielectric sphere. Core-shell particles have wide applications for which the theoretical investigation requires efficient methods for the Green’s function used to calculate pairwise interactions of ions. The ICM is based on an inverse Mellin transform from the coefficients of spherical harmonic series of the Green’s function such that the polarization charge due to dielectric boundaries is represented by a series of image point charges and an image line charge. The residue theorem is used to accurately calculate the density of the line charge. Numerical results show that the ICM is promising in fast evaluation of the Green’s function, and thus it is useful for theoretical investigations of core-shell particles. This routine can also be applicable for solving other problems with spherical dielectric interfaces such as multilayered media and Debye-Hückel equations. [ABSTRACT FROM AUTHOR]

Text The aim of this paper is to present a method to calculate the rate of growth of divergent series. In the case of the harmonic series, we study an alternate of other classical methods presented by L. Comtet, S.M. Zeyman, W. DeTemple and S.-H. Wang. Moreover, the method presented in this paper is suitable for obtaining similar results on other divergent series, as we will show in the final part of our investigation. Video For a video summary of this paper, please visit http://youtu.be/SFR7zY4pvG4 . [ABSTRACT FROM AUTHOR]

We present a comprehensive numerical analysis of spherical, spheroidal, and ellipsoidal harmonic series for gravitational field modeling near small moderately irregular bodies, such as the Martian moons. The comparison of model performances for these bodies is less intuitive and distinct than for a highly irregular object, such as Eros. The harmonic series models are each associated with a distinct surface, i.e., the Brillouin sphere, spheroid, or ellipsoid, which separates the regions of convergence and possible divergence for the parent infinite series. In their convergence regions, the models are subject only to omission errors representing the residual field variations not accounted for by the finite degree expansions. In the regions inside their respective Brillouin surfaces, the models are susceptible to amplification of omission errors and possible divergence effects, where the latter can be discerned if the error increases with an increase in the maximum degree of the model. We test the harmonic series models on the Martian moons, Phobos and Deimos, with moderate oblateness of $$<$$ 0.4. The possible divergence effects and amplified omission errors of the models are illustrated and quantified. The three models yield consistent results on a bounding sphere of Phobos in their common convergence region, with relative errors in potential of $$\sim $$ 0.01 and $$\sim $$ 0.001 % for expansions up to degree 10 and degree 20 respectively. On the surface of Phobos, the spherical and spheroidal models up to degree 10 both have maximum relative errors of $$\sim $$ 1 % in potential and $$\sim $$ 100 % in acceleration due ostensibly to divergence effect. Their performances deteriorate more severely on the more irregular Deimos. The ellipsoidal model exhibits much less distinct divergence behavior and proves more reliable in modeling both potential and acceleration, with respective maximum relative errors of $$\sim $$ 1 and $$\sim $$ 10 %, on both bodies. Our results show that for the Martian moons and other such moderately irregular bodies, the ellipsoidal harmonic series should be considered preferentially for gravitational field modeling. [ABSTRACT FROM AUTHOR]

Text: The Arakawa-Kaneko zeta functions interpolate the poly-Bernoulli numbers at the negative integers and their values at positive integers are connected to multiple zeta values. We give everywhere-convergent series expansions of these functions in terms of Bernoulli polynomials and Dirichlet series related to harmonic numbers, exhibiting their explicit analytic continuation. Differentiating the Barnes multiple zeta functions of order r with respect to their order produces Dirichlet series attached to Bernoulli polynomials. These series are invariant under an involution in which the order of the derivative is dual to the value of the first variable and the order of the zeta function is dual to the value of the second variable. This symmetry relation generalizes duality relations of Euler sums, and is featured in our series expansions of Arakawa-Kaneko zeta values. Video: For a video summary of this paper, please visit http://youtu.be/0vQqfgrkX2k. [ABSTRACT FROM AUTHOR]

The article presents a science experiment to enable students to visualize the harmonic series in resonant tubes through a smartphone. Topics discussed include the materials and mechanics of the experiment kit, the smartphone, platform, and applications used, and a comparison of theoretical and experimental harmonic frequency profiles for one-closed-end, for both-closed-end, and for both-open-end tubes.

With the advent of geodetic satellite missions mapping almost globally the Earth's gravitational field, new methods and theoretical approaches have been developed and investigated to fully exploit the potential of their new observables. Besides estimating values of numerical coefficients in harmonic series of the gravitational potential, new applications emerged such as data validation and combination. In this contribution, new integral transformations are presented which transform principal components of the terrestrial deflection of the vertical onto disturbing satellite-to-satellite tracking and gradiometric data at altitude. Using spherical approximation, necessary integral kernel functions are derived in both spectral and closed forms. The behaviour of isotropic kernel functions is studied and the new integral transformations are tested in a closed-loop simulation using synthetic terrestrial and satellite data synthesized from a global gravitational model. New integral transformations can be used for data validation and combination purposes. [ABSTRACT FROM AUTHOR]

The article aims to find expressions for the sums of the alternating harmonic series. Topics covered include the method of splitting the series into two converging alternating series, combining the sums into a single formula, and the sums of rearrangements of the signs of other harmonic-like series.

Abstract: A method is given for generating harmonic tensor product Bézier surfaces and the explicit expression of each point in the control net is provided as a linear combination of prescribed boundary control points. The matrix of scalar coefficients of these combinations works like a mould for harmonic surfaces. Thus, real-time manipulation of the resulting surfaces subject to modification of prescribed information is possible. [Copyright &y& Elsevier]

In this paper, we give some new and interesting identities involving harmonic and hyperharmonic numbers which are derived from the transfer formula for the associated sequences. [ABSTRACT FROM AUTHOR]

In this paper, a simple single-phase grid-connected photovoltaic (PV) inverter topology consisting of a boost section, a low-voltage single-phase inverter with an inductive filter, and a step-up transformer interfacing the grid is considered. Ideally, this topology will not inject any lower order harmonics into the grid due to high-frequency pulse width modulation operation. However, the nonideal factors in the system such as core saturation-induced distorted magnetizing current of the transformer and the dead time of the inverter, etc., contribute to a significant amount of lower order harmonics in the grid current. A novel design of inverter current control that mitigates lower order harmonics is presented in this paper. An adaptive harmonic compensation technique and its design are proposed for the lower order harmonic compensation. In addition, a proportional-resonant-integral (PRI) controller and its design are also proposed. This controller eliminates the dc component in the control system, which introduces even harmonics in the grid current in the topology considered. The dynamics of the system due to the interaction between the PRI controller and the adaptive compensation scheme is also analyzed. The complete design has been validated with experimental results and good agreement with theoretical analysis of the overall system is observed. [ABSTRACT FROM AUTHOR]