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In classical texture analysis the orientation distribution function (ODF) is expanded as an infinite series of Wigner D -functions D m n l (⋅) , with the classical texture coefficients c m n l serving as undetermined expansion parameters. Herein we present a systematic procedure by which the classical expansion of an ODF (with arbitrary crystallite and texture symmetry) truncated at the order l = L can be directly rewritten as a tensorial Fourier expansion truncated at the same order, where the same set of classical texture coefficients c m n l appear as undetermined parameters. The crucial step of this procedure is as follows: Let H l be the complexified space of l -th order harmonic tensors and 〈 ⋅ , ⋅ 〉 the Hermitian inner product on H l . Let R be a rotation and R ⊗ l : H l → H l be the l -th order tensor product of R with itself. For each 1 ≤ l ≤ L , determine an orthonormal irreducible tensor basis B m l (− l ≤ m ≤ l) in H l so that D m n l (R) = 〈 B m l , R ⊗ l B n l 〉 for each rotation R . We have determined explicit sets of irreducible basis tensors B m l in H l for 1 ≤ l ≤ 8 , which are displayed in tables in this paper. In addition to examples which illustrate the general procedure, we provide, for orthorhombic aggregates of cubic and hexagonal crystallites, explicit expressions of tensorial Fourier expansions of the ODF up to terms with l = 8 , in which the tensorial texture coefficients are given in terms of the classical texture coefficients and the basis tensors B m l . [ABSTRACT FROM AUTHOR]

Yamamoto’s integral was the integral associated with 2-posets, which was first introduced by Yamamoto. In this paper, we obtained the values of infinite series involving harmonic numbers and reciprocal of binomial coefficients by using some techniques of Yamamoto’s integral. We determine the value of infinite series of the form: ∑ m1,...,mn,ℓ1,...,ℓk≥1 H(a1)m1 · · · H(an)mn/mb11 · · · mbnn ℓc11 · · · ℓckk (m1+···+mn+ℓ1+···+ℓkℓk), in terms of a finite sum of multiple zeta values, for positive integers a1, . . ., an, b1, . . ., bn, c1, . . ., ck. [ABSTRACT FROM AUTHOR]

Analytical solutions of some contact problems are infinite functional series according to the system of basic functions. When constructing such solutions, it becomes necessary to represent the kernels of integral equations describing the process of interaction in the form of two-dimensional series on a given basis. Often the kernels have a rather complex appearance, therefore, the process of finding the decomposition coefficients is a rather complex and labor-intensive process, on which the accuracy and speed of obtaining final results depend. The paper proposes a calculation method that allows calculating the coefficients of decomposition of the kernels of plane contact problems according to a special orthonormal basis that takes into account the features of contacting bodies. Other approximate formulas are also derived for the special case when coating characteristics are constant. Based on the received presentation, conclusions and recommendations are formulated. [ABSTRACT FROM AUTHOR]

Ramanujan, in his lost notebook on page 188, delineated unique categories of remarkable infinite series, expressing them through the framework of Eisenstein series. The aim of this paper is to explore diverse differential identities concerning classical η-functions and h-functions. Furthermore, we introduce specific identities involving Eisenstein series of various levels and h-functions, which serve to establish connections between class one infinite series and h-functions. [ABSTRACT FROM AUTHOR]

In the present paper, K denotes a complete, non-trivially valued, ultrametric (or non-archimedean) field. Entries of sequences, infinite series and infinite matrices are in K. Following Kangro [2, 3, 4], we introduce the concept of boundedness with speed A or A-boundedness. We then obtain a characterization of the matrix class (mx.nF), where mx denotes the set of all A-bounded sequences in K. We conclude the paper with a remark about the matrix class (cx, rn), where cx denotes the set of all A-convergent sequences in K. [ABSTRACT FROM AUTHOR]

For a successful application of subspace migration algorithm to retrieve the exact location and shape of small anomaly in microwave imaging, one must begin the reconstruction process under the assumption that complete information about the homogeneous background medium, such as background permittivity and conductivity, is available. In many studies, the statistical value of the background medium was adopted, raising the possibility of an incorrect value being applied. Thus, simulation results have been examined in order to identify cases in which an inaccurate location and shape of anomaly were retrieved. However, the theory explaining this phenomenon has not been investigated. In this paper, we apply an alternative wavenumber instead of the true one and identify the mathematical structure of the subspace migration imaging function for retrieving two-dimensional small anomaly by establishing a relationship with an infinite series of Bessel functions of the first kind. The revealed structure explains the reason behind the retrieval of an inaccurate location and shape of anomaly. The simulation results with synthetic data are presented to support the theoretical result. [ABSTRACT FROM AUTHOR]

Philosophers have long discussed whether time really passes. Simultaneously, they have also discussed whether time could have begun from an infinite past. This paper clarifies the relationship between the reality of time's passage and an infinite past. I assert that time cannot have an infinite past if time really passes. This argument is based on a proposition that an infinite series of events cannot be completed if time really passes. A seemingly strong objection to this proposition is that no movement is possible if an infinite series cannot be completed. However, movement is clearly possible; thus, an infinite series can be completed. I argue that although it is sure that an infinite series can be completed in the static view, an infinite series cannot be completed (and thus, time could not have begun from an infinite past) in the dynamic view. [ABSTRACT FROM AUTHOR]

This paper deals with the three-dimensional analysis of stress distribution in a long circular cylinder imbedded in a poroelatic medium. The surface of the cylinder is subjected to a known pressure. The equations of the classical theory of elasticity are solved in terms of an infinite series. [ABSTRACT FROM AUTHOR]

This paper presented a new Ruscheweyh fractional derivative of fractional order in the complex conformable calculus sense. We applied the constructed complex conformable Ruscheweyh derivative (CCRD) on a certain base of polynomials (BPs) in different regions of convergence in Fréchet spaces (F-spaces). Accordingly, we investigated the relation between the approximation properties of the resulting base and the original one. Moreover, we deduced the mode of increase (the order and type) and the Tp-property of the polynomial bases defined by the CCRD. Some bases of special polynomials, such as Bessel, Chebyshev, Bernoulli, and Euler polynomials, have been discussed to ensure the validity of the obtained results. [ABSTRACT FROM AUTHOR]

In a recent paper [Cáceres MO, Comments on wave-like propagation with binary disorder. J. Stat. Phys. 2021;182(36):doi.org/10.1007/s10955-021-02699-0.], the evolution of a wave-like front perturbed by space-correlated disorder was studied. In addition, the generic solution of the field mean-value was presented as a series expansion in Terwiel's cumulants operators. This infinite series cuts due to the algebra of naked Terwiel's cumulants when these cumulants are associated to a space exponential-correlated symmetric binary disorder. We apply an equivalent approach to study the dispersion-relation for 1D surface gravity waves propagating on an irregular floor. The theory is based on the study of the mean-value of plane-wave-like Fourier modes for the propagation and damping of surface waves on a random bottom. [ABSTRACT FROM AUTHOR]

This paper introduces (graded) skew cellular algebras, which generalise Graham and Lehrer's cellular algebras. We show that all of the main results from the theory of cellular algebras extend to skew cellular algebras and we develop a "cellular algebra Clifford theory" for the skew cellular algebras that arise as fixed point subalgebras of cellular algebras. As an application of this general theory, the main result of this paper proves that the Hecke algebras of type G(\ell,p,n) are graded skew cellular algebras. In the special case when p = 2 this implies that the Hecke algebras of type G(\ell,2,n) are graded cellular algebras. The proofs of all of these results rely, in a crucial way, on the diagrammatic Cherednik algebras of Webster and Bowman. Our main theorem extends Geck's result that the one parameter Iwahori-Hecke algebras are cellular algebras in two ways. First, our result applies to all cyclotomic Hecke algebras in the infinite series in the Shephard-Todd classification of complex reflection groups. Secondly, we lift cellularity to the graded setting. As applications of our main theorem, we show that the graded decomposition matrices of the Hecke algebras of type G(\ell,p,n) are unitriangular, we construct and classify their graded simple modules and we prove the existence of "adjustment matrices" in positive characteristic. [ABSTRACT FROM AUTHOR]

The interaction between ingested turbulence and rotating blades is a key source of broadband noise in engineering applications. In this paper, a far-field noise model accounting for source correlation across the span of the blade and between blades is developed and applied to the study of homogeneous isotropic turbulence ingestion by a model cooling fan, wind turbine and aircraft propeller. The widely used theory of Amiet is revisited and it is shown that previous works produce conflicting results when attempting to account for blade-to-blade correlation. Central to the model is the calculation of the time between blade chops of the same turbulent eddy as heard by the observer. In this paper it is shown that, when derived correctly, Amiet's theory accounts for correlated sources between blades and, thus, can predict haystacking tones. Comparisons with the new rotational formulation and with experimental data enable us to show that Amiet's theory can be used to accurately predict turbulence ingestion noise from open rotors. In particular, it is found that the infinite-span assumption in strip theory and the neglect of correlation effects across the span do not undermine the accuracy of this theory. This is of great importance because, unlike Amiet's theory, models which treat rotational effects and source correlation exactly are expensive to evaluate routinely at high frequencies due to the slow convergence of infinite series with Bessel functions. [ABSTRACT FROM AUTHOR]

This paper investigates the properties of the change in persistence detection for observations with infinite variance. The innovations are assumed to be in the domain of attraction of a stable law with index κ ∈ (0 , 2 ] . We provide a new test statistic and show that its asymptotic distribution under the null hypothesis of non-stationary I (1) series is a functional of a stable process. When the change point in persistence is not known, the consistency is always given under the alternative, either from stationary I (0) to non-stationary I (1) or vice versa. The proposed tests can be used to identify the direction of change and do not over-reject against constant I (0) series, even in relatively small samples. Furthermore, we also consider the change point estimator which is consistent and the asymptotic behavior of the test statistic in the case of near-integrated time series. A block bootstrap method is suggested to determine critical values because the null asymptotic distribution contains the unknown tail index, which results in critical values depending on it. The simulation study demonstrates that the block bootstrap-based test is robust against change in persistence for heavy-tailed series with infinite variance. Finally, we apply our methods to the two series of the US inflation rate and USD/CNY exchange rate, and find significant evidence for persistence changes, respectively, from I (0) to I (1) and from I (1) to I (0) . [ABSTRACT FROM AUTHOR]

In this paper, a theorem dealing with the |N, pn|k summability factors of infinite series has been generalized to |A, pn, β; γ|k summability method by using (Φ, δ)- monotone sequences. This new theorem also includes some new results. [ABSTRACT FROM AUTHOR]

The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the new solution to the fractional Black–Scholes equation was the Daftardar-Geiji method. Additionally, the Picard–Lindelöf theorem was utilized for the existence and uniqueness of its solution. The fractional derivative employed was the Caputo operator. The search for a solution to the fractional Black–Scholes equation was essential due to the Black–Scholes equation's assumptions, which imposed relatively tight constraints. These included assumptions of a perfect market, a constant value of the risk-free interest rate and volatility, the absence of dividends, and a normal log distribution of stock price dynamics. However, these assumptions did not accurately reflect market realities. Therefore, it was necessary to formulate a model, particularly regarding the fractional Black–Scholes equation, which represented more market realities. The results obtained in this paper guaranteed the existence and uniqueness of solutions to the fractional Black–Scholes equation, approximate solutions to the fractional Black–Scholes equation, and very small solution errors when compared to the Black–Scholes equation. The novelty of this article is the use of the Daftardar-Geiji method to solve the fractional Black–Scholes equation, guaranteeing the existence and uniqueness of the solution to the fractional Black–Scholes equation, which has not been discussed by other researchers. So, based on this novelty, the Daftardar-Geiji method is a simple and effective method for solving the fractional Black–Scholes equation. This article presents some examples to demonstrate the application of the Daftardar-Gejji method in solving specific problems. [ABSTRACT FROM AUTHOR]

An interesting series based on Bessel function roots (zeros) is discussed and numerically analyzed. The novel-derived simplified general solutions are based on Lommel polynomials. This kind of series can have a large practical use in many scientific areas, such as solid mechanics, fluid mechanics, thermodynamics, electronics, physics, etc. Some practical examples connected with fluid mechanics are provided in this paper. The errors in Afanasiev solutions are corrected. In addition, the main solution for the series analyzed by Baricz and Angel is presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider invariant infinite series (α, β)-metrics. Then we describe all geodesic vectors of this spaces on the left invariant hypercomplex four dimensional simply connected Lie groups. [ABSTRACT FROM AUTHOR]

In this paper, we study the geometric properties of Finsler Σ-spaces. We prove that Infinite series Σ-spaces are Riemannian. [ABSTRACT FROM AUTHOR]

This paper presents an exact, analytical solution to the boundary value problem of the anti-plane (SH) waves scattering by an isosceles triangle hill on an elastic half-space by using the wavefunction expansion method. An appropriate region-matching technique is introduced to divide the half-space containing a triangle hill into two subregions. Then, the wavefield expression of each subregion is constructed in terms of an infinite series in two coordinate systems, respectively. Furthermore, a Graf's addition formula is derived to unify the coordinate system and solve the unknown coefficients in the wave functions. Finally, numerical results are calculated to illustrate the effects on ground motion due to the existence of an isosceles triangle hill. This paper revises the existing analytical methods, and aims to provide a benchmark for numerical method verification and a reference for engineering practice. [ABSTRACT FROM AUTHOR]

Throughout this paper, K denotes a complete, non-trivially valued, ultrametric (or non-archimedean) field. Entries of infinite matrices, sequences and infinite series are in K. In this paper, following ([4], [5]), we prove some more Steinhaus type theorems over K. [ABSTRACT FROM AUTHOR]

In a little-known paper Hurwitz gave an infinite series representation of the class number for positive definite binary quadratic forms. In this paper we give a similar formula in the indefinite case. We also give a simple proof of Hurwitz's formula and indicate some extensions. [ABSTRACT FROM AUTHOR]

In this paper, we define an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, we considered the case when a function has a decomposition in the Laurent series with the infinite principal part and finite regular part. Our central challenge is to improve this result having considered as a regular part an entire function satisfying the special condition of the growth regularity. As an application, we consider an opportunity to broaden the conditions imposed upon the second term not containing the time variable of the evolution equation in the abstract Hilbert space. [ABSTRACT FROM AUTHOR]

The Korteweg–de Vries (KDV) equation is one of the most well-known models in nonlinear physics, such as fluid physics, plasma, and ocean engineering. It is very important to obtain the exact solutions of this model in the process of studying these topics. In the present paper, using distinct function iteration relations in two ways, namely, squaring infinitely and extracting the square root infinitely, which have not been reported in other documents, we construct abundant types of new infinite-series exact solitary wave solutions using the auxiliary equation method. Most of these solutions have not been reported in previous papers. The numerical analysis of some solutions shows complex solitary wave phenomena. Some solutions can have stable solitary wave structures, while others may have singularities in certain space–time positions. The results show that the analysis model we use is very simple and effective for the construction of new infinite-series solutions and new solitary wave structures of nonlinear models. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method to obtain a solution to a single-parameter Bring quintic equation of the form, x5 +x = a, where a is real. The method transforms the given quintic equation to an infinite but convergent series expression in (x/a), which is further transformed to a quartic equation in a novel fashion. The coefficients of the quartic equation so obtained are some kind of infinite series expressions in a-4, which are termed as ultraradicals. The quartic equation is then solved and its one real solution is picked; further using this, the real solution of quintic equation, x5 + x = a; is extracted. The ultraradicals used in this method converge for |a| > 1; hence the method can be used when |a| > 1. [ABSTRACT FROM AUTHOR]

The present paper investigates the approximation of special monogenic functions (SMFs) in infinite series of hypercomplex Hasse derivative bases (HHDBs) in Fréchet modules (F-modules). The obtained results ensure the existence of such representation in closed hyperballs, open hyperballs, closed regions surrounding closed hyperballs, at the origin, and for all entire SMFs (ESMFs). Furthermore, we discuss the mode of increase (order and type) and the Tp-property. This study enlightens several implications for some associated HHDBs, such as hypercomplex Bernoulli polynomials, hypercomplex Euler polynomials, and hypercomplex Bessel polynomials. Based on considering a more general class of bases in F-modules, our results enhance and generalize several known results concerning approximating functions in terms of bases in the complex and Cliffiord settings. [ABSTRACT FROM AUTHOR]

Thermoelastic dissipation or thermoelastic damping (TED) can restrict the quality factor of micro/nanoring resonators seriously. This paper employs the non-Fourier model of Guyer–Krumhansl (GK model) to render a size-dependent formulation and analytical solution for approximating the amount of TED in micro/nanorings with circular cross-section by inclusion of nonlocal and single-phase-lagging effects. To fulfill this objective, the equation of heat conduction in the ring is first established according to GK model. Then, by placing the temperature distribution obtained from the heat conduction equation in the TED relation defined on the basis of thermal energy approach, an expression in the form of infinite series is given for TED, which includes non-classical parameters of GK model. Finally, after checking the validity of the model through a comparative study, several simulation results are prepared to emphasize on the influence of different factors such as non-classical parameters of GK model, geometry of ring, vibrational mode and ambient temperature on TED value. Numerical examples reveal that the mentioned factors along with the two- or three-dimensional heat transfer (2D or 3D) model have major influences on TED variations. [ABSTRACT FROM AUTHOR]

Currently, there are considerable challenges associated with the theoretical derivation and numerical calculation of Green's function for spherical layered earths. In order to study the coupling effect between wide-area interconnected systems and the earth, this earth model is widely used. Artificial intelligence methods have been applied in order to overcome these challenges. First, Green's function in the form of infinite Legendre series is derived using intelligent symbolic operations, and the weight function of Legendre series is recursively computed by means of a recursive algorithm. It is on this basis that the intelligent complex image method has been developed for solving spherical layered Green's functions, which transforms the summation of infinite series of Green's functions into the superposition of complex image potentials. Additionally, the upper error limit of the algorithm has been determined, and examples have been provided to demonstrate the accuracy of the algorithm. A solution based on multiple precision algorithms has been proposed to resolve the numerical singularity problem and the extremely slow convergence of the spherical layered Green's function at the earth's scale. It has been demonstrated that the intelligent complex image method has advantages in terms of computing speed and accuracy. A spherical layered Green's function at the earth scale is presented in this paper as an effective and intelligent solution. [ABSTRACT FROM AUTHOR]

Students’ development of reasoning through algebraic symbols is a crucial component of most mathematics courses. This paper reports the journey of one second-semester calculus student, Cedric, as he attempted to reason about and create algebraic representations for arbitrary partial sums and infinite series through two exploratory teaching interviews. We report Cedric’s symbolizing activity in terms of Eckman's (2023) expression framework, focusing on Cedric’s development and attribution of meaning to a personal expression template for denoting partial sums and series. Specifically, we describe how Cedric leveraged his initial meanings for partial sums to create personal expressions to reason about infinite series. We then describe Cedric's construction of a general personal expression template for partial sums and series, including cognitive and physical modifications he made to his personal expression template as he symbolized various series. Our results show potentially profitable task sequences and lines of questioning that instructors might utilize to help students be more successful in constructing productive meanings for algebraic symbols. [ABSTRACT FROM AUTHOR]

Sparse Ramanujan Sequences (SRS) are periodic series derived from Ramanujan Sums (RS). RS is an exponential summation used to derive infinite series expansions for arithmetic functions, whose application pervades different fields of study from classical mathematics to digital signal processing. SRS has properties similar to RS such as periodicity and orthogonality. SRS transforms allow the sparse representation of signals, hence they have many potential applications, such as denoising and compression. The application of SRS transform to reduce the Peak to Average Power Ratio (PAPR) in Orthogonal Frequency Division Multiplexing (OFDM) is proposed in this paper. The Complementary Cumulative Distribution Function (CCDF) and Bit Error Rate (BER) are used to assess the PAPR reduction performance of the proposed system. The results show that the proposed SRS transform-based OFDM system significantly reduces PAPR and BER with reduced computational complexity in comparison with the existing systems. [ABSTRACT FROM AUTHOR]

Quite recently, we have obtained two main theorems dealing with absolute weighted arithmetic mean summability factors of infinite series and trigonometric Fourier series [C. R. Math. Acad. Sci. Paris 359 (2021), pp. 323–328]. In this paper, we have generalized these theorems for a general summability method. We have also obtained some new and known results for certain absolute summability methods. [ABSTRACT FROM AUTHOR]

A formation of finite groups is said to be superradical if it satisfies the following requirements: is a normally hereditary formation; any group , where and are -subnormal -subgroups in , belongs to . The paper presents an infinite series of hereditary superradical formations of full characteristic that are not solvably saturated. This completes the negative answer to question 14.99 (b) in "Kourovka Notebook". [ABSTRACT FROM AUTHOR]

In this paper, we find necessary and suficient conditions under which the infinite series metric and Randers metric on a manifold M of dimension n 3 be projectively related. [ABSTRACT FROM AUTHOR]

Using the logarithmic law of resistance under the rivers' uniform flow involves implicit governing equations in the normal depth. Therefore, many authors have developed approximate solutions for determining the normal depths for wide rectangular and cosine-shaped sections. This paper presents new, more accurate direct solutions for predicting the normal depths of these sections using the Lambert W-Function and doubly infinite expanded series. The equations are expressed in dimensionless roughness and viscosity, which are functions of five physical parameters: flow discharge, roughness, kinematic viscosity, channel width, and angle of repose of the bed material. The novelty of this paper is as follows: (1) the proposed equations are in the form of fast converging power series, (2) the normal depth equations are developed for the rough and smooth flow regimes, and the transition region between them, and (3) the maximum relative error of the proposed solutions range from 0.01% to 0.8%, compared with 2% to 14% for the existing solutions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the statistical summability by the weighted mean method in a two-normed space and establish necessary and sufficient Tauberian conditions under which statistical convergence of a sequence (xn) follows from its statistical summability by weighted mean method in a two-normed space. In particular, our Tauberian conditions are satisfied if (xn) is statistically slowly oscillating in a two-normed space. The main theorem of this paper is the statistical extension of a Tauberian theorem in Çanak et al. [ABSTRACT FROM AUTHOR]

In this paper, we give a microscopic derivation of Gibbs measures for the focusing cubic nonlinear Schrödinger equation on the one-dimensional torus from many-body quantum Gibbs states. Since we are not making any positivity assumptions on the interaction, it is necessary to introduce a truncation of the mass in the classical setting and of the rescaled particle number in the quantum setting. Our methods are based on a perturbative expansion of the interaction, similarly as in [1]. Due to the presence of the truncation, the obtained series have infinite radius of convergence. We treat the case of bounded, L1 and delta function interaction potentials, without any sign assumptions. Within this framework, we also study time-dependent correlation functions. This is the first such known result in the focusing regime. [ABSTRACT FROM AUTHOR]

The paper deals with t-designs that can be partitioned into s-designs, each missing a point of the underlying set, called point-missing s-resolvable t-designs, with emphasis on their applications in constructing t-designs. The problem considered may be viewed as a generalization of overlarge sets which are defined as a partition of all the v + 1 k k-sets chosen from a (v + 1) -set X into (v + 1) mutually disjoint s- (v , k , δ) designs, each missing a different point of X. Among others, it is shown that the existence of a point-missing (t - 1) -resolvable t- (v , k , λ) design leads to the existence of a t- (v , k + 1 , λ ′) design. As a result, we derive various infinite series of 4-designs with constant index using overlarge sets of disjoint Steiner quadruple systems. These have parameters 4- (3 n , 5 , 5) , 4- (3 n + 2 , 5 , 5) and 4- (2 n + 1 , 5 , 5) , for n ≥ 2 , and were unknown until now. We also include a recursive construction of point-missing s-resolvable t-designs and its application. [ABSTRACT FROM AUTHOR]

In an effective theory of gravity, thermodynamic quantities of black holes receive corrections from the infinite series of higher derivative terms. At the next to leading order, these can be obtained by using only the leading order solution. In this paper, we push forward this property to the next to next to leading order. We propose a formula which yields the Euclidean action of asymptotically flat black holes at the next to next to leading order using only the solution up to and including the next to leading order. Other conserved quantities are derived from the Euclidean action via standard thermodynamic relation. We verify our formula in examples of D-dimensional pure gravity and Einstein-Maxwell theory extended by 4- and 6-derivative terms. Based on our formula, we also prove that for asymptotically flat black holes, the physical quantities are invariant under field redefinitions. [ABSTRACT FROM AUTHOR]

According to an influential line of argument, the past must be finite because no infinite series can be formed by successive addition. The present paper pinpoints the non sequitur at the heart of this argument, disentangles the ambiguities that disguise it, and dismantles the misleading picture of 'traversing the infinite' that gives the argument so much of its allure. Finally, the paper critically explores the related argument that a beginningless series of past events is impossible because there could be no explanation of its having been 'completed' at one time rather than another. [ABSTRACT FROM AUTHOR]

In this paper, we have proved a general theorem dealing with absolute Riesz |N,α,β,δ; γ|k summablility by applying an almost increasing sequence. Also, some known results are also deduced. [ABSTRACT FROM AUTHOR]

In a recent paper, [Sarigöl, M. A.: Characterization of summability methods with high indices, Math. Slovaca 63 (2013), 1053–1058], the equivalence ∣C, 0∣k ⟺ ∣R, pn∣k, k ≥ 1, was characterized for infinite single series. In the present paper, this result is extended to doubly summability a different approach. [ABSTRACT FROM AUTHOR]

On page 188 of his lost notebook, Ramanujan introduced distinct classes of exquisite infinite series, representing them in terms of Eisenstein series. The objective of this paper is to develop various differential identities related to Borwein's cubic theta functions and h-functions. Also, we present certain identities containing Eisenstein series of level 6 and h-functions that are used to establish relations involving class one infinite series and h-functions. Additionally, we present a straightforward approach to evaluate a discrete convolution sum by employing relationships involving Eisenstein series and h-functions. [ABSTRACT FROM AUTHOR]

The (p, k)-parametrization method, as introduced by Alaca, offers an innovative approach to derive a variety of new Eisenstein series identities that incorporate Borweins' theta functions. This paper presents new Eisenstein series identities that involve Borweins' cubic theta functions, achieved through the (p, k)-parameterization technique. [ABSTRACT FROM AUTHOR]

Kurokawa and Wakayama [Proc. Amer. Math. Soc. 132 (2004), pp. 935–943] defined a q-analogue of the Euler constant and proved the irrationality of certain numbers involving q-Euler constant. In this paper, we improve their results and prove the linear independence result involving q-analogue of the Euler constant. Further, we derive the closed-form of a q-analogue of the k-th Stieltjes constant \gamma _k(q). These constants are the coefficients in the Laurent series expansion of a q-analogue of the Riemann zeta function around s=1. Using a result of Nesterenko [C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), pp. 909–914], we also settle down a question of Erdős regarding the arithmetic nature of the infinite series \sum _{n\geq 1}{\sigma _1(n)}/{t^n} for any integer t > 1. Finally, we study the transcendence nature of some infinite series involving \gamma _1(2). [ABSTRACT FROM AUTHOR]

Recently, in [7], we proved a main theorem dealing with the φ − | C, α |k summability factors of infinite series. In the present paper, we have generalized this theorem for the φ−| C, α ∗ β |k summability methods by using a quasi-f-power increasing sequence instead of an almost increasing sequence. Some new results are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, first derivatives of the Whittaker function M κ , μ x are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Moreover, from the integral representation of M κ , μ x it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions). These infinite sums and integrals can be expressed in closed form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function are derived, along with finite and infinite integrals containing products of algebraic, exponential, logarithmic, and Bessel functions. Finally, reduction formulas for the Whittaker functions M κ , μ x and integral Whittaker functions Mi κ , μ x and mi κ , μ x are calculated. [ABSTRACT FROM AUTHOR]

The KPI equation is one of most well-known nonlinear evolution equations, which was first used to described two-dimensional shallow water wavs. Recently, it has found important applications in fluid mechanics, plasma ion acoustic waves, nonlinear optics, and other fields. In the process of studying these topics, it is very important to obtain the exact solutions of the KPI equation. In this paper, a general Riccati equation is treated as an auxiliary equation, which is solved to obtain many new types of solutions through several different function transformations. We solve the KPI equation using this general Riccati equation, and construct ten sets of the infinite series exact solitary wave solution of the KPI equation. The results show that this method is simple and effective for the construction of infinite series solutions of nonlinear evolution models. [ABSTRACT FROM AUTHOR]

In this paper, a general theorem dealing with φ − | C , α ; δ | k summability of an infinite series is proved under weaker conditions by using a quasi- β -power increasing sequence instead of an almost increasing sequence. [ABSTRACT FROM AUTHOR]

In this paper, we study several inverse problems associated with a fractional differential equation of the following form: \begin{document}$ (-\Delta)^s u(x)+\sum\limits_{k = 0}^N a^{(k)}(x) [u(x)]^k = 0, \ \ 0

Regarding the capacitive nature of input/output impedances of millimeter‐wave amplifiers, it is necessary to utilize wideband inductive elements for the design of input/output impedance matching circuits. In this paper, the design and optimization of an ultra‐wideband transmission line inductor with an inductance of less than 100 pH are introduced for using at frequencies above 50 GHz. The reason of using this inductor is its superior performance compared to that of a spiral inductor and single‐coil inductor. The inductor consists of three rectangular microstrip line segments with nonequal and variable characteristic impedances. The design is done by calculating the impedance matrix using planar circuit analysis (PCA) based on a planar waveguide model, segmentation/desegmentation methods, and an intelligent algorithm in MATLAB. With this design method, the effect of discontinuities, fringing fields at the edges of the microstrip, and the conductor as well as dielectric losses can be considered while minimizing the dispersion effect. In this study, to reduce the calculation time of the impedance matrix, the doubly infinite series are reduced to a singly infinite series by summing the inner sum for a rectangular segment whose one side is shorted and three other sides are open. [ABSTRACT FROM AUTHOR]