Sum Of Series Research Articles

Dirichlet series under standard convolutions: variations on Ramanujan’s identity for odd zeta values

Inspired by a famous identity of Ramanujan, we propose a general formula linearizing the convolution of Dirichlet series as the sum of Dirichlet series with modified weights; its specialization produces new identities and recovers several identities derived earlier in the literature, such as the convolution of squares of Bernoulli numbers by Dixit et al., or the Fourier expansion of the convolution of Bernoulli–Barnes polynomials by Komori et al.

SOME DIRECT AND INVERSE THEOREMS OF APPROXIMATION OF FUNCTIONS IN JACOBI-DUNKL DISCRETE HARMONIC ANALYSIS

In this paper, several direct and inverse theorems in terms of the best approximations of functions and moduli of smoothness are proved concerning the approximation of functions from the space \(\mathbb {L}_{2}^{(\alpha ,\beta )}\) by partial sums of Jacobi-Dunkl series. For this purpose, we use the generalized Jacobi-Dunkl translation operator which was defined by Vinogradov.

Harnessing S-duality in mathcal{N} = 4 SYM & supergravity as SL(2, ℤ)-averaged strings

We develop a new approach to extracting the physical consequences of S-duality of four-dimensional mathcal{N} = 4 super Yang-Mills (SYM) and its string theory dual, based on SL(2, ℤ) spectral theory.We observe that CFT observables \U0001d4aa, invariant under SL(2, ℤ) transformations of a complexified gauge coupling τ, admit a unique spectral decomposition into a basis of square-integrable functions. This formulation has direct implications for the analytic structure of mathcal{N} = 4 SYM data, both perturbatively and non-perturbatively in all parameters. These are especially constraining for the structure of instantons: k-instanton sectors are uniquely determined by the zero- and one-instanton sectors, and Borel summable series around k-instantons have convergence radii with simple k-dependence. In large N limits, we derive the existence and scaling of non-perturbative effects, in both N and the ‘t Hooft coupling, which we exhibit for certain mathcal{N} = 4 SYM observables. An elegant benchmark for these techniques is the integrated stress tensor multiplet four-point function, conjecturally determined by [1] for all τ for SU(N) gauge group; we elucidate its form, and explain how the SU(2) case is the simplest possible observable consistent with SL(2, ℤ)-invariant perturbation theory.These results have ramifications for holography. We explain how leftlangle mathcal{O}rightrangle , the ensemble average of \U0001d4aa over the mathcal{N} = 4 supersymmetric conformal manifold with respect to the Zamolodchikov measure, is cleanly isolated by the spectral decomposition. We prove that the large N limit of leftlangle mathcal{O}rightrangle equals the large N, large ‘t Hooft coupling limit of \U0001d4aa. Holographically speaking, leftlangle mathcal{O}rightrangle = \U0001d4aasugra, its value in type IIB supergravity on AdS5× S5. This result, which extends to all orders in 1/N, embeds ensemble averaging into the traditional AdS/CFT paradigm. The statistics of the SL(2, ℤ) ensemble exhibit both perturbative and non-perturbative 1/N effects. We discuss further implications and generalizations to other AdS compactifications of string/M-theory.

Some fuzzy Korovkin type approximation theorems via power series summability method

This article provides a power series summability-based Korovkin type approximation theorem for any sequence of fuzzy positive linear operators. Using the notion of fuzzy modulus of smoothness, we also derive an associated approximation theorem concerning the fuzzy rate of convergence of these operators. Furthermore, through an example, we illustrate that our summability- based Korovin type theorem has an advantage over the fuzzy Korovkin type theorem proved in the seminal paper by Anastassiou (Stud. Univ. Babeş-Bolyai Math L(4):3–10, 2005)

Center of Distances and Central Cantor Sets

We study a recently discovered metric invariant - the center of distances. The center of distances of a nonempty subset A of a metric space (X,,d) is defined by S(A) :={ alpha in [0,,+infty ): forall xin A exists yin A d(x,,y)=alpha } . Given a nonincreasing sequence (a_{n}) of positive numbers converging to 0, the set E(a_{n}) := left{ xin {mathbb {R}}: exists Asubset {mathbb {N}} x=,sum _{nin A}a_{n}right} is called the achievement set of the sequence (a_{n}). This new invariant is particularly useful in investigating achievability of sets on the real line. We concentrate on computing the centers of distances of central Cantor sets. Any central Cantor set C is an achievement set of exactly one fast convergent series sum a_{n}, and consequently S(C)supset left{ 0right} cup left{ a_{n}:nin {mathbb {N}}right} . We try to check which central Cantor sets have the minimal possible center of distances and which have not.

Partial sums of analytic functions defined by binomial distribution and negative binomial distribution

ABSTRACT The study of statistical distributions in a complex variable is one of the most vibrant areas of research. The complex analogue of many distributions has been studied. This article introduces the complex analogue of Binomial distribution and Negative Binomial distribution and studies certain geometrical properties of these analogues. It comprises the study of analytic functions defined by using the complex versions of Binomial distribution and Negative Binomial distribution functions. It includes the problems of finding the lower bounds of real parts of certain ratios of partial sums to the infinite series sums for these analytic functions. Such lower bounds are determined for the said analytic functions, for their first derivatives and for the Alexander transformation of these analytic functions.

Optimal Two-Sided Estimates on the Interval $$[\pi/2,\pi]$$ of the Sum of the Sine Series with Convex Coefficient Sequence

Optimal Two-Sided Estimates on the Interval $$[\pi/2,\pi]$$ of the Sum of the Sine Series with Convex Coefficient Sequence

On the negative order Cesáro summability of double series with respect to block-orthonormal systems

Abstract Some statements connected with the ( C , - 1 < α < 0 , - 1 < β < 0 ) {(C,-1<\alpha<0,-1<\beta<0)} summability almost everywhere of double series with respect to block-orthonormal systems are given. In particular, using the obtained results, we can determine Weyl multipliers for establishing negative order summability of double series with respect to Δ p , q {\Delta_{p,q}} -ONS.

Trigonometric multiplicative chaos and applications to random distributions

The random trigonometric series \(\sum\nolimits_{n = 1}^\infty {{\rho _n}\cos \left( {nt + {\omega _n}} \right)} \) on the circle \(\mathbb{T}\) are studied under the conditions ∑∣ρn∣2 = ∞ and ρn → 0, where {ωn} are independent and uniformly distributed random variables on \(\mathbb{T}\). They are almost surely not Fourier-Stieltjes series but determine pseudo-functions. This leads us to develop the theory of trigonometric multiplicative chaos, which produces a class of random measures. The kernel and the image of chaotic operators are fully studied and the dimensions of chaotic measures are exactly computed. The behavior of the partial sums of the above series is proved to be multifractal. Our theory holds on the torus \({\mathbb{T}^d}\) of dimension d ⩾ 1.

𝑁-tuple sum analogues for Ramanujan-type congruences

In this paper we prove supercongruence relations for truncated N N -tuple sums of basic hypergeometric series. As an application, we give double, triple, and quadruple sum analogues of some Ramanujan-type supercongruences.

On modified Anderson-Darling test statistics with asymptotic properties

This study proposes two modified Anderson-Darling test statistics that emphasize the upper or lower tails of distribution. Under the finite sample sizes, various models are used to estimate the upper tail percentage points of the proposed test statistic. The limiting distribution of the proposed test statistics is derived with a sum of infinite series. The polynomial approximation, the method of Durbin and Knott and the normal and non normal saddlepoint approximations are used to approximate the limiting distribution of the suggested test statistics. Simulations are used to investigate the accuracy of various approximations. The asymptotic power of the proposed statistics is obtained to compare with various statistics. The proposed test statistics are illustrated by the analysis of real data.

Autoregressive models for time series of random sums of positive variables: Application to tree growth as a function of climate and insect outbreak

We present a broad class of semi-parametric models for time series of random sums of positive variables. Our methodology allows the number of terms inside the sum to be time-varying and is therefore well suited to many examples encountered in the natural sciences. We study the stability properties of the models and provide a valid statistical inference procedure to estimate the model parameters. It is shown that the proposed quasi-maximum likelihood estimator is consistent and asymptotically Gaussian distributed. This work is complemented by simulation results and applied to time series representing growth rates of white spruce (Picea glauca) trees from a few dozen sites in Québec (Canada). This time series spans 41 years, including one major spruce budworm (Choristoneura fumiferana) outbreak between 1968 and 1991. We found significant growth reductions related to budworm-induced defoliation up to two years post-outbreak. Our results also revealed the positive effects of maximum summer temperature, precipitation, and the climate moisture index on white spruce growth. We also identified the negative effects of the climate moisture index in the spring and the maximum temperature of the previous summer. However, the model’s performance on this data set was not improved when the interactions between climate and defoliation on growth were considered. This study represents a major advance in our understanding of budworm–climate–tree interactions and provides a useful tool to project the combined effects of climate and insect defoliation on tree growth in a context of greater frequency and severity of outbreaks coupled with the anticipated increases in temperature.

The Sum of Reduced Harmonic Series Generated by Any Number of Positive Integer Factors

This paper is a free continuation of the author’s previous papers dealing with the sums of the reduced harmonic series generated of reciprocals of all products generated by all prime divisors of the numbers 2002 and 2022, that were inspired by one task on the sum of a special infinite series on the Berkeley Math Circle. We determined the sums of these series, i.e. the series of all the unit fractions that have denominators with only factors consisting of all prime divisors of the numbers 2022 and 2002, analytically and also by calculation in computer algebra system Maple. In this paper, we generalize our considerations and derive two formulas that obviously hold not only to series generated by n prime numbers, but also to series generated by n positive integers.

On logarithmic summability of conjugate series of Fourier series

On logarithmic summability of conjugate series of Fourier series

On the absolute Riesz summability of Fourier series

On the absolute Riesz summability of Fourier series

Cramer's rule for implicit linear differential equations over a non-Archimedean ring

We consider a linear nonhomogeneous $m$-th order differential equation in a ring of formal power series with coefficients from some field of characteristic zero. This equation has infinite many solutions in this ring -- one for each initial condition of the corresponding Cauchy problem. These solutions can be found using classical methods of differential equation theory. Let us suppose the coefficients of the equation and the coefficients of nonhomogeneity belong to some integral domain $K$. We are looking for a solution in the form of a formal power series with coefficients from this integral domain. The methods of classical theory do not allow us to find out whether there exists an initial condition that corresponds to the solution of the coefficients from $K$ and do not allow find this initial condition. To solve this problem, we use the method proposed by U. Broggi. This method allows to find a formal solution of the linear nonhomogeneous differential equation in the form of some special series. In previous articles, sufficient conditions for the existence and uniqueness of a solution were found for a certain class of rings $K$ with a non-Archimedean valuation. If these conditions hold, the formal power series obtained using the Broggi’s method is considered. Its coefficients are the sums of series that converge in the non-Archimedean topology considered. It is shown that this series is the solution from $K[[x]]$ of our equation. Note that this equation over a ring of formal power series can be considered as an infinite linear system of equations with respect to the coefficients of unknown formal power series. In this article it is proved that this system can be solved by some analogue of Cramer's method, in which the determinants of infinite matrices are found as limits of some finite determinants in the non-Archimedean topology.

Relativistic Hypercomputing: Problems and Prospects from the Physicist’s Point of View

The paper presents the main results on hypercomputing based on the use of relativistic effects. Two approaches to the problem are compared – formal-logical and physical. The basis of the physical approach is the study of the metric of curved space-time manifolds on which hypercomputing are realized, obtained either by applying the equivalence principle or by solving Einstein's equations. The properties of Malament-Hogarth spaces arising in these manifolds are discussed. The advantages of the physical approach are shown, which make it possible to verify the possibility of hypercomputing by the example of the problem of calculating the sum of the divergent Dirichlet series for the Riemann zeta function, which requires overcoming the so-called Turing barrier. It is stressed the possibility of using numerical algebras that differ from the field of real numbers, which promises significant progress in the development of modern physical theories first of all in cosmology. The issues of relativistic theory are considered separately. The relativistic solution of the problem of motion with constant acceleration by finding the gravitational potential field of an infinite homogeneous plane is discussed. The solution of this problem by applying the equivalence principle is also discussed. The results are compared with the well-known solutions of V. Fock and R. Tolman.

Near-Optimal Control for Offshore Structures with Nonlinear Energy Sink Mechanisms

To improve the safety and reliability of offshore structures subject to wave loading, the active vibration control problem is always one of significant issues in the field of ocean engineering. This paper deals with the near–optimal control problem of offshore structures with a nonlinear energy sink (NES) mechanism. By taking the dominant vibration mode of the offshore structure with the NES into account, a nonlinear dynamic model of the steel–jacket structure subject to wave loading is presented first. Then, using the parameter perturbation approach to solve a nonlinear two–point boundary value problem, an NES–based optimal controller with the form of infinite series sum is presented to suppress the vibration of the offshore structure. Third, an iteration algorithm is provided to obtain the near–optimal controller. Simulation results demonstrate that the NES–based near–optimal controller can mitigate the oscillation amplitude of offshore structures significantly. Moreover, the NES–based optimal controller outperforms the one based on active tuned mass damper.

On the solvability of nonlocal initial-boundary value problems for a partial differential equation of high even order

In the present paper, two non-local initial-boundary value problems have been formulated for a partial differential equation of high even order with a Bessel operator in a rectangular domain. The correctness of one of the considered problems has been investigated. To do this, applying the method of separation of variables to the problem under consideration, the spectral problem was obtained for an ordinary differential equation of high even order. The self-adjointness of the last problem was proved, which implies the existence of the system of its eigenfunctions, as well as orthonormality and completeness of this system. Further, the Green's function of the spectral problem was constructed, with the help of which it was equivalently reduced to the Fredholm integral equation of the second kind with symmetrical kernel. Using this integral equation and Mercer's theorem, the uniform convergence of some bilinear series depending on found eigenfunctions has been studied. The order of the Fourier coefficients was established. The solution of the considered problem has been written as the sum of a Fourier series with respect to the system of eigenfunctions of the spectral problem. The uniform convergence of this series and also the series obtained from it by term-by-term differentiation was proved. Using the method of spectral analysis, the uniqueness of the solution of the problem was proved. An estimate for the solution of the problem was obtained, from which its continuous dependence on the given functions follows.

A New Result in Form of Finite Triple Sums for a Series from Ramanujan’s Notebooks

We consider a function g(r,x,u) with x,u∈ℂ and r∈ℕ, which, over a symmetric domain, equals the sum of an infinite series as noted in the 16th Entry of Chapter 3 in Ramanujan’s second notebook. The function attracted new attention since it was established to be closely connected to the theory of labelled trees. However, to the best of our knowledge, a closed-form solution allowing, e.g., the rapid computation of g(r,x,u) in Mathematica without explicit use of recursions has been lacking until now. Our proposed formula transforms the part depending on the variable u into a more symmetric form, which then appears inside a finite triple sum consisting of binomials and Stirling numbers of the second kind.