Double Series

On the integrability of double cosine and sine series, I

In the first part of our theses we give sufficient conditions for the absolute convergence of the double Fourier series of f in terms of moduli of continuity, of bounded variation in the sense of Vitali or Hardy and Krause, and of the mixed partial derivative in case f is an absolutely continuous function. Our results extend the classical theorems of Bernstein and Zygmund from single to double Fourier series. In the second part we give sufficient conditions for the absolute convergence of the double Walsh-Fourier series of a function. These sufficient conditions are formulated in terms of (either global or local) dyadic moduli of continuity and s-bounded fluctuation.

We investigate some weighted integer partitions whose generating functions are double-series. We will establish closed formulas for these $q$-double series and deduce that their coefficients are non-negative. This leads to inequalities among integer partitions.

On an existence theory for the double series inverse

For the free models of statistical mechanics on torus, exact asymptotic expansions of the free energy, the internal energy and the specific heat in the vicinity of the critical point are found. It is shown that there is direct relation between the terms of the expansion and the Kronecker's double series. The latter can be expressed in terms of the elliptic theta-functions in all orders of the asymptotic expansion.

A reconfigurable artificial transmission line based on a novel unit cell configuration is presented. The double series (DS) unit cell consists of a series resonator in the series and the shunt branch of the unit cell. Basic theory of the DS line is given by using the transmission line approach. To tune the propagation constant and the characteristic impedance of the artificial transmission line varactors are used. A biasing network for the DS line is designed to apply a DC controlling voltage to the varactors. That way an either right- or left-handed transmission band can be obtained which is demonstrated by simulation and experiment.

We study composition operators on spaces of double Dirichlet series, focusing our interest on the characterization of the composition operators of the space of bounded double Dirichlet series $${\mathcal {H}}^\infty ({\mathbb {C}}_+^2)$$ . We also show how the composition operators of this space of Dirichlet series are related to the composition operators of the corresponding spaces of holomorphic functions. Finally, we give a characterization of the superposition operators in $${\mathcal {H}}^\infty ({\mathbb {C}}_+)$$ and in the spaces $${\mathcal {H}}^p$$ .

Leakage inductance is one of the key parameters of a transformer, and it is often intentionally integrated into transformers. Rogowski’s equation is generally used for leakage inductance calculation; however, it is only applicable to concentric winding transformers where windings have the same height. Consequently, it has limited applications. This paper proposes a transformer leakage inductance calculation method using a double Fourier series. The limitation of Rogowski’s leakage inductance equation was analyzed in practical applications, and a new model for calculating the leakage inductance of a double-group-overlapping winding transformer was derived. Experimental prototypes of transformers were developed, and their simulation models were built in Ansys. The correctness of the proposed calculation method for transformer leakage inductance using a double Fourier series was verified by comparing the calculation results with the simulation and measured ones. A sensitivity analysis was also conducted by studying the variations in different parameters that might affect the leakage inductance value. The proposed calculation model gives an intuitive and simple method with less calculation and design effort while maintaining reasonable accuracy for leakage inductance calculation. In addition, the featured double Fourier series approach has a wider range of applications than Rogowski’s equation.

We study the double shuffle relations satisfied by the double zeta values of level 2, and introduce the double Eisenstein series of level 2 which satisfy the double shuffle relations. We connect the double Eisenstein series to modular forms of level 2.

This paper is concerned with the convergence of double trigonometric series and Fourier series. Since the beginning of the 20th century, many authors have studied on those series. Also, Ferenc <TEX>$M{\acute{o}}ricz$</TEX> has studied the convergence of double trigonometric series and double Fourier series so far. We consider <TEX>$L^p(T^2)$</TEX>-convergence results focused on the Ferenc <TEX>$M{\acute{o}}ricz^{\prime}s$</TEX> studies from the second half of the 20th century up to now. In section 2, we reintroduce some of Ferenc <TEX>$M{\acute{o}}ricz^{\prime}s$</TEX> remarkable theorems. Also we investigate his several important results. In conclusion, we investigate his research trends and the simple minor genealogy from J. B. Joseph Fourier to Ferenc <TEX>$M{\acute{o}}ricz$</TEX>. In addition, we present the research minor lineage of his study on <TEX>$L^p(T^2)$</TEX>-convergence.

Double Agents

Effects of high temperature generator pressure on energetic and exergetic performance of double effect series flow type absorption cooling system

The sufficient and necessary conditions on the sequence Λ = {λn} are found for the uniformly convergence of Cesàro means of negative order of cubic partial sums of double Walsh-Fourier series of functions of bounded partial Λ-variation.

A solution is presented for the general, wave-interaction problem of steady, irrotational, homentropic flow of a perfect gas. It can be interpreted as a convergent process of successive approximations, based on the solution of linearized theory, for shock-free flow. It constitutes an approximate solution for flow with weak shocks. In part I, the equations of motion are transformed into linear equations in characteristic variables. Their solution is of different type according to whether the flow is near-sonic, hypersonic, or in between these extremes. Special attention is given to the types of boundary condition which occur in physical problems, and solution methods are devised to cope with these types in the medium Mach number range. The method of Riemann functions is used to calculate accurately the pressure distribution in the first interaction region of a jet expanding from a perfect nozzle. It is shown by the help of double power series that shock waves will always occur in the first period of such a jet, even for pressure ratios arbitrarily near unity. The Riemann function approach is also shown to be suitable for the approximate calculation of the flow past aerofoils of prescribed shape; when the requirements of accuracy are exacting, the method of double power series expansion presents the problem in a form suited to high-speed digital computers.

We present the Olsson.wl Mathematica package which aims to find transformations for some classes of multivariable hypergeometric functions. It is based on a well-known method developed by P. O. M. Olsson in J. Math. Phys. 5, 420 (1964) to derive the analytic continuations of the Appell F_1 double hypergeometric series using the linear transformations of the Gauss _2F_1 hypergeometric function. We provide a brief description of the method of Olsson and demonstrate the use of the commands of the Olsson.wl package using some examples. In particular, we reproduce various results of the literature on multivariable hypergeometric functions and show practical applications of this package in the derivation of novel formulas. In the context of high energy physics, we also demonstrate how it can be used to disentangle some known results about the analytic continuation of some series representations of the one-loop pentagon in multi-Regge kinematics and D=6-2ϵ. We also present companion package, called ROC2.wl, which is dedicated to the derivation of the regions of convergence of double hypergeometric series. This package can be used independently of Olsson.wl.