Double Sum Research Articles

New mock theta functions and formulas for basic hypergeometric series

AbstractIn recent years, mock theta functions in the modern sense have received great attention to seek examples of q-hypergeometric series and find their alternative representations. In this paper, we discover some new mock theta functions and express them in terms of Hecke-type double sums based on some basic hypergeometric series identities given by Z.G. Liu.

Stability analysis of discrete-time systems with time-varying delay via a delay-dependent matrix-separation-based inequality

This paper is concerned with the stability analysis of discrete-time systems with a time-varying delay. Different from the recent related works that reduce the conservatism by increasing the complexity of Lyapunov functions, this paper aims at conservatism-reduction by using relatively simple Lyapunov functions with an augmented double summation term. The key point for achieving this task is that a delay-dependent matrix-separation-based inequality is established to estimate the augmented-type summation term arising in the forward difference of Lyapunov functions. The proposed method provides a general form of inequalities, which realizes the effective reduction of estimation gap, and also introduces several tractable delay-dependent matrices, which achieves the full use of available delay-related information. As a result, for two types of time-varying delays, the usage of the proposed inequality under different separations leads to two less conservative and low complex stability criteria, whose advantages are demonstrated via three examples.

On squares of Fourier coefficients twist exponential functions with applications

Let f be a Hecke–Maass cusp form of weight zero for [Formula: see text] and [Formula: see text] be the nth Fourier coefficient. For almost all [Formula: see text] we have [Formula: see text] which improves the result of Acharya [Exponential sums of squares of Fourier coefficients of cusp forms, Proc. Indian Acad. Sci. Math. Sci. 130 (2020) 24], who showed an upper bound larger than [Formula: see text] For all [Formula: see text] of type [Formula: see text] we also show that [Formula: see text] where [Formula: see text] This result relies heavily on a generalized double sum (see Theorem 1.3).

Axial coupling constant in a magnetic background

The nucleon axial coupling constant ${g}_{A}$ is calculated in the presence of an external uniform magnetic field using finite energy sum rules. The correlation function of proton, neutron and axial-vector currents is calculated both the hadronic and the QCD sector. Once the axial contribution in the form factor is isolated, a double sum rule is considered, i.e., the usual QCD contour integration for the external moments of both the proton and neutron current. The effects of the external magnetic field come mainly through the in-medium current-nucleon coupling constants ${\ensuremath{\lambda}}_{N}(B)$ and the hadronic thresholds ${s}_{0}(B)$ provided by the nucleon-nucleon correlators. As a result, the axial coupling constant decreases in the presence of the magnetic field.

Nonfragile extended dissipativity state estimator design for discrete-time neural networks with time-varying delay

This paper studies the issue of nonfragile extended dissipativity state estimator design for discrete-time neural networks with time-varying delay. Firstly, a class of nonfragile proportional-integral state estimator with exponential type linear polynomial on time-varying delay is presented, which unifies the Arcak state estimator and the Luenberger state estimator as its special cases. Since the exponential type linear polynomial on time-varying delay is considered, more information of time delay can be utilized and the estimation error convergence rate can be adjusted. Secondly, an augmented Lyapunov–Krasovskii functional tailored for delayed discrete-time neural networks is presented, in which the double summation terms of the state vector and the information of the output estimation error are fully considered in the corresponding augmented vector, then a sufficient criterion that guarantees the nonfragile extended dissipativity state estimation for delayed discrete-time neural networks is derived. Furthermore, since an extended dissipativity performance level is introduced, the issues of passitivity state estimation, H∞ state estimation, L2-L∞ state estimation, and (Q,S,R)-γ dissipativity state estimation can be solved in a unified framework. Finally, simulation results are given to demonstrate the advantage of the presented method.

О суммировании 3j-символов Вигнера

Within the framework of the theory of irreducible tensor operators, using well -known general analytical results for double sums of products of two 3j-Wigner symbols, analytical expressions for single j m sums m for the valuesj1 j2 1 andj 2 parameters of the upper row 3j Wigner symbol are specified.The expressions obtained supplement the well-known analytical results of the theory of angular momentum and are in demand in solving, in particular, such problems of atomic physics as the construction of a nonrelativistic quantum theory of single and double bremsstrahlung when a photon is scattered by an atom (atomic ion).

Quantum modular forms from real-quadratic double sums

Abstract In 2015, Lovejoy and Osburn discovered 12 q-hypergeometric series and proved that their Fourier coefficients can be understood as counting functions of ideals in certain quadratic fields. In this paper, we study their modular and quantum modular properties and show that they yield three vector-valued quantum modular forms on the group $\Gamma_0 (2)$.

First moment of central values of quadratic Hecke L-functions in the Gaussian field

We evaluate the smoothed first moment of central values of a family of quadratic Hecke [Formula: see text]-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size [Formula: see text] under the generalized Riemann hypothesis. The same approach also allows us to obtain asymptotic formulas for all [Formula: see text], [Formula: see text] for a smoothed double character sum involving [Formula: see text], where [Formula: see text] denotes the quadratic symbol in the Gaussian field.

Mean value of real Dirichlet characters using a double Dirichlet series

AbstractWe study the double character sum $\sum \limits _{\substack {m\leq X,\\m\mathrm {\ odd}}}\sum \limits _{\substack {n\leq Y,\\n\mathrm {\ odd}}}\left (\frac {m}{n}\right )$ and its smoothly weighted counterpart. An asymptotic formula with power saving error term was obtained by Conrey, Farmer, and Soundararajan by applying the Poisson summation formula. The result is interesting because the main term involves a non-smooth function. In this paper, we apply the inverse Mellin transform twice and study the resulting double integral that involves a double Dirichlet series. This method has two advantages—it leads to a better error term, and the surprising main term naturally arises from three residues of the double Dirichlet series.

General spherical harmonic bra-ket overlap integrals of trigonometric functions

Closed formulas in terms of double sums of Clebsch–Gordan coefficients are computed for the evaluation of bra-ket spherical harmonic overlap integrals of a wide class of trigonometric functions. These analytical expressions can find useful application in problems involving non-separable wave equations, e.g. general-relativistic perturbation theory, electromagnetism, quantum theory, etc, wherein the overlap integrals arise from the coupling among different angular modes. We provide some examples related to linear perturbations of spinning black holes in general relativity and modified gravity, in which the analytical formulas for the overlap integrals are particularly useful to compute the black-hole spectrum.

Poly-Cauchy Numbers with Higher Level

In this article, mainly from the analytical aspect, we introduce poly-Cauchy numbers with higher levels (level s) as a kind of extensions of poly-Cauchy numbers with level 2 and the original poly-Cauchy numbers and investigate their properties. Such poly-Cauchy numbers with higher levels are yielded from the inverse relationship with an s-step function of the exponential function. We show such a function with recurrence relations and give the expressions of poly-Cauchy numbers with higher levels. Poly-Cauchy numbers with higher levels can be also expressed in terms of iterated integrals and a combinatorial summation. Poly-Cauchy numbers with higher levels for negative indices have a double summation formula. In addition, Cauchy numbers with higher levels can be also expressed in terms of determinants.

An Improved Unknown Input Observer Design for Disrete-Time Fuzzy Bilinear Systems

Design methods of an observer for a discrete-time Takagi-Sugeno fuzzy bilinear system with unknown inputs is proposed. Takagi-Sugeno fuzzy bilinear system describes a large class of nonlinear systems and this system with unknown inputs even represents a realistic model for most practical systems. An unknown input observer that estimates the state of Takagi-Sugeno fuzzy bilinear system with unknown input is proposed based on multiple Lyapunov matrices. For fuzzy systems, a parallel distributed observer (PDO), which is constructed with of local linear observers and the appropriate grade of the membership functions, has been used, but its design conditions are eventually conservative. To reduce such conservativeness in observer design conditions, non-PDO with multiple Lyapunov matrices technique is proposed in this paper. Such a design method is based on a multiple Lyapunov function with double sum of the membership functions. In order to show the effectiveness of our proposed approach, a simple example is given. Finally, concluding remarks are given.

Hecke-type double sums and mock theta functions

Mock theta functions were first introduced by Ramanujan in his last letter to Hardy. In the literature, Hecke-type double sums for the classical mock theta functions play an important role in proving mock theta conjectures. In this paper, with the aid of

Multi-sum Rogers-Ramanujan type identities

We use an integral method to establish a number of Rogers-Ramanujan type identities involving double and triple sums. The key step for proving such identities is to find some infinite products whose integrals over suitable contours are still infinite products. The method used here is motivated by Rosengren's proof of the Kanade-Russell identities.

New Designing Mixed Double Moving Average-Cumulative Sum Control Chart for Detecting Mean Shifts with Symmetrically and Asymmetrically Distributed Observations

Herein, we present a new control chart called the mixed double moving average-cumulative sum control chart (DMA-CUSUM: MDC) used for detecting shifts in the process mean when symmetrically and asymmetrically distributed. The performance of the MDC chart is compared with shewhart, cumulative sum (CUSUM), double moving average (DMA) and mixed cumulative sum-double moving average (CUSUM-DMA: MCD) control charts by using average run length (ARL) and median run length (MRL) with the monte carlo simulation (MC). The research results show that the proposed (MDC) control chart was more efficient than the Shewhart, CUSUM, DMA and MCD charts for all distributions tested. We apply the MDC chart to real sets of data: I) the tensile data of single carbon fiber and II) the survival times of guinea pigs infected with virulent bacilli.

Finite summation formulas for the multivariable $A$-function

The object of this paper is to evaluate some finite double summations relations for the multivariable $A$-function using the summation of a double hypergeometric series. The formulas derived in this paper are most general in character, we also provide a few particular cases for derived summation formulas.

Simulation of stationary and nonstationary wind velocity field along a long-span bridge using a numerical truncation method

The spectral representation method (SRM) has been extensively utilized to generate inhomogeneous stochastic wind velocity fields of long-span bridges. However, the simulation process usually needs a large number of computing resources, particularly when plenty of simulation points/samples are required. To address this issue, this study proposes a numerical truncation approach with the aid of a closed-form solution of Cholesky decomposition regarding the coherence matrix. First, the double summation of SRM is optimized as two separate inner and outer summations. Then, the closed-form solution of Cholesky decomposition is cut off to simplify the inner summation by setting an appropriate threshold. Since this inner summation dispenses with the participation of the truncated elements, the computational efforts are reduced significantly. As for the outer summation, it can be fast completed via introducing the fast Fourier transform technique. Finally, numerical examples are used to verify the effectiveness of the proposed method in terms of accuracy and efficiency. The results show that this method possesses a satisfactory accuracy and high efficiency with a given threshold. Therefore, it presents a high potential in simulating large-scale stochastic fluctuating wind velocity fields.

New proofs of some double sum Rogers–Ramanujan type identities

Recently, Rosengren utilized an integral method to prove a number of conjectural identities found by Kanade and Russell. Using this integral method, we give new proofs to some double sum identities of Rogers–Ramanujan type. These identities were earlier proved by approaches such as combinatorial arguments or using q-difference equations. Our proofs are based on streamlined calculations, which relate these double sum identities to some known Rogers–Ramanujan type identities with single sums. Moreover, we prove a conjectural identity of Andrews and Uncu which was earlier confirmed by Chern.

The fourth moment of Dirichlet L-functions at the central value

The asymptotic formula of the fourth moment of Dirichlet L-functions at the central value was predicted in a conjecture by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, and the prime moduli case was proved by M. P. Young in 2011. This work establishes this asymptotic formula for general moduli. The work relies on the study of a special divisor sum function, called \({\mathcal {D}}_q\)-function, which plays a key role in deducing the main terms. Another key ingredient is an application of new bounds for double sums in Kloosterman sums, applied in the estimate of the error terms.

Descriptor Representation-Based Guaranteed Cost Control Design Methodology for Polynomial Fuzzy Systems

This paper presents a descriptor representation-based guaranteed cost design methodology for polynomial fuzzy systems. This methodology applies the descriptor representation for presenting the closed-loop system of the polynomial fuzzy model with a parallel distributed compensation (PDC) based fuzzy controller. By the utility of descriptor representation, the guaranteed cost control (GCC) design analysis can utilize polynomial fuzzy slack matrices for obtaining less conservative results. The proposed GCC design is presented as the sum-of-squares (SOS) conditions. The application of polynomial fuzzy slack matrices leads to the double fuzzy summation issue in the control design. Accordingly, the copositive relaxation works out the problem well and is adopted in the control design analysis. The GCC design minimizes the upper limit of a predesignated cost function. According to the performance function, two simulation examples are provided to demonstrate the validity of the proposed GCC design. In these two examples, the proposed design obtains superior results.