Double Sum Research Articles

The Husimi function of a semiconfined harmonic oscillator model with a position-dependent effective mass

In this paper, the phase space representation for a semiconfined harmonic oscillator model with a position-dependent effective mass is constructed. We have found the Husimi distribution function for the stationary states of the oscillator model under consideration for both cases without and with the applied external homogeneous field. The obtained function is expressed through the double sum of the parabolic cylinder function. Different special cases and the limit relations are discussed, too.

Entanglement of electrons and nuclei: A most compact representation of the molecular wave function

Ultrafast pumping displaces both electrons and nuclei from equilibrium so that the wave function is a double sum of separable terms for the dynamics of the electrons and nuclei. We convert the double sum into a single one by a matricization of the wave function and then generate an exact separable expression for the entangled molecular wave function. A most compact approximation with a minimum number of terms is obtained via Singular Value Decomposition. LiH and N2 are used as an illustration: the two differ in their adiabatic electronic dynamics in the energy range accessible by a UV pulse.

Asymptotic Evaluation of a Lattice Sum Associated with the Laplacian Matrix

The Laplacian matrix is of fundamental importance in the study of graphs, networks, random walks on lattices, and arithmetic of curves. In certain cases, the trace of its pseudoinverse appears as the only non-trivial term in computing some of the intrinsic graph invariants. Here we study a double sum Fn which is associated with the trace of the pseudo inverse of the Laplacian matrix for certain graphs. We investigate the asymptotic behavior of this sum as n → ∞. Our approach is based on classical analysis combined with asymptotic and numerical analysis, and utilizes special functions. We determine the leading-order term, which is of size n2 log n, and develop general methods to obtain the secondary main terms in the asymptotic expansion of Fn up to errors of $$\mathcal {O}({\rm{log}} \; n)$$ and $$\mathcal {O}(1)$$ as n → ∞. We provide some examples to demonstrate our methods.

Three-parameter mock theta functions

Mock theta functions were first introduced by Ramanujan. Historically, mock theta functions can be represented as Eulerian forms, Appell-Lerch sums, Hecke-type double sums, and Fourier coefficients of meromorphic Jacobi forms. In this paper, in view of the q-Zeilberger algorithm and the Watson–Whipple transformation formula, we establish five three-parameter mock theta functions in Eulerian forms, and express them by Appell–Lerch sums. Especially, the main results generalize some two-parameter mock theta functions. For example, setting (m,q,x)→(1,q1/2,xq−1/2) in∑n=0∞(−q2;q2)nqn2+(2m−1)n(xqm,x−1qm;q2)n+1, we derive the universal mock theta function g2(x,q).

Wearable Dual Polarized Electromagnetic Knee Imaging System.

With the increasing uptake of sport activities, onsite detection of associated knee injuries at early stages is in high demand to avoid severe ligament tear and long treatment period. Portable electromagnetic imaging (EMI) systems have the potential to meet that demand, but there are challenges. For example, EMI is based on the contrast in the dielectric properties due to the accumulated fluid after knee injury. However, that fluid can be in any shape and orientation. Therefore, to capture enough data for processing, EMI should operate as a dual-polarized wearable system with compact antennas. Thus, the proposed system is a textile brace worn on the knee and consists of an 8-element dual-polarized aperture antenna array, which is matched with the knee. Each of the utilized antennas is fed by two orthogonal coaxial feed, occupies a small size of 36 ×36 ×3.1 mm3, and is backed by a full ground plane for unidirectional radiation. The antenna covers the band 0.7-3.3 GHz (130%), with front to back ratio of more than 10 dB. The textile wool-felt is used as the substrate to enable building flexible brace system. The system's capability to reconstruct knee images with different injuries is verified on realistic knee models and phantoms. The double stage delay, multiply and sum algorithm (DS-DMAS) is used to reconstruct those images, which demonstrate the efficiency of the dual-polarized system and its superiority over single-polarized systems.

Supremum of the Euclidean Norms of the Multidimensional Wiener Process and Brownian Bridge: Sharp Asymptotics of Probabilities of Large Deviations

For T > 0, we prove theorems concerning sharp asymptotics of the probabilities$$ \mathbf{P}\left\{\underset{t\in \left[0,T\right]}{\sup}\sum \limits_{j=1}^n{w}_j^2(t)>{u}^2\right\},\kern1em \mathbf{P}\left\{\underset{t\in \left[0,T\right]}{\sup}\sum \limits_{j=1}^n{w}_{j0,T}^2(t)>{u}^2\right\}, $$as u → ∞, where wj(t), j = 1, . . . , n, are independent Wiener processes and wj0,T (t), j = 1, . . . , n, are independent Brownian bridges on the segment [0, T]. Our research method is the double sum method for the Gaussian processes and fields. We also give an application of the obtained results to the statistical tests for the homogeneity hypothesis of k one-dimensional samples.

On the Maximum of a Gaussian Process with Unique Maximum Point of its Variance

Gaussian random processes which variances reach theirs maximum values at unique points are considered. Exact asymptotic behaviors of probabilities of large absolute maximums of theirs trajectories have been evaluated using Double Sum Method under the widest possible conditions.

The binary Gold function and its c-boomerang connectivity table

Here, we give a complete description of the entire c-Boomerang Connectivity Table for the Gold function over finite fields of even characteristic, by using double Weil sums. As a by-product, we generalize a result of Boura and Canteaut (IACR Trans. Symmetric Cryptol. 2018(3) : 290–310, 2018) for the classical boomerang uniformity (see also the extended abstract by Eddahmani and Mesnager at the Boolean Functions and their Applications (BFA 2021) conference).

Optimal condition-based warranty policy for multi-state products with three guarantees service

ABSTRACT Within a certain period of time after products are sold, many manufacturers offer free repairing, replacing and refunding for the defective products, referred to as three guarantees service. Under the assumption that the working states of the product can be partitioned into three non-overlapping sets and its degradation can be modeled by homogeneous Markov chain, a three-stage and conditioned-based warranty model is proposed. Under the model, the whole warranty period is divided into a compulsory replacement period, a condition-based maintenance period, and a minimal repair period. Over the condition-based maintenance period, depending on the state set that the working state before the failure belongs to, failed products are rectified minimally, imperfectly, replaced by an identical one and the warranty terms are renewed, respectively. Markov process and renewal process theories are used to analyze the failure process and the warranty servicing cost over the whole warranty period. Based on double Riemann sum, a recursive algorithm for finding the optimal partition method of working states and the warranty period is proposed. A numerical example is given to illustrate the validity of the warranty strategy.

An improved stability criterion for discrete-time time-delayed Lur'e system with sector-bounded nonlinearities

The absolute stability problem of discrete-time time-delayed Lur'e systems with sector bounded nonlinearities is investigated in this paper. Firstly, a modified Lyapunov-Krasovskii functional (LKF) is designed with augmenting additional double summation terms, which complements more coupling information between the delay intervals and other system state variables than some previous LKFs. Secondly, some improved delay-dependent absolute stability criteria based on linear matrix inequality form (LMI) are proposed via the modified LKF and the relaxed free-matrix-based summation inequality technique application. The stability criteria are less conservative than some results previously proposed. The reduction of the conservatism mainly relies on the full use of the relaxed summation inequality technique based on the modified LKF. Finally, two common numerical examples are presented to show the effectiveness of the proposed approach.

A note on the coefficients of power sums of arithmetic progressions

In this note we show a simple formula for the coefficients of the polynomial associated with the sums of powers of the terms of an arbitrary arithmetic progression. This formula consists of a double sum involving only ordinary binomial coefficients and binomial powers. Arguably, this is the simplest formula that can probably be found for the said coefficients. Furthermore, we give an explicit formula for the Bernoulli polynomials involving the Stirling numbers of the first and second kind.

Two q-congruences involving double basic hypergeometric sums

Two q-congruences involving double basic hypergeometric sums

Analytic Continuations of Character and Alternating Tornheim Zeta Functions

A method developed by Crandall is used to analytically continue character and alternating analogues of a double sum in three complex variables, known as the Tornheim or Mordell–Tornheim–Witten zeta function. We evaluate these functions at triples of nonpositive integer arguments in terms of generalized Bernoulli numbers, and determine those for which a given function is zero. Some special cases are related to tangent numbers.

Some new mock theta functions

Mock theta functions can be represented by Eulerian forms, Hecke-type double sums, Appell–Lerch sums, and Fourier coefficients of meromorphic Jacobi forms. In view of some q-series identities, we establish three two-parameter mock theta functions, and express them in terms of Appell-Lerch sums. In addition, we find three mock theta functions in Eulerian forms. Then in light of their Hecke-type double sums, we provide the Hecke-type double sums for some third order mock theta functions, and establish the relations between these functions and some classical mock theta functions.

On double sum generating functions in connection with some classical partition theorems

We focus on writing double sum representations of the generating functions for the number of partitions satisfying some gap conditions. Some example sets of partitions to be considered are partitions into distinct parts and partitions that satisfy the gap conditions of the Rogers–Ramanujan, Göllnitz–Gordon, and little Göllnitz theorems. We refine our representations by imposing a bound on the largest part and find finite analogues of these new representations. These refinements lead to many q-series and polynomial identities. Additionally, we present a different construction and a double sum representation for the products similar to the ones that appear in the Rogers–Ramanujan identities.

A Novel Path-Following-Method-Based Polynomial Fuzzy Control Design.

This article presents a novel path-following-method-based polynomial fuzzy control design. By examining the stabilization problem, the nonconvex stabilization criterion represented in terms of bilinear sum-of-squares (SOS) constraints is proposed to complement the existing convex stabilization criteria. Based on the polynomial Lyapunov function and considering the operation domain, the stabilization control is designed with a systematic region of attraction (ROA) analysis method. Since the proposed stabilization criterion remains in nonconvex form, the conservativeness caused by the transformation from nonconvex (bilinear SOS) constraints into convex (SOS) constraints can be avoided. Moreover, the restriction on the Lyapunov function candidates for the convex transformation in the literature does not exist in the proposed nonconvex stabilization criterion. The stabilization analysis for polynomial fuzzy control systems is concerned with the double fuzzy summation problem that can be treated as the copositivity problem. Therefore, the SOS-based copositive relaxation technique is applied for the proposed stabilization criterion. Since the proposed nonconvex stabilization criterion is represented in terms of bilinear SOS constraints, the path-following method is employed for solving the bilinear SOS problem. Finally, design examples are provided to demonstrate that the proposed nonconvex stabilization criterion complements the existing convex stabilization criteria.

An Euler-MacLaurin formula for polygonal sums

We prove an Euler-Maclaurin formula for double polygonal sums and, as a corollary, we obtain approximate quadrature formulas for integrals of smooth functions over polygons with integer vertices. Our Euler-Maclaurin formula is in the spirit of Pick's theorem on the number of integer points in an integer polygon and involves weighted Riemann sums, using tools from Harmonic analysis. Finally, we also exhibit a classical trick, dating back to Huygens and Newton, to accelerate convergence of these Riemann sums.

Straightforward path to Zernike polynomials

Starting from Weierstrass’ approximation theorem, Zernike polynomials are obtained by a few straightforward steps involving only the recast of the aberration function as a double sum in the polar coordinates followed by the weighted orthogonalization of a power series. The origin of the name Fringe Zernike polynomials is also explained.

Simulation of ergodic multivariate stochastic processes: An enhanced spectral representation method

The spectral representation method (SRM) has been extensively used in the simulation of multivariate stationary Gaussian stochastic processes. In this study, an enhanced SRM for the simulation of ergodic multivariate stochastic processes is developed. Through shifting a frequency increment, the enhanced SRM offers a faster convergence rate of probabilistic characteristics in comparison with conventional SRM. The computational efficiency is enhanced as the Cholesky decomposition is only required at single-indexed frequencies. Additionally, a two-dimensional fast Fourier transform (FFT) algorithm is proposed to expedite simultaneously estimation of the double summation of cosine functions in both frequency and space dimensions. Accordingly, compared to the traditional FFT-based SRM, in which FFT is only used for the summation in the frequency dimension, the simulation efficiency is further enhanced significantly. The numerical example concerning the simulation of wind velocity field demonstrates that the proposed method offers a faster convergence rate and a higher level of efficiency.

Ruin Probability for a Gaussian Process with Variance Attaining its Maximum on Discrete Sets

Ruin probability for a Gaussian locally stationary process is considered in the case where the process variance attains its maximum in a finite number of points. The double sum method is applied to calculate exact asymptotics of the corresponding probability. Also, we consider a family of processes with variance that has a countable set of maximum points containing a limit point.