Double Sum Research Articles

Fast Minimum Error Entropy for Linear Regression

The minimum error entropy (MEE) criterion finds extensive utility across diverse applications, particularly in contexts characterized by non-Gaussian noise. However, its computational demands are notable, and are primarily attributable to the double summation operation involved in calculating the probability density function (PDF) of the error. To address this, our study introduces a novel approach, termed the fast minimum error entropy (FMEE) algorithm, aimed at mitigating computational complexity through the utilization of polynomial expansions of the error PDF. Initially, the PDF approximation of a random variable is derived via the Gram–Charlier expansion. Subsequently, we proceed to ascertain and streamline the entropy of the random variable. Following this, the error entropy inherent to the linear regression model is delineated and expressed as a function of the regression coefficient vector. Lastly, leveraging the gradient descent algorithm, we compute the regression coefficient vector corresponding to the minimum error entropy. Theoretical scrutiny reveals that the time complexity of FMEE stands at O(n), in stark contrast to the O(n2) complexity associated with MEE. Experimentally, our findings underscore the remarkable efficiency gains afforded by FMEE, with time consumption registering less than 1‰ of that observed with MEE. Encouragingly, this efficiency leap is achieved without compromising accuracy, as evidenced by negligible differentials observed between the accuracies of FMEE and MEE. Furthermore, comprehensive regression experiments on real-world electric datasets in northwest China demonstrate that our FMEE outperforms baseline methods by a clear margin.

New convolved Fibonacci collocation procedure for the Fitzhugh–Nagumo non-linear equation

Abstract This article is dedicated to propose a spectral solution for the non-linear Fitzhugh–Nagumo equation. The proposed solution is expressed as a double sum of basis functions that are chosen to be the convolved Fibonacci polynomials that generalize the well-known Fibonacci polynomials. In order to be able to apply the proposed collocation method, the operational matrices of derivatives of the convolved Fibonacci polynomials are introduced. The convergence and error analysis of the double expansion are carefully investigated in detail. Some new identities and inequalities regarding the convolved Fibonacci polynomials are introduced for such a study. Some numerical results, along with some comparisons, are provided. The presented results show that our proposed algorithm is efficient and accurate.

Rogers–Ramanujan type identities involving double, triple and quadruple sums

Rogers–Ramanujan type identities involving double, triple and quadruple sums

Eulerian series as Hecke-type double sums through the transformation formulas

Recently, establishing the Hecke-type double sums forms for Eulerian series has received a great deal of attention to seek special cases of [Formula: see text]-hypergeometric series. In this paper, with the aid of some basic hypergeometric series identities given by Chen and Wang, and Liu, we discover many new Eulerian series and express them in terms of Hecke-type double sums and Appell–Lerch sums. Based on the modern sense for the mock theta functions, we find that some of these series are mock theta functions and the others are mixed mock theta functions. Furthermore, from the viewpoint of Hecke-type double sums, we find three string function identities with levels [Formula: see text], [Formula: see text], and [Formula: see text].

Directional Spectrum Estimation For Sea States Generated By The Single Summation Method

The influence of directional spreading of waves is significant for wave-induced loads, wave breaking and nonlinearity of the waves. For physical model testing performed at test facilities such as the Ocean and Coastal Engineering Laboratory at Aalborg University, it is crucial to validate if the test conditions match the target sea states by measurement and analysis of the generated directional wave field. Most of the existing methods assumes a double summation sea state to be present which is valid in the prototype. However, waves in the laboratory are usually generated by single summation. The current paper presents a method to analyse short-crested waves generated by the single summation method. Compared to similar methods oblique reflections are considered instead of only in-line reflections. The results show that the method successfully decomposes the incident and reflected wave fields in the time domain. Thus, for example the incident wave height distribution may be obtained. The sensitivity of the new method to additional reflective directions, noise, calibration errors and positional errors of the wave gauges was found small.

An efficient polar cubic equation of state for predictive modeling of phase behavior and critical phenomena of mixtures

AbstractA polar cubic equation of state (EOS) is developed by incorporating the dipolar theory of Jog and Chapman into the Soave‐Redlich‐Kwong (SRK) EOS. We propose simplifying assumptions in the dipolar term of Jog and Chapman to reduce the double and triple sums in the theory to single sums. The simplified version of the dipolar theory can significantly improve computational speed and can be used with either Cubic EOS or SAFT‐based EOS. The proposed model, which we call Simplified Polar SRK (SP‐SRK), is parametrized in a similar fashion to classical cubic EOS to exactly reproduce , and will self‐consistently reduce to the base SRK EOS in the absence of polar interactions. Binary VLE data with a non‐polar reference hydrocarbon is used to extract the polarity of the respective functional group. The model shows superior performance in capturing the phase behavior of polar mixtures compared to the base SRK.

Some Mean Convergence Theorems for the Maximum of Normed Double Sums of Banach Space Valued Random Elements

Some Mean Convergence Theorems for the Maximum of Normed Double Sums of Banach Space Valued Random Elements

Double and triple character sums and gaps between the elements of subgroups of finite fields

For an odd prime [Formula: see text], let [Formula: see text] be the finite field of [Formula: see text] elements. The main purpose of this paper is to establish new results on gaps between the elements of multiplicative subgroups of finite fields. For any [Formula: see text], we also obtain new upper bounds of the following double character sum [Formula: see text] and a triple character sum [Formula: see text] with [Formula: see text] and multiplicative subgroups [Formula: see text] of order [Formula: see text] and [Formula: see text], respectively.

On the density hypothesis for $L$-functions associated with holomorphic cusp forms

We study the range of validity of the density hypothesis for the zeros of L -functions associated with cusp Hecke eigenforms f of even integral weight, and prove that N_{f}(\sigma, T) \ll T^{2(1-\sigma)+\varepsilon} holds for \sigma \geq 1407/1601 . This improves upon a result of Ivić, who had previously shown the zero-density estimate in the narrower range \sigma\geq 53/60 . Our result relies on an improvement of the large value estimates for Dirichlet polynomials based on mixed moment estimates for the Riemann zeta function. The main ingredients in our proof are the Halász–Montgomery inequality, Ivić’s mixed moment bounds for the zeta function, Huxley’s subdivision argument, Bourgain’s dichotomy approach, and Heath-Brown’s bound for double zeta sums.

Numerical simulation and experimental study of dip angle spray film thickness under elliptical double Gaussian sum model

To address the problem of modeling the growth rate of coating film thickness when spraying at inclination angle, based on Gaussian sum model, it is proposed to use the elliptic double Gaussian sum model to establish the cumulative model of coating film thickness when spraying at static inclination angle of the spray gun. The differential geometry amplification theorem is used to establish the coating growth rate model with the spraying inclination angle as the variable; after that, the static inclination spraying experiments are carried out, and the coating thickness data of the sampling points are recorded through the spraying experiments, and the Levenberg-Maquart algorithm is used for the least-squares fitting of the model, which results in the static spraying film thickness distribution model. Finally, compared with the elliptic double β model, the fitting accuracy of the elliptic double Gaussian sum model is 6.3% higher than that of the elliptic double β model when spraying at inclination angle by comparing the R-square values, and the elliptic double Gaussian sum model is more capable of obtaining a better fitting accuracy, which further confirms the validity and practicability of the model.

Sparse sets that satisfy the prime number theorem

For arbitrary real t>1 we examine the set {⌊x/nt⌋:n≤x}. Asymptotic formulas for the cardinality of this set and the number of primes in this set are given. The prime counting result uses an alternate Vaughan's decomposition for the von Mangoldt function, with triple exponential sums instead of double exponential sums. These sets are the sparsest known sets that satisfy the prime number theorem, in the sense that the number of primes is asymptotically given by the cardinality of the set divided by the natural logarithm of the cardinality of the set.

Explicit formulae for Bernoulli numbers

<p>By examining the connection coefficients, we systematically review and extend (with an extra integer parameter) several double sum expressions for the Bernoulli numbers. New summation formulae are also established explicitly.</p>

H∞ dynamic observer design for a class of Lipschitz nonlinear discrete‐time systems with time varying delays

AbstractThis study explores the development of dynamic observer (HDO) for discrete‐time nonlinear systems (DTNLS) with time‐varying delay (TVD) and disturbances. The approach is to construct an augmented Lyapunov–Krasovskii function (LKF) with double summation terms, using the generalized reciprocally convex matrix inequality (GRCMI), as well as the Jensen‐based inequality (JBI) and the Wirtinger‐based inequality (WBI). These lead to less conservative time‐dependent conditions, represented as a set of linear matrix inequalities (LMIs) that can be efficiently solved using the LMI or YALMIP toolboxes. In addition, the proposed observer includes the widely used proportional observer (PO) and proportional integral observer (PIO) as specific cases. Two examples are presented to demonstrate the validity and effectiveness of the results.

Proof of conjectures of Sun on double basic hypergeometric sums

In this paper, we study some q-congruences on double basic hypergeometric sums. With the help of ‘creative microscoping’ method and several summation and transformation formulas for basic hypergeometric series, we confirmed several conjectures posed by Zhi-Wei Sun.

Efficient simulation of fully nonstationary wind velocity field by an enhanced numerical truncation method

The spectral representation method (SRM) is widely utilized to simulate the fully nonstationary wind velocity field with time-varying coherence. The simulation samples are generated by double summation of harmonic functions. Despite the fast Fourier transform (FFT) technique has been employed to accelerate the summation in the frequency dimension, the computational efficiency is still constrained by the number of simulation points and samples due to the time-consuming summation operation in the spatial dimension. In this study, a numerical truncation method is presented to further quicken the SRM-based simulation of a fully nonstationary wind velocity field. The fundamental idea is that the harmonic functions with tiny amplitude are inconsequential and can be omitted. A novel diagonal decoupling approach is first developed to rearrange the double summation. Further, the frequency truncation scheme is established to simplify the summation in the spatial dimension. Meanwhile, the FFT operations to calculate the summation in the frequency dimension are also considerably reduced. Finally, the superior performance of the proposed method on precision and efficiency is demonstrated by numerical examples.

Matrix elements of SO(3) in sl3 representations as bispectral multivariate functions

We compute the matrix elements of SO(3) in any finite-dimensional irreducible representation of sl3. They are expressed in terms of a double sum of products of Krawtchouk and Racah polynomials which generalize the Griffiths–Krawtchouk polynomials. Their recurrence and difference relations are obtained as byproducts of our construction. The proof is based on the decomposition of a general three-dimensional rotation in terms of elementary planar rotations and a transition between two embeddings of sl2 in sl3. The former is related to monovariate Krawtchouk polynomials and the latter, to monovariate Racah polynomials. The appearance of Racah polynomials in this context is algebraically explained by showing that the two sl2 Casimir elements related to the two embeddings of sl2 in sl3 obey the Racah algebra relations. We also show that these two elements generate the centralizer in U(sl3) of the Cartan subalgebra and its complete algebraic description is given.

Distribution of recursive matrix pseudorandom number generator modulo prime powers

Given a matrix A ∈ G L d ( Z ) A\in \mathrm {GL}_d(\mathbb {Z}) . We study the pseudorandomness of vectors u n \mathbf {u}_n generated by a linear recurrence relation of the form u n + 1 ≡ A u n ( mod p t ) , n = 0 , 1 , … , \begin{equation*} \mathbf {u}_{n+1} \equiv A \mathbf {u}_n \pmod {p^t}, \qquad n = 0, 1, \ldots , \end{equation*} modulo p t p^t with a fixed prime p p and sufficiently large integer t ⩾ 1 t \geqslant 1 . We study such sequences over very short segments of length which has not been accessible via previously used methods. Our technique is based on the method of N. M. Korobov [Mat. Sb. (N.S.) 89(131) (1972), pp. 654–670, 672] of estimating double Weyl sums and a fully explicit form of the Vinogradov mean value theorem due to K. Ford [Proc. London Math. Soc. (3) 85 (2002), pp. 565–633]. This is combined with some ideas from the work of I. E. Shparlinski [Proc. Voronezh State Pedagogical Inst., 197 (1978), 74–85 (in Russian)] which allows us to construct polynomial representations of the coordinates of u n \mathbf {u}_n and control the p p -adic orders of their coefficients in polynomial representations.

On some double Nahm sums of Zagier

Zagier provided eleven conjectural rank two examples for Nahm's problem. All of them have been proved in the literature except for the fifth example, and there is no q-series proof for the tenth example. We prove that the fifth and the tenth examples are in fact equivalent. Then we give a q-series proof for the fifth example, which confirms a recent conjecture of Wang. This also serves as the first q-series proof for the tenth example, whose explicit form was conjectured by Vlasenko and Zwegers in 2011 and whose modularity was proved by Cherednik and Feigin in 2013 via nilpotent double affine Hecke algebras.

A Potent Collocation Approach Based on Shifted Gegenbauer Polynomials for Nonlinear Time Fractional Burgers’ Equations

This paper presents a numerical strategy for solving the nonlinear time fractional Burgers's equation (TFBE) to obtain approximate solutions of TFBE. During this procedure, the collocation approach is used. The proposed numerical approximations are supposed to be a double sum of the products of two sets of basis functions. The two sets of polynomials are presented here: a modified set of shifted Gegenbauer polynomials and a shifted Gegenbauer polynomial set. Some specific integers and fractional derivatives are explicitly given as a combination of basis functions to apply the proposed collocation procedure. This method transforms the governing boundary-value problem into a set of nonlinear algebraic equations. Newton's approach can be used to solve the resulting nonlinear system. An analysis of the precision of the proposed method is provided. Various examples are presented and compared to some existing methods in the literature to prove the reliability of the suggested approach.

Quantized minimum error entropy with fiducial points for robust regression

Minimum error entropy with fiducial points (MEEF) has received a lot of attention, due to its outstanding performance to curb the negative influence caused by non-Gaussian noises in the fields of machine learning and signal processing. However, the estimate of the information potential of MEEF involves a double summation operator based on all available error samples, which can result in large computational burden in many practical scenarios. In this paper, an efficient quantization method is therefore adopted to represent the primary set of error samples with a smaller subset, generating a quantized MEEF (QMEEF). Some basic properties of QMEEF are presented and proved from theoretical perspectives. In addition, we have applied this new criterion to train a class of linear-in-parameters models, including the commonly used linear regression model, random vector functional link network, and broad learning system as special cases. Experimental results on various datasets are reported to demonstrate the desirable performance of the proposed methods to perform regression tasks with contaminated data.