Sum Of Terms Research Articles

Q-Supercongruences from Jackson's [formula omitted] summation and Watson's [formula omitted] transformation

q-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some q-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's ϕ78 summation, Watson's ϕ78 transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials. More concretely, we give a q-analogue of a nice formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773–808] and two q-supercongruences involving double series.

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.

Dissipativity for Discrete-Time Switched Positive Delay Systems: A Dwell-Time-Dependent Linear Copositive Storage Functional Method

This article addresses the problems of dissipativity analysis and feedback dissipativity of discrete-time switched positive systems with time-varying delay. On the basis of the positivity property of the investigated system, the definition of linear <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(q,r)$</tex-math> </inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha$</tex-math> </inline-formula> -dissipativity, utilizing the linear copositive storage functionals with linear supply rates, is first introduced. The switching among subsystems is orchestrated by a state-dependent switching policy that obeys prespecified dwell-time constraints. Second, in order to reduce the conservatism brought by the identical summation term functional for all positive subsystems, we propose a dwell-time-dependent linear copositive storage functionals involving summation terms which are diverse appearing in the functional. Then, less conservative criteria are developed to guarantee the system is strict <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(q,r)$</tex-math> </inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha$</tex-math> </inline-formula> -dissipativity by resorting to the switching policy or the joint implementation of the switching policy and time-varying controllers for subsystems. Finally, the validity of the proposed techniques is illustrated through two examples, and the relationship among the minimum dwell time, performance index, and time-varying delay is also revealed.

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.

How to Open a Black Box Classifier for Tabular Data

A lack of transparency in machine learning models can limit their application. We show that analysis of variance (ANOVA) methods extract interpretable predictive models from them. This is possible because ANOVA decompositions represent multivariate functions as sums of functions of fewer variables. Retaining the terms in the ANOVA summation involving functions of only one or two variables provides an efficient method to open black box classifiers. The proposed method builds generalised additive models (GAMs) by application of L1 regularised logistic regression to the component terms retained from the ANOVA decomposition of the logit function. The resulting GAMs are derived using two alternative measures, Dirac and Lebesgue. Both measures produce functions that are smooth and consistent. The term partial responses in structured models (PRiSM) describes the family of models that are derived from black box classifiers by application of ANOVA decompositions. We demonstrate their interpretability and performance for the multilayer perceptron, support vector machines and gradient-boosting machines applied to synthetic data and several real-world data sets, namely Pima Diabetes, German Credit Card, and Statlog Shuttle from the UCI repository. The GAMs are shown to be compliant with the basic principles of a formal framework for interpretability.

Correlation of Sum of ST Segment Depression in Leads V1 to V4 in Acute Inferior Myocardial Infarction with Angiographic Severity of Coronary Artery Disease

Background: Inferior wall acute myocardial infarction accounts for 40 to 50% of all acute STEMI. Patients of acute inferior myocardial infarction with ST depression in lateral leads often have greater incidence of triple vessel disease and proximal RCA lesion but in case of patients of acute inferior myocardial infarction with ST depression in precordial leads (leads V1 to V4), is a subject of determination whether it is associated with more sever coronary artery disease. The aim of this study is to determine the association of sum of ST depression in precordial leads with the probability of concomitant severity of coronary artery disease in patients with acute inferior STEMI. Methods: this cross sectional observational study was done at National institute of cardiovascular diseases (NICVD), Dhaka, Bangladesh. Total 90 patients of acute inferior myocardial infarction with precordial lead ST depression admitted at CCU within 12 hours of onset of chest pain were the study population. They were divided in two groups on the basis of sum of ST depression in precordial leads (Sum of ST depression d”4mm and &gt;4mm), 36 patients in group I and 54 patients in group II. Coronary angiography was performed during the index hospitalization period. Gensini score and Reardon score were measured. Results: Overwhelming majority of the patients was male (83.3% and 90.7%) with mean age of 51.0±9.7 and 51.0±9.9 years in group I and II respectively. Smoking, hypertension and diabetes mellitus were the most frequent risk factors in both groups. Serum troponin level was significantly high in group II (6.2±2.2 vs. 13.6±17.7, p &lt;0.05). Sum of ST depression in precordial leads was 2.84±0.66 and 7.53±3.51 in group I and group II respectively, and was statistically significant. Consideration of mean of ST depression in individual leads shows significantly higher mean in leads V2, V3 and V4 among group II patients. SVD was more frequent in group I but group II patients had higher statistically significant incidence of DVD and TVD. Gensini score (20.26±13.0 vs. 36.98±16.9) and Reardon score (4.63±2.2 vs. 6.83±2.2) was high in group II patients. Positive correlation had been depicted between summation value of ST segment depression with that of Gensini score &amp; Reardon score (Gensini score r=0.61 and Reardon score r= 0.52). Conclusion: Involvement Triple vessel and double vessel disease was remarkably higher with increasing sum of ST segment depression in precordial leads. The severity of ST segment depression in terms of summation of ST depression in leads V1 to V4 is directly proportional to the extent and severity of coronary artery disease. Bangladesh Heart Journal 2022; 37(2): 116-121

A Theoretical Estimate of the Temperature Dependence of Electron Concentration and Lorenz Number in the Spherically Symmetric Zone of Semiconductors

An accurate approach is proposed for the temperature dependence of electron concentration and Lorenz number in the spherically symmetric zone of semiconductors. The evaluation includes more accurate analytical calculations over the study for two parameters of Fermi functions. Recently, a new analytical approach for the calculation of the two parameters of Fermi functions has been reported in terms of summations of binomial coefficients and incomplete gamma functions. The method is applied to the case of the Ge and GaAs semiconductors, which can determine the electron concentration and Lorenz number as a function of temperature variation. The results obtained by the suggested and numerical methods are satisfactory for a wide range of temperatures. The method descriptions are very well for the investigation of other thermoelectric effects over the whole temperature ranges.

First- and second-order linear difference equations with constant coefficients: suggestions for making the theory more accessible

The aim of this paper is to present a teaching proposal for the theoretical part relating to the first- and second-order linear difference equations with constant coefficients suitable for the first-year students at various types of universities. In contradistinction to the methods often applied (memorization of algorithms without a proper conceptual understanding), we provide an alternative methodological way-out for lecturers that is partially based on posing suitable questions and constructing the knowledge. Our approach is therefore characterized by the minimal input knowledge imposed on students. In fact, we assume that students are acquainted only with the summation of terms of geometric sequences and with solving linear and quadratic equations.

Non-fragile output-feedback synchronization for delayed discrete-time complex-valued neural networks with randomly occurring uncertainties

This paper is step forward to establish an exponential synchronization criterion for discrete-time complex-valued neural networks (CVNNs) having time-varying delays subject to randomly occurring uncertain weighting parameters, in order to overcome the fluctuation when the output-feedback controller imposes on its dynamics. To achieve this, Jensen’s weighted summation inequalities (WSIs) and an extended reciprocal convex matrix inequality (ERCMI) are extended into the domain of complex field. By introducing some augmented vectors, a Lyapunov–Krasovskii functional (LKF) is constructed to attain an improved delay-dependent linear matrix inequalities (LMIs) constraint for the exponential synchronization phenomenon of the desired master–slave neuronal system model. For instance, the upper bound of the quadratic summation terms occurred in the finite difference of the LKF have been obtained from its linearization that has been made by the developed complex-valued WSIs and complex-valued ERCMI. The proposed results are less restrictive with the minimum number of decision variables than those obtained using existing inequalities. The designed output-feedback control gain has been determined by solving a set of complex-valued LMIs and it has been enforced with a prescribed exponential decay rate. Finally, in sight of MATLAB software, the established results have been examined via a numerical example supported by the simulation results.

Complementary system vs conventional trifocal intraocular lens: comparison of optical quality metrics and unwanted light distribution.

To evaluate the ARTIS Symbiose complementary intraocular lens (IOL) system, consisting of the MID and PLUS models, in comparison with a conventional trifocal IOL (AcrySof IQ PanOptix). The David J. Apple Center for Vision Research, Department of Ophthalmology, University of Heidelberg, Heidelberg, Germany. Laboratory investigation. Modulation and phase transfer functions were obtained in polychromatic light using an optical bench setup. Simulated visual acuity (VA) values were derived from optical quality metrics weighted by neural contrast sensitivity. United States Air Force (USAF) chart images were acquired and processed. Furthermore, the light distribution beyond the center of a polychromatic point spread function was assessed. The peak simulated VA values of ARTIS Symbiose MID were at 0 diopters (D) of defocus (-0.02 logMAR) and at -1.5 D (0.00 logMAR); of ARTIS Symbiose PLUS, they were at 0 D of defocus (-0.01 logMAR) and at -2.5 D (0.01 logMAR). AcrySof IQ PanOptix demonstrated 3 peaks: at 0 D of defocus (-0.02 logMAR), at -1.75 D (0.03 logMAR), and at -2.5 D (0.02 logMAR). The summation of USAF chart images in the simulated binocular IOL system produced a slightly better image quality at -1.0 D and -1.5 D than AcrySof IQ PanOptix. The IOLs yielded comparable light spread across the studied range except for a localized intensity spike of the ARTIS Symbiose IOLs. The complementary IOL system may yield better monocular intermediate VA compared with the conventional trifocal IOL. However, the effect of binocular summation in terms of VA and the perception of photic phenomena still needs to be investigated.

Bayesian Non-Parametric Thermal Thresholds for Helicoverpa armigera and Their Integration into a Digital Plant Protection System

The development of temperature-driven pest risk thresholds is a prerequisite for the buildup and implementation of smart plant protection solutions. However, the challenge is to convert short and abrupt phenology data with limited distributional information into ecological relevant information. In this work, we present a novel approach to analyze phenology data based on non-parametric Bayesian methods and develop degree-day (DD) risk thresholds for Helicoverpa armigera (Hübner) (Lepidoptera: Noctuidae) to be used in a decision support system for dry bean (Phaseolus vulgaris L.) production. The replication of each Bayesian bootstrap generates a posterior probability for each sampling set by considering that the prior unknown distribution of pest phenology is Dirichlet distribution. We computed R = 10,000 temperature-driven pest phenology replicates, to estimate the 2.5%, 50% and 95.5% percentiles (PC) of each flight generation peak in terms of heat summations. The related DD thresholds were: 114.04 (PC 2.5%) 131.8 (PC 50%) and 150.9 (PC 95.5%) for the first, 525.8 (PC 2.5%), 551.7 (PC 50%) and 577.6 (PC 95.5%) for the second and 992.7 (PC 2.5%), 1021.5 (PC 50%) and 1050 (PC 95.5%) for the third flight, respectively. The thresholds were evaluated by estimating the posterior differences between the predicted (2021) and observed (2022) phenology metrics and are in most cases in acceptable levels. The bootstrapped Bayesian risk thresholds have the advantage to be used in modeling short and noisy data sets providing tailored pest forecast without any parametric assumptions. In a second step the above thresholds were integrated to a sub-module of a digital weather-driven real time decision support system for precise pest management for dry bean crops. The system consists of a customized cloud based telemetric meteorological network, established over the border area of the Prespa National Park in Northern Greece, and delivers real time data and pest specific forecast to the end user.

Finite-time dissipative synchronization of discrete-time semi-Markovian jump complex dynamical networks with actuator faults

This paper is concerned with the problem of finite-time synchronization for a class of discrete-time semi-Markovian jumping complex dynamical networks (CDNs) with actuator faults based on reliable control. The main aim of this paper is to design a state feedback controller such that the resulting closed-loop system is finite-time synchronized under a prescribed dissipativity performance level even in the presence of actuator failures. Moreover, a stochastic nature followed by Bernoulli distribution is described in the considered networks due to the occurrence of probabilistic nature in time-varying delays. By composing a suitable Lyapunov–Krasovskii functional containing triple summation terms with the aid of Kronecker product properties, Lyapunov stability theory and free weighting matrix approach, sufficient criteria are established in terms of linear matrix inequalities that assure finite-time synchronization and meet the dissipativity performance to the addressed CDNs. The usefulness of the presented design scheme is finally verified by numerical examples.

Acoustic radiation force and torque on a spheroid near an interface

Acoustic radiation force and torque on a spheroid of arbitrary size near a rigid or pressure release interface are calculated using expansions of the pressure field in terms of both spherical and spheroidal wave functions. The spheroidal expansion coefficients are obtained by satisfying the boundary conditions at the interface and at the surface of the spheroid. Conditions at the interface are satisfied using a virtual (image) spheroid, while satisfying conditions at the spheroid surface require the use of addition theorems for spheroidal wave functions. The radiation force and torque are expressed as a summation of terms involving products of the coefficients in spherical wave expansions of the incident and scattered fields [Ilinskii et al., J. Acoust. Soc. Am. 144, 568–576 (2018)]. Far-field asymptotes are used to relate the spheroidal and spherical harmonic expansion coefficients analytically. Results from the present analytical model are compared with those from a finite-element model. The effect of the interface on the radiation force and torque relative to that on a spheroid in a free field is discussed. [B.E.S. is supported by the Applied Research Laboratories Chester M. McKinney Graduate Fellowship in Acoustics.]

Discretization Approach for the Homogenization of Three-Dimensional Solid-Solid Phononic Crystals in the Quasi-Static Limit: Density and Elastic Moduli

With the application of a homogenization theory, based on the Fourier formalism (which provides efficient and exact formulas by which to determine all the components of the effective stiffness and mass density tensors, valid in the regime of large wavelengths), a new approach to calculate the effective quasi-static response in three-dimensional solid-solid phononic crystals is reported. The formulas derived in this work for calculating the effective elastic parameters show a dependence, in terms of summations over the vectors, of the reciprocal lattice by the discretization of the volume of the inclusion in small parts (e.g., small cubes), to obtain a system of equations from which we define the effective response. In particular, we present the numerical results calculated for several cubic lattices with solid constituents and different shapes of inclusions in the unit cell versus the filling fraction, as well as for fixed values of it. By this means, we analyzed the effect of the type of Bravais lattice of the materials, and the geometry of the inclusions that constitute the three-dimensional phononic array, on the resulting effective anisotropy. Finally, our theory confirms other well-known results with previous homogenization theories as a particular case study. In this regard, the examples and results shown here can be useful for the design of metamaterials with predetermined elastic properties.

Position-space representation of charged particles\u2019 propagators in a constant magnetic field as an expansion over Landau levels

We have obtained propagators in the position space as an expansion over Landau levels for the charged scalar particle, fermion, and massive vector boson in a constant external magnetic field. The summation terms in the resulting expressions consisted of two factors, one being rotationally invariant in the 2-dimensional Euclidean space perpendicular to the direction of the field, and the other being Lorentz-invariant in the 1+1-dimensional space-time. The obtained representations are unique in the sense that they allow for the simultaneous study of the propagator from both space-time and energetic perspectives which are implicitly connected. These results contribute to the development of position-space techniques in QFT and are expected to be of use in the calculations of loop diagrams.

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.

Analytical Solutions to Definite Integrals for Combinations of Legendre, Bessel and Trigonometric Functions Encountered in Propagation and Scattering Problems in Spherical Coordinates

Mie theory is a rigorous solution to scattering problems in spherical coordinate system. The first step in applying Mie theory is expansion of some arbitrary incident field in terms of spherical harmonics fields in terms of spherical which in turn requires evaluation of certain definite integrals whose integrands are products of Bessel functions, associated Legendre functions and periodic functions. Here we present analytical results for two specific integrals that are encountered in expansion of arbitrary fields in terms of summation of spherical waves. The analytical results are in terms of finite summations which include Lommel functions. A concise analytical expression is also derived for the special case of Lommel functions that arise, rendering expensive numerical integration or other iterative techniques unnecessary.

A spectral projection based method for the numerical solution of wave equations with memory

In this paper, we compare two approaches to numerically approximate the solution of second-order Gurtin–Pipkin type of integro-differential equations. Both methods are based on a high-order Discontinous Galerkin approximation in space and the numerical inverse Laplace transform. In the first approach, we use functional calculus and the inverse Laplace transform to represent the solution. The spectral projections are then numerically computed and the approximation of the solution of the time-dependent problem is given by a summation of terms that are the product of projections of the data and the inverse Laplace transform of scalar functions. The second approach is the standard inverse Laplace transform technique. We show that the approach based on spectral projections can be very efficient when several time points are computed, and it is particularly interesting for parameter-dependent problems where the data or the kernel depends on a parameter.

Output-feedback photovoltaic WADC by H∞ criterion using auxiliary function

The wide-area damping controller (WADC) is vital in suppressing wide-area oscillations. However, the power system is now developing very fast, traditional output-feedback linear matrix inequality (LMI) criteria adopted in this field may not be very applicable. This paper proposes an output-feedback H∞ robust criterion to design a WADC installed in the photovoltaic plant considering the wide-area signal delay. We use the auxiliary function technique, and adopt Lyapunov functions containing triple summation, augmented variables and cross terms to deduce a stability analysis criterion followed by the corresponding controller design criterion. The low conservatism and control performance of the proposed criteria are verified in a 68-bus 16-machine test system. In addition, as system parameters often change, the control performance of integrating the proposed controller design criterion into a gain scheduling algorithm to handle the system parameter variance is validated.

A consistent nonlinear mild-slope equation model

A new nonlinear frequency-domain model based on the mild-slope equation is outlined. The model is an enhancement over previous work in that a closer correspondence between scaling of nonlinearity and horizontal variation of bathymetry is made relative to earlier models. This results in additional terms in the nonlinear summation terms of the model, as amplitude gradient terms are required in order to formulate a consistent model. From the resulting elliptic model, a parabolic approximation is developed in order to efficiently model the equations. Comparisons between the present model, previously-formulated models, and experimental data show that the present model does evidence improvement in performance over previous models.