Cardinal Functions Research Articles

Solving Time-Fractional Partial Differential Equation Using Chebyshev Cardinal Functions

We propose a numerical scheme based on the Galerkin method for solving the time-fractional partial differential equations. To this end, after introducing the Chebyshev cardinal functions (CCFs), using the relation between fractional integral and derivative, we represent the Caputo fractional derivative based on these bases and obtain an operational matrix. Applying the Galerkin method and using the operational matrix for the Caputo fractional derivative, the desired equation reduces to a system of linear algebraic equations. By solving this system, the unknown solution is obtained. The convergence analysis for this method is investigated, and some numerical simulations show the accuracy and ability of the technique.

A critical reflection on the gap-filling function of general principles of law

It is well established in international legal scholarship that general principles of law (GPL) have an important function of gap-filling. However, there are many controversies about the concept and the usages of gap-filling. It is unclear what gap-filling means in legal doctrines. Moreover, the judgments of the International Court of Justice rarely mention the gap-filling function of GPL. With these controversies, one may even ponder that the gap-filling function may have been over-claimed by legal scholars. This article takes stock of these questions and critically engages with the implications of having the gap-filling function as a cardinal function of GPL. It provides four aspects of critiques on the gap-filling discourse on the basis of the genesis of gap-filling, relevant legal doctrines and judicial practices. By providing these critiques, this article argues against an established episteme in international law that the gap-filling function is a major function of GPL. Moreover, it notes that scholars tend to attribute the gap-filling function to GPL because they have a vision of the lex ferenda of international law.

CONSTRUCTION AND VERIFICATION OF A DIGITAL EQUALIZER MODEL

An approach to the development of an equalizer by building its mathematical model based on a microcontroller is proposed. All operations, including signal processing and equalizer kernel calculation, are performed by a single microcontroller. Thanks to the created mathematical model of the equalizer, the calculation of the kernel is reduced to multiple uses of relatively simple operations, which saves time and memory of the program. The equalizer provides satisfactory processing quality at a small filter order which is selected as a digital filter with final impulse response (FIR) because of its linear phase-frequency response, a guarantee of stability at the complex amplitude-frequency response, and also its associativity and linearity allowing it easily reproduce a complex amplitude-frequency response. Schematic implementation is based on parallel bandpass filters and a low-pass filter followed by adding filtered and amplified signals. It is the distributive property of the FIR filter that makes it possible to obtain a new kernel that includes all the amplified ranges by the sum of the corresponding kernel elements, instead of adding amplified ranges. The associativity and linearity of the FIR filter gives the opportunity to easily implement different types of filters on the basis of a low-pass filter, for the calculation of which the cardinal sine function is used together with the window function, which in combination gives qualitative frequency characteristics. The low-pass filter kernel and equalizer model are verified in the GNU Octave environment, which is an open-source analogue of Matlab. The model is checked by setting the frequency response of the test equalizer, and for individual filters the allowable width of the transition band and the maximum value of pulsation in the suppression band are set. The low-pass filter kernel is created with an arbitrary cutoff frequency, and the filter consists of 129 elements, which were multiplied by the Kaiser window with a value of parameter equal to 4.5. As a result of verification of the mathematical model in the GNU Octave environment, the width of the transition band and the maximum value of pulsation in the suppression band meet the specified conditions. The simulated equalizer kernel fully corresponds to the specified frequency response. Verification of the mathematical model confirmed its efficiency and compliance of the obtained characteristics of the equalizer with the specified requirements.

Numerical simulation of the fractional diffusion equation

During this paper, a specific type of fractal-fractional diffusion equation is presented by employing the fractal-fractional operator. We present a reliable and accurate operational matrix approach using shifted Chebyshev cardinal functions to solve the considered problem. Also, an operational matrix for the considered derivative is obtained from basic functions. To solve the introduced problem, we convert the main equation into an algebraic system by extracting the operational matrix methods. Graphs of exact and approximate solutions along with error graphs are presented. These figures show how the introduced approach is reliable and accurate. Also, tables are established to illustrate the values of solutions and errors. Finally, a comparison of the solutions at a specific time is given for each test problem.

Green Coconut Water Supplementation Attenuates Flutamide-induced Testicular Damage in Male Prepubertal Wistar Rats

Background: The enhancement of male reproductive health function remains a paramount desire of every infertile men. Green coconut water (GCW) is the liquid in the inner cavity of an immature coconut fruit. Studies have demonstrated that GCW has endocrine property that regulates the reproductive system. Aim and Objectives: This study was designed to investigates the effects of GCW on the cardinal sperm function parameters such as; sperm count, sperm motility and sperm morphology, and the histo-architecture of the testis in flutamide-treated pre-pubertal wistar rats. Materials and Methods: Thirty-six male prepubertal wistar rats, weighing between 40-70g were divided into six study groups A-F. Group A, B and C received distilled water, flutamide at 25mg/kg and GCW at 20ml/kg respectively for 6 weeks. Group D and E received 25mg/kg flutamide for 2 weeks then GCW at 10ml/kg and 20ml/kg respectively 4 weeks, while Group F received 25mg/kg flutamide and GCW at 20ml/kg concomitantly for 6 weeks. At the end of the experiment, the animals were euthanized; caudal epididymis and the testis were collected for semen analysis and histological evaluation. Results: GCW supplementation was showed to significantly increased sperm count, sperm motility and also brings about percentage decrease in sperm morphology, as well as ameliorates histo-pathological degenerations caused by flutamide on the testis of the experimental rats. Conclusion: The findings from this study suggest that GCW supplementation may effectively enhance male fertility.

Fractal–fractional Klein–Gordon equation: A numerical study

In this work, we solve a new kind of the fractional Klein–Gordon problem numerically. In fact, we study the mentioned problem under fractal–fractional operator with the Riemann–Liouville frame with Mittag-Leffler kernel. We use an efficient operational matrix (OM) technique employing the shifted Chebyshev cardinal functions (CCFs) to get the approximate solutions of the considered equation. Moreover, an OM for the considered derivative is gained using the basic functions. To get the approximate solutions of the presented equation we change the principal model into an algebraic system. To see the numerical results of the problem, we provide the related graphs of the exact and approximate solutions along with the absolute errors of each example. The accuracy and reliability of the numerical solutions can be found form the figures. Also, for each example Tables displaying the values of solutions and errors are reported.

Comparison of Brillouin Light Scattering and Density of States in a Supported Layer: Analytical and Experimental Study

We provide a detailed analytical calculation of the Brillouin light scattering (BLS) intensity of a layer on a substrate, taking into account both photoelastic and moving boundary (ripple effect) mechanisms, and give a comparison between BLS intensity and density of states (DOS) to determine the dispersion curves of longitudinal guided modes in the supported layer. In particular, in the case where the mismatch between the elastic parameters of the substrate and the adsorbed layer is high, such as in a PMMA layer on a Si substrate, we derive closed-form expressions of BLS and DOS and demonstrate a simple relationship between these two quantities. A very good agreement between experimental and theoretical BLS spectra was found and compared to theoretical DOS spectra. In particular, we show that while the peaks in the DOS present a uniform behavior, the BLS spectra follows a sine cardinal (sinc) function shape around a given frequency fixed by the chosen laser wavelength. The theoretical calculation is performed within the framework of the Green’s function approach.

Closure index of semitopological groups

In this article, we define a new cardinal function called the closure index of a semitopological group. The closure index ci(G) of a semitopological group G satisfying the T1-separation axiom lies between their character and pseudocharacter. For locally compact topological groups, these three cardinal functions coincide. We extend this result to the class of feathered topological groups and determine the value of the closure index for other classes of topological groups with compact-like properties. We also present several examples of topological groups with compact-like properties for which the three cardinal functions are pairwise distinct.We also find some classes of topological groups of countable pseudocharacter that have countable closure index.

A Few Queries Go a Long Way: Information-Distortion Tradeoffs in Matching

We consider the One-Sided Matching problem, where n agents have preferences over n items, and these preferences are induced by underlying cardinal valuation functions. The goal is to match every agent to a single item so as to maximize the social welfare. Most of the related literature, however, assumes that the values of the agents are not a priori known, and only access to the ordinal preferences of the agents over the items is provided. Consequently, this incomplete information leads to loss of efficiency, which is measured by the notion of distortion. In this paper, we further assume that the agents can answer a small number of queries, allowing us partial access to their values. We study the interplay between elicited cardinal information (measured by the number of queries per agent) and distortion for One-Sided Matching, as well as a wide range of well-studied related problems. Qualitatively, our results show that with a limited number of queries, it is possible to obtain significant improvements over the classic setting, where only access to ordinal information is given.

Continuous-Wave Second-Harmonic Generation in Orientation-Patterned Gallium Phosphide Waveguides at Telecom Wavelengths

A new process to produce orientation-patterned gallium phosphide (OP-GaP) on GaAs with almost perfectly parallel domain boundaries is presented. Taking advantage of the chemical selectivity between phosphides and arsenides, OP-GaP is processed into suspended shallow-ridge waveguides. Efficient second-harmonic generation from telecom wavelengths is achieved in both Type-I and Type-II polarisation configurations. The highest observed conversion efficiency is 200% W–1 cm–2, with a bandwidth of 2.67 nm in a 1 mm-long waveguide. The variation of the conversion efficiency with wavelength closely follows a squared cardinal sine function, in excellent agreement with theory, confirming the good uniformity of the poling period over the entire length of the waveguide.

Transforming Ordinal Riskless Utility into Cardinal Risky Utility: A Comment on Chung, Glimcher, and Tymula (2019)

Chung, Glimcher, and Tymula (2019) observed both consumers’ choices over commodity bundles and choices under risk. They assumed a cardinal riskless utility function V representing consumer choices and a cardinal risky utility function U.The two were inconsistent. This note shows that the two functions can be reconciled if we assume that V is ordinal. Then one utility function U can accommodate both risky and riskless choices. (JEL C91, D12, D81)

Transfusable neutrophil progenitors as cellular therapy for the prevention of invasive fungal infections.

The use of mature neutrophil (granulocyte) transfusions for the treatment of neutropenic patients with invasive fungal infections (IFIs) has been the focus of multiple clinical trials. Despite these efforts, the transfusion of mature neutrophils has resulted in limited clinical benefit, likely owing to problems of insufficient numbers and the very short lifespan of these donor cells. In this report, we employed a system of conditionally immortalized murine neutrophil progenitors that are capable of continuous expansion, allowing for the generation of unlimited numbers of homogenous granulocyte-macrophage progenitors (GMPs). These GMPs were assayed in vivo to demonstrate their effect on survival in 2 models of IFI: candidemia and pulmonary aspergillosis. Mature neutrophils derived from GMPs executed all cardinal functions of neutrophils. Transfused GMPs homed to the bone marrow and spleen, where they completed normal differentiation to mature neutrophils. These neutrophils were capable of homing and extravasation in response to inflammatory stimuli using a sterile peritoneal challenge model. Furthermore, conditionally immortalized GMP transfusions significantly improved survival in models of candidemia and pulmonary aspergillosis. These data confirm the therapeutic benefit of prophylactic GMP transfusions in the setting of neutropenia and encourage development of progenitor cellular therapies for the management of fungal disease in high-risk patients.

Bias Versus Non-Convexity in Compressed Sensing

Cardinality and rank functions are ideal ways of regularizing under-determined linear systems, but optimization of the resulting formulations is made difficult since both these penalties are non-convex and discontinuous. The most common remedy is to instead use the ell ^1- and nuclear norms. While these are convex and can therefore be reliably optimized, they suffer from a shrinking bias that degrades the solution quality in the presence of noise. This well-known drawback has given rise to a fauna of non-convex alternatives, which usually features better global minima at the price of maybe getting stuck in undesired local minima. We focus in particular penalties based on the quadratic envelope, which have been shown to have global minima which even coincide with the “oracle solution,” i.e., there is no bias at all. So, which one do we choose, convex with a definite bias, or non-convex with no bias but less predictability? In this article, we develop a framework which allows us to interpolate between these alternatives; that is, we construct sparsity inducing penalties where the degree of non-convexity/bias can be chosen according to the specifics of the particular problem.

Third‐kind Chebyshev cardinal functions for variable‐order time fractional RLW‐Burgers equation

In this article, the Caputo type variable‐order fractional differentiation is utilized for defining a novel fractional form of the regularized long wave Burgers problem which arises in several physical applications. The third‐kind Chebyshev cardinal functions are generated and some matrix relationships related to them are derived. A numerical procedure dependent on these polynomial cardinal functions is developed for the introduced equation. This scheme converts the primary problem solving into finding an algebraic system solution with low computations using approximating the solution of the equation by the expressed cardinal functions. The accuracy of the presented procedure is investigated in some examples. The presented scheme is straightforward and powerful and can be adopted for many other nonlinear fractional problems.

Set Star-Menger and Set Strongly Star-Menger Spaces

AbstractMotivated by the Arhangel’skii “s-Lindelöf cardinal function” definition, Kočinac and Konca defined and studied set covering properties and set star covering properties. In this paper, we present results on the star covering properties called set star-Menger and set strongly star-Menger. We investigate the relationship among set star-Menger, set strongly star-Menger and other related properties and study the topological properties of set star-Menger and set strongly star-Menger properties.

A smoothing proximal gradient algorithm for matrix rank minimization problem

In this paper, we study the low-rank matrix minimization problem, where the loss function is convex but nonsmooth and the penalty term is defined by the cardinality function. We first introduce an exact continuous relaxation, that is, both problems have the same minimizers and the same optimal value. In particular, we introduce a class of lifted stationary points of the relaxed problem and show that any local minimizer of the relaxed problem must be a lifted stationary point. In addition, we derive lower bound property for the nonzero singular values of the lifted stationary point and hence also of the local minimizers of the relaxed problem. Then the smoothing proximal gradient (SPG) algorithm is proposed to find a lifted stationary point of the continuous relaxation model. Moreover, it is shown that any accumulating point of the sequence generated by SPG algorithm is a lifted stationary point. At last, numerical examples show the efficiency of the SPG algorithm.

On some set properties

Motivated by the Arhangel?skii ?s-Lindel?f cardinal function? definition, Kocinac, Konca, Bonanzinga, Maesano and Singh defined and studied the set covering properties. In this paper, we study some set versions of compact-type properties and give partial solution to a problem posed by Singh in [11].

Modified Fitness Dependent Optimizer for Solving Numerical Optimization Functions

The Fitness Dependent Optimizer (FDO) is a recent metaheuristic algorithm that was developed in 2019. It is one of the metaheuristic algorithms that has been used by researchers to solve various applications especially for engineering design problem. In this paper, a comprehensive survey conducted about FDO and its applications. Consequently, despite of having competitive performance of FDO, it has two major problems including low exploitation and slow convergence. Therefore, a modification of FDO (MFDO) is proposed for solving FDO issues. MFDO used two methods to enhance the performance of FDO: firstly, optimizing the range of weight factor between 0 and 0.2 which is used for finding fitness weight. Secondly, using sine cardinal mathematical function to update fitness weight and pace which is referred to the speed of the bees. To evaluate the performance of MFDO, 19 classical benchmark functions and CEC2019 benchmark functions are used. MFDO compared against all the enhancement of FDO and also it is compared with Grey Wolf Optimization (GWO), Chimp Optimization Algorithm (ChOA), Genetic Algorithm (GA), and Butterfly Optimization Algorithm (BOA). Statistical results proved that MFDO achieved significant performance compared to other algorithms. Finally, MFDO is used to solve three applications: Welded Beam Design (WDB), Pressure Vessel Design (PVD), and Spring Design Problem. Results proved that MFDO outperformed well in solving these applications against FDO, Gravitational Search Algorithm (GSA), GA, and Grasshopper Optimization Algorithm (GOA).

A New Numerical Approach for Solving Fractional Optimal Control Problems with the Caputo–Fabrizio Fractional Operator

In this study, the necessary optimality conditions are achieved for a class of fractional optimal control problems (FOCPs) involving the Caputo–Fabrizio (CF) fractional derivative. We offer a new numerical method based on the Chebyshev cardinal functions for solving the problem. Our goal was to reduce the original problem into a nonlinear system of algebraic equations. For doing this, we approximate the state and control variables in terms of the Chebyshev cardinal functions. The operational matrices (OMs) of left and right CF fractional integrals are also derived for the Chebyshev cardinal functions. Error estimation and convergence analysis of the method are also introduced. The numerical results illustrate the effectiveness and the rapid convergence of the proposed technique. Due to the high accuracy of the solutions, the suggested method is very useful for numerical techniques in control theory.

Numerical approximation of a variable-order time fractional advection-reaction-diffusion model via shifted Gegenbauer polynomials

<abstract><p>The fractional advection-reaction-diffusion equation plays a key role in describing the processes of multiple species transported by a fluid. Different numerical methods have been proposed for the case of fixed-order derivatives, while there are no such methods for the generalization of variable-order cases. In this paper, a numerical treatment is given to solve a variable-order model with time fractional derivative defined in the Atangana-Baleanu-Caputo sense. By using shifted Gegenbauer cardinal function, this approach is based on the application of spectral collocation method and operator matrices. Then the desired problem is transformed into solving a nonlinear system, which can greatly simplifies the solution process. Numerical experiments are presented to illustrate the effectiveness and accuracy of the proposed method.</p></abstract>