The comparison of group means in item response models constitutes an important issue in empirical research. The present article discusses an extension of Haebara linking by proposing a flexible class of robust linking functions for comparisons of many groups. These robust linking functions are particularly suited to item response data that are generated under partial invariance. In a simulation study, it is shown that the newly proposed robust Haebara linking approach outperforms existing approaches of Haebara linking. In an empirical application using PISA data, it is illustrated that country means can be sensitive to the choice of linking functions.

This paper proposes a definition of fractional line integral, generalising the concept of fractional definite integral. The proposal replicates the properties of the classic definite integral, namely the fundamental theorem of integral calculus. It is based on the concept of fractional anti-derivative used to generalise the Barrow formula. To define the fractional line integrals the Gr\"unwald-Letnikov and Liouville directional derivatives are introduced and their properties described. The integral is defined first for broken line paths and afterwards to any regular curve

A major task in Functional Time Series Analysis is measuring the dependence within and between processes, for which lagged covariance and cross-covariance operators have proven to be a practical tool in well-established spaces. This article deduces estimators and asymptotic upper bounds of the estimation errors for lagged covariance and cross-covariance operators of processes in Cartesian products of abstract Hilbert spaces for fixed and increasing lag and Cartesian powers. We allow the processes to be non-centered, and to have values in different spaces when investigating the dependence between processes. Also, we discuss features of estimators for the principle components of our covariance operators.

In this paper, we study the third order semilocal convergence of the Newton-like method for finding the approximate solution of nonlinear operator equations in the setting of Banach spaces. First, we discuss the convergence analysis under ω-continuity condition, which is weaker than the Lipschitz and Hölder continuity conditions. Second, we apply our approach to solve Fredholm integral equations, where the first derivative of involved operator not necessarily satisfy the Hölder and Lipschitz continuity conditions. Finally, we also prove that the R-order of the method is 2p + 1 for any p $\in$ (0,1].

In this study, a mixed rule of degree of precision nine has been developed and implemented in the field of electrical sciences to obtain the instantaneous current in the RLC- circuit for particular value .The linearity has been performed with the Volterra’s integral equation of second kind with particular kernel . Then the definite integral has been evaluated through the mixed quadrature to obtain the numerical result which is very effective. A polynomial has been used to evaluate Volterra’s integral equation in the place of unknown functions. The accuracy of the proposed method has been tested taking different electromotive force in the circuit and absolute error has been estimated.

Abstract: The objective of Peter Chew Triangle Diagram is to clearly illustrate the topic solution of triangle and provide a complete design for the knowledge of AI age. Peter Chew's triangle diagram will suggest a better single rule that allows us to solve any problem of topic solution of triangle simple , directly and more accurately. There are two important rules for solving the topic solution of triangle today [1,2], namely the sine rule and the cosine rule. The sine rule is used to find a non-included angle when are given two sides and a non-included angle or the opposite side angle given when are given two angles and one side. The cosine rule normally is used to find the included angle when are given three sides or the third side when are given two sides and the included angle. Peter Chew Method[3] allow us to find the third side simple and directly when given two sides and a non-included angle. Peter Chew rule [4] allow us to find a non included angle simple, directly and more accurately when given 2 sides and an included angle. Aply Peter Chew Triangle Diagram to Education 4.0 Calculator , Peter Chew Triangle Diagram Calculator allows the Calculator to solve any problems in the topic solution of triangle simple, directly and more accurate. This can be effective in increasing students' interest in using technology while learning mathematics and will help in the learning of mathematics, especially when similar covid-19 issues arise in the future.

In this work, we studied the Green’s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

This work uses a new approach to investigate the Riemann hypothesis, drawing conclusions about its trueness. It is based on the more general method presented in a very recent publication, by the same author, showing in detail how to approach that very well-known and interesting problem. This is achieved by means of the theory of dynamical systems (Poincaré index associated to equilibria of 2-dimensional systems) and the study of the zeros of the Dirichlet eta function, defined by a Dirichlet series. By using the well-known fact that the zeros of the eta function include all zeros of the Riemann zeta function in the (open) critical strip, excluded the critical line, ((0,1/2) ∪ (1/2,1)) × (−∞,∞), the development proceeds using only the eta function. In addition, the open and simply connected region (1/2,1) × (0,∞) is used along the text, taking into account the symmetries of zeros of the functions under analysis in the critical strip. The basic line of proof is to find the mathematical expression for the Poincaré index of the vector field associated to the eta function, assuming the existence of a zero of the eta function outside the critical line (in (1/2,1) × (0,∞)), and investigating the resulting unfoldings. Eventually, an inconsistency occurs and the proof ends by contradiction.

In this paper, we proposed the numerical method called the variational iteration orthogonal collocation method (VIOCM), for the approximate solution of the deadly Corona virus model using Mamadu-Njoseh polynomials as basis functions. The proposed method is an elegant mixture of the variational iteration method (VIM) and the orthogonal collocation method (OCM). It was observed that the proposed method converges rapidly to the exact solution even as N increases. This suggests that the use of orthogonal polynomials as trial functions for the SEIR model is indeed an effective approximant as it produces the analytic solution at just few iterations. Resulting numerical evidences were compared with the standard variational iteration method as available in literature. All computational frameworks were executed with MAPLE software.

This study focuses on a new class of functions called sub-b-s-preinvex, that is a generalization of sub-b-s-convex and preinvex functions, and discusses some of their properties. A new sub-b-s-preinvex programming is introduced and the sufficient conditions of optimality under this type of function is established.

In this paper, some mixed type bounds on the spectral radius $\rho(A\circ B)$ for the Hadamard product of two nonnegative matrices ($A$ and $B$) and the minimum eigenvalue $\tau(C\star D)$ of the Fan product of two $M$-matrices ($C$ and $D$) are researched. These bounds complement some corresponding results on the simple type bounds. In addition, a new lower bound on the minimum eigenvalue of the Fan product of several $M$-matrices is also presented: $$ \tau(A_{1}\star A_{2}\cdots\star A_{m})\geq \min_{1\leq i\leq n}\{\prod^{m}_{k=1}A_{k}(i,i)-\prod^{m}_{k=1}[A_{k}(i,i)^{P_{k}}-\tau(A_{k}^{(P_{k})})]^\frac{1}{P_{k}}\}, $$ where $A_{1},\ldots, A_{k}$ are $n\times n$ $M$-matrices and $P_{1},\ldots, P_{k}>0$ satisfy $\sum^{m}_{k=1}\frac{1}{P_{k}}\geq 1$. Some special cases of the above result and numerical examples show that this new bound improves some existing results.

In the present article we describe a few simple and efficient finite volume type schemes on moving grids in one spatial dimension for the sake of simplicity. The underlying finite volume scheme is conservative and it is accurate up to the second order in space. The main novelty consists in the motion of the grid. This new dynamic aspect can be used to resolve better the areas with high solution gradients or any other special features. No interpolation procedure is employed, thus an unnecessary solution smearing is avoided. Thus, our method enjoys excellent conservation properties. The resulting grid is completely redistributed according the choice of the so-called monitor function. Several more or less universal choices of the monitor function are provided. Finally, the performance of the proposed algorithm is illustrated on several examples stemming from the simple linear advection to the simulation of complex shallow water waves. We believe that the techniques described in our paper can be beneficially used to model tsunami wave propagation and run-up.

In this study, we present a numerical method for solving two--dimensional space--time fractional partial differential equations (FPDEs), where the solutions of the FPDEs are expanded in terms of the shifted Chebyshev polynomials. The numerical approximations are evaluated at the Chebyshev-Gauss-Lobatto points. In the case when the FPDE is nonlinear, we employ a Newton-Raphson approach to linearize the equation. Both the linear and nonlinear cases lead to a consistent system of linear algebraic equations. The scheme is tested on selected FPDEs and the numerical results show that the proposed numerical scheme is accurate and computationally efficient in terms of CPU times. To establish the accuracy of the method, we also present an error analysis which shows the convergence of the numerical method. These positive attributes make the proposed method a good approach for solving two--dimensional fractional partial differential equations with both space and time fractional orders.

Susceptible, infective, recovered, and hospitalized/isolated individuals are placed on the cells of a nxn square lattice, where each cell is occupied by a single individual or is vacant. At discrete time units (typically one day each) all susceptibles and infectives execute a random movement and when a coincidence of the two types occurs the susceptible is converted to infective status according to some probability. Infectives are labelled by the number of days since originally infected. At each time increment the age label of the infectives is increased by one unit. When the label reaches a number like 15 or 20 days the susceptibles recover with some probability or become isolated/hospitalized. Upon reaching some age the latter types either recover or die. Probabilities for the movements and conversions from one status to another are implemented by random numbergeneration. Simulations were carried out to investigate the effect of several probability and age parameters, the size of population (proportional to nxn) and density (related to fraction of occupied cells), and the size of the movements. Mid-term gradual conversion of susceptibles to isolated was explored as intervention policy.

In this paper, we study the dynamics of following system of nonlinear difference equations x_n+1 = x_n-1y_n - 1, y_n+1 = y_n-1z_n - 1, z_n+1 = z_n-1x_n - 1. Especially we investigate the periodicity, boundedness and stability of related system of difference equations.

In this paper, by using the q-integer, we introduce a new generalization of Fibonacci and Lucas sedenions called q-Fibonacci and q-Lucas sedenions. We present some fundamental properties of these type of sedenions such as Binet formulas, exponential generating fuctions, summation formulas, Catalan's identity, Cassini's identity and d'Ocagne's identity.

In Functional Analysis as well as Topology, we frequently encounter sets, say $X$, that contain elements close to each other. These sets display a defining "finite-ness" property: for all open cover $\mathcal{O}$ of $X$, there exists a finite subcollection $\mathcal{U}\subseteq\mathcal{O}$ such that $\mathcal{U}$ covers $X$. Such spaces $X$ are called $\textbf{compact}$, and the above "finite-ness" property afford us great convenience, because we can always investigate $X$ by investigating its finite open cover $\mathcal{U}$Arzela-

The aim of this survey is to review some recent developements in devising efficient preconditioners for sequences of linear systems A x = b. Such a problem arise in many scientific applications, such as discretization of transient PDEs, solution of eigenvalue problems, (Inexact) Newton method applied to nonlinear systems, rational Krylov methods for computing a function of a matrix. Full purpose preconditioners such as the Incomplete Cholesky (IC) factorization or approximate inverses are aimed at clustering eigenvalues of the preconditioned matrices around one. In this paper we will analyze a number of techniques of updating a given IC preconditioner (which we denote as P0 in the sequel) by a low-rank matrix with the aim of further improving this clustering. The most popular low-rank strategies are aimed at removing the smallest eigenvalues (deflation) or at shifting them towards the middle of the spectrum. The low-rank correction is based on a (small) number of linearly independent vectors whose choice is crucial for the effectiveness of the approach. In many cases these vectors are approximations of eigenvectors corresponding to the smallest eigenvalues of the preconditioned matrix P0 A. We will also review some techniques to efficiently approximate these vectors when incorporated within a sequence of linear systems all possibly having constant (or slightly changing) coefficient matrices. Numerical results concerning sequences arising from discretization of linear/nonlinear PDEs and iterative solution of eigenvalue problems show that the performance of a given iterative solver can be very much enhanced by the use of low-rank updates.

Applications in high-performance computing (HPC) may not use all available computational resources, leaving some of them underutilized. By co-scheduling, i.e. running more than one application on the same computational node, it is possible to improve resource utilization and overall throughput. Some applications may have conflicting requirements on resources and co-scheduling may cause performance degradation, so it is important to take it into account in scheduling decisions. In this paper, we formalized co-scheduling problem and proposed multiple scheduling strategies to solve it: an optimal strategy, an online strategy and heuristic strategies. These strategies vary in terms of the optimality of the solution they produce and a priori information about the system they require. We showed theoretically that the online strategy provides schedules with a competitive ratio that has a constant upper limit. This allowed us to solve the co-scheduling problem using heuristic strategies that approximate this online strategy. Numerical simulations showed how heuristic strategies compare to the optimal strategy for different input systems. We proposed a method for measuring input parameters of the model in practice and evaluated this method on HPC benchmark applications. We showed high accuracy of measurement method, which allows to apply proposed scheduling strategies in scheduler implementation.

Has anyone ever missed an event because he was confused in days and dates? Do we remember the date of any day without looking at a calendar? Is the current Gregorian Calendar efficient enough for use, and does it facilitate our life or make it more complicated? Have you ever thought about a much simpler way to calculate days and dates in a year? All these questions are answered in this paper, in which the author proposes original optimization algorithm that creates optimal perennial calendars. Results show that there is more than one way to create a perennial calendar, in which the number of days in each month does not change, neither the dates. Hence, all months have the same sequence of days and dates. In other meaning, Monday becomes the first day of every month, and Sunday becomes the last day. Consequently, the calendars become much easier to memorize and very simple to predict the days and dates in any year.

This paper answers the need to plan a cost-minimizing charging schedule for electric buses and proposes a three-stage procedure, which is oriented around the participation of electric buses aggregator in a day-ahead energy auction. First, optimization models are provided to determine charging availability expressed as minimum and maximum hourly energy requirements taking into account detailed, minutely characteristics and constraints of the charging equipment and the buses. Next, the auction model is formulated by considering aggregated bids submitted by the electric buses aggregator once the charging availability is determined. Hence, the day-ahead prices reflect the optimal schedules of auction participants, and the bus aggregator is safe against peak-hour charging. Finally, hourly auction-based schedules are disaggregated into optimal minutely charging schedules. Mixed-integer linear programming models are formulated for aggregation-disaggregation stages investigated in this paper, while the variables and constraints introduced into the auction model are linear. The proposed methodology has been verified on a recently published case study of a real-world bus service operated on The Ohio State University campus. We show that the auction-based charging of all 22 buses outperforms as-soon-as-possible schedules by 7% to even 28% of daily cost savings. Using the aggregated bids, buses can flexibly shift charges between high- and low-price periods while preserving constraints of the charging equipment and timetables.

The bilateral tempered fractional derivatives are introduced generalising previous works on the one-sided tempered fractional derivatives and the two-sided fractional derivatives. An analysis of the tempered Riesz potential is done and showed that it cannot be considered as a derivative.

In tracking-by-detection paradigm for multi-target tracking, target association is modeled as an optimization problem that is usually solved through network flow formulation. In this paper, we proposed combinatorial optimization formulation and used a bipartite graph matching for associating the targets in the consecutive frames. Usually, the target of interest is represented in a bounding box and track the whole box as a single entity. However, in the case of humans, the body goes through complex articulation and occlusion that severely deteriorate the tracking performance. To partially tackle the problem of occlusion, we argue that tracking the rigid body organ could lead to better tracking performance compared to the whole body tracking. Based on this assumption, we generated the target hypothesis of only the spatial locations of person’s heads in every frame. After the localization of head location, a constant velocity motion model is used for the temporal evolution of the targets in the visual scene. Qualitative results are evaluated on four challenging video surveillance dataset and promising results has been achieved.

By introducing a variable transformation $\xi=\frac{1}{2}(\sin \theta+1)$, a complex-form ordinary differential equation (ODE) for the small symmetrical deformation of an elastic torus is successfully transformed into the well-known Heun's ODE, whose exact solution is obtained in terms of Heun's functions. To overcome the computational difficulties of the complex-form ODE in dealing with boundary conditions, a real-form ODE system is proposed. A general code of numerical solution of the real-form ODE is written by using Maple. Some numerical studies are carried out and verified by both finite element analysis and H. Reissner's formulation. Our investigations show that both deformation and stress response of an elastic torus are sensitive to the radius ratio, and suggest that the analysis of a torus should be done by using the bending theory of a shell.

Interval-valued functions have been widely used to accommodate data inexactness in optimization and decision theory. In this paper, we study interval-valued vector optimization problems, and derive their relationships to interval variational inequality problems, of both Stampacchia and Minty types. Using the concept of interval approximate convexity, we establish necessary and sufficient optimality conditions for local strong quasi and approximate $LU$-efficient solutions to nonsmooth optimization problems with interval-valued multiobjective functions.

We construct a novel family of summation-integral type hybrid operators in terms of shape parameter $\alpha\in \lbrack 0,1]$ in this paper. Basic estimates, rate of convergence, and order of approximation are also studied using the Korovkin theorem and the modulus of smoothness. We investigate the local approximation findings for these sequences of positive linear operators utilising Peetre's K-functional, Lipschitz class, and second-order modulus of smoothness.

In this paper, we consider with a class of system of differential equations whose argument transforms are involution. In this an initial value problem for a differential equation with involution is reduced to an initial value problem for a higher order ordinary differential equation. Than either two initial conditions are necessary for a solution, the equation is then reduced to a boundary value problem for a higher order ODE.

In this article, we set up some adequate conditions for analytic functions associated with cosine hyperbolic function. We determine conditions on α such that 1+α (ϰ^(2+P(j-1) ) g^' (ϰ))/(Pg^j (ϰ)),for each j=0,1,2,3 are subordinated by Janowski function, then (g(ϰ))/ϰ^P <cosℏ⁡(ϰ),(ϰ∈C). By choosing specific values of functions g we get some adequate conditions for multivalent starlike function related with cosine hyperbolic.

Let $G$ be a group with identity $e$ and $R$ be a commutative $G$-graded ring with nonzero unity $1$. Graded semi-primary and graded $1$-absorbing primary ideals have been investigated and examined by several authors as generalizations of graded primary ideals. However, these three concepts are different. In this article, we characterize graded rings over which every graded semi-primary ideal is graded $1$-absorbing primary and graded rings over which every graded $1$-absorbing primary ideal is graded primary.

In this note we study the computation of the minimum polynomial of a matrix $A$ and how we can use it for the computation of the matrix $A^n$. We also describe the form of the elements of the matrix $A^{-n}$ and we will see that it is closely related with the computation of the Drazin generalized inverse of $A$. Next we study the computation of the exponential matrix and finally we give a simple proof of the Leverrier - Faddeev algorithm for the computation of the characteristic polynomial.

Background: Bangladeshi students from science, technology, engineering, and mathematics (STEM) often struggle with solving many mathematical problems in different pedagogic contexts. They mostly lack the considerable prior learning or strong basics required to cope with the teaching and learning materials used at the undergraduate levels, which leads many students to take readmissions every year. Objective: This research aims at investigating the factors affecting the success and deficit of university undergraduate mathematics students in Bangladesh. The mixed-method research incorporates quantitative and qualitative data analysis on the students' and teachers’ perspectives regarding the issues. The authors focus more on categorizing the reasons influencing effective mathematics pedagogies than on identifying new or unknown causes. Methodology: This study is outlined in three phases. The phases include i. Exploratory qualitative survey ii. Quantitative triangulation survey, iii. Explanatory semi-structured interviews. Findings: First, the qualitative survey exposes the important factors that highlight the student’s success and failure in mathematics. Next, the quantitative data confirm that there are some similarities and dissimilarities between students’ and teachers’ perceptions. Also, the coefficient correlation analysis shows male students lack consistency and passion for study resulting in poor performances. Conversely, female students emphasize the inability to connect mathematical theories to real-life usages, curriculum loads, and unavailable resources as the reasons for underperformance. Finally, the interview data demonstrate the students attribute their failure to inadequate practices, memorizing habits, poor teaching, low motivation, and external distractions. Also, students acknowledge the necessity of steady practice, clear understanding, regular study, and working strategies for successful mathematics education. Teachers emphasize students’ clear concepts, aptitude, motivation, and curiosity for successful learning. Conclusion: This conclusion proposes a fresh start with the local mathematics pedagogic practices by analyzing teacher-student feedback on the success and failure factors impacted by varied individual and contextual elements. The study offers inclusive feedback on the part of both stakeholders. However, an open discussion or interaction between students and teachers might be needed to enhance mutual trust and understanding between them.

This note addresses a fundamental problem in matrix theory on establishing and characterizing range equalities for matrix expressions that involve generalized inverses. We first establish a group of necessary and sufficient conditions for the matrix range equality ${\rm range}(D_1 - C_1A_1^{\dag}B_1) = {\rm range}(D_2 - C_2A_2^{\dag}B_2)$ to hold, where $(\cdot)^{\dag}$ denotes the Moore--Penrose inverse of matrix. We then give several groups of range equalities with extrusion properties for multiple matrix products associated with two matrices and their conjugate transposes and Moore--Penrose inverses.

We enhance Frobenius' method for solving linear ordinary differential equations about regular singular points. Key to Frobenius' approach is the exploration of the derivative with respect to a single parameter; this parameter is introduced through the powers of generalized power series. Extending this approach, we discover that tandem recurrence relations can be derived. These relations render coefficients for series occurring in logarithmic solutions. The method applies to the, practically important, exceptional cases in which the roots of the indicial equation are equal, or differ by a non-zero integer. We demonstrate the method on Bessel's equation and derive previously unknown tandem recurrence relations for coefficients of solutions of the second kind, for Bessel equations of all integer and half-integer order.

This paper reviews how dynamically adaptive wavelet methods can be designed to simulate atmosphere and ocean dynamics in both flat and spherical geometries. We highlight the special features that these models must have in order to be valid for climate modelling applications. These include exact mass conservation and various mimetic properties that ensure the solutions remain physically realistic, even in the under-resolved conditions typical of climate models. Particular attention is paid to the implementation of complex topography in adaptive models. Using \textsc{wavetrisk} as an example, we explain in detail how to build a semi-realistic global atmosphere or ocean model of interest to the geophysical community. We end with a discussion of the challenges that remain to developing a realistic dynamically adaptive atmosphere or ocean climate models. These include scale-aware subgrid scale parameterizations of physical processes, such as clouds. Although we focus on adaptive wavelet methods, many of the topics we discuss are relevant for adaptive mesh refinement (AMR).

In this paper, the author utilizes Saddle Theorem and variational methods to deduce existence of at least six stationary solutions for reaction-diffusion Gilpin-Ayala competition model (RDGACM). To obtain the global stabilization of the positive stationary solution of the RDGACM, the author designs a suitable impulsive event triggered mechanism (IETM) to derive the global exponential stability of the the positive stationary solution. It is worth mentioning that the new mechanism can exclude Zeno behavior and effectively reduce the cost of impulse control through event triggering mechanism. Besides, compared with existing literature, the restrictions on the parameters of the RDGACM are relaxed so that the methods used in existing literature can not be applied to the relaxed case of this paper, and so the author makes comprehensive use of Saddle Theorem, orthogonal decomposition of Sobolev space $H_0^1(\Omega)$ and variational methods to overcome the mathematical difficulty. Numerical examples show the effectiveness of the methods proposed in this paper.

An embryonic version of membrane theory can be date back to the Bernstein's work reported more than a hundred years ago. Such an originally old work has evolved conceptually and mathematically up until today, and it plays a central role in current membrane theory. Goldman-Hodgkin-Katz equation (GHK eq.) is one of the math-based monumental works, which constitutes the present membrane theory. Goldman theoretically derived GHK eq., but its physiological meaning was provided by the two renowned scientists, Hodgkin and Katz. These two employed an assumption that the electric field (EF) across the plasma membrane is constant to validate the GHK eq. physiologically. Proposal of Hodgkin-Huxley model (HH model) is another math-based monumental works developed from the membrane theory and now forms a fundamental part of the current membrane theory. GHK eq. and HH model are quite fundamental central concepts in the current physiology. Despite the broad acceptance of GHK eq. at present time, its prerequisite that the EF within the plasma membrane is constant is hardly believable. Especially when the action potential is generated, it sounds totally nonsense. Furthermore, the existence of constant EF within the plasma membrane is conceptually almost in conflict with the HH model. The authors will discuss those problematic issues the membrane theory inherits.

The statistical properties of a good estimator include robustness, unbiasedness, efficiency, and consistency. However, the commonly used estimators of dispersion have lack or are weak in one or more of these properties. In this paper, I proposed statistical mirroring as a good alternative estimator of dispersion around defined location estimates or points. In the main part of the paper, attention is restricted to Gaussian distribution and only estimators of dispersion around the mean that functionalize with all the observations of a dataset were considered at this time. The different estimators were compared with the proposed estimators in terms of alternativeness, scale and sample size robustness, outlier biasedness, and efficiency. Monte Carlo simulation was used to generate artificial datasets for application. The proposed estimators (of statistical meanic mirroring) turn out to be suitable alternative estimators of dispersion that is less biased (more resistant) to contaminations, robust to scale and sample size, and more efficient to a random distribution of variable than the standard deviation, variance, and coefficient of variation. However, statistical meanic mirroring is not suitable with a mean (of a normal distribution) close to zero, and on a scale below ratio level.

Natural circulation loop (NCL) systems are buoyancy-driven heat exchangers that are used in various industrial applications. The concept of passive heat exchange in NCL systems is attractive, because there is no need for an externally driven equipment (e.g., a pump) to maintain the fluid circulation. However, relying on buoyancy as the sole driving force may lead to several potential difficulties, one of which is generation of (possibly) time-varying nonlinearities in the dynamical system, where a difference in the time scales of heat transfer and fluid flow causes the flow to change from a steady-state regime to either an oscillatory regime or a flow-reversal regime, both of which are undesirable. In this paper, tools of symbolic time series analysis (e.g., probabilistic finite state automata (PFSA)) are proposed to identify selected regimes of operation in NCL systems, where the underlying principle is built upon the concept of pattern classification from measurements of fluid-flow dynamics. The proposed method is shown to be capable of identifying the current regime of operation from the initial time response under a given set of operational parameters. The efficacy of regime classification is demonstrated by testing on two data sets, generated from numerical simulation of a MATLAB SimuLink model that has previously been validated with experimental data. The results of the proposed PFSA-based classification are compared with those of a hidden Markov model (HMM) that serves as the baseline.

Branching patterns are ubiquitous in nature, consequently over the years many researchers have tried to characterize the complexity of their structures. Due to their hierarchical nature and resemblance to fractal trees, they are often thought to have fractal properties, however their non-homogeneity (i.e., lack of strict self-similarity) is often ignored. In this paper we review and examine the use of the box-counting and sandbox methods to estimate the fractal dimensions of branching structures. We highlight the fact that these methods rely on an assumption of self-similarity that is not present in branching structures due to their non-homogeneous nature. Looking at the local slopes of the log-log plots used by these methods reveals the problems caused by the non-homogeneity. Finally, we examine the role of the canopies (endpoints or limit points) of branching structures in the estimation of their fractal dimensions.

The key concepts in symmetry detection and similarity, identity measures are automorphism and isomorphism respectively. Therefore, methods for symmetry detection and similarity, identity measures should be functionally bijective, inverse, and invariance under a set of mathematical operations. Nevertheless, few or no existing method is functional for these properties. In this paper, a new methodological paradigm, called optinalysis, is presented for symmetry detections, similarity, and identity measures between isoreflective or autoreflective pair of mathematical structures. The paradigm of optinalysis is the re-mapping of isoreflective or autoreflective pairs with an optical scale. Optinalysis is characterized as invariant under a set of transformations and its isoreflective polymorphism behaves on polynomial and non-polynomial models.

Chellali et al. introduced two degree concepts, ve-degree and ev-degree (Chellali et al, 2017). The ve-degree of a vertex equals to number of different edges which are incident to a vertex from the closed neighborhod of v. Moreover the ev-degree of an edge e=ab equals to the number of vertices of the union of the closed neighborhoods of a and b. The most private feature of these degree concepts is, total number of ve-degrees and total number of ev-degrees equal to first Zagreb index of the graphs for triangle-free graphs. In this paper we introduce ve-degree entropy, ev-degree entropy and investigate the relations between these entropies and the first Zagreb index entropy. Finally we obtain the maximal trees with respect to ve-degree irregularity index.

Ultra-wideband wavelength division multiplexed networks enable operators to use more effectively the bandwidth offered by a single fiber pair and thus make significant savings, both in operational and capital expenditures. The main objective of this study is to minimize optical node resources, such as transponders, multiplexers and wavelength selective switches, needed to provide and maintain high quality of network services, in ultra-wideband wavelength division multiplexed networks, at low cost. A model based on integer programming is proposed, which includes a detailed description of optical network nodal resources. The developed optimization tools are used to study the ultra-wideband wavelength division multiplexed network performance when compared with the traditional C-band wavelength division multiplexed networks. The analysis is carried out for realistic networks of different dimensions and traffic demand sets.

In this paper, we study the entire or meromorphic solutions for differential-difference equations in f(z) , its shifts, its derivatives and derivatives of its shifts. and study some Hayman's results for differential-difference polynomials.

One of the most broadly used models for membrane fouling is the Hermia model, which separates this phenomenon into four blocking mechanisms, each with an associated parameter n. These mechanisms are complete blocking (n=2), intermediate blocking (n=1), standard blocking (n=3/2) and cake formation (n=0). The original model, which was obtained through experimental data, is given by an Ordinary Differential Equation (ODE) dependent on n. At the time, this ODE was only solved for these four values of n, which limits the effectiveness of the model when adjusted to experimental data. The aim of this paper is to not only mathematically prove the original Hermia model, but also to broaden the scope of this model for any real number n by using the original ODE, the equations of fluid mechanics and the properties of single and multivariable calculus. The final generalized Hermia model is given by a power-law for any n≠2 and is given by an exponential function at n=2 and can be fitted to ultrafiltration, microfiltration, nanofiltration and reverse osmosis data with acceptable values of R2 (&gt;0.93). Here it is also shown that the accumulated volume as a function of time follows the same type of ODE. The values of n between the four original discreate values could be physically interpreted as the existence of new blocking mechanisms.

In this note we construct explicit complex and real matrix representations for the generators of real Clifford algebra $C\ell_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. We find two classes of representation, the normal representation and exceptional one. The normal representation is a large class of representation which can only be expanded into $4m+1$ dimension, but the exceptional representation can be expanded as generators of the next period. In the cases $p+q=4m$, the representation is unique in equivalent sense. These results are helpful for both theoretical analysis and practical calculation. The generators of Clifford algebra are the faithful basis of $p+q$ dimensional Minkowski space-time or Riemann space, and Clifford algebra converts the complicated relations in geometry into simple and concise algebraic operations, so the Riemann geometry expressed in Clifford algebra will be much simple and clear.

One of essential procedure nowadays to analyze a dynamic process in Science and Engineering is Computational modeling. Most of dynamic processes are distributed parameter systems, i.e., systems with state variables depend on the number of independent variables (such as space and time) which are expressed by a set of Partial Differential Equations (PDEs). Lateral method of lines is a method which used for solving PDEs numerically by solving a set of boundary value problems sequentially. The aim is to report the development of PDE toolbox based on SCILAB. This paper provides users with a method that is very easy to understand in the applications related to dynamic process. The method uses a SCILAB template to develop numerical simulations in the new field. In this paper, a model of tropical fruit heat treatment for decimating insect is presented. The illustration will be given to show how the lateral method of line template works to solve the problem.

A novel mathematical model is developed to calculate the temperature distribution on the surface and bulk of a steel plate under the laser hardening process. The model starts with the basic heat equation then it is developed into a volumetric form and is connected to the various solid existing phases. The proposed model is based on three influencing parameters of the laser hardening process which are the velocity of the laser spot and irradiation time. The results are compared with the available experimental data reported in the literature. The volumetric model provides an assessment of temperature distribution in both the vertical and horizontal axis. Laser irradiation at sufficiently high fluence can be used to create a solid-state phase change on the surface. Primary calculations show that the temperature profile has a Gaussian distribution in horizontal x and y-axis and presents an exponentially decreasing in the horizontal and vertical depth directions.

Many authors proposed analytical methods for solving fully fuzzy Sylvester matrix equation (FFSME) based on Vec-operator and Kronecker product. However, these methods are restricted to nonnegative fuzzy numbers and cannot be extended to FFSME with near-zero fuzzy numbers. The main intention of this paper is to develop a new numerical method for solving FFSME with near-zero trapezoidal fuzzy numbers that provides a wider scope of trapezoidal fully fuzzy Sylvester matrix equation (TrFFSME) in scientific applications. This numerical method can solve the trapezoidal fully fuzzy Sylvester matrix equation with arbitrary coefficients and find all possible finite arbitrary solutions for the system. In order to obtain all possible fuzzy solutions, the TrFFSME is transferred to a system of non-linear equations based on newly developed arithmetic fuzzy multiplication between trapezoidal fuzzy numbers. The fuzzy solutions to the TrFFSME are obtained by developing a new two-stage algorithm. To illustrate the proposed method numerical example is solved.

Existing proofs of the 4-color theorem succeeded by establishing an unavoidable set of reducible configurations. By this device, their authors showed that a minimum counterexample cannot exist. G.D. Birkhoff proved that a minimum counterexample must satisfy a connectivity property that is referred to in modern parlance as internal 6-connectivity. We show that a minimum counterexample must also satisfy a coloring property, one that we call Kempe-locking. We define the terms Kempe-locking configuration and fundamental Kempe-locking configuration. We provide a heuristic argument that a fundamental Kempe-locking configuration must be of low order and then perform a systematic search through isomorphism classes for such configurations. We describe a methodology for analyzing whether an arbitrary planar triangulation is Kempe-locked; it involves deconstructing the triangulation into a stack of configurations with common endpoints and then creating a bipartite graph of coloring possibilities for each configuration in the stack to assess whether certain 2-color paths can be transmitted from the configuration's top boundary to its bottom boundary. All Kempe-locked triangulations we discovered have two features in common: (1) they are Kempe-locked with respect to only a single edge, say $xy$, and (2) they have a Birkhoff diamond with endpoints $x$ and $y$ as a proper subgraph. On the strength of our various investigations, we are led to a plausible conjecture that the Birkhoff diamond is the only fundamental Kempe-locking configuration. If true, this would establish that the connectivity and coloring properties of a minimum counterexample to the 4-color theorem are incompatible. It would also point to the singular importance of a particularly elegant 4-connected triangulation of order 9 that consists of a triangle enclosing a pentagon enclosing a single vertex.

The properties of iterative splitting methods with serial versions have been analyzed since recent years, see [1] and [3]. We extend the iterative splitting methods to a class of parallel versions, which allow to reduce the computational time and keep the benet of the higher accuracy with each iterative step. Parallel splitting methods are nowadays important to solve large problems, which can be splitted in subproblems and computed independently with the dierent processors. We present a novel parallel iterative splitting method, which is based on the multi-splitting methods, see [2], [10] and [15]. Such a exibilisation with multisplitting methods allow to decompose large iterative splitting methods and recover the benet of their underlying waveform-relaxation (WR) methods. We discuss the convergence results of the parallel iterative splitting methods, while we could reformulate such an error to a summation of the individual WR methods. We discuss the numerical convergence of the serial and parallel iterative splitting methods and present dierent numerical applications to validate the benet of the parallel versions.

We consider several unification problems in mathematics. We refer to transcendental numbers. Also, we present some ways to unify the main non-associative algebras (Lie algebras and Jordan algebras) and associative algebras.

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants such as Catalan’s constant and π

Machine learning forks into three main branches such as supervised learning, unsupervised learning, and reinforcement learning where reinforcement learning is much potential to artificial intelligence (AI) applications because it solves real problems by progressive process in which possible solutions are improved and finetuned continuously. The progressive approach, which reflects ability of adaptation, is appropriate to the real world where most events occur and change continuously and unexpectedly. Moreover, data is getting too huge for supervised learning and unsupervised learning to draw valuable knowledge from such huge data at one time. Bayesian optimization (BO) models an optimization problem as a probabilistic form called surrogate model and then directly maximizes an acquisition function created from such surrogate model in order to maximize implicitly and indirectly the target function for finding out solution of the optimization problem. A popular surrogate model is Gaussian process regression model. The process of maximizing acquisition function is based on updating posterior probability of surrogate model repeatedly, which is improved after every iteration. Taking advantages of acquisition function or utility function is also common in decision theory but the semantic meaning behind BO is that BO solves problems by progressive and adaptive approach via updating surrogate model from a small piece of data at each time, according to ideology of reinforcement learning. Undoubtedly, BO is a reinforcement learning algorithm with many potential applications and thus it is surveyed in this research with attention to its mathematical ideas. Moreover, the solution of optimization problem is important to not only applied mathematics but also AI.

This paper puts forward an alteration for Tensor Calculus utliized in a coordinate system which is under a dynamic distortion drawing inspiration from similar fields such as the Calculus of Moving Surfaces (CMS). The paper establishes transformation laws for Tensors within these regions and establishes Operators such as the Tensorial Field Derivative which enforce a Tensorial Transformation on Tensors defined within a Dynamically Moving coordinate system. This variation of Tensor Calculus can be utilized to observe how disciplines such as QFT and Continuum Mechanics would change to accomodate for a distorting coordinate system and can be utliized to develop new theoretical models which account for this temporal distortion particularly within Biological Settings.

This document focuses attention on the fundamental solution of an autonomous linear retarded functional differential equation (RFDE) along with its supporting cast of actors: kernel matrix, characteristic matrix, resolvent matrix, and the Laplace transform. The fundamental solution is presented in a form of the convolutional powers of the kernel matrix in the manner of a convolutional exponential matrix function. The fundamental solution combined with a solution representation gives an exact expression in explicit form for the solution of a RFDE. Algebraic graph theory is applied to the RFDE in the form of a weighted loop-digraph to illuminate the system structure and system dynamics and to identify the strong and weak components. Examples are provided in the document to elucidate the behavior of the fundamental solution. The paper introduces fundamental solutions of other functional differential equations.

Ultrasonic based inspection of thin-walled structures often requires prior knowledge of their mechanical properties. Their accurate estimation could be achieved in a non-destructive manner employing, e.g., elastic guided waves. Such procedures require efficient approaches for experimental data extraction and processing, which is still a challenging task. An advanced automated technique for material properties identification of an elastic waveguide is proposed in this investigation. It relies on the information on dispersion characteristics of guided waves, which are extracted by applying the matrix pencil method to the measurements obtained via laser Doppler vibrometry. Two objective functions have been successfully tested, and the advantages of both approaches are discussed (accuracy vs computational costs). The numerical analysis employing the synthetic data generated via the mathematical model as well as experimental data shows that both approaches are stable and accurate. The influence of the presence of various modes in the extracted data is investigated. One can conclude that the influence of the corruptions related to the extraction of dispersion curves is not critical if the majority of guided waves propagating in the considered frequency range are presented. Possible extensions of the proposed technique for damaged and multi-layered structures are also discussed.

The purpose of the research is to investigate the unique solvability of a nonlocal boundary value problem for a loaded equation of parabolic-hyperbolic type in a special domain. Methods: Using representations of a general regular solution, the existence and uniqueness of the problem posed is proved. Results: A new method for proving the unique solvability of nonlocal boundary value problems for one loaded equation of mixed type in a special domain has been formed and developed. Conclusions: The obtained results make it possible to solve some problems of genetics, immunology, transonic gas dynamics and thermal physics, which are associated with displacement.

We propose a model for medium-term hydropower scheduling (MTHS) with variable head and uncertainty in inflow, reserve capacity, and energy price. With an increase of intermittent energy sources in the generation mix, it is expected that a flexible hydropower producer can obtain added profits by participating in other markets than just the energy market. To capture this added potential the hydropower system should be modeled with a higher level of details. In this context, we apply an algorithm based on stochastic dual dynamic programming (SDDP) to solve the nonconvex MTHS problem, and show that the use of Strengthened Benders (SB) cuts to represent the expected future profit (EFP) function provide accurate scheduling results for slightly nonconvex problems. A method to visualize the EFP function in a dynamic programming setting is provided, serving as a useful tool for a priori inspection of the EFP shape and its nonconvexity.

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered as generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand-Naimark-Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs.

We evaluate some binomial cubic Fibonacci sums.

In this paper a new monolithic Eulerian formulation in the framework of non-classical continuum is presented for the analysis of fluid-strucutre interaction problems. In this respect, Cosserat continuum theory taking into account the micro-rotational degrees of freedom of the particles is considered. Continuum description of the model and variational formulation of the governing flow dynamics for non-classical -fluid-structure interaction nCFSI is presented. The model is analyzed by computing a well known benchmark problem by Hecht and Pironneau [16]. The algorithmic description is presented and implemented using FreeFEM++. Code is validated with the benchmark solution of Turek and Hron [38] in case of flow around a flag attached with cylinder. New microstructral behavior of the solution is studied and numerical simulations and results are shown in the form of figures. Some interesting feature of the flow is observed and microstructural characteristics are discussed.

This paper aims to introduce the notions of various types of Qc-contractions for which we establish some fixed circle and fixed disc theorems in the setting of metric spaces. Some illustrative examples are also provided to support our results. Moreover, we present some fixed circle and fixed disc results of integral type contractive self-mappings.

In this paper, inspired by Jleli and Samet [journal of inequalities and applications 38 (2014) 2 1–8] we introduce two new classes of auxiliary functions and utilize the same to define (q, y)R-weak 3 contractions. Utilizing (q, y)R-weak contractions, we prove some fixed point theorems in the setting 4 of relational metric spaces. We employ some examples to substantiate the utility of our newly proved 5 results. Finally, we apply one of our newly proved results to ensure the existence and uniqueness of 6 solution of a Volterra-type integral equation.

Although most biological processes are characterized by a strong temporal asymmetry, several popular mathematical models neglect this issue. Maximum entropy methods provide a principled way of addressing time irreversibility, which leverages powerful results and ideas from the3literature of non-equilibrium statistical mechanics. This article provides a comprehensive overview of these issues, with a focus in the case of spike train statistics. We provide a detailed account of the5mathematical foundations and work out examples to illustrate the key concepts and results from non-equilibrium statistical mechanics

The purpose of this paper is to prove strong convergent theorems for Browder's type iterations and Halpern's type iterations of a family of nonexpansive mappings in a complete geodesic space with curvature bounded above by a positive number. Moudafi's viscosity type methods are also discussed without the nice projection property.

The last series of Raven's standard progressive matrices (SPM-LS) test were studied with respect to its psychometric properties in a series of recent papers. In this paper, the SPM-LS dataset is analyzed with regularized latent class models (RLCM). For dichotomous item response data, an alternative estimation approach for RLCMs is proposed. For polytomous item responses, different alternatives for performing regularized latent class analysis are proposed. The usefulness of the proposed methods is demonstrated in a simulated data illustration and for the SPM-LS dataset. For the SPM-LS dataset, it turned out the regularized latent class model resulted in five partially ordered latent classes.

I want to find an extension of the Hausdorff Measure and Fractal Measure which allows me to compute a unique, satisfying average for functions defined on non-fractal, measurable sets in the sense of Carathedory which has no gauge function.

Let $f:V\rightarrow V $ be a Cohomological Expanding Mapping¹ of a smooth complex compact homogeneous manifold with dim_C(V) = k ≥ 1 and Kodaira Dimension $\leq 0$. We study the dynamics of such mapping from a probabilistic point of view, that is, we describe the asymptotic behavior of the orbit $ O_{f} (x) = \{f^{n} (x), n \in \mathbb{N} \quad \mbox{or}\quad \mathbb{Z}\}$ of a generic point. Using pluripotential methods, we construct a natural invariant canonical probability measure of maximum Cohomological Entropy $ \mu_{f} $ such that ${\chi_{2l}^{-m}} (f^m)^\ast \Omega \to \mu_f \qquad \mbox{as} \quad m\to\infty$ for each smooth probability measure $\Omega $ on V . Then we study the main stochastic properties of $ \mu_{f}$ and show that $ \mu_{f}$ is a measure of equilibrium, smooth, ergodic, mixing, K-mixing, exponential-mixing and the unique measure with maximum Cohomological Entropy. We also conjectured that $\mu_f:=T_l^+ \wedge T_{k-l}^-$, $\dim_\H(\mu_f)= \Psi h_{\chi}(f) $ and $\dim_\H( \mbox{Supp} T_l^+) \geq 2(k-l) + \frac{\log \chi_{2l}}{\psi_l}.$

In this paper, we present a relapse based demonstrating way to deal with investigate various arrangement MTC information. A commonplace use of this displaying approach incorporates three stages: first, define a model that approximates the connection between quality articulation and trial factors, with boundaries consolidated to address the exploration premium; second, utilize least-squares and assessing condition methods to gauge boundaries and their relating standard blunders; third, register test insights, P-qualities and NFD as proportions of factual criticalness. The benefits of this methodology are as per the following. To begin with, it tends to the exploration interest in a particular, precise way, and maximally uses all the information and other important data. Second, it represents both orderly and irregular varieties related with the information, and the consequences of such examination give not just quality explicit data applicable to the exploration objective, yet additionally its dependability, in this way helping agents to settle on better choices for subsequent investigations. Third, this methodology is truly adaptable, and can undoubtedly be stretched out to different sorts of MTC considers or other microarray explores by detailing various models dependent on the test plan of the investigations.

The angular diameter is the angle subtended by a generic object – an apple or a star – to the eye of an observer, and it describes how large the object appears from a given viewpoint. The angular diameter represents a powerful tool for distance calculations starting from a directly measurable information and it finds application in several contexts varying from cosmography to architecture. In this article, the author proposes a novel equation to calculate the apparent diameter of whatever object. This equation defines the relationship between the object’s apparent diameter with respect to the travelled distance starting from the initial distance R₀ at which the observed object is located. Based on the preliminary tests conducted, the model seems to faithfully portray this relation with respect to measured values, also at the astronomical scale, thus considering the Earth-Moon distance, where, the absolute error detected is about 0.56%. Tests highlight also a dependency between the results accuracy and the measurement conditions suggesting a high level of sensibility linked to the initial magnification effect produced by the retina or the artificial lens employed.

Nowadays, speed up development and use of digital devices such as smartphones have put people at risk of internet crimes. The evidence of present crimes in a computer file can be easily unreachable by changing the prefix of a file or other algorithms. In more complex cases, either file divided into different parts or the parts of a file that has information about the file type are deleted, where the file fragment recognition issue is discussed. The known files are divided into different fragments, and different classification algorithms to solve the problems of file fragment recognition. A confusion matrix measures the accuracy of type recognition. In the present study, first, the file is divided into different fragments. Then, the file fragment features, which are obtained from Binary Frequency Distribution (BFD), are reduced by 2 feature reduction algorithms; Sequential Forward Selection algorithm (SFS) as well as Sequential Floating Forward Selection algorithm (SFFS) to delete sparse features that result in increased accuracy and speed. Finally, the reduced features are given to 3 classifier algorithms, Multilayer Perceptron (MLP), Support Vector Machines (SVM), and K-Nearest Neighbor (KNN) for classification and comparison of the results. In this paper, we proposed the algorithm of file type recognition that can recognize 6 types of useful files ( pdf, txt, jpg, doc, html, exe), which may distinguish a type of file fragments with higher accuracy than the similar works done.

This paper focuses on quantifying the uncertainty in the specific absorption rate values of brain induced by the uncertain positions of the electroencephalography electrodes placed on patient's scalp. To avoid running a large number of simulations, an artificial neural network architecture for uncertainty quantification involving high-dimensional data is proposed in this paper. The proposed method is demonstrated to be an attractive alternative to conventional uncertainty quantification methods because of its considerable advantage in the computational expense and speed.

We derive new infinite series involving Fibonacci numbers and Riemann zeta numbers. The calculations are facilitated by evaluating linear combinations of polygamma functions of the same order at certain arguments.

In this paper, Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem and variational methods are employed to derive the existence and the unique existence results of (globally) exponentially stable positive stationary solution of delayed reaction-diffusion cell neural networks under Dirichlet zero boundary value, including the global stability criteria \textbf{in the classical meaning}. Next, sufficient conditions are proposed to guarantee the global stability invariance of ordinary differential systems under the influence of diffusions. New theorems show that the diffusion is a double-edged sword in judging the stability of diffusion systems. Besides, an example is constructed to illuminate that any non-zero constant equilibrium point must be not in the phase plane of dynamic system under Dirichlet zero boundary value, or it must lead to a contradiction. Next, under Lipschitz assumptions on active function, another example is designed to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks via a Saddle point theorem. Finally, a numerical example illustrates the feasibility of the proposed methods.

In this four-month-long study (from April 1, 2020 to August 1, 2020), we have collected, modeled, and analyzed COVID-19 data from the top five most infected counties per top six most infected states in the United States (30 counties total). More specifically, we collected data on each state’s total COVID-19 cases, deaths, tests conducted, and their counties’ population, density, percentage of seniors, number of hospitals, total COVID-19 cases, and total COVID-19 related deaths. In this study, we have models illustrating the growth of COVID-19 cases and deaths per county, growth of COVID-19 cases and deaths per state (which is really the sum of our chosen five counties), and growth of COVID-19 tests conducted per state. In addition, our study also contains models illustrating the statistics of several variables that might have affected a county’s COVID-19 data, which has been mentioned above: population, density, percentage of seniors, and number of hospitals. An interesting finding we have noticed upon modeling the 30 counties’ density and total COVID-19 cases as an xy scatter plot is that there is a considerably strong relationship between the two variables. Los Angeles County (which was an extreme outlier), in particular, supports the idea that a county’s most populous city can greatly affect its entire county’s COVID-19 cases; if the largest city is extremely dense, it appears that the entire county has a greater total COVID-19 case count.

Inverse problems of recovering surface fluxes on the boundary of a domain from pointwise observations are considered. Sharp conditions on the data ensuring existence and uniqueness of solutions in Sobolev classes are exposed. They are smoothness conditions on the data, geometric conditions on the location of measurement points, and the boundary of a domain. The proof relies on asymptotics of fundamental solutions to the corresponding elliptic problems and the Laplace transform. The problem is reduced to a linear algebraic system with a nondegerate matrix.

In this article, author performs computational analysis for the generalized zeta functions by using computational software Mathematica. To achieve the purpose recently obtained difference equations are used. These difference equations have a computational power to compute these functions accurately while they can not be computed by using their known integral represenations. Several authors investigated such functions and their analytic properties, but no work has been reported to study the graphical representations and zeors of these functions. Author performs numerical computations to evaluate these functions for different values of the involved parameters. Taylor series expansions are also presented in this research.

Software is playing the most important role in recent vehicle innovation, and consequently the amount of software has been rapidly growing last decades. Safety-critical nature of ships, one sort of vehicles, makes Software Quality Assurance (SQA) has gotten to be a fundamental prerequisite. Just-In-Time Software Defect Prediction (JIT-SDP) aims to conduct software defect prediction (SDP) on commit-level code changes to achieve effective SQA resource allocation. The first case study of SDP in maritime domain reported feasible prediction performance. However, we still consider that the prediction model has still rooms for improvement since the parameters of the model are not optimized yet. Harmony Search (HS) is a widely used music-inspired meta-heuristic optimization algorithm. In this article, we demonstrated that JIT-SDP can produce the better performance of prediction by applying HS-based parameter optimization with balanced fitness value. Using two real-world datasets from the maritime software project, we obtained an optimized model that meets the performance criterion beyond baseline of previous case study throughout various defect to non-defect class imbalance ratio of datasets. Experiments with open source software also showed better recall for all datasets despite we considered balance as performance index. HS-based parameter optimized JIT-SDP can be applied to the maritime domain software with high class imbalance ratio. Finally, we expect that our research can be extended to improve performance of JIT-SDP not only in maritime domain software but also in open source software.

Wireless cellular communication technology has developed into a very resourceful commodity worldwide. Today, people of all races can hardly live without means of voice and data cellular communication technology. Imprecise propagation loss estimation leads to high power waste, high co-channel interference and poor service quality in cellular communication system networks. This paper proposes a realistic adaptive fine-tuning method for distinctive propagation loss estimation over a microcellular communication radio links based on signal power measurements from Long Term Evolution radio broadband networks, taking non-line of sight (NLOS) and line of sight (LOS) environments into consideration. The methodology is verified by measurements taken in non-line of sight and line of sight signal propagation scenarios. The results showed that the estimated propagation losses using the proposed realistic adaptive tuning models were more accurate than the existing Cost -231 modelling estimation approach

An approach to estimating the objective function value of minimization maximum lateness problem is proposed. It is shown how to use transformed instances to define a new continuous objective function. After that, using this new objective function, the approach itself is formulated. We calculate the objective function value for some polynomially solvable transformed instances and use them as interpolation nodes to estimate the objective function of the initial instance. What is more, two new polynomial cases, that are easy to use in the approach, are proposed. In the end of the paper numeric experiments are described and their results are provided.

A simplified nonlinear dispersive system of BBM-type, initially derived by D. Mitsotakis, is employed here in order to model the generation and propagation of surface water waves over variable bottom. The simplification consists in applying the so-called Boussinesq approximation. Using the finite element method and the FreeFem++ software, we solve numerically this system for three different complexities for the bathymetry function: a flat bottom case, a variable bottom in space, and a variable bottom both in space and in time. The last case is illustrated with the Java 2006 tsunami event. This article is designed rather as a tutorial paper even if it contains the description of completely new adaptation techniques.

The switching game Lights Out and its variants were studied extensively as recreational mathematics problems. The game board of the ordinary Lights Out is a rectangular grid of lights, where each light is either on or off. By clicking a light, the clicked light and its adjacent rectilinear neighbors are toggled. Given an arbitrary initial configuration of lights, the final mission is to “solve” this game by switching off all the lights. Most studies on Lights Out and its variants focused on the solvability of given games or the number of solvable games, but when the game is viewed in a coding-theoretical perspective, more interesting questions with special meanings in coding theory will naturally pop up, such as finding the minimal number of lit lights among all solvable games except the solved game, or finding the minimal number of lit lights that the player can achieve from a given unsolvable game, etc. However, these problems are usually hard to be solved in general in terms of algorithmic complexity. This study considers a natural extension of the Lights Out game, which enlarges the toggle pattern in a way that all the lights in the same row and those in the same column of the clicked light are toggled. This variant of Lights Out is a two-state version of a switching game called Alien Tiles. In this paper, we investigate the properties of the two-state Alien Tiles, and discuss several coding-theoretical problems about this game. Then, we apply this game as an error correction code and investigate its optimality. We also give a brief overview on algorithmic complexity and coding theory for readers who are not familiar with these topics. The purpose of this paper is to propose ways of playing switching games in a think-outside-the-box manner, which benefit the recreational mathematics community.

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0&lt;ρ&lt;1, 0&lt;t≤T), u(ξ)=αu(0)+φ (α is a constant and 0&lt;ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann-Liouville derivative; naturally, in the case of the Riemann - Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

Let f : V → V be a Cohomological Expanding Mapping¹ of a smooth complex compact homogeneous manifold with $ dim_{\mathbb{C}}(\Vc)=k \ge 1$ and Kodaira Dimension $\leq 0$. We study the dynamics of such mapping from a probabilistic point of view, that is, we describe the asymptotic behavior of the orbit $ O_{h} (x) = \{h^{n} (x), n \in \mathbb{N} \quad \mbox{or}\quad \mathbb{Z}\}$ of a generic point. Using pluripotential methods, we have constructed in our previous paper \cite{Armand4} a natural invariant canonical probability measure of maximal Cohomological Entropy $ \nu_{h} $ such that ${\chi_{2l}^{-m}} (h^m)^\ast \Omega \to \nu_h \qquad \mbox{as} \quad m\to\infty$ for each smooth probability measure $\Omega $ in $\Vc$ . We have also studied the main stochastic properties of $ \nu_{h}$ and have shown that $ \nu_{h}$ is a smooth equilibrium measure , ergodic, mixing, K-mixing, exponential-mixing. In this paper we are interested on equidistribution problems and we show in particular that $ \nu_{h}$ reflects a property of equidistribution of periodic points by setting out the Third and Fourth Main Results in our study. Finally we conjecture that $$\nu_h:=T_l^+ \wedge T_{k-l}^-,$$ $$\dim_\HH(\nu_h)= \Psi h_{\chi}(h) , $$ $$\dim_\HH( \mbox{Supp} T_l^+) \geq 2(k-l) + \frac{\log \chi_{2l}}{\psi_l},$$ $$|\langle \nu_m^x-\nu_h,\zeta\rangle|\leq M \Big[1+\log^+{1\over D(x,\Tc)}\Big]^{\beta/2}\|\zeta\|_{\Cc^\beta} \gamma^{-\beta m/2}$$ and $$|\langle \nu_m^x-\nu_h,\zeta\rangle|\leq M \Big[1 +\log^+{1\over D(x,E_\gamma)}\Big]^{\beta/2}\|\zeta\|_{\Cc^\beta} \gamma^{-\beta m/2}.$$

COVID-19 outbreak started in the Chinese city of Wuhan and spread around the globe in months due to its high contagious level. Hence, this disease posed a great challenge to researchers and mathematicians. Numerous mathematical models have been suggested to visualize the spreading speed and trends of this pandemic. In this paper, a comparison of a widely established and accepted method, the Susceptible-Exposed-Infected-Quarantined-Recovered-Death-Insusceptible (SEIQRDP) model with a newly-proposed fractional-order SEIQRDP is studied. Densely populated Countries of Asia (Pakistan, India, and Bangladesh) have been chosen as data sets and both algorithms have been applied to their data. The same comparison technique has also been used on the data of two polar countries, New Zealand and Russia, that validated our findings. Error comparison of both algorithms has been recorded in a tabulated form which shows multi-fold erroneous trends over WHO data of selected countries.

Scattering hyperspectral technology is a nondestructive testing method with many advantages. Here, we propose a method to improve the accuracy of egg freshness, research the influence of incident angles of light source on the accuracy and explain its mechanism. A variety of weak classifiers classify eggs based on the spectra after preprocessing and feature wavelength extraction to obtain three classifiers with the highest accuracy. The three classifiers are used as metamodels of stacking ensemble learning to improve the highest accuracy from 96.25% to 100%. Moreover, the highest accuracy of scattering, reflection, transmission and mixed hyperspectral of eggs are 100.00%, 88.75%, 95.00% and 96.25%, respectively, indicating that the scattering hyperspectral for egg freshness detection is better than that of the others. In addition, the accuracy is inversely proportional to the angle of incidence due that the smaller the incident angle, the camera collects a larger proportion of scattering light, which contains more biochemical parameters of an egg than that of reflection and transmission. These results are very important for improving the accuracy of non-destructive testing and selecting the incident angle of the light source, and have potential applications in online non-destructive testing.

Video segmentation is crucial in a variety of practical applications especially in computer visions. Most of recent works in video segmentation are focusing on Deep learning based video segmentation, there are rooms for improvement in respect of the evolutionary algorithms. This paper aims to propose the novel method to video segmentation by using the optimization of segmentation parameters based on ensemble-based random forest and gradient boosting decision tree. The experimental results show Pareto front of segmentation parameters (hue, brightness, luminance, and saturation). Our optimization model yields accuracy: 85% +/-8.85 % (micro average: 85.00 %), average class precision: 84.88%, and average class recall: 85%. We also show the video segmentation results based on our optimization method and compare our results with Kinect-based video segmentation.

The collision avoidance mechanism adopted by the IEEE 802.11 standard is not optimal. The mechanism employs a binary exponential backoff (BEB) algorithm in the medium access control (MAC) layer. Such an algorithm increases the backoff interval whenever a collision is detected to minimize the probability of subsequent collisions. However, the expansion of the backoff interval causes degradation of the radio spectrum utilization (i.e., bandwidth wastage). That problem worsens when the network has to manage the channel access to a dense number of stations, leading to a dramatic decrease in network performance. Furthermore, a wrong backoff setting increases the probability of collisions such that the stations experience numerous collisions before achieving the optimal backoff value. Therefore, to mitigate bandwidth wastage and, consequently, maximize the network performance, this work proposes using reinforcement learning (RL) algorithms, namely Deep Q Learning (DQN) and Deep Deterministic Policy Gradient (DDPG), to tackle such an optimization problem. As for the simulations, the NS-3 network simulator is used along with a toolkit known as NS3-gym, which integrates a reinforcement-learning (RL) framework into NS-3. The results demonstrate that DQN and DDPG have much better performance than BEB for static and dynamic scenarios, regardless of the number of stations. Moreover, the performance difference is amplified as the number of stations increases, with DQN and DDPG showing a 27% increase in throughput with 50 stations compared to BEB. Furthermore, DQN and DDPG presented similar performances.

Abstract:Shown is the derivation of Lorentz-Einstein k-factor in SRT as an amplitude-term of oscillation-differential equations of second order.This case is shown for classical Lorentz-factor as solution of an equation for undamped oscillation as well as the developed theorem as a second solution for advanced SRT of fourth order with an equation for damped oscillation-states.This advanced term allows a calculation for any velocities by real rest mass

This paper reviews the unilateral and bilateral, one- and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one- and two-dimensional cases with the bilateral Laplace transform. General two-dimensional linear systems are introduced and the corresponding transfer function defined.

Global optimization is an imperative development of local optimization because there are many problems in artificial intelligence and machine learning requires highly acute solutions over entire domain. There are many methods to resolve the global optimization, which can be classified into three groups such as analytic methods (purely mathematical methods), probabilistic methods, and heuristic methods. Especially, heuristic methods like particle swarm optimization and ant bee colony attract researchers because their effective and practical techniques which are easy to be implemented by computer programming languages. However, these heuristic methods are lacking in theoretical mathematical fundamental. Fortunately, minima distribution establishes a strict mathematical relationship between optimized target function and its global minima. In this research, I try to study minima distribution and apply it into explaining convergence and convergence speed of optimization algorithms. Especially, weak conditions of convergence and monotonicity within minima distribution are drawn so as to be appropriate to practical optimization methods.

In 1937,German mathematician L.Collatz proposed a conjeture:for any positive integer,if it is even,divide it by 2;if it is odd,multiply it by 3 and 1 to get an even number;keep going on and on, the final result is 1.This paper gives a sequence classification of odd numbers,and gives 1-1 mapping of odd numbers and odd number sequences,and uses these results to give a proof of this conjecture.

This work is intended to lead a study of necessary and sufficient optimality conditions for scalar optimization problems on Hadamard manifolds. In the context of this geometry, we obtain and present new function types characterized by the property of having all their second-order stationary points to be global minimums. In order to do so, we extend the concept convexity in Euclidean space to a more general notion of invexity on Hadamard manifolds. This is done employing notions of second-order directional derivative, second-order pseudoinvexity functions and the second-order Karush-Kuhn-Tucker-pseudoinvexity problem. Thus, we prove that every second-order stationary point is a global minimum if and only if the problem is either second-order pseudoinvex or second-order KKT-pseudoinvex depending on whether the problem regards unconstrained or constrained scalar optimization respectively. This result has not been presented in the literature before. Finally, examples of these new characterizations are provided in the context of \textit{"Higgs Boson like"} potentials among others.

In this paper, we construct the degenerate poly-Fubini polynomials, called the type 2 degenerate poly-Fubini polynomials, by using the modified degenerate polyexponential function and derive several properties on the degenerate poly-Fubini polynomials and numbers. In the last section, we introduce type 2 degenerate unipoly- Fubini polynomials attached to an arithmetic function, by using the modified degenerate polyexponential function and investigate some identities for those polynomials. Furthermore, we give some new explicit expressions and identities of degenerate unipoly polynomials related to special numbers and polynomials.

This paper presents a new approach to a different proof of "Fermat's Last Theorem.” For the proof of the theorem, it is proposed to use a more straightforward geometrical approach. A special family of curves, the Elba curves, is introduced to facilitate the proof. This approach makes the proof easier to comprehend by mathematicians and those interested in the subject.

Kronecker products of matrices have some striking operation properties, one of which is called the mixed-product property $(A \!\otimes\! B)(C \!\otimes\! D) = AC \!\otimes\! BD$. In view of this property, the two-term Kronecker product $A_1 \otimes A_2$ can be rewritten as $A_1 \otimes A_2 = (A_1 \otimes I_{m_2})(I_{m_1} \otimes A_2)$ of dilation forms of $A_1$ and $A_2$, and three-term Kronecker product $A_1 \otimes A_2 \otimes A_3$ can be rewritten as the products $A_1 \otimes A_2 \otimes A_3 = (A_1\otimes I_{m_2} \otimes I_{m_3})(I_{m_1} \otimes A_2 \otimes I_{m_3})(I_{m_1} \otimes I_{m_2} \otimes A_3)$ of the dilation forms of $A_1$, $A_2$, and $A_3$, respectively, where the matrices on the right-hand sides of the two factorizations are commutative. In this note, we approach the commutative Kronecker products on the right-hand sides of the two factorization equalities, and present a variety of new and useful analytical formulas for calculating the ranks, dimensions, orthogonal projectors, and ranges of matrices composed of these Kronecker products.

The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush--Kuhn--Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash's critical and equilibrium points coincide in the case of invex payoff functions.

In the present study, a simple approximation expression is given for the relationship between the period and amplitude of a simple pendulum under magnetic action. The analytical solution presented for the given problem. Two numerical quadrature methods Simpson's and Boole's method were utilized to demonstrate a new approximation of the problem. The results of the numerical quadrature have been compared to the exact solution. Absolute and relative mistakes of the problem have been presented. The Matlab program 2013R has created a numerical method that is used to analyze the outcome, It has been determined that the comparison's outcomes attest to the method's suitability and correctness. Moreover, the results show that numerical solution is suitable for the problem.

One of the typical forms of linear matrix expressions (linear matrix-valued functions) is given by $A + B_1X_1C_1 + \cdots + B_kX_kC_k$, where $X_1, \ldots, X_k$ are independent variable matrices of appropriate sizes, which include almost all matrices with unknown entries as its special cases. The domain of the matrix expression is defined to be all possible values of the matrix expressions with respect to $X_1, \ldots, X_k$. I this article, we approach some problems on the relationships between the domains of two linear matrix expressions by means of the block matrix method (BMM), the matrix rank method (MRM), and the matrix equation method (MEM). As application, we discuss some topics on the relationships among general solutions of some linear matrix equations and their reduced equations.

Based on the definition of double fuzzy relations, two types of new rough set models are constructed, which are multi-granulation double fuzzy rough sets called optimistic and pessimistic multi-granulation double fuzzy rough sets, and some of their properties were studied. In addition, we study the relationship between optimistic and pessimistic multi-granulation double fuzzy rough sets and double fuzzy rough sets.