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The administration of A/B exams usually involves the use of items. Issues arise when the pre-establishment of a question bank is necessary and the inconsistency in the knowledge points to be tested (in the two exams) reduces the exams &lsquo, fairness&rsquo, These are critical for a large multi-teacher course wherein the teachers are changed such that the course and examination content are altered every few years. However, a fair test with randomly participating students should still be a guaranteed subject with no item pool. Through data-driven decision-making, this study collected data related to a term test for a compulsory general course for empirical assessments, pre-processed the data and used item response theory to statistically estimate the difficulty, discrimination and lower asymptotic for each item in the two exam papers. Binary goal programing was finally used to analyze and balance the fairness of A/B exams without an item pool. As a result, pairs of associated questions in the two exam papers were optimized in terms of their overall balance in three dimensions (as the goals) through the paired exchanges of items. These exam papers guarantee their consistency (in the tested knowledge points) and also ensure the fairness of the term test (a key psychological factor that motivates continued studies). Such an application is novel as the teacher(s) did not have a pre-set question bank and could formulate the fairest strategy for the A/B exam papers. The model can be employed to address similar teaching practice issues.

The International Institute of Knowledge Innovation and Invention (IIKII) is an institute that promotes the exchange of innovations and inventions, and establishes a communication platform for international innovations and researches. In 2019, IIKII cooperated with the Institute of Electrical and Electronics Engineers (IEEE) Tainan Section Sensors Council to hold IEEE conferences such as IEEE ICIASE 2019, IEEE ECBIOS 2019, IEEE ICKII 2019, ICUSA-GAME 2019, and IEEE ECICE 2019. This Special Issue entitled “Selected Papers from IIKII 2019 conferences” aims to select excellent papers from IIKII 2019 conferences, including symmetry in physics, chemistry, biology, mathematics, and computer science, etc. It selected 21 excellent papers from 750 papers presented in IIKII 2019 conferences on the topic of symmetry. The main goals of this Special Issue are to encourage scientists to publish their experimental and theoretical results in as much detail as possible, and to discover new scientific knowledge relevant to the topic of symmetry.

This comment is devoted to the paper “Lie Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation” (Symmery, 2020, vol.12, 170), in which several results are either incorrect, or incomplete, or misleading.

The first sentence in the abstract should be replaced with the sentence "A sequence $(x_{k})$ is called arithmetically convergent if for each $\varepsilon > 0$ there is an integer $n_{0}$ such that $|x_{m} - x_{}| \geq n_{0}$, where the symbol $< m, n >$ denotes the greatest common divisor of the integers $m$ and $n$".

In order to recognize outstanding papers that have been published in Symmetry, the Editorial Board established an annual “Symmetry Best Paper Award”. We are pleased to announce the first “Symmetry Best Paper Award” for 2015. Nominations were selected by the Editorial Board Office, with all research articles published in 2013 and 2014 eligible for consideration. Following review and voting by the Symmetry Best Paper Award Committee, the following three papers have won Symmetry Best Paper Awards for 2015

In this article we argue that in South Africa the current format of legitimised participation and practice in the examination papers for Mathematical Literacy restricts successful apprenticeship in the discipline of scientific mathematics and limits empowered preparation for real-world functioning. The currency of the subject, then, is brought into question. We further argue that the positioning of the subject as a compulsory alternative to Mathematics and the differential distribution of these two subjects to differing groups of learners facilitates the (re)production and sustainment of educational disadvantage. We draw on Dowling’s theoretical constructs of differing domains of mathematical practice and positions and focus analysis on a collection of nationally set exemplar Grade 12 examination papers to identify legitimised forms of participation in the subject. We conclude by arguing for a reconceptualised structure of knowledge and participation in Mathematical Literacy and make preliminary recommendations in this regard.

The class of log-elliptical distributions is well used and studied in risk measurement and actuarial science. The reason is that risks are often skewed and positive when they describe pure risks, i.e., risks in which there is no possibility of profit. In practice, risk managers confront a system of mutually dependent risks, not only one risk. Thus, it is important to measure risks while capturing their dependence structure. In this short paper, we compute the multivariate risk measures, multivariate tail conditional expectation, and multivariate tail covariance measure for the family of log-elliptical distributions, which captures the dependence structure of the risks while focusing on the tail of their distributions, i.e., on extreme loss events. We then study our result and examine special cases, as well as the optimal portfolio selection using such measures. Finally, we show how the given multivariate tail moments can also be computed for log-skew elliptical models based on similar approaches given for the log-elliptical case.

Recently, various types of single valued neutrosophic (SVN) rough set models were presented based on the same inclusion relation. However, there is another SVN inclusion relation in SVN sets. In this paper, we propose a new type of SVN covering rough set model based on the new inclusion relation. Furthermore, the graph and matrix representations of the new SVN covering approximation operators are presented. Firstly, the notion of SVN β 2 -covering approximation space is proposed, which is decided by the new inclusion relation. Then, a type of SVN covering rough set model under the SVN β 2 -covering approximation space is presented. Moreover, there is a corresponding SVN relation rough set model based on a SVN relation induced by the SVN β 2 -covering, and two conditions under which the SVN β 2 -covering can induce a symmetric SVN relation are presented. Thirdly, the graph and matrix representations of the new SVN covering rough set model are investigated. Finally, we propose a novel method for decision making (DM) problems in paper defect diagnosis under the new SVN covering rough set model.

This work presents an optimization proposal to better the computational convergence time in convection-diffusion and driven-cavity problems by applying a simulated annealing (SA) metaheuristic, obtaining optimal values in relaxation factors (RF) that optimize the problem convergence during its numerical execution. These relaxation factors are tested in numerical models to accelerate their computational convergence in a shorter time. The experimental results show that the relaxation factors obtained by the SA algorithm improve the computational time of the problem convergence regardless of user experience in the initial low-quality RF proposal.

The rationalization of logistics activities and processes is very important in the business and efficiency of every company. In this respect, transportation as a subsystem of logistics, whether internal or external, is potentially a huge area for achieving significant savings. In this paper, the emphasis is placed upon the internal transport logistics of a paper manufacturing company. It is necessary to rationalize the movement of vehicles in the company’s internal transport, that is, for the majority of the transport to be transferred to rail transport, because the company already has an industrial track installed in its premises. To do this, it is necessary to purchase at least two used wagons. The problem is formulated as a multi-criteria decision model with eight criteria and eight alternatives. The paper presents a new approach based on a combination of the Simple Additive Weighting (SAW) method and rough numbers, which is used for ranking the potential solutions and selecting the most suitable one. The rough Best–Worst Method (BWM) was used to determine the weight values of the criteria. The results obtained using a combination of these two methods in their rough form were verified by means of a sensitivity analysis consisting of a change in the weight criteria and comparison with the following methods in their conventional and rough forms: the Analytic Hierarchy Process (AHP), Technique for Ordering Preference by Similarity to Ideal Solution (TOPSIS) and MultiAttributive Border Approximation area Comparison (MABAC). The results show very high stability of the model and ranks that are the same or similar in different scenarios.

In order to solve the Sylvester equations more efficiently, a new four parameters positive and skew-Hermitian splitting (FPPSS) iterative method is proposed in this paper based on the previous research of the positive and skew-Hermitian splitting (PSS) iterative method. We prove that when coefficient matrix A and B satisfy certain conditions, the FPPSS iterative method is convergent in the parameter&rsquo, s value region. The numerical experiment results show that compared with previous iterative method, the FPPSS iterative method is more effective in terms of iteration number IT and runtime.

In this paper, it is explicitly determined the r-dynamic chromatic number of the direct product of any given path with either a complete graph or a wheel graph. Illustrative examples are shown for each one of the cases that are studied throughout the paper. Junta de Andalucía FQM-016

In this paper, variational discretization directed against the optimal control problem governed by nonlinear parabolic equations with control constraints is studied. It is known that the a priori error estimates is $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h+k)$ using backward Euler method for standard finite element. In this paper, the better result $|||u-u_h|||_{L^\infty(J; L^2(\Omega))} = O(h^2+k)$ is gained. Beyond that, we get a posteriori error estimates of residual type.

This paper establishes a model of continuous-state branching processes with time inhomogeneous competition in Levy random environments. Some results on extinction are presented, including the distribution of the extinction time, the limiting distribution conditioned on large extinction times and the asymptotic behavior near extinction. This paper also provides a new time-space transformation which can be used for further exploration in similar models. The results are applied to an epidemic model to describe the dynamics of infectious population and a virus model to describe the dynamics of viral load.

In this paper, we investigate the existence of stable standing waves for the nonlinear Schrodinger equation with inverse-power potential and combined power-type and Choquard-type nonlinearities $ i \partial_t\psi+\triangle \psi+\frac{\gamma}{|x|^\alpha}\psi+\lambda_1|\psi|^p\psi +\lambda_2(I_\beta\ast|\psi|^q)|\psi|^{q-2}\psi = 0,\; \; (t,x)\in [0,T^\star)\times \mathbb{R}^N. $ By using concentration compactness principle, when one nonlinearity is focusing and $ L^2 $-critical, the other is defocusing and $ L^2 $-supercritical, we prove the existence and orbital stability of standing waves. We extend the results of Li-Zhao in paper [ 14 ] to the $ L^2 $-critical and $ L^2 $-supercritical nonlinearities.

The aim of this paper is to generalize the results regarding fractals and prove escape conditions for general complex polynomial. In this paper we state the orbit of a newly defined iterative scheme and establish the escape criteria in fractal generation for general complex polynomial. We use established escape criteria in algorithms to generate Mandelbrot and Multi-corns sets. In addition, we present some graphs of quadratic, cubic and higher Mandelbrot and Multi-corns sets and discuss how the alteration in parameters make changes in graphs.

In this paper, several oscillation criteria for a class of higher order dynamic equations with superlinear neutral term are established. The proposed results provide a unified platform that adequately covers both discrete and continuous equations and further sufficiently comments on oscillatory behavior of more general class of equations than the ones reported in the literature. We conclude the paper by demonstrating illustrative examples.

This work relates to the structural act theory. The structural theory includes the description of acts over certain classes of monoids or having certain properties, for example, satisfying some requirement for the congruence lattice. The congruences of universal algebra is the same as the kernels of homomorphisms from this algebra into other algebras. Knowledge of all congruences implies the knowledge of all the homomorphic images of the algebra. A left $S$–act over monoid $S$ is a set $A$ upon which $S$ acts unitarily on the left. In this paper, we consider $S$–acts over linearly ordered and over well-ordered monoids, where a linearly ordered monoid $S$ is a linearly ordered set with a minimal element and with a binary operation $ \ max$, with respect to which $S$ is obviously a commutative monoid; a well-ordered monoid $S$ is a well-ordered set with a binary operation $ \ max$, with respect to which $S$ is also a commutative monoid. The paper is a continuation of the work of the author in co-authorship with M.S. Kazak, which describes $S$–acts over linearly ordered monoids with a linearly ordered congruence lattice and $S$-acts over a well-ordered monoid with distributive congruence lattice. In this article, we give the description of S-acts over a well-ordered monoid such that the corresponding congruence lattice is modular.

Covering rough set is a classical generalization of rough set. As covering rough set is a mathematical tool to deal with incomplete and incomplete data, it has been widely used in various fields. The aim of this paper is to extend the covering rough sets to interval neutrosophic sets, which can make multi-attribute decision making problem more tractable. Interval neutrosophic covering rough sets can be viewed as the bridge connecting Interval neutrosophic sets and covering rough sets. Firstly, the paper introduces the definition of interval neutrosophic sets and covering rough sets, where the covering rough set is defined by neighborhood. Secondly, Some basic properties and operation rules of interval neutrosophic sets and covering rough sets are discussed. Thirdly, the definition of interval neutrosophic covering rough sets are proposed. Then, some theorems are put forward and their proofs of interval neutrosophic covering rough sets also be gived. Lastly, this paper gives a numerical example to apply the interval neutrosophic covering rough sets.

This paper exploits the Markov reward method to availability and importance evaluation for CNC machine tools. The presented paper regards CNC machine tools as a multi-state system that state transitions process caused by elements and subsystem’s failures and corresponding maintenance activities during the lifetime. Simultaneously, considering the time-varying characteristics of the failure rates of subsystems, the non-homogeneous Markov reward model is introduced for evaluation of availability and subsystem importance identification for aging multi-state CNC machine tools under minimal repair. Corresponding procedures for the failure rates, state transition matrix, and reward matrix definition are suggested for the availability and importance measures. A numerical example is presented in order to illustrate the approach.

This paper proposes a semi-automatic methodology to assist the user in creating surveys about FMEA and Risk Analysis, based on a customized use of the tools for semantic analysis and in particular a home-developed syntactic parser called Kompat Cognitive. The core of this work has been the analysis of the specific FMEA-related jargon and its common modalities of description within scientific papers and patents in order to systematize the linguistic analysis of the reference documents within the proposed step-divided procedure. The main goals of the methodology are to assist not skilled in the art users about FMEA during the analysis of generic and specific features, by considering large moles of contributions in restricted amounts of time. The methodology has then been tested on the same pool of 286 documents, divided between 177 and 109 patents, manually analyzed in our previous survey, in order to replicate part of its classifications through the proposed new modality. In this way we evaluated the abilities of the methodology both to automatically suggesting the main features of interest and to classify the documents according to them.

Travel time estimation & reliability evaluation of any means of transportation in every type of travel mode- land, rail, sea and air has been of immense interest of the researchers; primarily due to growing economic concern in the field of logistics & passenger movement. In situations like quantitative data inaccessibility or data imprecision, fuzzy set based possibilistic approach is recognized as a practical choice in obtaining the reliability estimates. This paper proposes and advocates possibilistic approach for travel time reliability computation of any type transportation vehicle under fuzzy type of data. The proposed approach is a novel way of computing the travel time & obtaining the related reliability value. Initially, the paper proposes the general methodology for travel time reliability evaluation. Individual travel time components of a transportation vehicle are considered as fuzzy; as a result, travel time is modelled as a fuzzy variable. Travel time reliability of a transportation vehicle has been defined with the help of possibilistic measures. The proposed procedure is then demonstrated with an application to marine vessel carrying the bulk. After illustration of the proposed methodology, sensitivity analysis is carried out. The paper ends with the comments on comparative features of the three cases.

As the crime is increasing day by day around the world. To stop the crime many security gadgets and machines have been developed by various security agencies. The tripod turnstile machine also helps in stopping the crime and restricting the entry of unauthorized person on the premises. In this paper, the aim is to analyze the reliability measures of the two tripod turnstile machines which work in a parallel configuration. In the case of non-functionality of the machine, unauthorized persons may enter into the premises. Due to security and safety reason, organizations install these machines. Hence these machines must be highly reliable in the operation to avoid any unwanted event. In this paper, for the considered system, the transition state diagram has been drawn with the help of the Markov model. The Chapman-Kolmogorov differential equations are developed from the transition state diagram and solved using Laplace transformation. The failure and repair rates are assumed to be constant and follow negative exponential distribution. The system upstate and downstate probabilities are determined. The explicit expression of the system Availability, Reliability and MTTF are also obtained. The sensitivity analysis is also performed to determine which machine affects the system reliability the most.

In this paper a new approach to the use of kernel operators derived from fractional order differential equations is proposed. Three different types of kernels are used, power law, exponential decay and Mittag-Leffler kernels. The kernel's fractional order and fractal dimension are the key parameters for these operators. The main objective of this paper is to study the effect of the fractal-fractional derivative order and the order of the nonlinear term, 1

In this paper, several existence results of strongly essential set of the solution set for Ky Fan's section problems and vector Ky Fan's point problems are obtained. Firstly, two kinds of strongly essential sets of Ky Fan's section problems are defined, and some further results on existence of the strongly essential component of solutions set of Ky Fan's section problems are proved, which generalize the conclusion in [ 22 ], and further generalize the conclusions in [ 21 , 28 ]. Secondly, based on the above results, two classes of stronger perturbations of vector-valued inequality functions are proposed respectively, and several existence results of the strongly essential component of set of vector Ky Fan's points are obtained. By comparing several metrics, we give some strong and weak relationships among the various metrics involved in the text. The main results of this paper actually generalize the relevant conclusions in the current literature. Finally, as an application, we obtain an existence result of the strongly essential component of weakly Pareto-Nash equilibrium for multiobjective games.

When it comes across problems in creating Internet of Things (IOT) architecture, the major problem that arises is an automatic stipulation of resources. At the same time in today’s era, it is very important to integrate this problem with better Quality of Services (QoS) because of which the cloud computing is taking a shift. As being well acquainted that in fog computing, network’s bandwidth is limited, therefore it becomes quite important to build a joint architecture with resource allocation problem giving it a better quality of services with enhanced efficiency and low latency communication. Priority of QoS is determined by Systematic Ladder Process (SLP) and decision parameter evaluation by RECK algorithm. In this paper, there will be a design of a better framework for IOT resource allocation scheme with better efficiency and better QoS. The paper too highlights the comparison of the previous works of the resource allocation algorithms and schemes with RECK algorithm.

The goal of this paper is to explore the interrelated relationships between the quality of corporate governance, Stakeholder’s pressure, and Sustainability Reporting. We claim that sustainable development strategies are the result of the interaction of two variables: the standard of corporate governance and the pressure of stakeholders rather than the outcome of a single variable. The results indicate that the company size and corporate governance rating are positively correlated with sustainability reporting Scores disclosed by the Turkish companies. The analysis of 125 firm-year data suggest that the company provides more SRS if it has a better corporate governance rating; or larger total assets. The paper also reveals that social, environmental and economic indicators, as pillars of sustainable development, are not the result of corporate governance practices or of stakeholder pressure alone, but represent a holistic reflection of the accumulated effects of various factors related to both corporate governance practices and stakeholder theory. This paper extends the work of Taşkirmaz, and Gamze (2017) which examined the relationship between corporate governance and corporate social responsibility. Our results increase understanding of sustainable practices of the circular economy in Turkish industrial companies and indicate future trends for these applications. This research will help managers of manufacturing companies in formulating their plans to maximize the use of available resources and improve efficiency in the context of the circular economy.

The paper deals with the existence of solutions for(p,Q)(p,Q)coupled elliptic systems in the Heisenberg group, with critical exponential growth at infinity and singular behavior at the origin. We derive existence of nonnegative solutions with both components nontrivial and different, that is solving an actual system, which does not reduce into an equation. The main features and novelties of the paper are the presence of a general coupled critical exponential term of the Trudinger-Moser type and the fact that the system is set inℍn{{\mathbb{H}}}^{n}.

Motivated by the stereological problem of volume estimation from parallel section profiles, the so-called Newton-Cotes integral estimators based on random sampling nodes are analyzed. These estimators generalize the classical Cavalieri estimator and its variant for non-equidistant sampling nodes, the generalized Cavalieri estimator, and have typically a substantially smaller variance than the latter. The present paper focuses on the following points in relation to Newton-Cotes estimators: the treatment of dropouts, the construction of variance estimators, and, finally, their application in volume estimation of convex bodies.Dropouts are eliminated points in the initial stationary point process of sampling nodes, modeled by independent thinning. Among other things, exact representations of the variance are given in terms of the thinning probability and increments of the initial points under two practically relevant sampling models.The paper presents a general estimation procedure for the variance of Newton-Cotes estimators based on the sampling nodes in a bounded interval. Finally, the findings are illustrated in an application of volume estimation for three-dimensional convex bodies with sufficiently smooth boundaries. Motivated by the stereological problem of volume estimation from parallel section profiles, the so-called Newton-Cotes integral estimators based on random sampling nodes are analyzed. These estimators generalize the classical Cavalieri estimator and its variant for non-equidistant sampling nodes, the generalized Cavalieri estimator, and have typically a substantially smaller variance than the latter. The present paper focuses on the following points in relation to Newton-Cotes estimators: the treatment of dropouts, the construction of variance estimators, and, finally, their application in volume estimation of convex bodies.Dropouts are eliminated points in the initial stationary point process of sampling nodes, modeled by independent thinning. Among other things, exact representations of the variance are given in terms of the thinning probability and increments of the initial points under two practically relevant sampling models.The paper presents a general estimation procedure for the variance of Newton-Cotes estimators based on the sampling nodes in a bounded interval. Finally, the findings are illustrated in an application of volume estimation for three-dimensional convex bodies with sufficiently smooth boundaries.

A normalized version of the ubiquitous two-by-two contingency matrix is associated with a variety of marginal, conjunctive, and conditional probabilities that serve as appropriate indicators in diagnostic testing. If this matrix is enhanced by being interpreted as a probabilistic Universe of Discourse, it still suffers from two inter-related shortcomings, arising from lack of length/area proportionality and a potential misconception concerning a false assumption of independence between the two underlying events. This paper remedies these two shortcomings by modifying this matrix into a new Karnaugh-map-like diagram that resembles an eikosogram. Furthermore, the paper suggests the use of a pair of functionally complementary versions of this diagram to handle any ternary problem of conditional probability. The two diagrams split the unknowns and equations between themselves in a fashion that allows the use of a divide-and-conquer strategy to handle such a problem. The method of solution is demonstrated via four examples, in which the solution might be arithmetic or algebraic, and independently might be numerical or symbolic. In particular, we provide a symbolic arithmetic derivation of the well-known formulas that express the predictive values in terms of prevalence, sensitivity and specificity. Moreover, we prove a virtually unknown interdependence among the two predictive values, sensitivity, and specificity. In fact, we employ a method of symbolic algebraic derivation to express any one of these four indicators in terms of the other three. The contribution of this paper to the diagnostic testing aspects of mathematical epidemiology culminates in a timely application to the estimation of the true prevalence of the contemporary world-wide COVID-19 pandemic. It turns out that this estimation is hindered more by the lack of global testing world-wide rather than by the unavoidable imperfection of the available testing methods.

In this paper, we introduce a version of the Moebius function and other special functions on a particular class of intervals for groupoids, and study them to obtain results analogous to those obtained in the usual lattice, combinatorics and number theory setting, but of course much more general due to the viewpoint taken in this paper.

In this paper, first, we find the integrability theorems for the invariant and slant distributions which appeared in the concept of semi-slant submanifolds. Utilizing these theorems, we prove that a semi-slant submanifold reduces to be a warped product semi-slant submanifold, provided some necessary and sufficient conditions concerning the shape operators. Also, it is shown that a few earlier results are exceptional cases of this paper results.

The numerical method is proposed in this paper to solve a general class of continuous-time linear programming problems in which the functions appeared in the coefficients and the time-dependent matrices are assumed to be piecewise continuous. In order to make sure that all the subintervals of time interval will not contain the discontinuities of the involved functions, a methodology for not equally partitioning the time interval is proposed. The main issue of this paper is to obtain an analytic formula of the error bound, where the strong duality theorem for the primal and dual pair of continuous-time linear programming problems with time-dependent matrices and piecewise continuous functions is a by-product. We shall propose two kinds of computational procedure to evaluate the error bounds. One needs to solve the dual problem of the discretized linear programming problem, and another one does not need to solve the dual problem. The detailed differences between these two computational procedures will be also presented. Finally we present a numerical example to demonstrate the usefulness of the numerical method.

The aim of the paper is to propose a new Reflected Power function distribution (RPFD). We provide the various properties of the new model in detail such as moments, vitality function and order statistics. We characterize the RPFD based on conditional moments (Right and Left Truncated mean) and doubly truncated mean. We also study the shape of the new distribution to be applicable in many real life situations. We estimate the parameters for the proposed RPFD by using different methods such as maximum likelihood method, modified maximum likelihood method, percentile estimator and modified percentile estimator. The aim of the study is to increase the application of the Power function distribution (PFD). Using two different data sets from real life, we conclude that the RPFD perform better as compare to different competitor models already exist in the literature. We hope that the findings of this paper will be useful for researchers in different field of applied sciences.

Environmental worries in production and inventory models have received large attention in inventory management literature. In this paper, an economic production model is proposed with two-echelon supply chain when trade credit is offered by the supplier. This paper proposes human errors of Type I and Type II due to fatigue and inexperience of the inspector during screening. It considers the use of energy for production along with greenhouse gases (GHG) emission from production and transportation operations. The developed model optimizes the environmental and economic performances of the supply chain. Our aim is to explore the effects of human errors during inspection on the emission cost, transportation cost and delay in payment on the replenishment of order sizes and the expected total profit of the retailer. A mathematical model is developed and numerical examples are provided to illustrate the solution procedure.

In this paper, we discuss the existence of solutions to a boundary value problem of differential equations of variable order, which is a piecewise constant function. Our results are based on the Schauder fixed point theorem. Then, under some assumptions on the nonlinear term, we obtain a generalized Lyapunov-type inequality to the two-point boundary value problem considered. To the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for boundary value problems in term of variable order. In addition, some examples of the obtained inequalities are given.

In this paper, we study $R_{\lambda}$-semiconservative $FK$-spaces for Riesz-method defined by the Riesz matrix $(R)$ and give some characterizations. We show that if $\ell_{A}$ is $\ell$-replaceable, then $A$ can not be $R_{\lambda}$-semiconservative and also if $X_{A}$ is $R_{\lambda}$-conull $FK$-space then it must be $R_{\lambda}$-semiconservative space. In addition, we determine a new $r(\lambda)$ and $rb(\lambda)$ type duality of a sequence space $X$ containing $\varphi$. The paper aims to develop some new subspaces which each one has its own value on topological sequence spaces theory. These subspaces are called as $R_{\lambda}S; R_{\lambda}W; R_{\lambda}F^{+}; $ and $R_{\lambda}B^{+}$ for a locally convex $FK$-space X containing $\varphi$. The subspaces mentioned in the work requires some serious studies and they arose independently from the literature which was done at the recent stage of the development of summability through functional analysis.

This paper is to study the following coupled version of compound KdV and MKdV equations with two components $ \left\{ \begin{array}{l} {u_t} + \alpha v{v_x} + \beta {u^2}{u_x} + {u_{xxx}} + \lambda u{u_x} = 0,\beta \gt 0,\\ {v_t} + \alpha {(uv)_x} + 2v{v_x} = 0, \end{array} \right. $ which clearly has Hamiltonian form. The orbital stability and instability of solitary waves with nonzero asymptotic value have been few studied. In this paper, we mainly consider the orbital stability and instability of solitary waves with zero or nonzero asymptotic value for this equations. Precisely, we first obtain two explicitly exact solitary waves with zero asymptotic value and four explicitly exact solitary waves with nonzero asymptotic value. Secondly, we conclude some results on the orbital stability of solitary waves with zero or nonzero asymptotic value. To this aim, in order to overcome the difficulty in studying orbital stability of solitary waves with nonzero asymptotic value, we use a translation transformation to transfer this problem into solitary waves with zero asymptotic value for a reduced nonlinear coupled equations. Then by applying the classical orbital stability theory presented by Grillakis et al. and Bona et al., we obtain the orbital stability and instability of solitary waves with zero asymptotic value for the new equations. We finally derive some results on orbital stability of solitary waves with zero or nonzero asymptotic value. In addition, we also obtain the stability results for the coupled compound KdV and MKdV equations with the degenerate condition $v = 0$, called the compound KdV and MKdV equation, which have been studied by Zhang et al.

In the late 1960s and early 1970s, a new class of problems appeared in the theory of optimal control. It was determined that the structure of a number of systems or processes is not homogeneous and can change over time. Therefore, new mathematical models of heterogeneous structure have been developed. Research methods for this type of system vary widely, reflecting various scientific schools and thought. One of the proposed options was to develop an approach that retains the traditional assumptions of optimal control theory. Its basis is Krotov's sufficient optimality conditions for discrete systems, formulated in terms of arbitrary sets and mappings. One of the classes of heterogeneous systems is considered in this paper: discretecontinuous systems (DCSs). DCSs are used for case where all the homogeneous subsystems of the lower level are not only connected by a common functional but also have their own goals. In this paper a generalization of Krotov's sufficient optimality conditions is applied. The foundational theory is the Krotov method of global improvement, which was originally proposed for discrete processes. The advantage of the proposed method is that its conjugate system of vector-matrix equations is linear; hence, its solution always exists, which allows us to find the desired solution in the optimal control problem for DCSs.

This paper proposes an optimal coordination scheduling methodology for a wind-thermal power system by considering the intermittent behavior of wind power and load demands. The optimum amount of spinning reserves (SRs) needed to handle the uncertain nature of hybrid power system with wind generators is determined by using the probability analysis. The proposed model demonstrates the error in wind power and load forecasts by using the probability distribution functions for over and under estimations of load and wind power. In this work, two objectives, i.e., total operating cost and risk level are considered, which handles both economical and reliability issues, simultaneously. This problem is solved by using the meta-heuristic based Cuckoo search algorithm. The reactive power capability limits of synchronous generators and variable speed doubly fed induction generators are considered in this paper. Simulations have been carried out on IEEE 300 bus system.

System safety is a discipline of applying engineering and management principles, criteria, and techniques to achieve acceptable or tolerable risk within the constraints of operational effectiveness, suitability, time, and cost throughout all phases of the system life. System safety engineering is the program to identify hazards, and to eliminate hazards or reduce the associated risks when the hazards cannot be eliminated. System safety management involves plans and activities taken to identify hazards; assess and mitigate associated risks; track, control, close, and document risks encountered in the design, development, test, manufacturing, installation, operation and maintenance, and the disposal of systems, subsystems, and equipment. In this paper, the concept and principle of system safety in the transit system is discussed. The paper also introduces the safety standards, safety life-cycle, Safety Integrity Levels (SILs), safety analysis techniques and safety cases etc.

The principal aim of this paper is to introduce $\mathtt{k}$-fractional derivative operator by using the definition of $\mathtt{k}$-beta function. This paper establishes some results related to the newly defined fractional operator such as the Mellin transform and the relations to $\mathtt{k}$-hypergeometric and $\mathtt{k}$-Appell's functions. Also, we investigate the $\mathtt{k}$-fractional derivative of $\mathtt{k}$-Mittag-Leffler and the Wright hypergeometric functions.

In this paper, we consider the solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. The expression of the solution for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is discussed based on the definition and the property of the Caputo differential operators. By the fixed point theorem in Banach algebra due to Dhage, an existence theorem for the boundary value problem of nonlinear fractional differential equations with mixed perturbations of the second type is given under mixed Lipschitz and Caratheodory conditions. As an application, an example is presented to illustrate the main results. Our results in this paper extend and improve some well-known results. To some extent, our work fills the gap on some basic theory for the boundary value problems of fractional differential equations with mixed perturbations of the second type involving Caputo differential operator.

In this paper, we obtain the upper bounds for the n-th (n ≥ 1) coefficients for meromorphic bi-subordinate functions of complex order by using Faber polynomial expansions. The results, which are presented in this paper, would generalize those in related works of several earlier authors.

In the paper, we solve one extremal problem of the theory of approximation of functional classes by linear methods. Namely, questions are investigated concerning the approximation of classes of differentiable functions by $\lambda$-methods of summation for their Fourier series, that are defined by the set $\Lambda =\{{{\lambda }_{\delta }}(\cdot )\}$ of continuous on $\left[ 0,\infty \right)$ functions depending on a real parameter $\delta$. The Kolmogorov-Nikol'skii problem is considered, that is one of the special problems among the extremal problems of the theory of approximation. That is, the problem of finding of asymptotic equalities for the quantity $$\mathcal{E}{{\left( \mathfrak{N};{{U}_{\delta}} \right)}_{X}}=\underset{f\in \mathfrak{N}}{\mathop{\sup }}\,{{\left\| f\left( \cdot \right)-{{U}_{\delta }}\left( f;\cdot;\Lambda \right) \right\|}_{X}},$$ where $X$ is a normalized space, $\mathfrak{N}\subseteq X$ is a given function class, ${{U}_{\delta }}\left( f;x;\Lambda \right)$ is a specific method of summation of the Fourier series. In particular, in the paper we investigate approximative properties of the three-harmonic Poisson integrals on the Weyl-Nagy classes. The asymptotic formulas are obtained for the upper bounds of deviations of the three-harmonic Poisson integrals from functions from the classes $W^{r}_{\beta,\infty}$. These formulas provide a solution of the corresponding Kolmogorov-Nikol'skii problem. Methods of investigation for such extremal problems of the theory of approximation arised and got their development owing to the papers of A.N. Kolmogorov, S.M. Nikol'skii, S.B. Stechkin, N.P. Korneichuk, V.K. Dzyadyk, A.I. Stepanets and others. But these methods are used for the approximations by linear methods defined by triangular matrices. In this paper we modified the mentioned above methods in order to use them while dealing with the summation methods defined by a set of functions of a natural argument.

The problem of restoring input signals is one of the intense developing research areas and is the intersection of the mathematical modeling theory, the automatic control theory and the inverse problems theory. This paper focuses on solving the identification problem of the input signal that corresponds to a given (desired) output in the case of no feedback. An approach to the approximate solution of polynomial Volterra equations of the first kind of the Nth degree that arise when modeling nonlinear dynamics by the apparatus of Volterra integro-power series is described. These equations appear when a nonlinear dynamic process is modeled using the integro-power Volterra series.One class of nonlinear dynamical black box type systems is considered. Unlike a scalar input, the form of the integral model is complicated by the inclusion of terms that take into account the simultaneous change of individual components of the input signal vector. Integral models with constant Volterra kernels were considered earlier. This paper assumes the symmetric Volterra kernels are representable as the product of a finite number of continuous functions. The identification problem is solved using the Newton-Kantorovich method. A numerical solution of the corresponding linear integral Volterra equation of the first kind is proposed as an initial approximation. The obtained formulas for calculations are based on quadrature methods (right rectangles). The effectiveness of the proposed algorithms is illustrated for the reference dynamic system and confirmed by numerical results.

This paper aims to present a new pathwise approximation method, which gives approximate solutions of order $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ for stochastic differential equations (SDEs) driven by multidimensional Brownian motions. The new method, which assumes the diffusion matrix non-degeneracy, employs the Runge-Kutta method and uses the Itô-Taylor expansion, but the generating of the approximation of the expansion is carried out as a whole rather than individual terms. The new idea we applied in this paper is to replace the iterated stochastic integrals Iα by random variables, so implementing this scheme does not require the computation of the iterated stochastic integrals Iα. Then, using a coupling which can be found by a technique from optimal transport theory would give a good approximation in a mean square. The results of implementing this new scheme by MATLAB confirms the validity of the method.

In this paper, we consider the improved estimation of average production of peppermint at block level of Barabanki district of Uttar Pradesh State (India). We suggest certain estimators for population-mean. Here, population refers to production population as study variable and auxiliary-variable refers to Area of field. We study the sampling properties naming bias and MSE of estimators, which are presently proposed by us in the paper. We compare our proposed estimators with other ones existing in literature. For the support of the theoretical findings, we carry out a numerical study for the natural population on primary data collected from Banikodar Block of Barabanki District situated in Uttar Pradesh State.

In this paper the linear representation of a graph is defined. A linear representation of a graph is a subgroup of $GL(p,\mathbb{R})$, the group of invertible matrices of order $ p $ and real coefficients. It will be demonstrated that every graph admits a linear representation. In this paper, simple and finite graphs will be used, framed in the graphs theory's area