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In recent years, as detailed in several monographs, derivations of the fractional differential equations and fractional integral equations are based on random functional or stochastic equations, with the output that physical interpretation of the resulting fractional derivatives and fractional integrals has been elusive. In many different sciences and problems such as biological systems, environmental quality and natural resources engineering, and so on, stochastic equations and in some cases random functional have appeared. Despite the widespread use of stochastic fractional integro-differential equations (SFIDE), the analytical solution of this equation is not easy and in some cases it is impossible. Therefore, the existence of an efficient and appropriate numerical method can solve this problem. In this study and based on fractional derivative in the Caputo sense, we investigate a class of SFIDE due to Brownian motion by using fractional B-spline basis functions (FB-spline) and with the help of the collocation-Galerkin method as well as the trapezoidal integral law. In other words, a continuous operator problem (here was named as SFIDE) is transformed into a discrete problem by limited sets of basis functions with common assumptions and approximation methods. In the follow-up a system of linear equations is generated, which makes the analysis of the method be efficient. As an important advantage of this combined method is its flexible and easy implementation. Another advantage of the method is its ability to be implemented for different types of linear, non-linear and system of SFIDE, which are discussed in the body of manuscript. An accurate upper bound is obtained and some theorems are established on the stability and convergence analysis. The computational cost is estimated from the sum of the number arithmetic operations and a lower bound for approximation of this equation is formulated. Finally, by examining several examples, the computational performance of the proposed method effectively verifies the applicability and validity of the suggested scheme. [ABSTRACT FROM AUTHOR]

In this article, we study the piecewise multistep collocation method for a class of functional integral equations with non-vanishing delays. Based on the notions of the tractability index and the ν -smoothing property of a Volterra integral operator, our numerical analysis and optimal convergence properties are investigated. Here, multistep collocation method which depends on the numerical solution in a fixed number of previous time steps is described by the constructive technique and dividing the definition domain into several subintervals according to the primary discontinuous points associated with the delay function. Numerical experiments confirm the theoretical expectations. [ABSTRACT FROM AUTHOR]

We use physics-informed neural networks (PINNs) to numerically solve the discrete-time nonlinear observer-based state estimation problem. Integrated within a single-step exact observer linearization framework, the proposed PINN approach aims at learning a nonlinear state transformation operator by solving a system of functional equations. The performance of the proposed approach is assessed via two illustrative case studies, for which the observer linearizing transformation operator can be derived analytically. We also perform an uncertainty quantification analysis for the proposed scheme. The performance and numerical approximation accuracy of the proposed scheme is compared with conventional power-series numerical implementation. • Reconstructing unobservable states is crucial for complex systems. • For this task, we use Physics-Informed neural networks. • Our method finds the nonlinear operator, transforming nonlinear to linear design. • We validate the method using two benchmark problems with singularities. [ABSTRACT FROM AUTHOR]

In this article, we study the maximal robust invariant set estimation problem for discrete-time perturbed nonlinear systems within the optimal control framework. The maximal robust invariant set of interest is a set of all states such that every possible trajectory starting from it never violates a specified state constraint, regardless of actual disturbances. The maximal robust invariant set is shown to be the zero level set of the unique bounded solution to a Bellman type equation, which is a functional equation being widely used in discrete-time optimal control. Consequently, the maximal robust invariant set estimation problem is reduced to a problem of solving a Bellman type equation. This is the main contribution of this article. The uniqueness of bounded solutions enables us to solve the derived Bellman type equation using numerical methods such as the value iteration and policy iteration, which provide an approximation of the maximal robust invariant set. Finally, two examples demonstrate the performance of our Bellman equation based method. [ABSTRACT FROM AUTHOR]

We introduce featom, an open source code that implements a high-order finite element solver for the radial Schrödinger, Dirac, and Kohn-Sham equations. The formulation accommodates various mesh types, such as uniform or exponential, and the convergence can be systematically controlled by increasing the number and/or polynomial order of the finite element basis functions. The Dirac equation is solved using a squared Hamiltonian approach to eliminate spurious states. To address the slow convergence of the κ = ± 1 states due to divergent derivatives at the origin, we incorporate known asymptotic forms into the solutions. We achieve a high level of accuracy (10 − 8 Hartree) for total energies and eigenvalues of heavy atoms such as uranium in both Schrödinger and Dirac Kohn-Sham solutions. We provide detailed convergence studies and computational parameters required to attain commonly required accuracies. Finally, we compare our results with known analytic results as well as the results of other methods. In particular, we calculate benchmark results for atomic numbers (Z) from 1 to 92, verifying current benchmarks. We demonstrate significant speedup compared to the state-of-the-art shooting solver dftatom. An efficient, modular Fortran 2008 implementation, is provided under an open source, permissive license, including examples and tests, wherein particular emphasis is placed on the independence (no global variables), reusability, and generality of the individual routines. Program Title: featom CPC Library link to program files: https://doi.org/10.17632/962fzmm7f7.1 Licensing provisions: MIT Programming language: Fortran with command-line interfaces Nature of problem: Solution of the Schrödinger, Dirac, and Kohn–Sham equations of density functional theory for isolated atoms. Solution method: A high-order finite element method is used to assemble a generalized eigenproblem that is then solved for eigenvalues and orbitals. The Poisson equation is solved using the finite element method also. Self-consistent field equations are solved using Pulay mixing. Additional comments including restrictions and unusual features: Our solution methodology for the Dirac equation does not suffer from spurious states. We maintain high accuracy even for κ = ± 1 states. Unlike other methods, our approach is not limited to Coulombic or self-consistent potentials and can handle non-uniform meshes, including exponential meshes. Restrictions: Spherical symmetry. [ABSTRACT FROM AUTHOR]

• Show flaws of existing multiplicative transitivity models of interval-valued fuzzy preference relations (IVFPRs). • Develop an and-like-uninorm-based functional transitivity equation for IVFPRs. • Find a closed-form solution for the optimal interval-valued fuzzy weights of IVFPRs. • Formulate geometric consistency indices of IVFPRs and propose a new acceptability checking model. • Devise an and-like-uninorm-based method to aggregate local interval-valued fuzzy weights. The framework of interval-valued fuzzy preference relations (IVFPRs) is adequate and effective to model human preference evaluations under indeterminacy. This paper analyzes three recently presented multiplicative transitivity models of IVFPRs and exposes their drawbacks. An and-like-uninorm-based functional transitivity equation is developed to introduce a multiplicative consistency notion for IVFPRs. Based on the transitivity logarithmic equation, a geometric-consistency index is further proposed to compute the inconsistency level of an IVFPR. The paper builds a logarithmic least squares model with row indeterminacy constraints and equivalently transforms it into a quadratic programming model for finding a closed-form solution of the normalized interval-valued fuzzy weights of IVFPRs. A novel method is subsequently presented to check the acceptability of an IVFPR by examining its acceptable consistency and acceptable indeterminacy. An approach including an and-like-uninorm-based maximization model is introduced to aggregate local interval-valued fuzzy weights and an interval fuzzy analytic hierarchy process is designed step-by-step. An illustrative example and a comparison study are utilized to demonstrate the performance and merits of the presented models. Meanwhile, an outstanding undergraduate student selection problem in international exchange is provided to show the application of the proposed decision method. [ABSTRACT FROM AUTHOR]

In this work, we discuss the law of α -migrativity as applied to fuzzy implication functions in a meaningful way. A generalisation of this law leads us to Pexider-type functional equations connected with the law of importation, viz., the generalised law of importation I (C (x , α) , y) = I (x , J (α , y)) (GLI) and the generalised cross-law of importation I (C (x , α) , y) = J (x , I (α , y)) (CLI), where C is a generalised conjunction. In this article we investigate only (GLI). We begin by showing that the satisfaction of law of importation by the pairs (C, I) and/or (C, J) does not necessarily lead to the satisfaction of (GLI). Hence, we study the conditions under which these three laws are related. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical solutions of nonlinear functional-differential and functional equations (FDFEs) with constant delay. The block boundary value methods (BBVMs) are extended to solve the FDFEs. Under the suitable conditions, it is shown that the extended BBVMs are uniquely solvable and globally stable. Moreover, the method can be convergent of order p whenever the Lipschitz condition holds and this method is preconsistent and p -order consistent. With several numerical examples, the theoretical results and computational validity of the extended BBVMs are further confirmed. [ABSTRACT FROM AUTHOR]

Abstract The structures of continuous cross-migrative triangular norms with respective to some continuous triangular norm have been studied, and only partial results were obtained. This paper is devoted to presenting a complete characterization of the class of continuous triangular norms, being cross-migrative with respect to some continuous triangular norm. [ABSTRACT FROM AUTHOR]

Abstract In this paper we derive a macroscopic model for thermoporoelasticity from the pore scale linearized fluid-structure and energy equations. We consider the continuum mechanics thermodynamically compatible pore scale equations corresponding to realistic rock mechanics parameters. They are upscaled using two-scale asymptotic expansions. For the upscaled equations a Lyapunov functional (a generalization of Biot's free energy) is constructed and the well-posedness of the model is discussed. Possible applications to large time numerical simulations are pointed out. [ABSTRACT FROM AUTHOR]

Nonlinear observer design for systems whose state-space evolves on Lie groups is considered. The proposed method is similar to previously developed nonlinear observers in that it involves propagating the state estimate using a process model and corrects the propagated state estimate using an innovation term on the tangent space of the Lie group. In the proposed method, the innovation term is constructed by passing the gradient of an invariant cost function, resolved in a basis of the tangent space, through a linear time-invariant system. The introduction of the linear system completes the extension of linear complementary filters to nonlinear Lie group observers by allowing higher order filtering. In practice, the proposed method allows for greater design freedom and, with the appropriate selection of the linear filter, the ability to filter bias and noise over specific bandwidths. A disturbance observer that accounts for constant and harmonic disturbances in group velocity measurements is also considered. Local asymptotic stability about the desired equilibrium point is demonstrated. A numerical example that demonstrates the desirable properties of the observer is presented in the context of pose estimation. [ABSTRACT FROM AUTHOR]

This paper presents continuous-time multiagent systems for distributed minimax optimization subject to bounded constraints. All agents in the system are divided into two groups for minimization and maximization. The multiagent system features competitive intergroup interactions and cooperative intragroup interactions, both of which are based on the output information of agents. First, a proportional-integral (PI) intragroup interaction rule is utilized for consensus within each group in the system. With this interaction rule, the system is proved to be convergent to an optimal solution to the problem, under a certain requirement on the intergroup interactions. Second, another discontinuous intragroup interaction rule is introduced. It is proved that the system with such an interaction is still convergent to an optimal solution if the proportional gain exceeds a derived lower bound, without the previous requirement on the intergroup interactions. As a special case, the systems are further applied for distributed optimization. Finally, simulation results are presented to substantiate the theoretical results. [ABSTRACT FROM AUTHOR]

In this technical note, we give (unverifiable) necessary and sufficient conditions, which are similar to the ones in the literature, for the discrete-time multioutput observer error linearization problem. Using these, we derive verifiable sufficient conditions. Since our proofs are constructive, a desired state transformation can also be found in the theorem. [ABSTRACT FROM AUTHOR]

Equilibrium-independent dissipativity (EID) is a recently introduced system property that requires a system to be dissipative with respect to any forced equilibrium configuration. This paper is a detailed examination of EID with quadratic supply rates for a common class of nonlinear control-affine systems. We provide an algebraic characterization of EID for such systems in the spirit of the Hill–Moylan lemma, where the usual stability condition is replaced by an incremental stability condition. Based on this characterization, we state results concerning internal stability, feedback stability, and absolute stability of EID systems. Finally, we study EID for discrete-time systems, providing the relevant definitions and an analogous Hill–Moylan-type characterization. Results for both continuous-time and discrete-time systems are illustrated through examples on physical systems and convex optimization algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new state observer design technique for nonlinear systems. It combines the well-known Kazantzis–Kravaris–Luenberger observer and the recently introduced parameter estimation-based observer, which become special cases of it—extending the realm of applicability of both methods. A second contribution of this paper is the proof that these designs can be recast as particular cases of immersion and invariance observers—providing in this way a unified framework for their analysis and design. Simulation results of a physical system that illustrates the superior performance of the proposed observer compared to other existing observers are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of distributed Nash equilibrium (NE) seeking over networks, a setting in which players have limited local information on the others’ decisions. We start from a continuous-time gradient-play dynamics that converges to an NE under strict monotonicity of the pseudogradient and assumes perfect information. We consider how to modify it in the case of partial, or networked information between players. We propose an augmented gradient-play dynamics with correction, in which players communicate locally only with their neighbors to compute an estimate of the other players’ actions. We derive the new dynamics based on the reformulation as a multiagent coordination problem over an undirected graph. We exploit incremental passivity properties and show that a synchronizing, distributed Laplacian feedback can be designed using relative estimates of the neighbors. Under a strict monotonicity property of the pseudogradient, we show that the augmented gradient-play dynamics converges to consensus on the NE of the game. We further discuss two cases that highlight the tradeoff between properties of the game and the communication graph. [ABSTRACT FROM AUTHOR]

Abstract In this work we construct and discuss special solutions of a homogeneous problem for the Laplace equation in a domain with cone-shaped boundaries. The problem at hand is interpreted as that describing oscillatory linear wave movement of a fluid under gravity in such a domain. These solutions are found in terms of the Mellin transform and by means of the reduction to some new functional-difference equations solved in an explicit form (by quadrature). The behavior of the solutions at large distances is studied by use of the saddle point technique. The corresponding eigenoscillations of a fluid are then interpreted as generalized eigenfunctions of the continuous spectrum. Highlights • Explicit formulas for the eigenmodes of the continuous spectrum are considered. • New type of the functional-difference equations are solved. • The far field asymptotics of the eigenfunctions are obtained. • Some physical analysis of the solution for the problem at hand is also given. [ABSTRACT FROM AUTHOR]

An integral equation is derived for the generating function for binary tree values, the values reflecting sorting effort. The analysis does not assume uniformly distributed branching ratios, and therefore is applicable to a family of sorting algorithms discussed by Hoare, Singleton, and van Emden. The solution to the integral equation indicates that using more advanced algorithms in the family makes only minor reductions in the expected sorting effort, but substantially reduces the variance in sorting effort. Statistical tests of the values of several thousand trees containing up to 10,000 points have given first, second, and third moments of the value distribution function in satisfactory agreement with the moments computed from the generating function. The empirical tests, as well as the analytical results, are in agreement with previously published results for the first moment in the cases of uniform and nonuniform distribution of branching ratio, and for the second moment in the case of uniform distribution of branching ratio. [ABSTRACT FROM AUTHOR]

Several letters to the editor are presented in response to articles in the previous issues including "A Comparison of the Primal-Simplex and Primal-Dual Algorithms for Linear Programming," by R.K. Mueller and L. Cooper, "Algorithm and Formula," by T. Wangsness and J. Franklin, and a letter which further expounded the terms algorithm and program.

This paper revisits the distributivity for uninorms over nullnorms from a new perspective, transforming it into the distributivity equation involving uninorms over t-norms/t-conorms. In this paper, we show that this new perspective leads to not only all current solutions, but also many new solutions that do not appear in previous studies. [ABSTRACT FROM AUTHOR]

In this paper we analyze the notion of a finite mean from an axiomatic point of view. We discuss several axiomatic alternatives, with the aim of establishing a universal definition reconciling all of them and exploring theoretical links to some branches of Mathematics as well as to multidisciplinary applications. [ABSTRACT FROM AUTHOR]

Motivated by a problem posed in a competition textbook, we give the general solution of a functional equation related to the addition theorem of the hyperbolic tangent function. [ABSTRACT FROM AUTHOR]

The migrative functional equations provide a very powerful tool for constructing and characterizing new fuzzy logic connectives by convex combination, and have particularly important applications in image processing. So far, the migrativity between conjunctive logic connectives has been extensively studied, the obtained results do not, however, work well for triangular conorms. This paper is devoted to an in-depth investigation on three types of migrativity for continuous triangular conorms S with respect to N -ordinal sum implications I , which have distinctive features from ordinary ordinal sum implications. We will provide first detailed characterizations on the (α , I) -migrativity of S for each case according to the position relation of α in the range of N , by giving the corresponding ordinal sum decompositions of the t-conorm and implication solutions. Then the (α , I) -migrativity is extended to internal and global cases, and their characterizations are given under some additional constraints. • The migrativity of t-conorms over fuzzy implications opens a new perspective for the study on migrative properties of fuzzy logic connectives. • This paper is devoted to an in-depth investigation on three types of migrativity for continuous t-conorms over N-ordinal sum implications. • The results show that the ordinary ordinal sums of t-conorms match well the opposite ordinal sums of implications through the migrative equation. [ABSTRACT FROM AUTHOR]

In fuzzy logic, the fuzzy reasoning rules generalized hypothetical syllogism (GHS), generalized modus tollens (GMT) and generalized reduction to absurdity (GRA) play an indispensable role in selecting appropriate fuzzy implications in specific tasks. In order to enhance the set of fuzzy implications satisfying these three properties to a greater extent, in this work, we investigate in detail under what conditions the T -power based implications will satisfy (GHS), (GMT) and (GRA). Then the necessary and sufficient conditions are given. The results obtained can not only provide a theoretical basis for selecting fuzzy implication satisfying these properties, but also help us further insight the behaviour of this new kind of fuzzy implication. [ABSTRACT FROM AUTHOR]

We present the Mathematica package QMeS-Derivation , available on GitHub. It derives symbolic functional equations from a given master equation. The latter include functional renormalisation group equations, Dyson-Schwinger equations, Slavnov-Taylor and Ward identities and their modifications in the presence of momentum cutoffs. The modules allow to derive the functional equations, take functional derivatives, trace over field space, apply a given truncation scheme, and do momentum routings while keeping track of prefactors and signs that arise from fermionic commutation relations. The package furthermore contains an installer as well as Mathematica notebooks with showcase examples. Program Title: QMeS-Derivation CPC Library link to program files: https://doi.org/10.17632/dzb2z4tshd.1 Developer's repository link: https://github.com/QMeS-toolbox/QMeS-Derivation Licensing provisions: GPLv3 Programming language: Mathematica Nature of problem: Deriving symbolic functional equations starting from a quantum master equation. Solution method: Taking functional derivatives of the modified Slavnov-Taylor identities, the Dyson-Schwinger, functional renormalisation group equation or user defined master equations, taking the trace in field space and applying a truncation, doing a momentum routing for one-loop diagrams. Additional comments including restrictions and unusual features: QMeS operates theory independent and is based on a small number of rules, i.e. the (anti-)commutation relation of (fermions)bosons and general functional derivative rules. [ABSTRACT FROM AUTHOR]

In this paper, we derive an analytic formula for the American knock-out options with rebate. Rather than using a probabilistic method, we use the Laplace–Carson Transform(LCT) method to induce a simple functional equation associated with the complex problem of option pricing Partial Differential equation with free boundary. The transformed value of free boundary could be solved by applying Newton’s method. Lastly, numerical Laplace inversion techniques are used to solve for the wanted free boundary value and the options value. [ABSTRACT FROM AUTHOR]

This paper is concerned with distributed computation of several commonly used centrality measures in complex networks. In particular, we propose deterministic algorithms, which converge in finite time, for the distributed computation of the degree, closeness and betweenness centrality measures in directed graphs. Regarding eigenvector centrality, we consider the PageRank problem as its typical variant, and design distributed randomized algorithms to compute PageRank for both fixed and time-varying graphs. A key feature of the proposed algorithms is that they do not require to know the network size, which can be simultaneously estimated at every node, and that they are clock-free. To address the PageRank problem of time-varying graphs, we introduce the concept of persistent graph, which eliminates the effect of spamming nodes. Moreover, we prove that these algorithms converge almost surely and in the sense of $L^p$. Finally, the effectiveness of the proposed algorithms is illustrated via extensive simulations using a classical benchmark. [ABSTRACT FROM PUBLISHER]

Let { Z (t) ; t ⩾ 0 } be a Markov branching process (MBP). There exists a well-known sequence { C (t) ; t ⩾ 0 } such that W (t) ≔ Z (t) / C (t) a.s. converges to a non-degenerate random variable W as t → ∞. This paper attempts to study the asymptotic behavior of P (Z (t) = k t) and P (0 ⩽ Z (t) ⩽ k t) with k t e − λ t → 0 as t → ∞ for MBPs, which helps to study large deviations of Z (t + s) / Z (t). Moreover, we obtain the local limit theorem of this process as an additional finding. During the argumentation, the Cramér method is applied to analyze the large deviation of the sum of random variables. [ABSTRACT FROM AUTHOR]

Cancellation conditions play a central role in the representational theory of measurement for a weak order ≾ on a finite two-dimensional Cartesian product set X = X 1 × X 2 . The order has an additive real-valued representation if and only if it satisfies a sequence of cancellation conditions C ( 2 ) , C ( 3 ) , . . .. Given fixed cardinalities m and n for X 1 and X 2 , there is a largest K , denoted by f ( m , n ) , such that some ≾ on X satisfies C ( 2 ) to C ( K − 1 ) but violates C ( K ) . In a pivotal paper, Fishburn, mentioning some results reported by Krantz, Luce, Suppes and Tversky, shows that f ( 3 , 3 ) = 3 , f ( 3 , 4 ) = f ( 4 , 4 ) = 4 . He gives lower and upper bounds for f ( m , n ) , including 4 ≤ f ( 3 , 5 ) ≤ 7 , and asks for the exact values of f ( m , n ) for some small ( m , n ) such as ( 3 , 5 ) , ( 4 , 5 ) and ( 5 , 5 ) . In this article, we obtain f ( 3 , 5 ) = 4 and make explicit a minimal chain of cancellation violating sequences adequate for the detection of all not additively representable weak orders for ( m , n ) = ( 3 , 3 ) , ( 3 , 4 ) and ( 3 , 5 ) . [ABSTRACT FROM AUTHOR]

In 1959, Duncan Luce published the famous paper entitled “ On the possible psychophysical laws. ” The results presented here were inspired by that paper and by the ensuing controversy centered on the invariance concept of “meaningfulness.” We give a formal definition of this concept. As we define it, the condition of meaningfulness is quite potent. In its context, relatively weak additional conditions may suffice for the derivation of precise scientific or geometric laws. We give several examples of such derivations, including the Pythagorean Theorem and Beer’s Law. [ABSTRACT FROM AUTHOR]

The article presents a computer algorithm for solving exponential integral equations. The number of convergent values required for solving integral equations can be expressed in the form of a non-increasing function. It will be noted that near a transition point, the continued fraction evaluation is considerably more time consuming than the series evaluation. The imbalance could easily be corrected by moving up the transition point. No provisions have been made to test for the overflow or underflow conditions. The parameters that have been used in the algorithm have been described.

An integral equation representation is given for parabolic partial differential equations. When the equations are defined in unbounded domains, as in the initial value (Cauchy) problem, the solution of the integral equation by the method of successive approximation has inherent advantages over other methods. Error bounds for the method are of order h3/2 and h7/2 (h is the increment size) depending on the finite difference approximations involved. [ABSTRACT FROM AUTHOR]

Equations of the form du / di = g(u(t), u(h(t))) arise in a number of scientific contexts. The authors point out some interesting properties of the solution of u′(t) = -u(t - 1 - k sin ωt) + sin at. These properties were obtained by means of numerical solution. [ABSTRACT FROM AUTHOR]

In this paper we propose an evolving network model, which is a randomized version of the pseudofractal graphs by introducing an evolutionary parameter 0 < p < 1. Our network model grows exponentially over time, and can be generated in an iterative manner: at each time step, with probability p each existing edge recruits independently a new node and connects to it with both endpoints. We first briefly discuss the network size, which can correspond to a supercritical branching process. Then, it shows that the asymptotic degree distribution in our network model can be uniquely determined by a functional equation of its probability generating function. [ABSTRACT FROM AUTHOR]

The article offers suggestions for improving statistical process analysis for a significant client. It highlights the application of an engineering design process to communicate information to the target audience through images and mathematical equations. Also mentioned is the aspect of understanding the amount of education readers and particular subject matter.

Three-dimensional contact problems without friction have been studied using the boundary face method (BFM). In this paper, a non-conforming contact discretization approach is used to enforce the contact conditions between the two contact surfaces. This method is based on node-to-surface (NTS), and there is no need that the identical discretization is performed along the contact surfaces of both bodies. The contact equations are written explicitly with both tractions and displacements which are retained as unknowns in boundary integral equation (BIE). An iterative procedure is presented to determine the correct contact zone by obtaining a solution compatible with the contact conditions (no interpenetrations between the domains and no tensile on the final contact zone). Several numerical examples have been presented to illustrate the applicability of the method. [ABSTRACT FROM AUTHOR]

The electromagnetic-thermal dosimetry model of the human brain exposed to electromagnetic (EM) radiation is developed. The EM model based on the surface integral equation (SIE) formulation is derived using the equivalence theorem for the case of a lossy homogeneous dielectric body. The thermal dosimetry model of the brain is based on the form of Pennes׳ equation for heat transfer in biological tissue. The numerical solution of the EM model is carried out using the Method of Moments (MoM) while the bioheat equation is solved using the finite element method (FEM). Developed EM-thermal model has been applied for the internal dosimetry of the human brain to assess the absorbed EM energy and the consequent temperature rise due to the exposure of 900 MHz plane wave. Due to the variability of various parameters, the sensitivity of the maximum, minimum and the average steady-state temperature, on the various thermal parameters have been examined, as well as the influence of the parameters variation on the temperature distribution in case of EM exposure. The proposed model may be found useful in the rapid assessment of the temperature distribution in the human brain, prior to having to deal with a tedious development of a more complex models. [ABSTRACT FROM AUTHOR]

The Dirichlet problem for the wave equation is a classical example of a problem which is ill-posed. Nevertheless, it has been used to model internal waves oscillating harmonically in time, in various situations, standing internal waves amongst them. We consider internal waves in two-dimensional domains bounded above by the plane z = 0 and below by z = − d ( x ) for depth functions d . This paper draws attention to the Abel and Schröder functional equations which arise in this problem and use them as a convenient way of organising analytical solutions. Exact internal wave solutions are constructed for a selected number of simple depth functions d . [ABSTRACT FROM AUTHOR]

Considering the hesitancy degrees on pair-wise comparisons of alternatives as interval-valued intuitionistic fuzzy (IVIF) sets (IVIFSs), we develop a new fuzzy mathematical programming method for solving heterogeneous multiattribute decision-making problems based on the Linear Programming Technique for Multidimensional Analysis of Preference. In this method, IVIFSs, intuitionistic fuzzy sets (IFSs), trapezoidal fuzzy numbers, linguistic variables, intervals and real numbers are used to represent multiple types of attribute values and the attribute weights are not completely known. The preference relations between alternatives given by decision maker are expressed with IVIFSs of ordered pairs of alternatives. The consistency and inconsistency indices are defined as IVIFSs on the basis of comparisons of alternatives with IVIF truth degrees. The attribute weights and fuzzy ideal solution (FIS) are estimated through constructing a fuzzy mathematical programming model, which is solved by the technically developed method of IVIF mathematical programming. Hereby the distances of alternatives to the FIS are computed to rank the alternatives. Some generalization and discussion on the constructed IVIF mathematical programming model are also presented. A green supplier selection example is provided to illustrate the effectiveness of the proposed model and method. The comparison analyses verify the superiorities of the proposed method. [ABSTRACT FROM AUTHOR]

A CARMA ( p , q ) process is defined by suitable interpretation of the formal p th order differential equation a ( D ) Y t = b ( D ) D L t , where L is a two-sided Lévy process, a ( z ) and b ( z ) are polynomials of degrees p and q , respectively, with p > q , and D denotes the differentiation operator. Since derivatives of Lévy processes do not exist in the usual sense, the rigorous definition of a CARMA process is based on a corresponding state space equation. In this note, we show that the state space definition is also equivalent to the integral equation a ( D ) J p Y t = b ( D ) J p − 1 L t + r t , where J , defined by J f t : = ∫ 0 t f s d s , denotes the integration operator and r t is a suitable polynomial of degree at most p − 1 . This equation has well defined solutions and provides a natural interpretation of the formal equation a ( D ) Y t = b ( D ) D L t . [ABSTRACT FROM AUTHOR]

Recently, Vemuri and Jayaram (2012) have proposed a novel generating method of fuzzy implications, called the ⊛ -composition. Further, they have also proposed Mutual Exchangeability (ME), a generalization of the Exchange Principle (EP) to a pair of fuzzy implications and have shown that (ME) plays a central role in the preservation of basic properties, functional equations and families of fuzzy implications w.r.t. the ⊛ -composition (Vemuri and Jayaram, 2014). Due to the important role played by (ME), in this work, we investigate the pairs ( I , J ) of fuzzy implications that satisfy (ME). Towards this, we show first that there exist pairs ( I , J ) of fuzzy implications that satisfy (ME) and determine some necessary conditions on such fuzzy implications. Following this, for a given I ∈ I , we find the set J I of all fuzzy implications J such that the pair ( I , J ) satisfies (ME). Keeping in view the variety of fuzzy implications and the complexity of the functional equation (ME), we restrict our investigations to four important families of fuzzy implications, namely, ( S , N ) -, R -, f - and g -implications. Further, we discuss a generalization of the Cauchy multiplicative equation, whose solutions help us in obtaining the set J I for some families of fuzzy implications. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to suggest a new analytical technique called the fractional series expansion method for solving linear fractional differential equations (FDEs). Design/methodology/approach – This method is based on the idea of Kantorovich method, convergent series, and the modified Riemann-Liouville derivative. Findings – This work suggests a new analytical technique. The FDEs are described in Jumarie’s sense. Originality/value – It finds a new method for solving linear FDEs. The solution procedure is elucidated by two examples. [ABSTRACT FROM AUTHOR]

We construct an explicit formula for the fractal interpolation function associated to an IFS with variable parameters. The solution is given in terms of the base p representation of numbers. This construction is a consequence of the formulation of the problem in a general functional equation setting. We introduce compatibility conditions as essential hypotheses to ensure problems in the functional system form are well-defined. [ABSTRACT FROM AUTHOR]

This paper presents a method for computing validated error bounds on the truncation error associated with the linear approximation of an (un)stable invariant manifold by its eigenspace. The method is based on studying a certain functional equation which describes a chart map for the invariant manifold. The truncation error is represented as a bounded analytic function on an explicitly given neighborhood. Moreover studying this functional equation leads to a method for bounding derivatives of the truncation error as well. The methods developed in the present work are well suited for studying higher dimensional invariant manifolds, and validated numerical results are provided for manifolds of dimension up to one hundred. As an application of these ideas we present some computer assisted proofs of transverse homoclinic connecting orbits. [ABSTRACT FROM AUTHOR]

Recently, Vemuri and Jayaram proposed a novel method of generating fuzzy implications, called the ⊛ -composition, from a given pair of fuzzy implications (Vemuri and Jayaram, 2014). However, as with any generation process, the ⊛ -composition does not always generate new fuzzy implications. In this work, we study the generative power of the ⊛ -composition. Towards this end, we study some specific functional equations all of which lead to the solutions of the iterative functional equation I ( x , I ( x , y ) ) = I ( x , y ) involving fuzzy implications which has been studied extensively for different families of fuzzy implications in this very journal, see Balasubramaniam (2007), Xie and Qin (2010) and Xie et al. (2012). In this work, unlike in other existing works, we do not restrict the solutions to a particular family of fuzzy implications. Thus we take an algebraic approach towards solving these functional equations. Viewing the ⊛ -composition as a binary operation ⊛ on the set I of all fuzzy implications one obtains a monoid structure ( I , ⊛ ) on the set I . From the Cayley’s theorem for monoids, we know that any monoid is isomorphic to the set of all right translations. We determine the complete set K of fuzzy implications w.r.t. which the right translations also become semigroup homomorphisms on the monoid ( I , ⊛ ) and show that K not only answers our questions regarding the generative power of the ⊛ -composition but also contains many as yet unknown solutions of the iterative functional equation I ( x , I ( x , y ) ) = I ( x , y ) . [ABSTRACT FROM AUTHOR]

In this note we consider continuous-time systems \mathdotx(t)= A(t)x(t)+B(t)u(t), $y(t)=C(t)x(t)+D(t)u(t)$ as well as discrete-time systems $x(t+1)=A(t)x(t)+B(t)u(t)$, $y(t)= C(t)x(t)+D(t)u(t)$ whose coefficient matrices $A$, $B$, $C$ and $D$ are not exactly known. More precisely, all that is known about the systems is their nonzero pattern, i.e., the locations of the nonzero entries in the coefficient matrices. We characterize the patterns that guarantee controllability and observability, respectively, for all choices of nonzero time functions at the matrix positions defined by the pattern, which extends a result by Mayeda and Yamada for time-invariant systems. As it turns out, the conditions on the patterns for time-invariant and for time-varying discrete-time systems coincide, provided that the underlying time interval is sufficiently long. In contrast, the conditions for time-varying continuous-time systems are more restrictive than in the time-invariant case. [ABSTRACT FROM AUTHOR]

Highlights: [•] Impulsive functional systems of fractional order are investigated. [•] We establish a new comparison principle using differential inequalities. [•] We investigate impulsive effects on the stability of solutions. [•] We extend the Lyapunov function method to impulsive functional fractional equations. [•] We apply our results to a single species model of Lotka–Volterra type. [Copyright &y& Elsevier]

Abstract: In this article we discuss the distributive equation of implications over t-representable t-norms generated from nilpotent t-norms in interval-valued fuzzy sets theory. As a byproduct result we show all solutions of some functional equation related to this case. [Copyright &y& Elsevier]

A model of an arbitrary spin coupled to a bath of spins 12 in a star configuration is considered. The exact reduced dynamics of the central spin is found for the case of noncorrelated initial conditions of the system and the bath. The exact solution is used to test two approximation techniques, namely, the Nakajima-Zwanzig projection operator technique and the time-convolutionless projection operator technique corresponding to the second order of the coupling constant. Two types of projection operators are used for deriving the master equations, and the results are compared with the exact solution for a central spin equal to one. It is shown that the approximate master equations reproduce the exact dynamics on time scales 1/(AN), where A is the coupling constant and N is the number of spins in the bath. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned, in the main, with a problem of pursuit–evasion in the context of a stochastic finite-state system. Two cases are considered: (a) non-competitive pursuit in which the target does not try to evade the pursuer; and (b) a competitive case in which the aim of the target is to maximize the time of interception, and that of the pursuer is to minimize it. Employing dynamic programming, it is shown that determination of optimal policies for the target and pursuer reduce to solution of a functional equation involving the expected time of interception vector. Furthermore, it is shown that the functional equation is a contraction mapping. Optimal solution is obtained through iterated contraction. Convergence of iteration is established through the use of the Banach fixed-point theorem. [Copyright &y& Elsevier]