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Summarizes the construction of paper folding. Method for approximating rational subdivisions or arbitrary angles and line segments; Angle-folding algorithm; Approximating angles, regular polygons and star polygons; Dissection of angles into equal parts.

Although flexible neural networks (FNNs) have been used more successfully than classical neural networks (CNNs) in many industrial applications, nothing is rigorously known about their properties. In fact they are not even well known to the systems and control community. In the first part of this paper, existing structures of and results on FNNs are surveyed. In the second part FNNs are examined in a theoretical framework. As a result, theoretical evidence is given for the superiority of FNNs over CNNs and further properties of the former are developed. More precisely, several fundamental properties of feedforward and recurrent FNNs are established. This includes the universal approximation capability, minimality, controllability, observability, and identifiability. In the broad sense, the results of this paper help that general use of FNNs in systems and control theory and applications be based on firm theoretical foundations. Theoretical analysis and synthesis of FNN-based systems thus become possible. The paper is concluded by a collection of topics for future work. [ABSTRACT FROM PUBLISHER]

In this paper, the propagation of pulse wave in a deformable elastic vessel filled with inviscid blood is studied. Starting from the stress–strain relationship of blood vessel wall, momentum conservation equation and the Naiver–Stokes equation, the basic equations describing the wall motion and blood flow are established. By utilizing reductive perturbation technique and long wave approximation theory, the basic equations are simplified into a classical third-order mKdV equation with variable coefficients. In order to describe the propagation characteristics of pulse wave more accurately, a fifth-order variable-coefficient mKdV equation is derived. Then, the tanh-function method is applied to find the localized traveling wave solutions of these equations. Based on these localized traveling wave solutions, we further investigate the effects of higher order terms and initial vessel deformation on the characteristics of pulse wave propagation, blood flow velocity and the volume of blood flow. The results show that the higher-order nonlinear and dispersion terms lead to the distortion of the wave, while the initial deformation of the tube wall will influence the wave amplitude and wave width. [ABSTRACT FROM AUTHOR]

The paper addresses the problem of determining the field of a reflected wave in a multiscale inhomogeneous medium, which consists of a background deterministic large-scale irregularity and multiscale random irregularities with scales both larger and smaller than the wavelength. A layer with a linear permittivity profile is taken as the background irregularity. The small-scale irregularities are responsible for wide-angle scattering including backscattering. The first approximation of the perturbation theory is used to account for scattering from these irregularities. As the zero approximation and Green's function in determining the field backscattered by small-scale irregularities, we utilize the integral representation of the field, obtained in our earlier work by combining the method of double weighted Fourier transform (DWFT) and the Fock proper-time method. Asymptotic methods are used to reduce the sevenfold integral representation of a field to lower-order integrals. Conditions for validity of such representations are obtained. Formulas of the frequency coherence functions of waves reflected and backscattered from the turbulent plasma layer are given. The paper presents the results of the simulation of pulse sounding of a randomly inhomogeneous reflecting plasma layer demonstrating the effect of large-scale irregularities on backscattering by small-scale irregularities. [ABSTRACT FROM AUTHOR]

The simple question of how much paper is left on my toilet roll is studied from a mathematical modelling perspective. As is typical with applied mathematics, models of increasing complexity are introduced and solved. Solutions produced at each step are compared with the solution from the previous step. This process exposes students to the typical stages of mathematical modelling via an example from everyday life. Two activities are suggested for students to complete, as well as several extensions to stimulate class discussion. [ABSTRACT FROM PUBLISHER]

In this paper, we prove the existence of the initial-nonlinear nonlocal solutions for a parabolic system in a weighted Sobolev space. The methods applied are the Faedo–Galerkin approximation and the general theory of weak compactness in appropriate weighted Sobolev spaces together with using the Poincaré-type operator for dealing nonlinear nonlocal conditions. Furthermore, the boundedness and positivity of solutions depending on the boundedness and positivity of given data are also discussed by using suitable test functions. [ABSTRACT FROM AUTHOR]

In this paper, by using regular summability methods we modify the Bernstein–Chlodovsky operators in order get more general and powerful results than the classical aspects. We study Korovkin-type approximation theory on weighted spaces. As a special case, it is possible to Cesàro approximate (arithmetic mean convergence) to the test function e 2 (x) = x 2 although it fails for the classical Bernstein–Chlodovsky operators. At the end of the paper, we extend our results to the multi-dimensional case. [ABSTRACT FROM AUTHOR]

This paper develops a new method for the design of a recursive allpass phase-compensating system (PCS) with an arbitrarily prescribed stability margin (SM). The design methodology comes from the generalised stability-triangle (GST) that defines a closed stability region for the second-order system, and the closed stability region is described by a pair of parameterised inequalities. Since the GST is developed for the second-order recursive system, to design a high-order PCS, we first construct a high-order PCS using the cascade of a set of second-order systems (sections), and then apply the GST to each of the second-order sections. The two coefficients of each second-order section are first transformed into two new variables by employing a coefficient-transformation technique, and then all the new variables are optimised in such a way that a given phase specification is best approximated. Thanks to both the coefficient transformations and the general stability condition from the GST, the resulting high-order PCS is not only stable, but also satisfies a prescribed SM design specification. This paper includes two examples to illustrate the usefulness of the GST-based PCS design. [ABSTRACT FROM AUTHOR]

This paper investigates an adaptive cooperative fault-tolerant supervisory control problem for nonlinear strict-feedback leader-following systems with unknown control coefficients and actuator faults under the fixed directed graph. Radial basis function neural networks are used to approximate system uncertainties. The Nussbaum gain technique is introduced to address unknown signs of control gains. Then, based on the dynamic surface control method, a distributed adaptive fault-tolerant control scheme is presented to compensate for the actuator faults of followers. For neural networks approximation theory, the unknown smooth function can only be approximated on a compact set due to the approximation errors. By theoretic analysis, the supervisory-based adaptive controllers are proposed to guarantee that system state signals can converge to compact sets and the leader-following systems can achieve the practical output consensus. Finally, the simulation results are provided to show the validity of the proposed consensus scheme. [ABSTRACT FROM AUTHOR]

From the eigenvalue problem theory, we see that the convergence rate of the biharmonic eigenvalues obtained by the mixed method in I. Bab u ˇ ska and J. Osborn, [Eigenvalue Problems, Handbook of Numerical Analysis, Vol. II, North-Holland, Amsterdam, 1991.] is h 2 k − 2 for k ≥ 2. In this paper, we give a presentation of the lowest-order virtual element method for the approximation of Kirchhoff plate vibration problem. This discrete scheme is based on a conforming H 1 (Ω) × H 1 (Ω) formulation, following the variational formulation of Ciarlet–Raviart method, which allows us to make use of simpler and lower-regularity virtual element space. By using the classical spectral approximation theory in functional analysis, we prove the spectral approximation and optimal convergence order h 2 for the eigenvalues. Finally, some numerical experiments are presented, which show that the proposed numerical scheme can achieve the optimal convergence order. [ABSTRACT FROM AUTHOR]

This paper aims to obtain decompositions of higher dimensional functions into sums of non-tangential boundary limits of the corresponding Hardy space functions on tubes for the index range . In the one-dimensional case, Deng and Qian recently obtained such a Hardy space decomposition result: for any function , there exist functions and such that , where and are, respectively, the non-tangential boundary limits of some Hardy space functions in the upper-half and lower-half planes. In the present paper, we generalize the one-dimensional Hardy space decomposition result to the higher dimensions and discuss the uniqueness issue of such decomposition. [ABSTRACT FROM AUTHOR]

In a previous paper, by combining three techniques, i.e. Subdomain Decomposition Iterative Method (SDIM), Adaptive Cross Approximation (ACA), and Forward-Backward Spectral Acceleration (FBSA), from the Method of Moments (MoM), a high-efficiency calculation of the propagation and scattering in ducting maritime environments has been proposed. In this paper, this algorithm is updated by adding a perfectly conducting object above the sea surface, assumed to be highly conducting, which makes the environment very complex. Then, to quantify the effect of the object on the total scattered field, the coherent and incoherent powers, with and without object, are simulated by considering a surface of 800,000 unknowns (length of 6 km and a frequency of 5 GHz). [ABSTRACT FROM AUTHOR]

In the first part of the paper we deal with a second-order evolution variational inequality involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem. Next, we show that the solution is unique and the convergence is strong. In the second part of the paper, we consider a dynamic visco-elastic problem of contact mechanics. We assume that the contact process is governed by a normal damped response condition with a unilateral constraint and the body is non-clamped. The mechanical problem in its weak formulation reduces to a variational-hemivariational inequality that can be solved by finding a solution of a corresponding abstract problem related to one studied in the first part of the paper. Hence, we apply obtained existence result to provide the weak solvability of contact problem. [ABSTRACT FROM AUTHOR]

In this paper, we address the analytical investigation into a model for adhesive contact introduced in a paper by Freddi and Fremond, which includes nonlocal sources of damage on the contact surface, such as the elongation. The resulting PDE system features various nonlinearities rendering the unilateral contact conditions, the physical constraints on the internal variables, as well as the contributions related to the nonlocal forces. For the associated initial-boundary value problem, we obtain a global-in-time existence result by proving the existence of a local solution via a suitable approximation procedure and then by extending the local solution to a global one by a nonstandard prolongation argument. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate weakly/properly/robust efficient solutions of a nonsmooth semi-infinite multiobjective programming problem, in terms of convexificators. In some of the results, we assume the feasible set to be locally star-shaped. The appearing functions are not necessarily smooth/locally Lipschitz/convex. First, constraint qualifications and the normal cone to the feasible set are studied. Then, as a major part of the paper, various necessary and sufficient optimality conditions for solutions of the problem under consideration are presented. The paper is closed by a linear approximation problem to detect the solutions and by studying a gap function. [ABSTRACT FROM PUBLISHER]

We study methods for solving a class of the quasivariational inequalities in Hilbert space when the changeable set is described by translation of a fixed, closed and convex set. We consider one variant of the gradient-type projection method and an extragradient method. The possibilities of the choice of parameters of the gradient projection method in this case are wider than in the general case of a changeable set. The extragradient method on each iteration makes one trial step along the gradient, and the value of the gradient at the obtained point is used at the first point as the iteration direction. In the paper, we establish sufficient conditions for the convergence of the proposed methods and derive a new estimate of the rates of the convergence. The main result of this paper is contained in the convergence analysis of the extragradient method. [ABSTRACT FROM AUTHOR]

In missing data analysis, the assumption of the missing data mechanism is crucial. Under different assumptions, different statistical methods have to be developed accordingly; however, in reality this kind of assumption is usually unverifiable. Therefore a less stringent, and hence more flexible, assumption is preferred. In this paper, we consider a generally applicable missing data mechanism. Under this general missing data mechanism, we introduce the conditional likelihood and its approximate version as the base for estimating the unknown parameter of interest. Since this approximate conditional likelihood uses the completely observed samples only, it may result in large estimation bias, which could deteriorate the statistical inference and also jeopardise other statistical procedure. To tackle this problem, we propose to use some resampling techniques to reduce the estimation bias. We consider both the Jackknife and the Bootstrap in our paper. We compare their asymptotic biases through a higher order expansion up to. We also derive some results for the mean squared error (MSE) in terms of estimation accuracy. We conduct comprehensive simulation studies under different situations to illustrate our proposed method. We also apply our method to a prostate cancer data analysis. [ABSTRACT FROM AUTHOR]

This paper presents a novel approximation technique based on metaheuristics and weighted residual function (WRF) for tackling singular boundary value problems (BVPs) arising in engineering and science. With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic optimisation algorithms, singular BVPs can be approximated as an optimisation problem with boundary conditions as constraints. The target is to minimise the WRF (i.e. error function) constructed in approximation of BVPs. The scheme involves generational distance metric for quality evaluation of the approximate solutions against exact solutions (i.e. error evaluator metric). Four test problems including two linear and two non-linear singular BVPs are considered in this paper to check the efficiency and accuracy of the proposed algorithm. The optimisation task is performed using three different optimisers including the particle swarm optimisation, the water cycle algorithm, and the harmony search algorithm. Optimisation results obtained show that the suggested technique can be successfully applied for approximate solving of singular BVPs. [ABSTRACT FROM AUTHOR]

The Simplified P N ( S P N ) approximation is often used to model radiation transport phenomena, but it converges to the true solution of the transport equation only in one-dimensional slab geometry. In all other geometries, it incurs a model error that needs to be quantified. In this paper, we estimate the radiation transport model error due to the S P N approximation and employ S N transport solutions (with high S N order) as reference transport solutions. Because the S P N solution does not contain the full angular information of the transport solution, an angular intensity must be reconstructed from the S P N solution in order to compute the S P N model error. We propose two such reconstruction schemes. Model error estimates are given for various quantities of interests, i.e., scalar radiation intensity, radiation flux, and boundary leakage. An adjoint-based approach is proposed to evaluate the model error and is compared against forward and residual techniques. Two-dimensional numerical experiments are presented. [ABSTRACT FROM AUTHOR]

The debate between proponents and opponents of causal foundationalism has recently surfaced as a disparity between causal structuralism and causal anti-foundationalism in the structural realist camp. The paper outlines and dissolves the problem of disparity for (informational) structural realism. I follow John Collier (also Carl T. Bergstrom and Martin Rosvall) to specify causation in terms of the transmission of information. Unlike them, I built upon the reverse quantum data-processing inequality to show how this approach models causation as a symmetric process at the level of fundamental physics (but not special sciences). I show how this suggestion reduces the disparity about causation to a problem of application to diverse contexts. [ABSTRACT FROM AUTHOR]

The high-precision method for modeling inhomogeneous media and the precision limit of finite-difference time-domain (FDTD) are studied theoretically and numerically. Firstly, based on the transform-domain idea, a new layered modeling technique for inhomogeneous media is developed, with the control equation of the modeling precision given. Compared with other modeling methods, the paramount advantage of this method is that the modeling process is not restrained by the magnitude range of the data, and that the modeling precision can be controlled easily. Then, the shift operator finite-difference time-domain is adopted to determine the field values in inhomogeneous media, and a high-precision computational method is proposed. This method theoretically proves that the relationship between the FDTD grid size and the calculated field value satisfies a definite function, which is then verified by numerical examples. The significance of the results is that this relationship will provide an effective approach to predict field values during the modeling grid size tending to zero, which can be regarded in theory as an approximation to the true value of the electromagnetic (EM) fields. The advantages of this approach are that it not only inherits the advantages of the FDTD method, but also gives a way out of the difficulty that the EM algorithms available are difficult to provide a high-precision prediction of radio propagation; and it can be seen from the validation process in this paper that this approach could also be used in the algorithms based on the mesh modeling and in the inhomogeneous media for improving their calculation precision. [ABSTRACT FROM AUTHOR]

The aim of this paper is a use of Stein's method and w-functions to determine a non uniform bound on the geometric approximation for a non negative integer-valued random variable. Some applications of the obtained results are provided to approximate the negative hypergeometric, Pólya and negative Pólya distributions. [ABSTRACT FROM AUTHOR]

In this paper, the problem of estimating unknown parameters of a two-parameter Kumaraswamy-Exponential (Kw-E) distribution is considered based on progressively type-II censored sample. The maximum likelihood (ML) estimators of the parameters are obtained. Bayes estimates are also obtained using different loss functions such as squared error, LINEX and general entropy. Lindley's approximation method is used to evaluate these Bayes estimates. Monte Carlo simulation is used for numerical comparison between various estimates developed in this paper. [ABSTRACT FROM AUTHOR]

This paper investigates a predefined performance control problem for adaptive tracking of uncertain nonlinear time-delay systems in nonstrict-feedback form. Nonstrict-feedback nonlinearities, time-varying delays and external disturbances are assumed to be unknown. Based on the exponential decaying design functions denoting the preassigned bounds of transient and steady-state tracking errors, some variable separation lemmas are derived to design an approximation-based robust adaptive control scheme in the presence of nonstrict-feedback time-delayed nonlinearities. The proposed control system guarantees that a tracking error remains within a predesigned bound for allt≥ 0 and converges to a preselected neighbourhood of the origin. Compared with the existing results in the literature, the main contribution of this paper is to provide a solution on the guaranteed performance control in the presence of unknown nonstrict-feedback nonlinearities related to all delayed state variables. Simulation results illustrate the effectiveness of the proposed methodology. [ABSTRACT FROM PUBLISHER]

The Griddy Gibbs sampling was proposed by Ritter and Tanner [Facilitating the Gibbs Sampler: the Gibbs Stopper and the Griddy–Gibbs Sampler. J Am Stat Assoc. 1992;87(419):861—868] as a computationally efficient approximation of the well-known Gibbs sampling method. The algorithm is simple and effective and has been used successfully to address problems in various fields of applied science. However, the approximate nature of the algorithm has prevented it from being widely used: the Markov chains generated by the Griddy Gibbs sampling method are not reversible in general, so the existence and uniqueness of its invariant measure is not guaranteed. Even when such an invariant measure uniquely exists, there was no estimate of the distance between it and the probability distribution of interest, hence no means to ensure the validity of the algorithm as a means to sample from the true distribution. In this paper, we show, subject to some fairly natural conditions, that the Griddy Gibbs method has a unique, invariant measure. Moreover, we provideestimates on the distance between this invariant measure and the corresponding measure obtained from Gibbs sampling. These results provide a theoretical foundation for the use of the Griddy Gibbs sampling method. We also address a more general result about the sensitivity of invariant measures under small perturbations on the transition probability. That is, if we replace the transition probabilityPof any Monte Carlo Markov chain by another transition probabilityQwhereQis close toP, we can still estimate the distance between the two invariant measures. The distinguishing feature between our approach and previous work on convergence of perturbed Markov chain is that by considering the invariant measures as fixed points of linear operators on function spaces, we do not need to impose any further conditions on the rate of convergence of the Markov chain. For example, the results we derived in this paper can address the case when the considered Monte Carlo Markov chains are not uniformly ergodic. [ABSTRACT FROM AUTHOR]

Many problems in engineering and science involve calculation of difficult Laplace integrals of the form:In previous papers [Hanna and Davis (2011), Davis and Hanna (2013), referred to as Papers I and II, respectively], the authors introduced two new complementary analytical methods for theimprovedasymptotic (x→ ∞) approximation of these integralswhen the upper limit A equal to ∞[the general Watson lemma (WL) problem for anyAifx→ ∞]. A procedure is developed here that extends application of the previous improvement methods to thedifficult problem of Laplace integrals having a finite upper limit (FUL) = A. In addition, problems havinggrowingexponential behavior, certain infinite Fourier integrals and problems having large (tα) factors, are also considered. The main result is that, with some modifications, the exponential, expansion-point, and combination procedures developed forinfiniteintegrals (Papers I and II) can be easily applied directly to FUL Laplace integrals. This is accomplished with the aid of a simple new “generalized incomplete gamma function (IGF)” algorithm which itself utilizes an improvement procedure. The new FUL procedure requires onlyF(t),F′(t), and a few terms of the Taylor expansion. A simple EXCEL program which implements the new procedure is discussed in detail in Appendix A and is freely available to users athttp://www.d.umn.edu/~rdavis/CEC/. Many numerical comparisons presented here indicate that good engineering accuracy is achieved for these improved approximations at virtually all positiveAvalues, over a very wide range inx, for variousα, β,andF(t) functions. Where comparisons are possible (A = ∞), the new results are far superior to those of the best Watson’s lemma results. [ABSTRACT FROM AUTHOR]

In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operatorFover a closed and convex setC. We assume that the setCcan be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method, the main idea of which is to project at each step onto a particular half-space constructed using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper, we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima’s method has so far been considered only in the Euclidean setting with different conditions onF. We provide several examples for the case whereCis the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results. [ABSTRACT FROM AUTHOR]

The paper deals with the expected maxima of continuous Gaussian processesthat are Hölder continuous in-norm and/or satisfy the opposite inequality for the-norms of their increments. Examples of such processes include the fractional Brownian motion and some of its “relatives” (of which several examples are given in the paper). We establish upper and lower bounds forand investigate the rate of convergence to that quantity of its discrete approximation. Some further properties of these two maxima are established in the special case of the fractional Brownian motion. [ABSTRACT FROM PUBLISHER]

We consider a large number of Bosons with interaction potential. In our earlier papers (Grillakis et al. inComm. Math. Phys.(2010) and inAdv. Math.(2011), as well as Grillakis and Machedon inComm. Math. Phys., (2013)) we considered a set of equations for the condensateϕand pair excitation functionkand proved that they provide a Fock space approximation to the exact evolution of a coherent state for. In Grillakis and Machedon,J. Fixed Point Theory Appl., (2013), in the hope of treating higher values ofβ<1, we introduced a coupled refinement of our original equations. In that paper, we showed the coupled equations conserve the number of particles and energy. In the current paper, we prove that the coupled equations do indeed provide a Fock space approximation for, at least locally in time. In order to do that, we reformulate the coupled equations in a way reminiscent of BBGKY and apply harmonic analysis techniques in the spirit of those used by Chen and Holmer inJ. Euro. Math. Soc.(2016) to prove the necessary estimates. In turn, these estimates provide bounds for the pair excitation functionk. While our earlier papers provide background material, the methods of this paper paper are mostly new, and the presentation is self-contained. [ABSTRACT FROM AUTHOR]

In this paper, we have proved the degree of approximation of functions belonging to L[0, ∞) by Harmonic-Euler means of its Fourier-Laguerre series at x = 0. The aim of this paper is to concentrate on the approximation properties of the functions in L[0, ∞) by Harmonic-Euler means of its Fourier-Laguerre series associated with the function f. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a combinatorial optimization problem, called the converse connectedp-centre problem which is the converse problem of the connectedp-centre problem. This problem is a variant of thep-centre problem. Given an undirected graphwith a nonnegative edge length function ℓ, a vertex set, and an integerp,, letdenote the shortest distance fromvtoCofGfor each vertexvin, and theeccentricityofCdenote. The connectedp-centre problem is to find a vertex setPinV,, such that the eccentricity ofPis minimized but the induced subgraph ofPmust be connected. Given an undirected graphand an integer, the converse connectedp-centre problem is to find a vertex setPinVwith minimum cardinality such that the induced subgraph ofPmust be connected and the eccentricity. One of the applications of the converse connectedp-centre problem has the facility location with load balancing and backup constraints. The connectedp-centre problem had been shown to be NP-hard. However, it is still unclear whether there exists a polynomial time approximation algorithm for the converse connectedp-centre problem. In this paper, we design the first approximation algorithm for the converse connectedp-centre problem with approximation ratio of,. We also discuss the approximation complexity for the converse connectedp-centre problem. We show that there is no polynomial time approximation algorithm achieving an approximation ratio of,, for the converse connectedp-centre problem unless. [ABSTRACT FROM PUBLISHER]

The present paper is concerned with the necessary and sufficient conditions of optimality for third-order polyhedral optimization described by polyhedral discrete and differential inclusions (PDIs). In the first part of the paper, the discrete polyhedral problemis reduced to convex minimization problem and the necessary and sufficient condition for optimality is derived. Then the necessary and sufficient conditions of optimality for discrete-approximation problemare formulated using the transversality condition and approximation method for the continuous polyhedral problemgoverned by PDI. On the basis on the obtained results in Section 3, we prove the sufficient conditions of optimality for the problem. It turns out that the concerned method requires some special equivalence theorem, which allow us to make a bridge betweenandproblems. [ABSTRACT FROM PUBLISHER]

This paper deals with the investigation of the analytical approximate solutions for two-term fractional-order diffusion, wave-diffusion, and telegraph equations. The fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], (1,2), and [1,2], respectively. In this paper, we extended optimal homotopy asymptotic method (OHAM) for two-term fractional-order wave-diffusion equations. Highly approximate solution is obtained in series form using this extended method. Approximate solution obtained by OHAM is compared with the exact solution. It is observed that OHAM is a prevailing and convergent method for the solutions of nonlinear-fractional-order time-dependent partial differential problems. The numerical results rendering that the applied method is explicit, effective, and easy to use, for handling more general fractional-order wave diffusion, diffusion, and telegraph problems. [ABSTRACT FROM PUBLISHER]

In the modelling of many important problems in science and engineering we face stiff stochastic differential equations (SDEs). In this paper, a new class of split-step double balanced (SSDB) approximation methods is constructed for numerically solving systems of stiff Itô SDEs with multi-dimensional noise. In these methods, an appropriate control function has been used twice to improve the stability properties. Under global Lipschitz conditions, convergence with order one in the mean-square sense is established. Also, the mean-square stability (MS-stability) properties of the SSDB methods have been analysed for a one-dimensional linear SDE with multiplicative noise. Therefore, the MS-stability functions of SSDB methods are determined and in some special cases, their regions of MS-stability have been compared to the stability region of the original equation. Finally, simulation results confirm that the proposed methods are efficient with respect to accuracy and computational cost. [ABSTRACT FROM AUTHOR]

In this paper, we investigate several modified exponential finite-difference methods to approximate the solution of the one-dimensional viscous Burgers' equation. Burgers' equation admits solutions that are positive and bounded under appropriate conditions. Motivated by these facts, we propose nonsingular exponential methods that are capable of preserving some structural properties of the solutions of Burgers' equation. The fact that some of the techniques preserve structural properties of the solutions is thoroughly established in this work. Rigorous analyses of consistency, stability and numerical convergence of these schemes are presented for the first time in the literature, together with estimates of the numerical solutions. The methods are computationally improved for efficiency using the Padé approximation technique. As a result, the computational cost is substantially reduced in this way. Comparisons of the numerical approximations against the exact solutions of some initial-boundary-value problems for different Reynolds numbers show a good agreement between them. [ABSTRACT FROM AUTHOR]

In this paper, we analyse piecewise deterministic Markov processes (PDMPs), as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of PDMPs. There one is interested in computing certain quantities of interest such as the probability of ruin or the value of an insurance company. Instead of explicitly solving the related integro-(partial) differential equation (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules; this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. We prove a convergence result for our PDMPs approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and compare deterministic and Monte Carlo integration. [ABSTRACT FROM AUTHOR]

Let (), let with closed range. In this paper, utilizing the gap between closed subspaces and the perturbation bounds of metric projections, we present some new perturbation results of the Moore-Penrose metric generalized inverse. As applications of our results, we also investigate the best approximate solution problem for the ill-posed operator equation Tx=y under some conditions. The main results have three parts, part one covers the null space preserving case, part two covers the range preserving case, and part three covers the general case. Examples in connection with the theoretical results will be also presented. [ABSTRACT FROM AUTHOR]

In this paper, firstly, we discuss the hölder continuity of the integral operators and associated with the Helmholtz equation and show the relation between and . Secondly, we prove that the operator has a unique fixed point by the Banach Contraction Mapping Principle. Finally, we give the iterative approximation of fixed point of the modified integral operator [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of estimating the scale parameter of the inverse Rayleigh distribution based on general progressively Type-II censored samples and progressively Type-II censored samples. The pivotal quantity method is used to derive the estimator of the scale parameter. Besides, considering that the maximum likelihood estimator is tough to obtain for this distribution, we derive an explicit estimator of the scale parameter by approximating the likelihood equation with Taylor expansion. The interval estimation is also studied based on pivotal inference. Then we conduct Monte Carlo simulations and compare the performance of different estimators. We demonstrate that the pivotal inference is simpler and more effective. The further application of the pivotal quantity method is also discussed theoretically. Finally, two real data sets are analyzed using our methods. [ABSTRACT FROM AUTHOR]

King's Point Optimal (PO) test of a simple null hypothesis is useful in a number of ways, for example it can be used to trace the power envelope against which existing tests can be compared. However, this test cannot always be constructed when testing a composite null hypothesis. It is suggested in the literature that approximate PO (APO) tests can overcome this problem, but they also have some drawbacks. This paper investigates if King's PO test can be used for testing a composite null in the presence of nuisance parameters via a maximized Monte Carlo (MMC) approach, with encouraging results. [ABSTRACT FROM AUTHOR]

In this paper, a discontinuous least-squares (DLS) finite-element method is introduced. The novelty of this work is twofold, to develop a DLS formulation that works for general polytopal meshes and to provide rigorous error analysis for it. This new method provides accurate approximations for both the primal and the flux variables. We obtain optimal-order error estimates for both the primal and the flux variables. Numerical examples are tested for polynomials up to degree 4 on non-triangular meshes, i.e. on rectangular and hexagonal meshes. [ABSTRACT FROM AUTHOR]

In this paper, we give the wavelet packet approximation theorem for type norm which can measure difference of the (weak) derivatives. We will show that with equal distribution of the vanishing moments between the scaling function and the wavelet packets , if the sample values of a smooth function as scaling function coefficients at a fine scale are used, then the wavelet packet series approximates the smooth function under consideration with increasing verity in type norm as the number of vanishing moments M increases or the scale J gets finer. [ABSTRACT FROM AUTHOR]

In this paper, the partial fractions expansions for the generalized hypergeometric function will be presented. Our results generalized several well-known results in the literature. [ABSTRACT FROM AUTHOR]

The paper deals with strong global approximation of stochastic differential equations (SDEs) driven by two independent processes: a nonhomogeneous Poisson process and a Wiener process. We assume that the jump and diffusion coefficients of the underlying SDE satisfy jump commutativity condition (see Chapter 6.3 in [21]). We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson and Wiener processes. We consider classes of methods that use equidistant or nonequidistant sampling of the Poisson and Wiener processes. We provide a construction of optimal methods, based on the classical Milstein scheme, which asymptotically attain the established minimal errors. The analysis implies that methods based on nonequidistant mesh are more efficient, with respect to asymptotic constants, than those based on the equidistant mesh. [ABSTRACT FROM AUTHOR]

In this paper, through appropriate rational approximation, we prove that a function f in , with particular interest in the index range can be decomposed into a sum in the sense of , where is a half strip domain in the complex plane, g and h are the non-tangential limits of functions in and , respectively. For the case we show that a rational function in can be decomposed into a sum of rational functions in and . [ABSTRACT FROM AUTHOR]

The stochastic alpha-beta-rho (SABR) model is widely used in fixed income and foreign exchange markets as a benchmark. The underlying process may hit zero with a positive probability and therefore an absorbing boundary at zero should be specified to avoid arbitrage opportunities. However, a variety of numerical methods choose to ignore the boundary condition to maintain the tractability. This paper develops a new principle of not feeling the boundary to quantify the impact of this boundary condition on the distribution of underlying prices. It shows that the probability of the SABR hitting zero decays to 0 exponentially as the time horizon shrinks. Applying this principle, we further show that conditional on the volatility process, the distribution of the underlying process can be approximated by that of a time-changed Bessel process with an exponentially negligible error. This discovery provides a theoretical justification for many almost exact simulation algorithms for the SABR model in the literature. Numerical experiments are also presented to support our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a constant stress accelerated life test terminated by a hybrid Type-I censoring at the first stress level. The model is based on a general log-location-scale lifetime distribution with mean life being a linear function of stress and with constant scale. We obtain the maximum likelihood estimators (MLE) and the approximate maximum likelihood estimators (AMLE) of the model parameters. Approximate confidence intervals, likelihood ratio tests and two bootstrap methods are used to construct confidence intervals for the unknown parameters of the Weibull and lognormal distributions using the MLEs. Finally, a simulation study and two illustrative examples are provided to demonstrate the performance of the developed inferential methods. [ABSTRACT FROM AUTHOR]

This paper is devoted to the construction and analysis of compact finite difference methods for a class of time-fractional convection-reaction-diffusion equations with variable coefficients. Based on some new techniques coupled with the - approximation formula of the time-fractional derivative and a fourth-order compact finite difference approximation to the spatial derivative, a compact finite difference method is proposed for the equations with spatially variable convection and reaction coefficients. The local truncation error and the solvability of the method are discussed in detail. The unconditional stability of the resulting scheme and also its convergence of second order in time and fourth order in space are rigorously proved using a discrete energy analysis method. The proposed method is further extended to the more general case when the convection and reaction coefficients are variable both spatially and temporally. A high-order combined compact finite difference method is also proposed. Numerical results demonstrate the effectiveness of the methods. [ABSTRACT FROM AUTHOR]

When classical rough set (CRS) theory is used to analyze spatial data, there is an underlying assumption that objects in the universe are completely randomly distributed over space. However, this assumption conflicts with the actual situation of spatial data. Generally, spatial heterogeneity and spatial autocorrelation are two important characteristics of spatial data. These two characteristics are important information sources for improving the modeling accuracy of spatial data. This paper extends CRS theory by introducing spatial heterogeneity and spatial autocorrelation. This new extension adds spatial adjacency information into the information table. Many fundamental concepts in CRS theory, such as the indiscernibility relation, equivalent classes, and lower and upper approximations, are improved by adding spatial adjacency information into these concepts. Based on these fundamental concepts, a new reduct and an improved rule matching method are proposed. The new reduct incorporates spatial heterogeneity in selecting the feature subset which can preserve the local discriminant power of all features, and the new rule matching method uses spatial autocorrelation to improve the classification ability of rough set-based classifiers. Experimental results show that the proposed extension significantly increased classification or segmentation accuracy, and the spatial reduct required much less time than classical reduct. [ABSTRACT FROM AUTHOR]

In this paper, a new structured fractional order-[proportional integral] () controller design method is demonstrated. The designed controller is implemented in decoupled Two-Input Two-Output (TITO) Coupled Tank System (CTS) for liquid level control. An inverted decoupling technique is used to decouple the TITO process into two independent SISO (single-input single-output) processes. The designed controller is implemented for each SISO process. A frequency domain approach is used to tune the parameters of the controller. While designing, to approximate the fractional integral terms of the controller, Oustaloup's Recursive Approximation (ORA) method is used in the stable frequency bands. To validate the proposed approach, the performance of the controller is compared with the existing controllers designed based on the same specifications. From the simulation and experimental studies, it is shown that proposed gives better performance than , PI and PID controllers. [ABSTRACT FROM AUTHOR]