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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.

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]

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]

To increase the efficiency of photovoltaic (PV) systems, maximum power point (MPP) tracking of the solar arrays is needed. Under partially shaded conditions (PSCs), the solar arrays power–current (P–I) characteristic has multiple MPP. This paper presents various methods and approaches of tracking the MPP from a PV generator operating under PSC. Some comparisons, advantages, drawbacks and critical analysis of each method are discussed. It was found that, indirect methods use empirical data or mathematical expressions of numerical approximations to estimate the MPP from the PV generator’s voltage, current and irradiance. Direct methods offer the advantage of obtaining the actual maximum power from the PV generator’s voltage and current. Artificial intelligence methods do not need exact mathematical models. They can perform under parameter variation, load and supply voltage disturbances. Finally, novel methods require less number of iterations to converge, independent to the initial conditions. All these algorithms can be included in some of the DC/DC converters and MPP trackers for stand-alone or grid-connected systems. [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]

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 article presents the author's comments on an article "Optimum Sample Size for a Problem in Choosing the Population With the Largest Mean," by statistician Paul N. Somerville, published in the June 1970 issue of "Journal of the American Statistical Association." In the article Somerville described a procedure for selecting the population with the largest mean from normal populations with unknown means and a common known variance. The author claims that his procedure gives a unique minimax solution and that the minimax is a reasonable solution to the problem. According to the author, Somerville has not made it clear that his procedure gives only the local minimax solution. This makes practical applications of his procedure severely limited for it is difficult to think of a situation in which an experimenter would like to determine a local minimax value.

This paper tackles the problem of optimising safety stocks in a two-echelon assembly system. It presents and discusses several approximation models for the assembly lead-time under the assumption of normality of the assembly demand and normality of components' nominal lead times. These approximation models are subsequently used to optimise safety stocks throughout a two-echelon assembly system. They are then tested on a particular two-echelon N-identical component assembly system. The obtained results are compared with the results of a discrete event simulation. Finally, it is shown that lead-times and safety stock results already obtained for a two-echelon distribution system can also be derived without difficulty from those of two-echelon assembly systems. [ABSTRACT FROM AUTHOR]

Aiming at improving the efficiency of calculating the cutter location point with toroid cutter based on the projection method in NC machining for surface, a new algorithm is proposed to calculate the cutter location point directly by torus surface approximating the surface to be machined. According to the geometric information of the points on the surface, the geometrical conditions of the two tangential tori are figured out, and then the contact point is obtained by solving multivariate non-linear equations. Parameters of the tangent point on the surface to be machined are calculated in the next step. Finally, the cutter location point is calculated by a small adjustment. The proposed algorithm is applied to calculate the cutter location point with toroid cutter in surface machining and compared with the existing algorithm. The results show that the computing time of the proposed algorithm in this paper saved about 63-78%. [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]

The optimal feedrate planning on five-axis parametric tool path with multi-constraints remains challenging due to the variable curvature of tool path curves and the nonlinear relationships between the Cartesian space and joint space. The methods for solving this problem are very limited at present. The optimal feedrate associated with a programmed tool path is crucial for high speed and high accuracy machining. This paper presents a novel feedrate optimisation method for feedrate planning on five-axis parametric tool paths with preset multi-constraints including chord error constraint, tangential kinematic constraints and axis kinematic constraints. The proposed method first derives a linear objective function for feedrate optimisation by using a discrete format of primitive continuous objective function. Then, the preset multi-constraints are converted to nonlinear constraint conditions on the decision variables in the linear objective function and are then linearised with an approximation strategy. A linear model for feedrate optimisation with preset multiple constraints is then constructed, which can be solved by well-developed linear programming algorithms. Finally, the optimal feedrate can be obtained from the optimal solution and fitted to the smooth spline curve as the ultimate feedrate profile. Experiments are conducted on two parametric tool paths to verify the feasibility and applicability of the proposed method that show both the planning results and computing efficiency are satisfactory when the number of sampling positions is appropriately determined. [ABSTRACT FROM AUTHOR]

This paper studies the problem of clinical appointment scheduling when taking revisits into account. We consider two classes of patients: (1) routine patients who have made an appointment weeks in advance and (2) same-day patients who call in at the very beginning of the day, before the first clinical consultation begins. After the first appointment and consultation, patients might need an additional examination and a second consultation to confirm their health status. This paper aims to create an advanced scheduling method for both routine patients and same-day patients to optimise the expected weighted sum of three performance measures: patients’ waiting time, physician’s idle time and overtime. A stochastic programme model is constructed and solved by sample average approximation and benders’ decomposition. Numerical tests show that revisits significantly affect the three performance measures; to improve the hospital system’s operation management, both scheduling of appointment times and daily workload plans are taken into account. [ABSTRACT FROM PUBLISHER]

Sequencing mixed-model assembly lines is a well researched topic in the literature. However, many methods that have been developed to solve this problem fail to cope with either the large size or the specific characteristics of real-life problems. In this paper, a heuristic is proposed that is derived from Vogel's approximation method for transportation planning. The heuristic is able to handle large and supposedly difficult problem instances. Sophisticated test scenarios considering real-life aspects were generated to evaluate the performance of the heuristic for realistic problem instances. It is shown that the proposed heuristic significantly outperforms priority rule-based methods and requires only reasonable computational effort. [ABSTRACT FROM AUTHOR]

This paper presents some models of exchange rate with jumps, namely jump diffusion exchange rate models. Jump diffusion models are quite common in computational and theoretical finance. It is known that exchange rates sometimes exhibit jumps during some time periods. Therefore, it is important to take into account the presence of these jumps in exchange rate modeling in general. However, even the simplest jump diffusion model introduces some analytical difficulty in terms of finding a solution to the model. The models we analyze in this paper make use of Approximation Theory in order to come up with closed form solutions to the underlying variables. This approach leads to the branch of differential equations called functional differential equations and more specifically the so-called delay differential equations. Our approach leads to a second order delay differential equation. Though, in principle, these types of functional differential equations can be solved analytically in some cases, the task, in general, is quite enormous. We circumvent this technical difficulty by deriving an approximate solution using a power series expansion of the second order. Therefore, we derive a complete solution to the models and also investigate the model's predictions of the exchange rate. We introduce two jump diffusion models. The first model examines the case where there are jumps with a constant magnitude. The second model considers the case of jumps of different sizes. These are relatively simpler cases to be analyzed. We will present some computational aspects in terms of the difficulty often encountered in estimating these types of models. The difficulty increases for the type of exchange rate models being considered in this paper. Taking advantage of the specification of the models we have estimated the parameters using a two-step M-estimation strategy that combines full information maximum likelihood estimation in the first step and the simulated method of moments in the second step. [ABSTRACT FROM AUTHOR]

Pythagorean fuzzy sets are advancements of the intuitionistic fuzzy sets and overcome their limitations. In this paper, we exploit the concept of full congruence relation of Pythagorean fuzzy sets and define the lower and upper approximations of Pythagorean fuzzy set. Using the concept of approximations of Pythagorean fuzzy set we introduce the concept of rough Pythagorean fuzzy set. [ABSTRACT FROM AUTHOR]

An analytical model for evaluating the throughput of two-machine lines, characterised by intermediate buffer with finite capacity, deterministic processing times and multiple failure modes for each machine is presented in this paper. Both operation-dependent failure and time-dependent failure are captured in a unique model as extension of the existing literature that was dealing with either one of them. Each machine has two failure modes, one is operation-dependent and the other is time-dependent. Time to failure and time to repair are assumed to be geometrically distributed. The presented method calculates the steady-state probabilities of the manufacturing system with a computational effort that depends only on the number of failure modes and not on the buffer capacity. A performance comparison of the proposed model with existing techniques is also reported, the aim is to show the error introduced by an analytical model that considers the operation-dependent failure mode as approximation of the time-dependent one. [ABSTRACT FROM PUBLISHER]

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]

This paper presents a novel arrangement of Routh table array for deriving an approximate model of a higher orderz-domain uncertain system. The demand for this computation is to procure a lower order model which is easy to be exercised in comparison to their original large scale systems. Additionally, the derived model should preserve fewer dynamic characteristic of the comprehensive higher order systems. The mentioned new arrangement is achieved from the arena of different combinations of numerator and denominator polynomials. The combinations are validated by their practice over the conventional example from the literature. This precise blend is then applied to a real-time system for its rational acceptability. Both the models play a significant role in establishing the algorithm. Besides this, the limitation encountered during the foundation course of the arrangement is also taken into consideration. The paper also offers a future scope for fellow researchers. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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, 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]

This paper describes the rate of convergence of solutions of Robin boundary value problems of an elliptic equation to the solution of a Dirichlet problem as a boundary parameter decreases to zero. The results are found using representations for solutions of the equations in terms of Steklov eigenfunctions. Particular interest is in the case where the Dirichlet data is only in L2(∂Ω,dσ). Various approximation bounds are obtained and the rate of convergence of the Robin approximations in the H1 and L2 norms are shown to have convergence rates that depend on the regularity of the Dirichlet data. [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]

Unbalanced, unreliable (UR), unpaced, merging assembly lines are simulated in this study with varying line lengths, buffer storage capacities, imbalance degrees and unequal mean operation time configurations and uneven buffer capacity (BC) allocation. This paper contributes to the literature by suggesting that, in many cases, imbalance can improve merging lines' performance, as compared to a corresponding balanced merging line. It was found that an inverted bowl or descending patterns for mean operation times (MTs), and an inverted bowl (concentrating BC towards the centre of the line) or an ascending pattern for buffer allocation, result in higher throughput (TR). In terms of average buffer level (ABL), the best pattern is a monotone decreasing order regarding MTs and a monotone increasing order with respect to BC allocation. Additionally, it was found that when considering a profit function, the best performing patterns for UR lines tend to be the patterns that reduce ABL, even when considering very low inventory holding costs; contrary to the behaviour of the profit function in reliable lines, which suggests that either patterns that increase TR or reduce ABL can lead to a good performance, depending on the values of the unitary inventory holding costs. [ABSTRACT FROM AUTHOR]

Multi-product systems with finite buffers and sequence-dependent set-up times are quite common in modern manufacturing industry. In practice, the distribution of machine processing time could be arbitrary, while in existing literature it is often assumed to follow an exponential distribution. In this paper, we develop an analytical method to study the multi-product manufacturing systems with non-exponential processing times. An embedded Markov chain model is constructed and two approximation methods, Gamma estimation and linear approximation, are proposed. The model is validated with high accuracy by numerical experiments and practical data from an automotive assembly system. [ABSTRACT FROM PUBLISHER]

In this paper we show an approximation in law to the subfractional Brownian motion within the Skorohod topology. The construction of these approximations is based on random walks. [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]

Freud wrote a curious short paper at the end of 1927, the year in which his book, The future of an illusion was published. The paper is remarkable in that we as readers also have the evidence that Freud drew on, that is, a letter by an American physician unknown to Freud, detailing his religious experience. (This situation of having the same source material is not completely singular, as we also have access to Da Vinci's painting of the ‘Madonna and Child with Saint Anne’, the subject of a previous exercise by Freud (1910a) in interpreting religious material in a reductive manner.) Having available the exact text that Freud used for his analysis enables a close examination of Freud's use of the material. It becomes clear that in his analysis of the physician's experience, Freud makes a string of interpolations, breaking his own rules concerning wild analysis. It can also be seen from the paper how he ignores considerations of context and culture. [ABSTRACT FROM AUTHOR]

In this paper we develop and test experimental methodologies for selection of the best alternative among a discrete number of available treatments. We consider a scenario where a researcher sequentially decides which treatments are assigned to experimental units. This problem is particularly challenging if a single measurement of the response to a treatment is time-consuming and there is a limited time for experimentation. This time can be decreased if it is possible to perform measurements in parallel. In this work we propose and discuss asynchronous extensions of two well-known Ranking & Selection policies, namely, Optimal Computing Budget Allocation (OCBA) and Knowledge Gradient (KG) policy. Our extensions (Asynchronous Optimal Computing Budget Allocation (AOCBA) and Asynchronous Knowledge Gradient (AKG), respectively) allow for parallel asynchronous allocation of measurements. Additionally, since the standard KG method is sequential (it can only allocate one experiment at a time) we propose a parallel synchronous extension of KG policy - Synchronous Knowledge Gradient (SKG). Computer simulations of our algorithms indicate that our parallel KG-based policies (AKG, SKG) outperform the standard OCBA method as well as AOCBA, if the number of evaluated alternatives is small or the computing/experimental budget is limited. For experimentations with large budgets and big sets of alternatives, both the OCBA and AOCBA policies are more efficient. [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]

Nowadays, supplier selection process, a multicriteria decisionmaking problem, has become one of the most indispensable parts for every purchasing sector for the improvement of performances of business operations. Most of the literatures in this field have considered only the opinion of decision-makers. But in fact, each company has its own opinion about the suppliers. The purpose of this paper is to select the best supplier by integrating the opinions of both decision- makers and company's stake holders. In this literature, these opinions are taken as fuzzy soft sets. These two fuzzy soft sets are then integrated by the rough approximation theory. The attributes in this literature are taken in the form of linguistic variable. At the end of this paper, a case study is given to illustrate the proposed method for selecting the best supplier. [ABSTRACT FROM AUTHOR]

In this paper, trigonometrically fitted multi-step hybrid (TFMSH) methods for the numerical integration of oscillatory special second-order initial value problems are proposed and studied. TFMSH methods inherit the frame of multi-step hybrid (MSH) methods and integrate exactly the differential system whose solutions can be expressed as the linear combinations of functions from the set or equivalently the set , where w represents an approximation of the main frequency of the problem. The corresponding order conditions are given and two explicit TFMSH methods with order six and seven, respectively, are constructed. Stability of the new methods is examined and the corresponding regions of stability are depicted. Numerical results show that our new methods are more efficient in comparison with other well-known high quality methods proposed in the scientific literature. [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]

Let X = Lp(Ω,μ) (1M - TM in terms of the gap function. As an application of main results, we also investigate the best approximate solution problem of ill-posed operator equation. [ABSTRACT FROM AUTHOR]

In the present paper, the positivity of the second order of approximation of the difference operator approximates the second-order differential operator with nonlocal conditions is established. The equivalence of norms of these fractional spaces generated by this operator and Hölder spaces is proved. In applications, the stability of difference schemes for the approximate solution of the boundary value problem for elliptic differential equations is provided. [ABSTRACT FROM AUTHOR]

In the present paper, a time-dependent source identification problem subject to an integral overdetermination is considered. Stability estimates for this differential problem are established. Furthermore, a first and a second order of accuracy difference schemes are proposed for the approximate solution of this problem. Stability and almost coercive stability estimates for these difference schemes are established. Additionally, some illustrative numerical results are provided. [ABSTRACT FROM AUTHOR]

This paper addresses a daily caregiver scheduling and routing problem arising in home health care or home care service providers. The problem is quite challenging due to its uncertainties in terms of travel and service times derived from changes in road traffic conditions and customer health status in practice. We first model the problem as a stochastic programme with recourse, where the recourse action is to skip customers without services if the caregiver arrives later than their latest starting service time (i.e. hard time window requirements). Then, we formulate the problem as a set partitioning model and solve it with a branch-and-price (B&P) algorithm. Specifically, we devise an effective discrete approximation method to calculate the arrival time distribution of caregivers, incorporate it into a problem-specific label algorithm, and use a removal-and-insertion-based heuristic and the decremental state-space relaxation technique to accelerate the pricing process. Finally, we conduct numerical experiments on randomly generated instances to validate the effectiveness of the discrete approximation method and the proposed B&P algorithm. [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]

This article deals with locally optimal cutting positions and cutting directions for tool tip gouging avoidance in 5-axis sculptured surface machining. In order to measure the quality of tool positioning, this paper suggests a new concept of ‘machined region width’, which does not have the drawbacks of ‘machined strip width’. The method evaluates the optimal cutting position of a flat endmill or toroidal endmill, where optimality is with respect lo the avoidance of tool tip gouging. It is based on a second-order Taylor approximation of the design surface and multipoint tool positioning. The implementation and some illustrative examples are discussed. [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]

This paper derives transition and first hitting time densities and moments for the Ornstein–Uhlenbeck Process (OUP) between exponential thresholds. The densities are obtained by simplifying the process via Doob’s representation into Brownian motion between affine thresholds. The densities in this paper also offer easy-to-use and fast small-time approximations for the densities of OUP between constant thresholds given that exponential thresholds are virtually constant for a small time. This is of interest for estimation with high-frequency data given that extant approaches for constant thresholds impose a large demand on computing power. The moments of the transition distribution up to order n are derived within a closed-form recursive formula that offers valuable information for management. Expressions for the moments of the first hitting time distribution are also obtained in closed form by simplifying integrals via series expansions. [ABSTRACT FROM AUTHOR]

In this paper, we use the η-approximation method for a class of non-convex multiobjective variational problems with invex functionals. In this approach, for the considered multiobjective variational problem, the associated η-approximated multiobjective variational problem is constructed at the given feasible solution. The equivalence between (weakly) efficient solutions in the original multiobjective variational problem and its associated η-approximated multiobjective variational problem is established under invexity hypotheses. [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]

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]

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]