Numerical Analysis and Computational Mathematics

We introduce a new strategy in solving the truncated complex moment problem. To this aim we investigate recursive doubly indexed sequences and their characteristic polynomials. A characterization of recursive doubly indexed moment sequences is provided. As a simple application, we obtain a computable solution to the complex moment problem for cubic harmonic characteristic polynomials of the form z 3 + az + bz, where a and b are arbitrary real numbers. We also recapture a recent result due to Curto-Yoo given for cubic column relations in M (3) of the form Z 3 = itZ + uZ with t, u real numbers satisfying some suitable inequalities. Furthermore, we solve the truncated complex moment problem with column dependence relations of the form Z k+1 = 0≤n+m≤k anmZ n Z m (anm ∈ C).

Stock price movement prediction is challenging due to unpredictable fluctuations and the significant impact of market sentiment and news. Accurate prediction models can enhance investor decision-making and control over stock price movements. Creating a model for predicting high-accuracy stock price movements can improve investor control over stock prices. In this study, a wide range of technical indicators and various aspects of sentiment analysis in tweets were used to predict stock price movement. The impact of the maximum number of positive comments on Tesla stocks on price increases is investigated. Also, we proposed a method for adding sentiments to each tweet. Extracted advanced sentiment analysis features such as the number of positive comments, the number of negative comments, the average score of positive comments, the average score of negative comments, daily tweet volume, ratio of positive to negative tweets. Effect of time windows with variate size is investigate. A CNN-LSTM deep neural network is used to predict stock price movement and compared with LSTM and GRU models. According to the results, the proposed CNN-LSTM deep neural network has the best results to predict stock price movement over the 30-day interval.
INDEX TERMS: Long short-term memory (LSTM), convolutional neural network (CNN), gated recurrent unit (GRU), stock price movement prediction, sentiment analysis, Twitter.

We discuss a prototypical reaction-diffusion-flow problem in saturated/unsaturated porous media. The special features of our problem are: the reaction produces water and therefore the flow and transport are coupled in both directions and moreover, the reaction may alter the microstructure. This means we have a variable porosity in our model. For the spatial discretization we propose a mass conservative scheme based on the mixed finite element method (MFEM). The scheme is semiimplicit in time. Error estimates are obtained for some particular cases. We apply our finite element methodology for the case of concrete carbonation -one of the most important physico-chemical processes affecting the durability of concrete.

The Global Random Walk algorithm (GRW) performs a simultaneous tracking on a fixed grid of huge numbers of particles at costs comparable to those of a single-trajectory simulation by the traditional Particle Tracking (PT) approach. Statistical ensembles of GRW simulations of a typical advection-dispersion process in groundwater systems with randomly distributed spatial parameters are used to obtain reliable estimations of the input parameters for the upscaled transport model and of their correlations, input-output correlations, as well as full probability distributions of the input and output parameters.

We identify sufficient conditions under which evolution equations for probability density functions (PDF) of random concentrations are equivalent to Fokker-Planck equations. The novelty of our approach is that it allows consistent PDF approximations by densities of computational particles governed by Itô processes in concentration-position spaces. Accurate numerical solutions are obtained with a global random walk (GRW) algorithm, stable, free of numerical diffusion, and insensitive to the increase of the total number of computational particles. The system of Itô equations is specified by drift and diffusion coefficients describing the PDF transport in the physical space, provided by up-scaling procedures, as well as by drift and mixing coefficients describing the PDF transport in concentration spaces. Mixing models can be obtained similarly to classical PDF approaches or, alternatively, from measured or simulated concentration time series. We compare their performance for a GRW-PDF numerical solution to a problem of contaminant transport in heterogeneous groundwater systems.

An alternate formulation of the classical vehicle routing problem (VRP) is considered for distributing fresh potatoes from warehouse to the food outlet or shop. We propose a new-heuristic (metaheuristic) method to solve the problem, based on the Cross-Entropy method. In order to better estimate the objective function at each point in the domain, we incorporate Monte Carlo sampling. This creates many practical issues, especially the decision as to when to draw new samples and how many samples to use. We also develop a framework for obtaining exact solutions and tight lower bounds for the problem under various conditions by grouping the destination. This is used to assess the performance of the algorithm. Numerical results are presented for various problem instances to illustrate the ideas. Finally, solution obtained by Cross-Entropy compared with Branch and Bound Algorithm. It is proved that two methods have similar solution, with number of iteration was similar relatively.

An alternate formulation of the classical vehicle routing problem (VRP) is considered for distributing fresh potatoes from warehouse to the food outlet or shop. We propose a new-heuristic (metaheuristic) method to solve the problem, based on the Cross-Entropy method. In order to better estimate the objective function at each point in the domain, we incorporate Monte Carlo sampling. This creates many practical issues, especially the decision as to when to draw new samples and how many samples to use. We also develop a framework for obtaining exact solutions and tight lower bounds for the problem under various conditions by grouping the destination. This is used to assess the performance of the algorithm. Numerical results are presented for various problem instances to illustrate the ideas. Finally, solution obtained by Cross-Entropy compared with Branch and Bound Algorithm. It is proved that two methods have similar solution, with number of iteration was similar relatively.

We consider a spatially distributed population dynamics model with excitable predator-prey kinetics, where species propagate in space due to their taxis with respect to each other’s gradient in addition to, or instead of, their diffusive spread. Earlier, we have described new phenomena in this model in one spatial dimension, not found in analogous systems without taxis: reflecting and self-splitting waves. Here we identify new phenomena in two spatial dimensions: unusual patterns of meander of spirals, partial reflection of waves, swelling wave tips, attachment of free wave ends to wave backs, and as a result, a novel mechanism of self-supporting complicated spatiotemporal activity, unknown in reaction-diffusion population models.

Ventricular tachycardia and fibrillation are potentially lethal cardiac arrhythmias generated by high frequency, irregular spatio-temporal electrical activity. Re-entrant propagation has been demonstrated as a mechanism generating these arrhythmias in computational and in vitro animal models of these arrhythmias. Re-entry can be idealised in homogenous isotropic virtual cardiac tissues as spiral and scroll wave solutions of reaction-diffusion equations. A spiral wave in a bounded medium can be terminated if its core reaches a boundary. Ventricular tachyarrhythmias in patients are sometimes observed to spontaneously self-terminate. One possible mechanism for self-termination of a spiral wave is meander of its core to an inexcitable boundary. We have previously proposed the hypothesis that the spatial extent of meander of a re-entrant wave in the heart can be directly related to its probability of self-termination, and so inversely related to its lethality. Meander in two-dimensional ...

Discrete analogs of the classical Kontorovich-Lebedev transforms are introduced and investigated. It involves series with the modified Bessel function or Macdonald function K in (x), x > 0, n ∈ N, i is the imaginary unit, and incomplete Bessel functions. Several expansions of suitable functions and sequences in terms of these series and integrals are established. As an application, a Dirichlet boundary value problem in the upper half-plane for inhomogeneous Helmholtz equation is solved.

In this work, we present a fractional model of Cahn-Allen equation associated with newly introduced Atangana-Baleanu (AB) derivative of fractional order which uses Mittag-Leffler function as the nonsingular and non-local kernel. The existence and uniqueness of this modified fractional model are discussed by employing the fixed-point postulate. An efficient scheme homotopy perturbation transform technique (HPTT) which is an amalgamation of homotopy perturbation technique with Laplace transform is used to examine this time-fractional phase-field model numerically. Also, convergence and error analysis of the proposed technique is presented. The numerical simulations are analyzed graphically as well as in tabulated form.

We develop space-time adaptive and high-order methods for valuing American options using a partial differential equation (PDE) approach. The linear complementarity problem arising due to the free boundary is handled by a penalty method. Both finite difference and finite element methods are considered for the space discretization of the PDE, while classical finite differences, such as Crank-Nicolson, are used for the time discretization. The high-order discretization in space is based on an optimal finite element collocation method, the main computational requirements of which are the solution of one tridiagonal linear system at each time step, while the resulting errors at the gridpoints and midpoints of the space partition are fourth-order. To control the space error, we use adaptive gridpoint distribution based on an error equidistribution principle. A time stepsize selector is used to further increase the efficiency of the methods. Numerical examples show that our methods converge fast and provide highly accurate options prices, Greeks, and early exercise boundaries.

We consider Quadratic Spline Collocation (QSC) methods for linear second order elliptic Partial Differential Equations (PDEs). The standard Cannulation of these methods leads to non-optimal approximations. In order (0 derive optimal QSC approximations, high order perturbations of the PDE problem arc generated. These perturbations can be applied either to die POE problem operators or to the right sides, thus leading 10 two different fonnulations of optimal QSC methods. The convergence properties of the QSC methods arc studied. Optimal 0 (h J-j) global error estimates for the j.th partial derivative are obtained for a certain class of problems. Moreover, O(h4-j ) error bounds for the j-th partial derivative are obtained at certain sets of points. Results from numerical experiments verify the theoretical behaviour of Lhe QSC methods. Performance results also show that the QSC methods arc very effective from the computational point of view. The QSC methods have been implemented efficiently on parallel machines.

Since the beginnings of computational science, iterative methods were essential tools in the solution of nonlinear problems, such as nonlinear systems of equations and eigenvalue computations. It was also soon recognized that some of the most efficient linear solvers were based on iterative methods. Moreover, iterative methods are in the core of optimization algorithms. Throughout the last half a century, iterative methods have been the subject of intense research and development, and have been attracting increasing interest by scientists and engineers. The need of solving larger and larger problems, that were deemed to be infeasible in the past, gave a spark to the development of highperformance and parallel solution techniques. Iterative methods are in general much more suited for massively parallel computation than direct methods. New areas, such as domain decomposition, have developed alongside parallel iterative techniques. Preconditioning techniques incorporated into iterative methods allow the solution of very large sparse linear systems arising from multidimensional and other large-scale problems. New emerging application areas, such as computational biology, computational engineering, financial engineering create new opportunities and offer new challenges for iterative methods. About ninety participants from all over the world gathered in Toronto for the seventh IMACS international symposium on iterative methods in scientific computing, May 5-8, 2005. The symposium provided a forum to present and discuss the use of iterative methods in scientific computing problems, with emphasis in recent developments, future research directions, and new application areas. The symposium featured eight invited talks, seven minisymposia with 3-4 talks each, about 30 contributed talks, as well as 15 student papers, taking part in the student paper competition. Topics spanned from core iterative methods and eigenvalues to preconditioning, multigrid and parallel computation, as well as optimization, nonlinear equations and differential equations. Applications such as data and web mining, image processing, computational finance and computational fluid dynamics received special attention. The symposium received funding from the Fields Institute, the Faculty of Mathematics at the University of Waterloo and MITACS, so that 13 requests for funding by students and postdoctoral fellows were at least partially satisfied, and five awards were given to student paper competition winners. The Bahen Centre for Information Technology (BCIT) at the University of Toronto served as hosting venue. The high-ceiling hallways of BCIT were the ideal place for informal discussions and breaks, while the projectors in the rooms booked for the conference were working full-time during scientific sessions. Many participants indicated their appreciation of the high quality of the talks and interaction. On the social front, participants enjoyed the hospitality of downtown Chinatown and of the Greek village by the Danforth, as well as the ethnic diversity and colour of Baldwin street. Photos from the conference are found in and cc/imacs05/Photos1/ courtesy of David Kincaid and Omar Halabieh. This special issue of the Applied Numerical Mathematics journal is sponsored by IMACS and contains eleven selected papers presented at the conference. Among the eighteen papers submitted, these were the papers accepted for publication after the refereeing process. The range of topics covered by these papers is wide, spanning from core

The second author introduced with I. Törmä a two-player word-building game [Playing with Subshifts, Fund. Inform. 132 (2014), 131-152]. The game has a predetermined (possibly finite) choice sequence α 1 , α 2 , . . . of integers such that on round n the player A chooses a subset S n of size α n of some fixed finite alphabet and the player B picks a letter from the set S n . The outcome is determined by whether the word obtained by concatenating the letters B picked lies in a prescribed target set X (a win for player A) or not (a win for player B). Typically, we consider X to be a subshift. The winning shift W(X) of a subshift X is defined as the set of choice sequences for which A has a winning strategy when the target set is the language of X. The winning shift W(X) mirrors some properties of X. For instance, W(X) and X have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts generated by marked uniform substitutions, and show that these winning shifts, viewed as subshifts, also have a substitutive structure. Particularly, we give an explicit description of the winning shift for the generalized Thue-Morse substitutions. It is known that W(X) and X have the same factor complexity. As an example application, we exploit this connection to give a simple derivation of the first difference and factor complexity functions of subshifts generated by marked substitutions. We describe these functions in particular detail for the generalized Thue-Morse substitutions.

We consider expansive group actions on a compact metric space containing a special fixed point denoted by [Formula: see text], and endomorphisms of such systems whose forward trajectories are attracted toward [Formula: see text]. Such endomorphisms are called asymptotically nilpotent, and we study the conditions in which they are nilpotent, that is, map the entire space to [Formula: see text] in a finite number of iterations. We show that for a large class of discrete groups, this property of nil-rigidity holds for all expansive actions that satisfy a natural specification-like property and have dense homoclinic points. Our main result in particular shows that the class includes all residually finite solvable groups and all groups of polynomial growth. For expansive actions of the group [Formula: see text], we show that a very weak gluing property suffices for nil-rigidity. For [Formula: see text]-subshifts of finite type, we show that the block-gluing property suffices. The study o...

The second author introduced with I. Törmä a two-player word-building game [Fund. Inform. 132 (2014) 131–152]. The game has a predetermined (possibly finite) choice sequence α1, α2, … of integers such that on round n the player A chooses a subset Sn of size αn of some fixed finite alphabet and the player B picks a letter from the set Sn. The outcome is determined by whether the word obtained by concatenating the letters B picked lies in a prescribed target set X (a win for player A) or not (a win for player B). Typically, we consider X to be a subshift. The winning shift W(X) of a subshift X is defined as the set of choice sequences for which A has a winning strategy when the target set is the language of X. The winning shift W(X) mirrors some properties of X. For instance, W(X) and X have the same entropy. Virtually nothing is known about the structure of the winning shifts of subshifts common in combinatorics on words. In this paper, we study the winning shifts of subshifts genera...

The dynamical motion of laser-micromachined copper springs used for a meso-scale vibration-based power generator was successfully modeled using ANSYS to reveal 3 modes of multi-directional vibratory motion due to a pure vertical input vibration. A MATLAB simulation was also used to predict the voltage output of the micro power generator system with coupled electrical and mechanical damping effects. The simulated output matched experimental results closely. These capabilities are essential for the successful design and development of a miniature, low-frequency, and robust micro energy generator that could be potentially used to convert human mechanical energy into usefully electrical power to operate devices such as mobile phones and heart-pacers. Thus far, 1cm 1 meso-scale generators are demonstrated capable of producing up to 4V AC with instantaneous peak power of 80mW, at input frequencies ranging from 60 to 120Hz with ~200μηι input vibration amplitude. A generator capable of producing 2V DC output with 40μ\ν power after voltage rectification, and able to drive a commercial infrared wireless signal transmitter to send 140ms pulse trains with ~60sec power generation time was also demonstrated. The ANSYS model and MATLAB simulation results are presented and compared with the experimental results in this paper.

For sequentially monitoring and controlling average and variability of an online manufacturing process, x¯ and s control charts are widely utilized tools, whose constructions require the data to be real (precise) numbers. However, many quality characteristics in practice, such as surface roughness of optical lenses, have been long recorded as fuzzy data, in which the traditional x¯ and s charts have manifested some inaccessibility. Therefore, for well accommodating this fuzzy-data domain, this paper integrates fuzzy set theories to establish the fuzzy charts under a general variable-sample-size condition. First, the resolution-identity principle is exerted to erect the sample-statistics’ and control-limits’ fuzzy numbers (SSFNs and CLFNs), where the sample fuzzy data are unified and aggregated through statistical and nonlinear-programming manipulations. Then, the fuzzy-number ranking approach based on left and right integral index is brought to differentiate magnitude of fuzzy numbe...

Transfer entropy (TE) captures the directed relationships between two variables. Partial transfer entropy (PTE) accounts for the presence of all confounding variables of a multivariate system and infers only about direct causality. However, the computation of partial transfer entropy involves high dimensional distributions and thus may not be robust in case of many variables. In this work, different variants of the partial transfer entropy are introduced, by building a reduced number of confounding variables based on different scenarios in terms of their interrelationships with the driving or response variable. Connectivity-based PTE variants utilizing the random forests (RF) methodology are evaluated on synthetic time series. The empirical findings indicate the superiority of the suggested variants over transfer entropy and partial transfer entropy, especially in the case of high dimensional systems. The above findings are further highlighted when applying the causality measures on...

This study presents a comprehensive statistical analysis exploring the effects of Initial Public Offerings (IPOs) on the financial performance of Saudi Arabian companies. Utilizing advanced statistical techniques, the research examines key financial metrics-such as profitability, liquidity, leverage, and stability-before and after IPO events. Conducted by Dr. Mohamed Baha Eldin and Dr. Ibrahim Abu El-Azm, both distinguished statisticians from Cairo University, this rigorous analysis offers valuable insights into the financial dynamics of IPOs in an emerging market context. For expert statistical solutions and cutting-edge research, Dr.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Longest edge (nested) algorithms for triangulation refinement in two dimensions are able to produce hierarchies of quality and nested irregular triangulations as needed both for adaptive finite element methods and for multigrid methods. They can be formulated in terms of the longest edge propagation path (Lepp) and terminal edge concepts, to refine the target triangles and some related neighbors. We discuss a parallel multithread algorithm, where every thread is in charge of refining a triangle t and its associated Lepp neighbors. The thread manages a changing Lepp(t) (ordered set of increasing triangles) both to find a last longest (terminal) edge and to refine the pair of triangles sharing this edge. The process is repeated until triangle t is destroyed. We discuss the algorithm, related synchronization issues, and the properties inherited from the serial algorithm. We present an empirical study that shows that a reasonably efficient parallel method with good scalability was obtained.

We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let A be the positive symmetric matrix taking the outer ellipsoid to the inner one. If tr A = 1, there exists a bijection between the orthogonal group O(n) and the set of such labeled simplices. Similarly, if tr A 2 = 1, there are families of parallelotopes and of cross polytopes, also indexed by O(n).

We compute the spectra of the adjacency matrices of the semi-regular polytopes. A few different techniques are employed: the most sophisticated, which relates the 1-skeleton of the polytope to a Cayley graph, is based on methods akin to those of Lovász and Babai ([L], [B]). It turns out that the algebraic degree of the eigenvalues is at most 5, achieved at two 3-dimensional solids.

We present an original construction inspired by Legendre's conjecture, which ensures the existence of at least one prime number in each interval [n 2 , (n+1) 2 ] for all n ∈ N. By defining a filtering function k(m, n) = mn+ m + n, we identify and eliminate a subset of composite numbers in each interval. We demonstrate that the remaining numbers, when incremented by 1, always contain at least one prime. We provide a mathematical framework and computational experiments supporting this approach and highlight its consistency with prime distribution.

Sparse model identification by means of data is especially cumbersome if the sought dynamics live in a high dimensional space. This usually involves the need for large amount of data, unfeasible in such a high dimensional settings. This well-known phenomenon, coined as the curse of dimensionality, is here overcome by means of the use of separate representations. We present a technique based on the same principles of the Proper Generalized Decomposition that enables the identification of complex laws in the low-data limit. We provide examples on the performance of the technique in up to ten dimensions.

This paper introduces the Imaginary Laplace Transform (ImLT), a novel mathematical framework for analyzing systems governed by recursive imaginary coefficients. By extending the classical Laplace transform into a higher-order imaginary domain-through units such as √-i-we propose a structure for capturing deeply nested oscillatory behaviors. These behaviors appear in theoretical systems where conventional tools break down. We define the ImLT, explore its convergence, and introduce its inverse, paving the way for a new class of differential system analysis.

Let X0 and X1 be two order continuous Banach function spaces on a finite measure space, (E0, E1) a Banach space interpolation pair, and T : X0 + X1 → E0 + E1 an admissible operator between the pairs (X0, X1) and (E0, E1). is the interpolated operator by the first complex method of Calderón and m0, m1 and m θ are the vector measures coming from T | X 0 and T | X 1 and T θ , respectively, then we study the relationship between the optimal domain L 1 (m θ ) of T θ and the complex interpolation space [L 1 (m0), L 1 (m1)] [θ] of the optimal domains of T | X 0 and T | X 1 . Then, we apply the obtained result to study interpolation of p-th power factorable and bidual (p, q)power-concave operators.

In the current study, we design a new computational method to solve a class of Liénard's equations. This equation originates from advancements in radio and vacuum tube technology. To attain the proposed goal, we develop a method using a three-layer artificial neural network, consisting of an input layer, a hidden layer, and an output layer. We use the Morgan-Voyce even Fibonacci polynomials and sinh function as activation functions for the hidden layer and the output layer, respectively. Then, the neural network is trained using a classical optimization method. Finally, we analyze four examples using graphs and tables to demonstrate the accuracy and effectiveness of the numerical approach.

In this work, the stability results for a nonlinear mathematical model are derived, and the power system is realized by utilizing fractional calculus theory. The fixed point theorem is used to establish sufficient conditions for the existence of a mild solution and the stability of a nonlinear impulsive fractional stochastic integro-differential equation with state-dependent delays with Mainardi's function in a Hilbert space. Numerical simulations are provided to validate the obtained theoretical results. The proposed *Corresponding author

Particle swarm optimization (PSO) is a widely recognized bio-inspired algorithm for systematically exploring solution spaces and iteratively identifying optimal points. Through updating local and global best solutions, PSO effectively explores the search process, enabling the discovery of the most advantageous outcomes. This study proposes a novel Smith chartbased particle swarm optimization to solve convex and nonconvex multiobjective engineering problems by representing complex plane values in *Corresponding author

In this study, Allee type, single-species (prey), two-patch model with nonlinear harvesting rate, and species migration across two patches have been developed and analyzed. As we all know, the population of any species in an ecosystem is greatly dependent on the carrying capacity of the corresponding ecosystem; the main focus of our work is on how carrying capacity affects system dynamics in the presence and absence of randomness (deterministic and stochastic case, respectively). In the deterministic case,

This study presents the process of using extrapolation methods to solve the nonlinear Volterra-Fredholm integral equations of the second kind. To do this, by approximating the integral terms contained in equations by a quadrature rule, the nonlinear Volterra-Fredholm integral equations of the *Corresponding author

This study presents the process of using extrapolation methods to solve the nonlinear Volterra-Fredholm integral equations of the second kind. To do this, by approximating the integral terms contained in equations by a quadrature rule, the nonlinear Volterra-Fredholm integral equations of the *Corresponding author

The purpose of this study is the derivation of a closed-form formula for Green's function in elliptic coordinates that could be used for achieving an analytic solution for the second-order diffraction problem by elliptical cylinders subjected to monochromatic incident waves. In fact, Green's function represents the solution of the so-called locked wave component of the second-order velocity potential. The mathematical analysis starts with a proper analytic formulation of the second-order diffraction potential that results in the inhomogeneous Helmholtz equation. The associated boundary-value problem is treated by applying Green's theorem to obtain a closed-form solution for Green's function. Green's function is initially expressed in polar coordinates while its final elliptic form is produced through the proper employment of addition theorems.

Mathematical modeling and optimal control of customer's behavior ... but also fosters global connectivity and economic restructuring. However, despite its critical role in the global economy, e-commerce faces challenges, notably the hesitation of some consumers due to concerns about security and trust. To address this, we propose a novel mathematical model to examine customer behavior dynamics toward e-commerce, particularly the impact of the refusal behavior. Our study comprehensively examines the characteristics of our mathematical model, conducts a thorough stability analysis, and investigates the parameter sensitivity. Furthermore, control theory has been adopted to optimize the adoption of e-commerce using Pontryagin's maximum principle, with numerical simulations to evaluate the effectiveness of our proposed strategies.

This research presents a nonuniform Haar wavelet approximation of a singularly perturbed convection-diffusion problem with an integral boundary. The problem is discretized by approximating the second derivative of the solution with the help of a nonuniform Haar wavelets basis on an arbitrary nonuniform mesh. To resolve the multiscale nature of the problem,

Sinc numerical methods are essential approaches for solving nonlinear problems. In this work, based on this method, the sinc neural networks (SNNs) are designed and applied to solve the fractional optimal control problem (FOCP) in the sense of the Riemann-Liouville (RL) derivative. To solve the FOCP, we first approximate the RL derivative using Grunwald-Letnikov operators. Then, according to Pontryagin's minimum principle for FOCP and using an error function, we construct an unconstrained minimization problem. We approximate the solution of the ordinary differential equation obtained from the Hamiltonian condition using the SNN. Simulation results show the efficiencies of the proposed approach.

These days Mandelbrot set with transcendental function is an interesting area for mathematicians. New equations have been created for Mandelbrot set using trigonometric, logarithmic and exponential functions. Earlier, Ishikawa iteration has been applied to these equations and generate new fractals named as Relative Superior Mandelbrot Set with transcendental function. In this paper, the Mann iteration is being applied on Mandelbrot set with sine function i.e. sin(z n )+c and new fractals with the concept of Superior Transcendental Mandelbrot Set will be shown. Our goal is to focus on the less number of iterations which are required to obtain fixed point of function sin(z n )+c.

The Rubik's Cube is a complex combinatorial puzzle that has challenged researchers and enthusiasts alike. Traditional solving algorithms rely on predefined heuristics or brute-force searches, which can be computationally expensive or suboptimal. This paper presents an advanced genetic algorithm (GA) framework for discovering efficient solution sequences to the Rubik's Cube. By encoding cube states and move sequences as chromosomes, we use evolutionary principles such as selection, crossover, and mutation to evolve efficient solutions. Experimental results demonstrate improved performance over standard methods, offering both optimization potential and novel insights into heuristic-free cube solving.

Given a data set arising from a series of observations, an outlier is a value that deviates substantially from the natural variability of the data set as to arouse suspicions that it was generated by a different mechanism. We call an observation an extreme outlier if it lies at an abnormal distance from the "center" of the data set. We introduce the Monte Carlo SCD algorithm for detecting extreme outliers. The algorithm finds extreme outliers in terms of a subset of the data set called the outer shell. Each iteration of the algorithm is polynomial. This could be reduced by preprocessing the data to reduce its size. This approach has an interesting new feature. It estimates a relative measure of the degree to which a data point on the outer shell is an outlier (its "outlierness"). This measure has potential for serendipitous discoveries in data mining where unusual or special behavior is of interest. Other applications include spatial filtering and smoothing in digital image processing. We apply this method to baseball data and identify the ten most exceptional pitchers of the 1998 American League. To illustrate another useful application, we also show that the SCD can be used to reduce the solution time of the D-optimal experimental design problem.

We reduce the size of large semidefinite programming problems by identifying necessary linear matrix inequalities (LMI's) using Monte Carlo techniques. We describe three algorithms for detecting necessary LMI constraints that extend algorithms used in linear programming to semidefinite programming. We demonstrate that they are beneficial and could serve as tools for a semidefinite programming preprocessor. A necessary LMI is one whose removal changes the feasible region defined by all the LMI constraints. The general problem of checking whether or not a particular LMI is necessary is NPcomplete. However, the methods we describe are polynomial in each iteration, and the number of iterations can be limited by stopping rules. This provides a practical method for reducing the size of some large Semidefinite Programming problems before one attempts to solve them. We demonstrate the applicability of this approach to solving instances of the Löwner ellipsoid problem. We also consider the problem of classification of all the constraints of a semidefinite programming problem as redundant or necessary.

This paper presents a new modification of Harmony Search (HS) algorithm to improve its accuracy and convergence speed and eliminates setting parameters that have to be defined before optimization process and it is difficult to predict fixed values for all kinds of problems. The proposed algorithm is named Global Dynamic Harmony Search (GDHS). In this modification, all the key parameters are changed to dynamic mode and there is no need to predefine any parameters; also the domain is changed to dynamic mode to help a faster convergence. Two experiments, with large sets of benchmark functions, are executed to compare the proposed algorithms with other ones. In the first experiment, 15 benchmark problems are used to compare the proposed algorithm with other similar algorithms based on the Harmony Search method and in the second experiment, 47 benchmark problems are used to compare the performance of the GDHS with other algorithms from different families, including: GA, PSO, DE and ABC algorithms. Results showed that the proposed algorithm outperforms the other algorithms, considering the point that the GDHS does not require any predefined parameter.