N-dimensional Problem Research Articles

Positive and nodal solutions bifurcating from the infinity for a semilinear equation: solutions with compact support

The countable branches of nodal solutions bifurcating from the infinity for a sublinear semilinear equation are described with two different approaches. In the one-dimensional case we use plane phase methods of ordinary differential equations. The general N-dimensional problem can be studied by using topological methods and we sketch here some previous results by the second author in collaboration with João-Paulo Dias. One of the main motivations of the present paper was the ambiguity of the mathematical treatment of the Schrödinger equation for the infinite well potential. We study the classes of flat solutions (i.e. with zero normal derivative at the boundary) and solutions with compact support of the semilinear problem which allow to offer a kind of "alternative approach" to the infinite well potential for the Schrödinger equation.

Bat Algorithm: A Survey of the State-of-the-Art

The area of metaheuristic optimization algorithms has been attracting researchers for many years. These algorithms have built in capability to explore a large region of the solution space, are computationally robust, efficient and can avoid premature convergence. They have been extensively tested and applied on many hard optimization problems where conventional computing techniques perform unsatisfactorily. They are capable of solving general N-dimensional, linear, nonlinear and complex global optimization problems. One of the latest entrants in this field is the Bat algorithm which is based on the echolocation behaviour of bats. It has been proven to have good convergence properties on different benchmark functions and seems promising for dealing with optimization problems. The aim of this paper is to provide a survey of the state of the art on Bat algorithm. A concise effort has been made so that the readers get a rapid insight into some of the applications upon which bat algorithm has been applied till date in specialized fields of science and engineering. Some of the variants of the bat algorithm as reported in the literature have also been discussed.

On the free terms of the dual BIE for N-dimensional Laplace problems

Dual boundary integral equations for the N-dimensional Laplace problems with a smooth boundary are derived by using the contour approach surrounding the singularity. The potentials resulted from the four kernel functions in the dual formulation have different properties across the smooth boundary. For the generalization, we focus on the N-dimensional Laplace equation. The Hadamard principal value (H.P.V.) is derived naturally and is composed of two parts, the Cauchy principal value (C.P.V.) and an unbounded boundary term. The hypersingular integral is not a divergent integral since we can collect the C.P.V. and the unbounded term together. Besides, the weighting of the free term contributed by different kernels is also examined. Finally, a special case of the four-dimensional Laplace equation is implemented and the free term, for any dimension are obtained. The contributions of the free terms for the boundary normal derivative of potential due to the single (L kernel) and the double (M kernel) layer potentials are 1/N and (N−1)/N, respectively. It is an interesting phenomenon that the hypersingular kernel contributes more than the singular kernel, and, in addition, the former also yields an unbounded boundary term.

Approximation of solutions to multidimensional parabolic equations by approximate approximations

We propose a fast method for high order approximations of the solution of n-dimensional parabolic problems over hyper-rectangular domains in the framework of the method of approximate approximations. This approach, combined with separated representations, makes our method effective also in very high dimensions. We report on numerical results illustrating that our formulas are accurate and provide the predicted approximation rate 6 also in high dimensions.

A real options perspective on the future of the Euro

A break-up of the Eurozone is no longer regarded as implausible. This will be a costly and irreversible decision in conditions of continuing uncertainty; therefore it is amenable to analysis in the real options framework. We do so by solving as an n-dimensional optimal stopping problem with country-specific shocks and “convergence” of member economies. We compare a complete break-up with individual country departures. In calibrated solutions for a symmetric case we find a non-negligible but small option value. Furthermore, we find a new theoretical result on the non-monotonicity of abandonment threshold with respect to volatility.

Efficient computing of n-dimensional simultaneous Diophantine approximation problems

Abstract In this paper we consider two algorithmic problems of simultaneous Diophantine approximations. The first algorithm produces a full solution set for approximating an irrational number with rationals with common denominators from a given interval. The second one aims at finding as many simultaneous solutions as possible in a given time unit. All the presented algorithms are implemented, tested and the PariGP version made publicly available.

Diffraction problems for quasilinear parabolic systems with boundary intersecting interfaces

In this paper, we discuss the n-dimensional diffraction problem for weakly coupled quasilinear parabolic system on a bounded domain Ω, where the interfaces ( ) are allowed to intersect with the outer boundary ∂ Ω and the coefficients of the equations are allowed to be discontinuous on the interfaces. The aim is to show the existence of solutions by approximation method. The approximation problem is a diffraction problem with interfaces, which do not intersect with ∂ Ω. MSC:35R05, 35K57, 35K65.

Optimisation of antenna arrays using the cuckoo search algorithm

The goal of this article is to introduce and use the cuckoo search (CS) as an optimisation algorithm for the electromagnetics and antenna community. The CS is a new nature-inspired evolutionary algorithm (EA) for solving N-dimensional optimisation problems. Compared with other nature-inspired algorithms, the CS algorithm is easy to understand and implement and has minimum number of parameters to tune. Different examples are presented that illustrate the use of the CS algorithm, and the results are compared with results obtained using other optimisation methods. Preliminary results suggest that the CS algorithm can in some cases outperform other EAs, at least for the examples studied in this article.

Improved Evolutionary Extreme Learning Machines Based on Particle Swarm Optimization and Clustering Approaches

Extreme Learning Machine (ELM) is a new learning method for single-hidden layer feedforward neural network (SLFN) training. ELM approach increases the learning speed by means of randomly generating input weights and biases for hidden nodes rather than tuning network parameters, making this approach much faster than traditional gradient-based ones. However, ELM random generation may lead to non-optimal performance. Particle Swarm Optimization (PSO) technique was introduced as a stochastic search through an n-dimensional problem space aiming the minimization (or the maximization) of the objective function of the problem. In this paper, two new hybrid approaches are proposed based on PSO to select input weights and hidden biases for ELM. Experimental results show that the proposed methods are able to achieve better generalization performance than traditional ELM in real benchmark datasets.

The exact solutions for the interaction V(r) = αr 2d − 2 − βr d − 2 by Nikiforov–Uvarov method

The exact solutions for the two- and N-dimensional Schrodinger equation have been rederived for the potential V(r) = αr2d − 2 − βrd − 2 by Nikiforov–Uvarov method. Specific results are presented for (i) the hydrogen atom and (ii) an isotropic harmonic oscillator. The dimensionality of the problem is seen to enter into these relations in such a way that one can immediately verify the corresponding three-dimensional results. The local accidental degeneracies are also explained for the two- and N-dimensional problems.

Wavelet Shrinkage With Consistent Cycle Spinning Generalizes Total Variation Denoising

We introduce a new wavelet-based method for the implementation of Total-Variation-type denoising. The data term is least-squares, while the regularization term is gradient-based. The particularity of our method is to exploit a link between the discrete gradient and wavelet shrinkage with cycle spinning, which we express by using redundant wavelets. The redundancy of the representation gives us the freedom to enforce additional constraints (e.g., normalization) on the solution to the denoising problem. We perform optimization in an augmented-Lagrangian framework, which decouples the difficult n-dimensional constrained-optimization problem into a sequence of n easier scalar unconstrained problems that we solve efficiently via traditional wavelet shrinkage. Our method can handle arbitrary gradient-based regularizers. In particular, it can be made to adhere to the popular principle of least total variation. It can also be used as a maximum a posteriori estimator for a variety of priors. We illustrate the performance of our method for image denoising and for the statistical estimation of sparse stochastic processes.

Numerical solution of the variational PDEs arising in optimal control theory

An iterative method based on Picard's approach to ODEs' initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examples arising from optimal control problems for nonlinear systems and regular Lagrangians in finite dimension, and against the numerical solution obtained through standard mathematical software. An application to the (n + 1)-dimensional variational PDEs associated with the n-dimensional finite-horizon time-variant linear-quadratic problem is discussed, due to the key role the LQR plays in two-degrees-of freedom control strategies for nonlinear systems with generalized costs. Mathematical subject classification: Primary: 35F30; Secondary: 93C10.

Asymptotic behaviour of singular points of solutions of the problem of heavy n-dimensional body motion in the Lagrange case

The single-valuedness of solutions to the Lagrange case of a heavy n-dimensional body motion problem is proved. All entire solutions are found.Bibliography: 23 titles.

The construction of numerical integration rules of degree three for product regions

Numerical integration formulas in n-dimensional Euclidean space of degree three are discussed. In this paper, for the product regions a method is presented to construct numerical integration formulas of degree three with 2 n real points and positive weights. The presented problem is a little different from those dealt with by other authors. All the corresponding one-dimensional integrals can be different from each other and they are also nonsymmetrical. In this paper an n-dimensional numerical integration problem is turned into n one-dimensional moment problems, which simplifies the construction process. Some explicit numerical formulas are given. Furthermore, a more generalized numerical integration problem is considered, which will shed light on the final solution to the third degree numerical integration problem.

Time-dependent density-functional and reduced density-matrix methods for few electrons: Exact versus adiabatic approximations

To address the impact of electron correlations in the linear and non-linear response regimes of interacting many-electron systems exposed to time-dependent external fields, we study one-dimensional (1D) systems where the interacting problem is solved exactly by exploiting the mapping of the 1D N-electron problem onto an N-dimensional single electron problem. We analyze the performance of the recently derived 1D local density approximation as well as the exact-exchange orbital functional for those systems. We show that the interaction with an external resonant laser field shows Rabi oscillations which are detuned due to the lack of memory in adiabatic approximations. To investigate situations where static correlations play a role, we consider the time-evolution of the natural occupation numbers associated to the reduced one-body density matrix. Those studies shed light on the non-locality and time-dependence of the exchange and correlation functionals in time-dependent density and density-matrix functional theories.

A MultiCommodity Grid Logistics Model for Data Transportation in Federated Datagrid

Data Sharing is the process of making data resources available across geographically distributed institutions working on a common purpose. When there is need or request, these data resources are transported so as to co-allocate them at desired destinations. Distributed computing, particularly Grid computing environment involves transporting huge volume of data across geographically spread sources and destinations managed by an efficient data management architecture called the Datagrid. This affinity towards movement of data is to have the data and task collocated to eliminate network traffic and to improve performance by reducing latency. Participants of the datagrid are autonomous organizations, organized in a federated fashion that can request for resources from the federation as long as they are authentic and authorized and can fall into or fall out of the grid at their own will. Data Transportation mechanism in grid can be compared to the Multi-index transportation problem i.e. an n-dimensional problem (here n = 3) and usually referred to as multi-commodity transportation problem with an additional index for handling multiple resource types. This work proposes a novel method to transport resources taking various challenges posed by the dynamic grid environment into account.

Self-organizing maps and boundary effects: quantifying the benefits of torus wrapping for mapping SOM trajectories

In this study the impact of a planar and toroidal self-organizing map (SOM) configuration are investigated with respect to their impact on SOM trajectories. Such trajectories are an encoding of processes within an n-dimensional input data set and offer an important means of visualizing and analyzing process complexity in large n-dimensional problem domains. However, discontinuity associated with boundaries in the standard, planar SOM results in error that limits their analytical use. Previous studies have recommended the use of a toroidal SOM to reduce these errors, but fall short of a fully quantified analysis of the benefits that result. In this study, the comparative analysis of fifteen pairs of identically initiated and trained SOMs, of planar and toroidal configuration, allows the error in trajectory magnitude to be quantified and visualized; both within the SOM and data space. This offers an important insight into the impact of planar SOM boundaries that goes beyond the general, statistical measures of clustering efficacy associated with previous work. The adoption of a toroidal SOM can be seen to improve the distribution of error in the trajectory sets, with the specific spatial configuration of SOM neurons associated with the largest errors changing from those at the corners of the planar SOM to a more complex and less predictable pattern in the toroidal SOM. However, this improvement is limited to the smallest 60% of errors, with torus and planar SOMs performing similarly for the largest 40%.

The method of upper and lower solutions for diffraction problems of quasilinear elliptic reaction–diffusion systems

The aim of this paper is to show the existence of solutions of the n-dimensional diffraction problem for weakly coupled quasilinear elliptic reaction–diffusion system. The coefficients of the equations under consideration are allowed to be discontinuous. We extend the method of upper and lower solutions for reaction–diffusion equations with continuous coefficients to the elliptic diffraction problem. An application of these results is given to the steady-state problem of Lotka–Volterra cooperation model with two cooperating species.

Ant Algorithm for AP-N Aimed at Optimization of Complex Systems

Assignment Problem (AP), which is well known combinatorial problem, has been studied extensively in the course of many operational and technical researches. It has been shown to be NP-hard for three or more dimensions and a few non-deterministic methods have been proposed to solve it. This paper pays attention on new heuristic search method for the n-dimensional assignment problem, based on swarm intelligence and comparing results with those obtained by other scientists. It indicates possible direction of solutions of problems and presents a way of behaviour using ant algorithm for multidimensional optimization complex systems. Results of researches in the form of computational simulations outcomes are presented.

A level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration

We present a level set based shape and topology optimization method for maximizing the simple or repeated first eigenvalue of structure vibration. Considering that a simple eigenvalue is Frechet differentiable with respect to the boundary of a structure but a repeated eigenvalue is only Gateaux or directionally differentiable, we take different approaches to derive the boundary variation that maximizes the first eigenvalue. In the case of simple eigenvalue, material derivative is obtained via adjoint method, and variation of boundary shape is specified according to the steepest descent method. In the case of N-fold repeated eigenvalue, variation of boundary shape is obtained as a result of a N-dimensional algebraic eigenvalue problem. Constraint of a structure's volume is dealt with via the augmented Lagrange multiplier method. Boundary variation is treated as an advection velocity in the Hamilton---Jacobi equation of the level set method for changing the shape and topology of a structure. The finite element analysis of eigenvalues of structure vibration is accomplished by using an Eulerian method that employs a fixed mesh and ersatz material. Application of the method is demonstrated by several numerical examples of optimizing 2D structures.