Narrow Results

Parallel and vector processing algorithms of Preconditioned Iterative methods for solving sparse linear systems have been studied for the efficient use of multivector computers. Several 3D orderings, satisfying the properties of compatibility and partial compatibility [6], are introduced and implemented, using the Nested Dissection technique [11]. These preconditioners, as well as the Tridiagonal Factorization preconditioner [5], are realized on the Cray-2 using the microtasking facility. Results of numerical experiments are described to characterize these algorithms on the Cray-2.

Solving large, linear systems is among the most important and most frequently encountered problems in computational mathematics and computer science. This paper presents efficient parallel Jacobi and Gauss-Seidel algorithms, in spite of the apparent inherent sequentiality of the latter, for the iterative solution of large linear systems on hypercube machines. To evaluate their performance, expressions for the speedup factor of the algorithms are derived. The results show that the hypercubes are highly effective in solving large systems of dense linear algebraic equations. Finally, the suitability of the hypercubes for solving sparse linear systems is discussed.

In this paper a parallel algorithm for solving systems of linear equation on the k-ary n-cube is presented and evaluated for the first time. The proposed algorithm is of O(N³/kⁿ) computation complexity and uses O(Nn) communication time to factorize a matrix of order N on the k-ary n-cube. This is better than the best known results for the hypercube, O(N log kⁿ), and the mesh, , each with approximately kⁿ nodes. The proposed parallel algorithm takes advantage of the extra connectivity in the k-ary n-cube in order to reduce the communication time involved in tasks such as pivoting, row/column interchanges, and pivot row and multipliers column broadcasts.

In this paper we consider five types of parallel preconditioners for solving large sparse nonsymmetric linear systems on the CRAY-T3E. They are ILU(0) in the wavefront ordering, ILU(0) in the multi-coloring ordering, SSOR in the wavefront ordering, the SPAI(SParse Approximate Inverse) preconditioner, and finally Multi-color Block SOR preconditioner. The ILU(0) is known to be robust and the wavefront ordering naturally exploits the parallelism but has a limited speedup due to the nonuniform lengths of the wavefronts. Multi-coloring is an efficient way of introducing the parallelism of order(N), where N is the order of the matrix but the convergence rate often deteriorates. The SPAI type preconditioner is inherently parallel and is gaining popularity. Finally, for the 5-point Laplacian matrix SOR method is known to have a nondeteriorating rate of convergence when the multi-coloring order is adopted. Also, Block SOR is expected to incur less communication overheads in a message-passing machine. Hence, Multi-Color Block SOR method is expected to have a good performance. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to 1024×1024. MPI library was used for interprocess communications. The results show that ILU(0) in the multi-coloring ordering gives the best performance.

In this paper, we present a mechanism of generation of a class of switched dynamical system without equilibrium points that generates a chaotic attractor. The switched dynamical systems are based on piecewise linear (PWL) systems. The theoretical results are formally given through a theorem and corollary which give necessary and sufficient conditions to guarantee that a linear affine dynamical system has no equilibria. Numerical results are in accordance with the theory.

Car-following is an approach to understand traffic behavior restricted to pairs of cars, identifying a “leader” moving in front of a “follower”, which at the same time, it is assumed that it does not surpass to the first one. From the first attempts to formulate the way in which individual cars are affected in a road through these models, linear differential equations were suggested by author like Pipes or Helly. These expressions represent such phenomena quite well, even though they have been overcome by other more recent and accurate models. However, in this paper, we show that those early formulations have some properties that are not fully reported, presenting the different ways in which they can be expressed, and analyzing them in their stability behaviors. Pipes’ model can be extended to what it is known as Helly’s model, which is viewed as a more precise model to emulate this microscopic approach to traffic. Once established some convenient forms of expression, two control designs are suggested herein. These regulation schemes are also complemented with their respective stability analyses, which reflect some important properties with implications in real driving. It is significant that these linear designs can be very easy to understand and to implement, including those important features related to safety and comfort.

In this paper the regulation problem of linear discrete-time systems with uncertain parameters under state and control constraints is studied. In the first part of the paper, two theorems concerning necessary and sufficient conditions for the existence of a solution to this problem are presented. Due to the constructive form of the proof of these theorems, these results can be used to the development of techniques for the derivation of a control law transferring to the origin any state belonging to a given set of initial states while respecting the state and control constraints.

In this article a Radial Basis Function Network (RBFN) approach for fast and efficient computation of inverse continuous time variant functions is presented. The approach is based on using a novel RBFN approach for computing inverse continuous time variant functions via a damped least squares formulation and also on a non-conventional implementation of an original approach for singularities prevention and conditioning improvement. The singularities avoidance approach in turn consists on establishing some characterizing matrices, in order to obtain a performance index and a null space vector, and then properly including it in the overall RBFN approach.

Complex systems, as interwoven miscellaneous interacting entities that emerge and evolve through self-organization in a myriad of spiraling contexts, exhibit subtleties on global scale besides steering the way to understand complexity which has been under evolutionary processes with unfolding cumulative nature wherein order is viewed as the unifying framework. Indicating the striking feature of non-separability in components, a complex system cannot be understood in terms of the individual isolated constituents’ properties per se, it can rather be comprehended as a way to multilevel approach systems behavior with systems whose emergent behavior and pattern transcend the characteristics of ubiquitous units composing the system itself. This observation specifies a change of scientific paradigm, presenting that a reductionist perspective does not by any means imply a constructionist view; and in that vein, complex systems science, associated with multiscale problems, is regarded as ascendancy of emergence over reductionism and level of mechanistic insight evolving into complex system. While evolvability being related to the species and humans owing their existence to their ancestors’ capability with regards to adapting, emerging and evolving besides the relation between complexity of models, designs, visualization and optimality, a horizon that can take into account the subtleties making their own means of solutions applicable is to be entailed by complexity. Such views attach their germane importance to the future science of complexity which may probably be best regarded as a minimal history congruent with observable variations, namely the most parallelizable or symmetric process which can turn random inputs into regular outputs. Interestingly enough, chaos and nonlinear systems come into this picture as cousins of complexity which with tons of its components are involved in a hectic interaction with one another in a nonlinear fashion amongst the other related systems and fields. Relation, in mathematics, is a way of connecting two or more things, which is to say numbers, sets or other mathematical objects, and it is a relation that describes the way the things are interrelated to facilitate making sense of complex mathematical systems. Accordingly, mathematical modeling and scientific computing are proven principal tools toward the solution of problems arising in complex systems’ exploration with sound, stimulating and innovative aspects attributed to data science as a tailored-made discipline to enable making sense out of voluminous (-big) data. Regarding the computation of the complexity of any mathematical model, conducting the analyses over the run time is related to the sort of data determined and employed along with the methods. This enables the possibility of examining the data applied in the study, which is dependent on the capacity of the computer at work. Besides these, varying capacities of the computers have impact on the results; nevertheless, the application of the method on the code step by step must be taken into consideration. In this sense, the definition of complexity evaluated over different data lends a broader applicability range with more realism and convenience since the process is dependent on concrete mathematical foundations. All of these indicate that the methods need to be investigated based on their mathematical foundation together with the methods. In that way, it can become foreseeable what level of complexity will emerge for any data desired to be employed. With relation to fractals, fractal theory and analysis are geared toward assessing the fractal characteristics of data, several methods being at stake to assign fractal dimensions to the datasets, and within that perspective, fractal analysis provides expansion of knowledge regarding the functions and structures of complex systems while acting as a potential means to evaluate the novel areas of research and to capture the roughness of objects, their nonlinearity, randomness, and so on. The idea of fractional-order integration and differentiation as well as the inverse relationship between them lends fractional calculus applications in various fields spanning across science, medicine and engineering, amongst the others. The approach of fractional calculus, within mathematics-informed frameworks employed to enable reliable comprehension into complex processes which encompass an array of temporal and spatial scales notably provides the novel applicable models through fractional-order calculus to optimization methods. Computational science and modeling, notwithstanding, are oriented toward the simulation and investigation of complex systems through the use of computers by making use of domains ranging from mathematics to physics as well as computer science. A computational model consisting of numerous variables that characterize the system under consideration allows the performing of many simulated experiments via computerized means. Furthermore, Artificial Intelligence (AI) techniques whether combined or not with fractal, fractional analysis as well as mathematical models have enabled various applications including the prediction of mechanisms ranging extensively from living organisms to other interactions across incredible spectra besides providing solutions to real-world complex problems both on local and global scale. While enabling model accuracy maximization, AI can also ensure the minimization of functions such as computational burden. Relatedly, level of complexity, often employed in computer science for decision-making and problem-solving processes, aims to evaluate the difficulty of algorithms, and by so doing, it helps to determine the number of required resources and time for task completion. Computational (-algorithmic) complexity, referring to the measure of the amount of computing resources (memory and storage) which a specific algorithm consumes when it is run, essentially signifies the complexity of an algorithm, yielding an approximate sense of the volume of computing resources and seeking to prove the input data with different values and sizes. Computational complexity, with search algorithms and solution landscapes, eventually points toward reductions vis à vis universality to explore varying degrees of problems with different ranges of predictability. Taken together, this line of sophisticated and computer-assisted proof approach can fulfill the requirements of accuracy, interpretability, predictability and reliance on mathematical sciences with the assistance of AI and machine learning being at the plinth of and at the intersection with different domains among many other related points in line with the concurrent technical analyses, computing processes, computational foundations and mathematical modeling. Consequently, as distinctive from the other ones, our special issue series provides a novel direction for stimulating, refreshing and innovative interdisciplinary, multidisciplinary and transdisciplinary understanding and research in model-based, data-driven modes to be able to obtain feasible accurate solutions, designed simulations, optimization processes, among many more. Hence, we address the theoretical reflections on how all these processes are modeled, merging all together the advanced methods, mathematical analyses, computational technologies, quantum means elaborating and exhibiting the implications of applicable approaches in real-world systems and other related domains.

The solutions of systems of fractional differential equations depend on the type of the fractional derivative used in the system. In this paper, we present in closed forms the solutions of linear systems involving the modified Atangana–Baleanu derivative that has been introduced recently. For the nonlinear systems, we implement a numerical scheme based on the collocation method to obtain approximate solutions. The applicability of the results is tested through several examples. We emphasize here that certain systems with the Atangana–Baleanu derivative admit no solutions which is not the case with the modified derivative.

We illustrate an approach to uncertain knowledge based on lower conditional probability bounds. We exploit the coherence principle of de Finetti and a related notion of generalized coherence (g-coherence), which is equivalent to the "avoiding uniform loss" property introduced by Walley for lower and upper probabilities. Based on the additive structure of random gains, we define suitable notions of non relevant gains and of basic sets of variables. Exploiting them, the linear systems in our algorithms can work with reduced sets of variables and/or constraints. In this paper, we illustrate the notions of non relevant gain and of basic set by examining several cases of imprecise assessments defined on families with three conditional events. We adopt a geometrical approach, obtaining some necessary and sufficient conditions for g-coherence. We also propose two algorithms which provide new strategies for reducing the number of constraints and for deciding g-coherence. In this way, we try to overcome the computational difficulties which arise when linear systems become intractable. Finally, we illustrate our methods by giving some examples.

We give the expression for the solution to some particular initial value problems in the space E¹ of fuzzy subsets of ℝ. We deduce some interesting properties of the diameter and the midpoint of the solution and compare the solutions with the corresponding ones in the crisp case.

Farkas type results are available for solutions to linear systems. These can also include restrictions such as nonnegative solutions or integer solutions. They show that the unsolvability can be reduced to a single constraint that is not solvable and this condition is implied by the original system. Such a result does not exist for integer solution to inequality system because a single inequality is always solvable in integers. But a single equation that does not have nonnegative integer solution exists. We present some cases when polynomial algorithms to find nonnegative integer solutions exist.

The long-range dependence of Internet traffic has been experimentally observed. One issue in handling long-range dependent traffic is how to simulate random traffic data with long-range dependence. The authors discuss a correlation-based simulator with a white noise input for generating long-range dependent traffic data. With the real TCP traffic traces, a simulation model of TCP arrival traffic is empirically developed and the experimental results are satisfactory.

The hyperstar network has been recently proposed as an attractive product network that outperforms many popular topologies in various respects. In this paper we explore additional capabilities for the hyperstar network through an efficient parallel algorithm for solving the LU factorization problem on this network. The proposed parallel algorithm uses O(n) communication time on a hyperstar formed by the cross-product of two n-star graphs. This communication time improves the best known result for the hypercube-based LU factorization by a factor of log(n), and improves the best known result for the mesh-based LU factorization by a factor of (n - 1)!.

Classical ways of cooling require some of these elements: phase transition, compressor, nonlinearity, valve and/or switch. A recent example is the 2018 patent of Linear Technology Corporation; they utilize the shot noise of a diode to produce a standalone nonlinear resistor that has $T$/2 noise temperature (about 150K). While such “resistor” can cool its environment when it is AC coupled to a resistor, the thermal cooling effect is only academically interesting. The importance of the invention is of another nature: In low-noise electronics, it is essential to have resistors with low-noise temperature to improve the signal-to-noise ratio. A natural question is raised: can we use a linear system with feedback to cool and, most importantly, to show reduced noise temperature? Exploring this problem, we were able to produce standalone linear resistors showing strongly reduced thermal noise. Our must successful test shows $T$/100 (about 3K) noise temperature, as if the resistor would have been immersed in liquid helium. We also found that there is an old solution offering similar results utilizing the virtual ground of an inverting amplifier at negative feedback. There, the “cold” resistor is generated at the input of an amplifier. On the other hand, our system generates the “cold” resistance at the output, which can have practical advantages.

Steenrod's problem of realizing modules has been solved by Arnold and Vogel for groups of a cyclic nature, and has counterexamples given by Carlsson, Vogel, Benson and Habegger, most concretely for Z/2×Z/2. The last two authors develop nontrivial methods to realize other modules for this group, but they do not extend to the general case Z/p×Z/p. In this paper we give some positive results for groups (Z/p)^m. Of interest we find that relevant sequences and their duals are both realizable. Also we establish an invariance of the Steenrod problem suggested by Benson and Habeggers' approach.

This paper is concerned with an input-output relation of linear systems when using the complex wavelet packet transform. In general, the linear relation between input-output is guaranteed by the following two conditions: (1) the mother wavelet has a better frequency resolution, and (2) the real and imaginary parts in the mother wavelet consist of a Hilbert transform pair. The complex wavelet satisfying the above conditions has been used for demonstrating the linear relation. In this paper, a complex wavelet packet transform is applied for improving the frequency resolution. The validity of our approach is shown through a numerical experiment.

We derive algorithms that compute a balanced state representation from the differential equation describing a finite-dimensional linear system.

In this paper we consider the abstract minimization problem: