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We describe the results of an expert system applied to the evaluation of samples of kaolin for industrial use in paper or ceramic manufacture. Different machine learning techniques—classification trees, support vector machines and Bayesian networks—were applied with the aim of evaluating and comparing their interpretability and prediction capacities. The predictive capacity of these models for the samples analyzed was highly satisfactory, both for ceramic quality and paper quality. However, Bayesian networks generally proved to be the most useful technique for our study, as this approach combines good predictive capacity with excellent interpretability of the kaolin quality structure, as it graphically represents relationships between variables and facilitates what‐if analyses.

The main purpose of this short paper is to present the Chernoff product formula from which we derive the exponential formula and the Lie-Trotter formula for C0-semigroups of linear operators on L∞.

The main purpose of this short paper is to introduce the evolution semigroup associated with an evolutionary process with exponential growth on L∞.

In this short paper, we present a Kato’s type inequality for symmetric diffusion operators.

In this short paper we present a necessary and sufficient condition for that the one‐dimensional mass transport equation has one unique L1 weak solution.

Neutralizing pH value of sugar cane juice is the important craft in the control process in the clarifying process of sugar cane juice, which is the important factor to influence output and the quality of white sugar. On the one hand, it is an important content to control the neutralized pH value within a required range, which has the vital significance for acquiring high quality purified juice, reducing energy consumption and raising sucrose recovery. On the other hand, it is a complicated physical‐chemistry process, which has the characteristics of strong non‐linearity, time‐varying, large time‐delay, and multi‐input. Therefore, there has not been a very good solution to control the neutralized pH value. Firstly, in this chapter, a neural network model for the clarifying process of sugar juice is established based on gathering 1200 groups of real‐time sample data in a sugar factory. Then, the HDP (Heuristic Dynamic Programming) method is used to optimize and control the neutralized pH value in the clarifying process of sugar juice. Simulation results indicate that this method has good control effect. This will build a good foundation for stabilizing the clarifying process and enhancing the quality of the purified juice and lastly enhancing the quality of white sugar.

In the noiseless context, it has long been known that the average encoded word length of an instantaneous or uniquely decipherable code can be made to lie between the source entropy and that value plus unity. We address the question of finding sufficient conditions on the code–word probabilities for it to be possible to make the average code–word length approximate the entropy by a smaller prescribed amount.

The paper is devoted to applications the quaternionic analysis and the two-dimensional stationary phase method for problems of electromagnetic waves propagation from moving modulated sources in dispersive chiral media. This paper is based in the article [2].

Complex-valued harmonic functions that are univalent and sense-preserving in the open unit disk Δ can be written in the form f = h+g¯, where h and g are analytic in Δ several mathematicians examined classes of complex harmonic functions in the unit disk with some coefficient conditions. In this paper we introduce some results which generalize problems considered in other previous papers. The main results concern starlikeness and convexity of harmonic functions satisfying a three parameter coefficient condition.

Making good decisions are very essential in any organization. However, the nature of the problems to be solved is becoming more complicated where as opposed to definiteness, elements of subjectivity and vagueness are now common. In this paper we propose a method of solving decision making problems using fuzzy number in order to cater the problem. We improvise the method of area dominance in ranking fuzzy number given by Tseng and Klein by introducing the concept of upper dominance. This proposed method can effectively rank combination of normal fuzzy numbers and non-normal fuzzy numbers. We implement the upper dominance concept in representing the confidence level of decision makers in solving problems. In addition, spread factor also incorporated in the ranking procedure in order to solve equality in ranking order. Several examples are given in the paper to illustrate the usefulness of this concept.

This paper has two purposes: the first is to determine the magnitude of the inconsistency in the overlap region of the water-aluminum and water-tin sub-ranges through investigation of calibration of the SPRTs. The other is to understand the characteristic of the inconsistency function in the region. In this paper, twenty one SPRTs were investigated. The results show that their inconsistencies are in the range from −1.26mK to 1.22mK except one of them is 4.08mK. The inconsistency function between the ITS-90 interpolations in the overlap region is unique cubic polynomial function with a movable real root, different from other inconsistency functions. By means of factoring, this paper gives an alternative form of the inconsistency function and makes an analysis of the roots and extrema of the function.

This paper explores the extension of the Legendre transform from scalar calculus to geometric calculus. In physics, the Legendre transform provides a change of variables to express equations of motion or other physical relationships in terms of the most convenient dynamical quantities for a given experimental or theoretical analysis. In classical mechanics and in field theory, the Legendre transform generates the Hamiltonian function of a system from the Lagrangian function or vice versa. In thermodynamics, the Legendre transform allows thermodynamic relationships to be written in terms of alternative sets of independent variables. In this paper, we review the properties of the Legendre transform in scalar calculus and show how an analogous transformation with similar properties may be constructed in geometric calculus.

This paper combines first-order system least-squares (FOSLS) and first-order system LL∗ (FOSLL∗) to create a Hybrid method. FOSLS minimizes the error, eh = uh− u, over a finite element subspace, V h, in the operator norm: uh = argminuh∈V h ‖L(uh− u)‖. Here, L is a first-order differential operator, obtained by possibly reformulating the original equations properly, such that c0‖e‖V h ≤ ‖L(uh− u)‖ ≤ c1‖e‖V h . Usually, V h is a subspace of H1, in which case this property of L is called H1-equivalence. This leads to many attractive numerical features of the resulting dicretized problem [5, 6, 11]. FOSLS also offers a locally sharp, globally reliable, and easily computable a posterior error estimate, which can be utilized for adaptive mesh refinement [3, 1]. The FOSLL∗ method seeks an approximation in the range of L∗, the adjoint of L: uh = L∗wh, with wh ∈ W h, a standard finite element space. FOSLL∗ minimizes the L2 norm of the error over L∗(W h), that is, wh = argminwh∈W h ‖L∗wh− u‖. For problems that are not smooth enough due to discontinuous coefficients, or nonsmooth boundary conditions, to name a few, H1-equivalence of L and its associated properties may be lost [12, 13, 17]. One of the solutions is to use FOSLL* [10, 15]. Unfortunately, FOSLL∗ does not have an easily computable error estimate. The Hybrid method attempts to retain the best properties of both FOSLS and FOSLL∗. This is accomplished by combining the FOSLS functional, the FOSLL∗ functional, and an intermediate term that draws them together. The FOSLS and intermediate terms in the Hybrid functional provide an effective a posteriori error measure. In this paper, we show that the Hybrid functional is coercive and continuous in a graph-like norm with modest coercivity and continuity constants, c0 = 1/3 and c1 = 3, which results in a better control of L2 error and mass conservation numerically, without going to a very fine grid. Numerical tests are provided to confirm the efficiency of the Hybrid method and effectiveness of the a posteriori error measure.

In this paper, effects of different modeling of cell-face velocities in convection term on accuracy and convergence behavior of the solution are studied. Three different upwind schemes, namely first order mass weighted upwind scheme, second order mass weighted upwind scheme and the modified version of physical influence scheme are used for modeling of convected velocities. Discussions are focused on the collocated scheme of Karimian and Schneider in the context of control volume finite element method. The modified version of this scheme, proposed by Alisadeghi and Karimian, has been used for modeling of mass conserving velocities in the present paper. Results of this method, then are compared with the results of pressure weighted interpolation method of Rhie and Chow. The steady-state flow in a two-dimensional driven cavity is studied for the comparison and validation of the numerical results. In general, the physical influence scheme gives the best accuracy among the three schemes.

In this paper, the linear fractional programming problem with interval coefficients in the objective function is considered. The aim of the paper is to show that a parametric approach, which is an iterative procedure, can be utilized to solve such problem. Questions of how to select the best coefficient through the intervals and also selecting the best value of decision variables are answered in this paper. Finally, the proposed method yields to the optimal solution. An example is given to show the efficiency of the method.

The paper is a part of student cooperation in AKTION project (Austria-Czech). Method of aposteriori error estimation based on weighted averaging to improve initial triangulation to get better solution of the planar elliptic boundary-value problem of second order and numerical illustrations of the method are presented in the paper.

In a recent paper [On Some Unified Integrals, Advances in Comput. Math. and Its Applications, Vol. 1, No. 3, PP. 151-153 (2012)], the author has evaluated three very interesting integrals involving hypergeometric function in terms of gamma function. In this paper, three new unified integrals involving Fox’s H-function have been evaluated. By specializing the parameters, we can easily obtain a large number of new and known integrals including one obtained earlier by Garg and Mittal. The results established in this paper are simple, interesting, easily established and may be useful.

In this paper, we solve the mixed boundary value problem on bounded multiply connected region by using the method of boundary integral equation. Our approach in this paper is to reformulate the mixed boundary value problem into the form of Riemann-Hilbert problem. The Riemann-Hilbert problem is then solved using a uniquely solvable Fredholm integral equation on the boundary of the region. The kernel of this integral equation is the adjoint generalized Neumann kernel. As an examination of the proposed method, some numerical examples for some different test regions are presented.

In this paper, we applied the Riemann-Liouville approach and the fractional Euler-Lagrange equations in order to obtain the fractional nonlinear dynamic equations involving the classical physical application "Spring-Mass-Dumper System" to both, integer order calculus (IOC) and fractional order calculus (FOC) approaches. After the numerical simulations, we outlined the main results through time histories and the investigation of phase portraits. Thereafter, we believe that the results showed on this paper are pertinent enough to be thoroughly analyzed and maybe it can be the very beginning of a stride towards more realistic and more precise results about fractional-order models when compared to the integer order models in these applications.

This paper is a continuation of the paper [1]. Some basic dynamical analysis will be done using the Routh-Hurwitz Criterion. Furthermore, this paper will determine the Hopf bifurcation of the three-dimensional modified Lorenz system by means of the first Lyapunov coefficient. The result shows that Hopf bifurcations occurred at two equilibrium points and are subcritical.

This paper focuses on some 3-Engel groups. Suppose x and y are elements of a group G. The commutativity degree of a group is the probability that two elements in the group commute and is denoted by P(G). Meanwhile, the nth commutativity degree of a group G is the probability that for any pairs of x and y in G, xn and y commute. In this paper, the nth commutativity degree of some 3-Engel groups is determined.

The ITS-90 Inconsistency in overlap region of the Mercury-Gallium and the Water-Argon Sub-ranges are investigated in this paper. The calibration data of thirteen SPRTs are used to determine the magnitude of the inconsistency. The result shows it no greater than 0.22mK. By mean of extending a logarithm of the platinum resistance ratio of the ITS-90 interpolation in a power series, this paper gives an inconsistency function of the form of interpolation error in the region . The function is only with approximate error less than 3.4×10−5 mK, almost no error, checked by the SPRTs.

Aiming at the situation that the turn point of the whitenization weight function must be real number, this paper constructed and solved the whitenization function when the turn point is an interval grey number. Firstly, this paper delimited the standardized methods of the interval grey number. Then using the standardized interval grey number replace the real number of the whitenization function. And the expression of the whitenization function with interval grey number is given. Secondly, discussing the two circumstances that only one interval grey number is in the piecewise curve of the whitening weight function or both two endpoints are interval grey numbers. And when the four turn points are all interval grey numbers, the expression of whitenization function is given. At last, an example is given to illustrate the validity and the feasibility of this algorithm.

The aim of this paper is to modify the suggested method by Noura et al. (A.A. Noura, F. Hosseinzadeh Lotfi, G.R. Jahanshahloo, S. Fanati Rashidi, Super-efficiency in DEA by effectiveness of each unit in society, Applied Mathematics Letters, 24, 2010, 623-626), that is a ranking method based on the effectiveness of each unit in society. They utilized the assigned weights for ranking DMUs, but this is not a conventional method for determining the weights. This paper proposes common weights approach for improving its method. A multi-objective linear fractional is derived and then it was converted to a multi-objective linear programming by Taylor series. The model is solved by Max Min method. Based on the obtained optimal solution, common weights are acquired and then DMUs will be ranked. And we introduce stochastic version of this model in DEA. The deterministic equivalent of this stochastic model will be obtained. The proposed method is illustrated by ranking Taiwan forests after reorganization.

In this paper we define and characterize the concepts of polynomial stability in mean square for stochastic cocycles in Hilbert spaces. The main concern of this paper is to give definitions, connections and characterizations for various concepts for the asymptotic properties of stability in mean square of solutions for stochastic equations in Hilbert spaces, in a nonuniform setting.

The paper is a part of student cooperation in AKTION project (Austria-Czech). Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. The paper deals with possibilities of numerical solution of Initial Value Problems of Ordinary Differential Equations (ODEs) - using the Taylor series method with automatic computation of higher Taylor series terms. The explicit and implicit scheme of Taylor series method is compared with numerical solvers implemented in MATLAB software [1]. The computation time and accuracy of our approach are compared with that of MATLAB ode solvers on a set of ODEs test examples [2].

The discrete element method is used extensively for the simulation of granular materials. The great majority of engineering applications require bulk design parameters to be determined which requires some averaging or coarse graining techniques to be applied to the particle data from DEM computations. Although numerous papers have been published on these techniques, we suggest that there are still significant challenges in the calculation of the continuum quantities from the particle data. This is chiefly because DEM computes at grain contact level and at the order of 10−5 or smaller computational time step, whereas important engineering events often occur at much larger timescale, so the question of what temporal and spatial averaging scales should be adopted is not so clear. This paper outlines an implementation of coarse graining methods for the projection of discrete quantities into continuum fields. Two DEM simulation examples with a quasi-static and a rapid dynamic flow conditions are used to study the influence of the averaging length scales. The results show that the combined temporal and spatial regimes can provide an effective method to evaluate the salient continuum quantities at the appropriate frequency of occurrence. They also show that the computed continuum parameters can be very sensitive to the averaging length scales such that erroneous conclusions can be reached depending on whether an appropriate scale has been chosen.

In a recent paper [A reducibility of the Kampe de Feriet function, Advances in Comput. Math. and its Applications, Vol. 1, No. 1, pp. 76-79 (2012)], the author has evaluated one very interesting reduction formula. In this paper, two new reduction formulas of Kampe de Feriet function have been evaluated by using Gauss's second summation theorem on the sum of a 3F2. The results established in this paper are simple, interesting, easily established and may be useful.

In this paper, we consider the class Kα* of the functions of the form f(z) = z+a2z2+⋯ which are analytic univalent in the disc D = {z:|z| 0 where g ∈ C. This paper focuses on the functional |a2a4-a32| for a function belonging to Kα* is obtained.

The present paper is a continuation of our paper [4]. The problem to be studied is the non-impact of algebraic property on the convergence properties of the μ-th root base of special monogenic polynomials. These convergence properties are the investigation of the relation between the effectiveness in closed balls of a given base and that of its μ-th root base, and also the relation between their rate of increase. Our result which is concerned with effectiveness in closed balls is quite different from its corresponding one for the square root case in complex settings.

The main goal of the paper is studying stability properties of the difference of solutions of a system of nonlinear impulsive differential equations with different initial data. The initial data differs both in initial time and initial position. In many real repeated experiments it is difficult to keep both the initial position and the initial time unchanged because of all kinds of disturbed factors. It requires the changing of initial time to be taken into consideration. In the paper various types of stability with respect to initial time difference are studied. The investigations are based on piecewise continuous Lyapunov functions and comparison results for scalar impulsive differential equations. A special type of appropriately generalized derivative of Lyapunov functions is defined and applied. It is proved that the equistability/uniform stability/Lipschitz stability property of the zero solution of the comparison scalar equation implies the corresponding stability properties with respect to difference initial time of the given nonlinear system. The obtained sufficient conditions significantly depend on the impulses.

Probabilistic Evolution Approach (PEA) is a method recently developed for the solution of explicit ODEs. There is now a comprehensive knowledge accumulation for PEA. It is well known that the PEA truncation approximants converge to a unique limit representing the solution of the considered ODE in a finite region of the complex planes of the unknowns. The complement of the union of these regions to entire complex plane(s) is the subject of analytic continuation. In this paper we focus on this issue for the Van der Pol Equation. Paper gives the importance rather to the formalism and conceptuality. True implementations are left to future works.

This paper explores the use of a parabolized model for hydrodynamic stability study of a two-dimensional flow with complex swirling motion. The approach for predicting transition to vortex breakdown based on the parabolized stability equations to be discussed in this paper reduces the hydrodynamic stability study on the solution of a two-point linear differential eigenvalue problem. The parallel implementation of the LPSE using spectral collocation technique is considered and the numerical stability of the governing equations is presented. The accuracy and efficiency of the proposed numerical scheme are tested for various values of the involved parameters.

A prime number is co-prime with all the primes as well. This paper utilizes this fact by generating larger prime numbers based on a set of smaller prime numbers. The prime numbers are ordered and each two consecutive primes are coupled to generate their co-prime number formula extending this process larger prime sequence is established. Will the process help us produce larger prime numbers faster and more efficiently? This paper investigates the described process.

As is well known investigation of many processes of natural sciences reduce to the solving of initial value problem for integro-differential equations which are one of the priority areas of modern mathematics. To define the exact solution of such problems is not always possible. Therefore the scientists constructed approximate methods for solving them. There are a number of papers devoted to finding approximate solutions of integro-differential equations. Unlike at papers investigated, here the numerical solution of initial value problem for Volterra integro-differential equations by the hybrid methods, constructed concrete methods with the degree p ≤ 6 and suggested algorithm for using them.

In this paper we study a mixed problem for the nonlinear Schrodinger equation that have a nonlinear adding, in which the coefficient is a generalized function. Here is proved a solvability theorem of the considered problem with use of the general solvability theorem of the article [28]. Furthermore here is investigated also the behaviour of the solution of the studied problem. aipproc class produce a paper with the correct layout for AIP Conference Proceedings 8.5in × 11in double column.

Seismic tomography becomes important tool recently for imaging complex subsurface. It is well known that imaging complex rich fault zone is difficult. In this paper, The application of time domain inverse scattering wave tomography to image the complex fault zone would be shown on this paper, especially an efficient time domain inverse scattering tomography and their run in cluster parallel computer which has been developed. This algorithm is purely based on scattering theory through solving Lippmann Schwienger integral by using Born's approximation. In this paper, it is shown the robustness of this algorithm especially in avoiding the inversion trapped in local minimum to reach global minimum. A large data are solved by windowing and blocking technique of memory as well as computation. Parameter of windowing computation is based on shot gather's aperture. This windowing technique reduces memory as well as computation significantly. This parallel algorithm is done by means cluster system of 120 processors from 20 nodes of AMD Phenom II. Benchmarking of this algorithm is done by means Marmoussi model which can be representative of complex rich fault area. It is shown that the proposed method can image clearly the rich fault and complex zone in Marmoussi model even though the initial model is quite far from the true model. Therefore, this method can be as one of solution to image the very complex mode.

The Gompertz distribution is commonly used in biology for modeling fatigue and mortality. This paper studies a class of models proposed by Adham and Walker, featuring a Gompertz type distribution where the dependence structure is modeled by a lognormal distribution, and develops a new multivariate formulation that facilitates several numerical and computational aspects. This paper also implements the FBST, the Full Bayesian Significance Test for pertinent sharp (precise) hypotheses on the lognormal covariance structure. The FBST’s e-value, ev(H), gives the epistemic value of hypothesis, H, or the value of evidence in the observed in support of H.

In this paper, we focus on the inverse problem associated with IFSP: Given a target measure μ, find an IFSP (w,p), such that its invariantmeasure μ is sufficiently close to μ in Monge-Kantorovich distance, i.e., dMK(μ,μ) < e. We extend themoment-basedmethod developed by Forte and Vrscay (1995) along two different directions. First, we search for a set of probabilities pi that not only minimizes the collage error but also maximizes the entropy of the iterated function system. Second, we include an extra term in the minimization process which takes into account the sparsity of the set of probabilities. In our new formulations, the minimization of collage error (for moments) can be understood as amulti-criteria problem: At least three different and conflicting criteriamust be considered, i.e., collage error, entropy and sparsity. In this paper, we employ scalarization to reduce the multi-criteria program to a single-criteria program by combining all objective functions with different trade-off weights. The...

The aim of this paper is to complete the classification of three-dimensional complex Leibniz algebras. The description of isomorphism classes of three-dimensional complex Leibniz algebras has been given by Ayupov and Omirov in 1999. However, we found that this list has a little redundancy. In this paper we apply a method which is more elegant and it gives the precise list of isomorphism classes of these algebras. We compare our list with that of Ayupov-Omirov and show the corrections which should be made.

This paper devoted to a computational prediction of blood flow in a stenotic artery using a suitable mathematical model. The artery was modelled as a rigid two-dimensional tube with three different levels of severity of the disease. In the mathematical model, the blood flow equations were governed by the stream function-vorticity equations. Then the cubic interpolation profile scheme with high order of accuracy was applied to discretise the advection term of the equation. Different cases were considered, where the Reynolds number of the blood flow were varied to predict pulsatile of blood flow. The present paper reports the analysis of flow separation area downstream the stenosed artery and the location and peak wall shear stress values. These findings may contribute for early detection and medical treatments to prevent further development of the disease and rupture of the artery.

The problem of robust preview control for uncertain discrete-time systems is investigated in this paper. According to preview control theory, an LQ optimal preview controller for the corresponding normal system is constructed. Based on the controller and robust control theory, some robust stability criterions of uncertain closed-loop systems were obtained by the method of Lyapunov functions. The main contribution of this paper is that the perturbation bound of the parameter uncertainties has been estimated. At last, an example shows the availability of the criterions.

The transition to chaotic phase synchronization for a periodically driven spiral-type chaotic oscillator is known to involve a dense set of saddle-node bifurcations. By following the synchronization transition through the cascade of period-doubling bifurcations in a forced Ro¨ssler system, this paper describes how these saddle-node bifurcations arise and how their characteristic cyclic organisation develops. We identify the cycles that are involved in the various saddle-node bifurcations and describe how the formation of multi-layered resonance cycles in the synchronization domain is related to the torus doubling bifurcations that take place outside this domain. By examining a physiology-based model of the blood flow regulation to the individual functional unit (nephron) of the kidney we demonstrate how a similar bifurcation structure may arise in this system as a response to a periodically varying arterial blood pressure. The paper finally discusses how an alternative transition to chaotic phase synchronization may occur in the mutual synchronization of two chaotically oscillating period-doubling systems.

Electro-hydraulic servomechanisms (EHSM) are important components of flight control systems and their role is to control the movement of the flying control surfaces in response to the movement of the cockpit controls. As flight-control systems, the EHSMs have a fast dynamic response, a high power to inertia ratio and high control accuracy. The paper is devoted to the study of the sensitivity for an electro-hydraulic servomechanism used for an aircraft aileron action. The mathematical model of the EHSM used in this paper includes a large number of parameters whose actual values may vary within some ranges of uncertainty. It consists in a nonlinear ordinary differential equation system composed by the mass and energy conservation equations, the actuator movement equations and the controller equation. In this work the focus is on the sensitivities of the eigenvalues of the linearized homogeneous system, which are the partial derivatives of the eigenvalues of the state-space system with respect the parameters...

The amount of data to be measured and processed becomes extremely large in modern digital systems. As a result, the performance of the method to processing the data is of prime importance. This paper presents an iteration free method using the Maximum relative Entropy method when observed data is introduced as a Gaussian prior. It is shown that the posterior is a Gaussian function, but with updated means. Such an approach is capable of performing optimization for a single sample, which makes it applicable to real time applications. A practical example is provided in the paper where updating is performed on vector coordinate variables, where the constraint is an average vector length. This produces a general vector normalization solution that takes into account uncertainty in the measurements. A special case of this updating is also shown in limit as the standard deviation goes to zero in the priors (essentially making them Dirac delta functions). This reproduces the vector normalization formula. In addition, comparisons with some Bayesian filtering methods will be discussed.

In another paper (Butterfield 2011), one of us argued that emergence and reduction are compatible, and presented four examples illustrating both. The main purpose of this paper is to develop this position for the example of phase transitions. We take it that emergence involves behaviour that is novel compared with what is expected: often, what is expected from a theory of the system's microscopic constituents. We take reduction as deduction, aided by appropriate definitions. Then the main idea of our reconciliation of emergence and reduction is that one makes the deduction after taking a limit of an appropriate parameter $N$. Thus our first main claim will be that in some situations, one can deduce a novel behaviour, by taking a limit $N\to\infty$. Our main illustration of this will be Lee-Yang theory. But on the other hand, this does not show that the $N=\infty$ limit is physically real. For our second main claim will be that in such situations, there is a logically weaker, yet still vivid, novel behaviour that occurs before the limit, i.e. for finite $N$. And it is this weaker behaviour which is physically real. Our main illustration of this will be the renormalization group description of cross-over phenomena., 24 pp, v2: one minor change. Contribution to the Frontiers of Fundamental Physics (FFP 11) Conference Proceedings

This paper outlines the initial investigations on the influence of chaotic dynamics to the performance of evolutionary algorithms. The focus of this paper is the embedding of chaotic system in the form of chaos number generator for Differential Evolution. The chaotic systems of interest are the discrete dissipative systems. The two-dimensional Dissipative Standard map was selected as a possible chaos number generator for Differential Evolution. Repeated simulations were performed on a selected benchmark function. Finally, the obtained results are compared with canonical Differential Evolution.

Our earlier paper provided an introduction to basic ideas relating topological techniques to chaos theory. In the present paper we provide additional details on a number of these techniques. We go into more detail on properties of inverse limits related to chaos. In particular, we provide a detailed outline of the result of Jubran on producing a chaotic embedding of the Whitehead continuum. This paper was partially motivated by a talk given by the second author at the 8th international summer school and conference Chaos 2011: Let's Face Chaos Through Nonlinear Dynamics (CAMTP, University of Maribor, Slovenia, 26 June - 10 July 2011).

This paper comments on a number of inaccuracies in the article by Brambley [E.J. Brambley, Fundamental problems with the model of uniform flow over acoustic linings, Journal of Sound and Vibration 322(2009)1026-1037] concerning the new concept of "well-posed partial differential equation" introduced in the paper. In particular, the neglect of specifying: the initial and boundary conditions; meaning of solution; conditions assuring existence and uniqueness of the solution; topology in the space of the solutions necessary for analyze the continuous dependence on the initial data and Lyapunov stability. It is shown that, due to the above inaccuracies, the concept introduced by Brambley is confusing, i.e. depending on the set of initial data, boundary conditions, meaning of solution and topology, the same equation can be ill-posed or it can be well-posed. In our paper the concept of well-posed problem, introduced by Hadamard long times ago, is refreshed and applied in the case of rectangular lined duct. Suffi...

One is reminded in this paper of the often overlooked fact that the geometric straight line, or GSL, of Euclidean geometry is not necessarily identical with its usual Cartesian coordinatisation given by the real numbers in R. Indeed, the GSL is an abstract idea, while the Cartesian, or for that matter, any other specific coordinatisation of it is but a mathematical model chosen upon certain reasons. And as is known, there are a a variety of mathematical models of GSL, among them given by nonstandard analysis, reduced power algebras, the topological long line, or the surreal numbers. As shown in this paper, the GSL can allow coordinatisations which are arbitrarily more rich locally and also more large globally, being given by corresponding linearly ordered sets of no matter how large cardinal. Thus one can obtain in relatively simple ways structures which are more rich locally and large globally than in nonstandard analysis, or in various reduced power algebras. Furthermore, vector space structures can be defined in such coordinatisations. This fact can have a major importance in physics, since such locally more rich and globally more large coordinatisations of the GSL do allow new physical insights, just as the introduction of various microscopes and telescopes have done. Among others, it can reassess special relativity with respect to its independence of the mathematical models used for the GSL. Also, it can allow the more appropriate modelling of certain physical phenomena. One of the long vexing issue of so called “infinities in physics” can obtain a clarifying reconsideration. It indeed all comes down to looking at the GSL with suitably constructed microscopes and telescopes, and apply the resulted new modelling possibilities in theoretical physics. One may as well consider that in string theory, for instance, where several dimensions are supposed to be compact to the extent of not being observable on classical scales, their mathematical modelling may benefit from the presence of infinitesimals in the mathematical models of the GSL presented here. However, beyond all such particular considerations, and not unlikely also above them, is the following one: theories of physics should be not only background independent, but quite likely, should also be independent of the specific mathematical models used when representing geometry, numbers, and in particular, the GSL.