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Let T and S be two self-adjoint positive invertible operators of a complex Hilbert space H. In this paper, we investigate some inequalities involving the quadratic forms of the weighted arithmetic and harmonic means of T and S. Application for operator entropies is also discussed. [ABSTRACT FROM AUTHOR]

Some preconditioners for accelerating the classical iterative methods are given in Zhang et al. [Y. Zhang and T.Z. Huang, A class of optimal preconditioners and their applications, Proceedings of the Seventh International Conference on Matrix Theory and Its Applications in China, 2006. Y. Zhang, T.Z. Huang, and X.P. Liu, Modified iterative methods for nonnegative matrices and M-matrices linear systems, Comput. Math. Appl. 50 (2005), pp. 1587-1602. Y. Zhang, T.Z. Huang, X.P. Liu, A class of preconditioners based on the (I+S(α))-type preconditioning matrices for solving linear systems, Appl. Math. Comp. 189 (2007), pp. 1737-1748]. Another kind of preconditioners approximating the inverse of a symmetric positive definite matrix was given in Simons and Yao [G. Simons, Y. Yao, Approximating the inverse of a symmetric positive definite matrix, Linear Algebra Appl. 281 (1998), pp. 97-103]. Zhang et al. 's preconditioners and Simons and Yao's are generalized in this paper. These preconditioners are all of low construction cost, which all could be taken as approximate inverse of M-matrices. Numerical experiments of these preconditioners applied with Krylov subspace methods show the effectiveness and performance, which also show that the preconditioners proposed in this paper are better approximate inverse for M-matrices than Simons'. [ABSTRACT FROM AUTHOR]

This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the non-stationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initial-boundary value problem has already been studied by the author who proved existence, uniqueness and continuous dependence on data for a suitable weak solution along some regularity results. A careful and detailed investigation of the variable time-step discretization is the goal of this paper. Thus, we deduce some estimates for the discretization error. These estimates depend only on data, impose no constraints between consecutive time-steps and show an optimal order of convergence. Finally, we prove another regularity result for the solution under stronger regularity assumptions on data. [ABSTRACT FROM AUTHOR]

The combination of geometrically nonlinear analysis with structural optimization against instability opens many possibilities for new formulations of the optimization problem. One of them is optimization against instability in the large. When a critical state does not exist and instability occurs at finite displacements either the lower critical loading is maximized or the generalized displacement for that load is minimized. This paper undertakes these new optimization problems providing formulations and numerical solutions for selected elements. The simple finite dimensional model is analysed first showing characteristic features of the problem, and then a real elastic element is optimized. [ABSTRACT FROM AUTHOR]

In the paper, a new trust region algorithm is given for nonlinear inequality constrained optimization problems. Motivated by a dual problem introduced by Han and Mangasarian [Han, S. P., Mangasarian, O. L. (1983). A dual differentiable exact penalty function. Math. Programming 25:293-306], which is a nonnegatively constrained maximization problem, we construct a trust region algorithm for solving the dual problem. At each iteration, we only need to minimize a quadratic subproblem with simple bound constraints. Under the condition that the iterate sequence generated by the algorithm is contained in some bounded closed set, any accumulation point of the sequence is a Karush- Kuhn-Tucker point of the original problem. [ABSTRACT FROM AUTHOR]

The aim of this paper is to generalize the results of Yamada et al. [Yamada, I., Ogura, N., Yamashita, Y., Sakaniwa, K. (1998). Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces. Numer. Funct. Anal. Optimiz. 19(l):165-190], and to provide complementary results to those of Deutsch and Yamada [Deutsch, F., Yamada, I. (1998). Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19(1&2):33-56] in which they consider the minimization of some function d over a closed convex set F, the nonempty intersection of N fixed point sets. We start by considering a quadratic function θ and providing a relaxation of conditions of Theorem 1 of Yamada et al. (1998) to obtain a sequence of fixed points of certain contraction maps, converging to the unique minimizer of θ over F. We then extend Theorem 2 and obtain a complementary result to Theorem 3 of Yamada et al. (1998) by replacing the condition lim n → ∞ (λn - λn+1)/λ²n+1 = 0 on the parameters by the more general condition lim n → ∞ λn / λn+1 = 1. We next look at minimizing a more general function θ than a quadratic function which was proposed by Deutsch and Yamada (1998) and show that the sequence of fixed points of certain maps converge to the unique minimizer of 9 over F. Finally, we prove a complementary result to that of Deutsch and Yamada (1998) by using the alternate condition on the parameters. [ABSTRACT FROM AUTHOR]

In many wavelets applications, a scalar product of given function with the scaling function has to be calculated. For deriving effective one point quadrature formulas, the relation among the first scaling moment and the second one is crucial. In this paper, new relations among scaling moments are deduced. [ABSTRACT FROM AUTHOR]

This paper presents a method for carrying out the calculus of the risk priority of failures in Failure Mode and Effect Analysis (FMEA). The novelty of the method consists of new management of data provided by the design team, normally given on qualitative scales, without necessitating an arbitrary and artificial numerical conversion. The practical effects of these issues are shown in an application example. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of analytic invariant curves of an iterative equationwhich is from Pielou's equation. By reducing the equation with the Schröder transformation to an auxiliary equation, the author discusses not only that the constantat resonance, i.e. at a root of the unity, but also thosenear resonance under the Brjuno condition. [ABSTRACT FROM AUTHOR]

In this paper, we give a review on rigorous and numerical results for amplitude equations. We focus on the Swift-Hohenberg equation and the Burgers' equation in order to determine the quality of the approximation and the impact of degenerate noise on the approximating equation. [ABSTRACT FROM AUTHOR]

The Reissner-Mindlin plate model for calculation of functionally graded materials has been proposed in literature by using shear correction coefficient of homogeneous model. However, this use is a priori not appropriate for the gradient material. Identification of the transverse shear factors is thus investigated in this paper. The transverse shear stresses are derived by using energy considerations from the expression of membrane stresses. Using the obtained transverse shear factor, a numerical analysis is performed on a simply supported FG square plate whose elastic properties are isotropic at each point and vary through the thickness according to a power law distribution. The numerical results of a static analysis are compared with available solutions from previous studies. [ABSTRACT FROM AUTHOR]

In this paper, we study the solution of a certain non-smooth eigenvalue problem, using operator-splitting methods to solve an equivalent, constrained minimization problem. We present the Marchuk-Yanenko and Peaceman-Rachford schemes for solving the problem and compare their performance numerically on some model problems. The Peaceman-Rachford scheme turns out to be superior to the Marchuk-Yanenko scheme in terms of accuracy and computational efficiency. [ABSTRACT FROM AUTHOR]

The aim of our paper is to provide a new class of five-phase anti-lock brake algorithms (that use wheel deceleration logic-based switching) and a simple mathematical background that explains their behavior. First, we completely characterize the conditions required for our algorithm to work. Next, we explain how to compute analytically an approximation of the Poincaré map of the system (without using numerical integration) and show how to calibrate the algorithm’s parameters to obtain the most efficient limit cycle. [ABSTRACT FROM AUTHOR]

Theon's ladder is an ancient algorithm for calculating rational approximations for . It features two columns of integers (called a ladder), in which the ratio of the two numbers in each row is an approximation to . It is remarkable for its simplicity. This algorithm can easily be generalized to find rational approximations to any square root. In this paper we show how Theon's original method is naturally generalized for the calculation of any root, , where 1

In the procedure of steady-state hierarchical optimization for large-scale industrial processes, it is often necessary that the control system responds to a sequence of step function-type control decisions with distinct magnitudes. In this paper a set of iterative learning controllers are de-centrally embedded into the procedure of the steady-state optimization. This generates upgraded sequential control signals and thus improves the transient performance of the discrete-time large-scale systems. The convergence of the updating law is derived while the intervention from the distinction of the scales is analysed. Further, an optimal iterative learning control scheme is also deduced by means of a functional derivation. The effectiveness of the proposed scheme and the optimal rule is verified by simulation. [ABSTRACT FROM AUTHOR]

Traditional 'pencil and paper' derivations of the numerical differentiation formulae for f ′[x0] and f″[x0] have been done independently as if there was no connection among the two derivations. This new approach gives a parallel development of the formulae. It requires the solution of a 'linear system' that includes symbolic quantities as coefficients and constants. It is shown how the power of a computer algebra system such as Mathematica can be used to elegantly solve this linear system for f′[x0] and f″[x0]. The extension to derivations of higher order numerical differentiation formulas for the central, forward or backward differences are also presented. [ABSTRACT FROM AUTHOR]

The article presents an example of a tricky linear algebra. It states that the trick starts when the instructor writes the number 65 on a paper and the instructor announces his psychic ability to predict sums in advance. Moreover, the numbers from 1-25 are then written consecutively in a 5-by-5 arrangement and a student is asked to choose any five numbers from this with the restriction that no two numbers can lie in the same column or row. It is further instructed that these numbers are then added together by the student before the instructor shows the paper with the number 65 written on it.

This paper is a report on an attempt to teach students in their first and second year of university how to write mathematics. The problems faced by these students are outlined and the system devised to emphasize the importance of communicating mathematics is explained. [ABSTRACT FROM AUTHOR]

We discuss a general class of trigonometric functions whose corresponding Fourier series can be used to calculate several interesting numerical series. Particular cases are presented. [ABSTRACT FROM AUTHOR]

In this study of two-step hybrid methods for second-order equations of the type y ″ = f ( y ), we apply P -series [Hairer, E., Lubich, C. and Wanner, G. (2002). Geometric Numerical Integration Structure-Preserving Algorithms for Ordinary Differential Equations . Springer Series in Computational Mathematics.] to formalise the approach of Chan [Chan, R. P. K. (2002). Two-step hybrid methods. Internal Publication .] to the order conditions, and present two characterizations of symmetry. Although order conditions can be obtained through the classical theory for the Nyström methods, it is of interest to derive particular simpler formulas for the class of two-step hybrid methods in order to facilitate the search for high-order methods. Moreover, the approach proves useful in analysing the symmetry of the hybrid methods. [ABSTRACT FROM AUTHOR]

Poincaré-Friedrichs inequalities are derived for piecewise H² functions on two dimensional domains. These inequalities can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerfcin methods. [ABSTRACT FROM AUTHOR]

A new approach to a solution of a nonlinear constrained mathematical programming problem and its Mond-Weir duals is introduced. An n-approximated problem associated with a primal nonlinear programming problem is presented that involves n-approximated functions constituting the primal problem. The equivalence between the original mathematical programming problem and its associated n-approximated optimization problem is established under invexity assumption. Furthermore, n-approximated dual problems in the sense of Mond-Weir are introduced for the obtained n-approximated optimization problem in this method. By the help of n-approximated dual problems some duality results are established for the original mathematical programming problem and its original duals. [ABSTRACT FROM AUTHOR]

The article presents a shortened classical proof of the quadratic reciprocity law. It states that in 1838 V. A. Lebesgue forwarded a proof by determining the number solutions in two different ways. Relative to this, a book by F. Lemmermeyer entitled "Reciprocity Laws: From Euler to Eisenstein" is also recommended as a basis for research on quadratic reciprocity law. Moreover, the author also employed various numerical calculations to identify the nature of quadratic reciprocity law. These numerical calculations manifest the nature of this law.

The adhesive contact between a sphere and a longitudinal wavy surface is simulated numerically. A modified simulation method is proposed using the Newton BI-CGSTAB method in a rectangular coordinate. The effective Tabor parameter is proposed. It is found that when the amplitude of the wavy surface is larger, the contact area is smaller and the pull-off force is smaller. Jump-in from noncontact phenomena occurs when the Tabor parameter is large. Jumping from one ridge to the next ridge occurs when the effect of the Tabor parameter is large and the amplitude of the wavy surface is not too small. Jumping from noncontact to full contact is affected by the amplitude and the wave number of the wavy surface and is also affected by the Tabor parameter. [ABSTRACT FROM PUBLISHER]

Statistical researcher M. Stone continues in his diligent search for flaws in the Bayesian theory of statistical inference. In the present paper he considers two examples in which Bayesian strong inconsistency can occur with a flat prior. Only a relatively few statisticians believe that a single theory of inference can be the answer for all of statistics. The committed Bayesian's, however, are prominent among such believers. There are, of course, substantial arguments against the Bayesian theory as a single theory of inference, a summary of these arguments may be may be found in D.A.S. Fraser. These arguments are not primarily concerned with the Bayesian method as a tool in the statistician's tool bag rather, they are concerned with the catholic claim for Bayesian theory and with the meaning and consequences of the theory in scientific contexts. The two examples considered by Stone are some what remote from Bayesian theory in a scientific context indeed, the theory will not rise or fall on the basis of the examples. Nevertheless the examples are extremely interesting they are concerned with implications of the theory, and they are presented with the attractive flair that we expect from Stone.

The inverse problems for a parabolic equation with an unknown source (space or time dependent) on the right-hand side are considered. The numerical method is based on the method of lines. The results of numerical experiments on test problems are given. [ABSTRACT FROM PUBLISHER]

We formulate general principles on the existence of homeomorphic absolutely continuous on lines (ACL) solutions for the Beltrami equations with degeneration and derive from them a series of criteria and, in particular, a generalization and strengthening of the well-known Lehto existence theorem. Furthermore, we prove that in all these cases there exist the so-called strong ring solutions satisfying additional moduli conditions which play a great role in the research of various properties of such solutions. [ABSTRACT FROM AUTHOR]

Modern Hindu-Arabic numeration is the end result of a long period of evolution, and is clearly superior to any system that has gone before, but is it optimal? We compare it to a hypothetical base 5 system, which we dub Predator arithmetic, and judge which of the two systems is superior from a mathematics education point of view. We find that complex calculations such as long multiplication can be carried out more efficiently in base 5 than in base 10, and that base 5 is in fact close to being optimal in this regard. We also show that base 5 is small enough so that the intuitiveness of simple grouping and the efficiency of fully ciphered numerals can be combined effectively in a single notation, something which Hindu-Arabic numeration tries but fails to achieve. Furthermore, as a consequence of these notational advantages, we show that the basic operations of arithmetic, addition and subtraction, also borrowing and carrying (regrouping), are easier to teach and to learn in base 5 than in base 10. Finally we show that, even though a shift from base 10 to base 5 may not be a realistic possibility, there are many ways in which the teaching of elementary arithmetic could be improved significantly, along the lines of Predator arithmetic, and which could be implemented at little cost within our current Hindu-Arabic system. [ABSTRACT FROM AUTHOR]

In this study we are concerned with the local convergence of a Newton-type method introduced by us [I.K. Argyros and D. Chen, On the midpoint iterative method for solving nonlinear equations in Banach spaces, Appl. Math. Lett. 5 (1992), pp. 7-9.] for approximating a solution of a nonlinear equation in a Banach space setting. This method has also been studied by Homeier [H.H.H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), pp. 227-230.] and Ozban [A.Y. Ozban, Some new variants of Newton's method, Appl. Math. Lett. 17 (2004), pp. 677-682.] in real or complex space. The benefits of using this method over other methods using the same information have been explained in [I.K. Argyros, Computational theory of iterative methods, in Studies in Computational Mathematics, Vol. 15, C.K. Chui and L. Wuytack, eds., Elsevier Science Inc., New York, USA, 2007.; I.K. Argyros and D. Chen, On the midpoint iterative method for solving nonlinear equations in Banach spaces, Appl. Math. Lett. 5 (1992), pp. 7-9.; H.H.H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003), pp. 227-230.; A.Y. Ozban, Some new variants of Newton's method, Appl. Math. Lett. 17 (2004), pp. 677-682.]. Here, we give the convergence radii for this method under a type of weak Lipschitz conditions proven to be fruitful by Wang in the case of Newton's method [X. Wang, Convergence of Newton's method and inverse function in Banach space, Math. Comput. 68 (1999), pp. 169-186 and X. Wang, Convergence of Newton's method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal. 20 (2000), pp. 123-134.]. Numerical examples are also provided. [ABSTRACT FROM AUTHOR]

Using W-transformation of Hairer and Wanner, high-order implicit Runge-Kutta methods based on Radau quadrature are studied. Methods constructed and presented include and extend the existing RadauIA, IB, IIA, IIB type RK methods. 2- and 3-stage general Radau RK methods with two parameters (δ, γ) are constructed. The RK methods with five parameters (α, β, γ, δ, τ), which include both Radau and Lobatto type methods, will be discussed at last. [ABSTRACT FROM AUTHOR]

The article offers information on a strong version of Liouville's theorem. This theorem states that every bounded holomorphic function on C is constant. The authors, through various mathematical problems and their corresponding solutions, present an elementary proof of this theorem. They were also able to obtain a corresponding statement for complex-value functions by applying Theorem 1.1. Moreover, different mathematical propositions and their corresponding numerical analysis and solutions are presented.

This article discusses a splitting extrapolation method for solving second-order parabolic equations with curved boundaries by using domain decomposition and d-quadratic isoparametric finite elements. This method possesses superconvergence, a high order of accuracy and a high degree of parallelism. First, we prove the multi-variable asymptotic expansion of fully discrete d-quadratic isoparametric finite element errors. Based on the expansion, we generate splitting extrapolation formulas. These formulas generate a numerical solution on a globally fine grid with higher accuracy by solving only a set of smaller discrete subproblems on different coarser grids. Therefore, a large-scale multidimensional problem with a curved boundary is turned into a set of smaller discrete subproblems on a polyhedron. Because these subproblems are independent of each other and have similar scales, our algorithm possesses a high degree of parallelism. In addition, this method is effective for solving discontinuous problems if we regard the interfaces of the problems as the interfaces of the initial domain decomposition. Our numerical results also show that the algorithm is effective for solving nonlinear parabolic equations. [ABSTRACT FROM AUTHOR]

We propose a time-splitting spectral method for the generalized Gross-Pitaevskii equations, which describe the dynamics of spinor F=1 Bose-Einstein condensates at a very low temperature. The new numerical method is explicit, unconditionally stable, and of spectral accuracy in space. Moreover, it conserves the position densities at the discretized level. We apply the method for studying both the dynamic generation of vortices and vortex lattice dynamics for the spinor F=1 Bose-Einstein condensates held in an Ioffe-Pritchard magnetic field. [ABSTRACT FROM AUTHOR]

Although Runge-Kutta and partitioned Runge-Kutta methods are known to formally satisfy discrete multisymplectic conservation laws when applied to multi-Hamiltonian PDEs, they do not always lead to well-defined numerical methods. We consider the case study of the nonlinear Schrödinger equation in detail, for which the previously known multisymplectic integrators are fully implicit and based on the (second order) box scheme, and construct well-defined, explicit integrators, of various orders, with local discrete multisymplectic conservation laws, based on partitioned Runge-Kutta methods. We also show that two popular explicit splitting methods are multisymplectic. [ABSTRACT FROM AUTHOR]

The article provides an optimization framework for polynomial zerofinders. The Newton's method is an iterative method for solving equations resulting to zero. There are two standard ways to derive the equations and these can be demonstrated algebraically via Taylor expansion and geometrically through tangent lines. Some of the resulting methods are applicable only to polynomial equations though others can also be used for general nonlinear equations. These approaches are not limited to the construction of zerofinders because they also help in carrying out a controlled enhancement of those zerofinders.

Ranking fuzzy numbers plays a very important role in decision-making problems. Existing centroid-index ranking methods have some drawbacks. In this article, a new centroid-index ranking method of fuzzy numbers is proposed. The proposed method is using the ideal of Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS). Some numerical examples show that the new method can overcome the drawbacks of the existing methods. Finally, a human selection problem is used to illustrate the efficiency of the proposed fuzzy ranking method. [ABSTRACT FROM AUTHOR]

In this article, we propose a method to identify contamination events (location and time of release) by enhancing a mathematical method originally proposed by Carasso et al. (Carasso, A., Sanderson, J.G. and Hyman, J.M., 1978, Digital removal of random media image degradation by solving the diffusion equation backward in time. SIAM Journal of Numerical Analysis, 15(4)). The method of the Marching-Jury Backward Beam/Plate Equation, applied earlier to groundwater problems, is enhanced and coupled to discrete Fourier transform processing techniques to solve a two-dimensional (2D) advection-dispersion transport problem with homogeneous and isotropic parameters backwards in time. (Atmadja, J. and Bagtzoglou, A.C., 2001a, Pollution source identification in heterogeneous porous media. Water Resources Research, 37(8), 2113–2125; Bagtzoglou, A.C. and Atmadja, J., 2003, The marching-jury backward beam equation and quasi-reversibility methods for hydrologic inversion: application to contaminant plume spatial distribution recovery. Water Resources Research, 39(2), 1038. Cornacchiulo, D. and Bagtzoglou, A.C., 2002, The marching-jury backward plate equation for contaminant plume spatial distribution recovery in two-dimensional heterogeneous media: Computational Issues, In: S.M. Hassanizadeh, R.J. Schotting, W.G. Gray and G.F. Pinder (Eds.) Computational Methods for Subsurface Flow and Transport (Netherlands: Elsevier Publishers) pp. 461–468. The difficulties associated with this ill-posed, inverse problem are well-recognized (Atmadja, J. and Bagtzoglou, A.C., 2001b, State-of-the-art report on mathematical methods for groundwater pollution source identification. Environmental Forensics, 2(3), 205–214). We, therefore, enhance the method by integrating an optimization scheme that takes as input parameters the stabilization parameter, the transport velocities, and the coefficient of diffusion. The objective function is set as an equally weighted sum of different mass and peak errors that can be calculated based on a combination of exhaustive contaminant coverage at specific points in time (e.g., lidar) and/or point data collected at a continuously monitored network of chemical sensors or biosensors, which may be stationary or mobile. [ABSTRACT FROM AUTHOR]

We complete our basic presentation of the α-calculus. A generalized Stokes theorem is given. Several explicit calculations are done for analytic functions. We propose an extension of Fourier transform theory. We also give several complementary results. [ABSTRACT FROM AUTHOR]

The aim of this article is to study necessary optimality conditions for a vector minimization program involving locally Lipschitz functions under certain general regularity conditions. We study problems involving only inequality and both inequality and equality constraints. [ABSTRACT FROM AUTHOR]

A coupled pair of first order nonlinear discrete hierarchical age-structured models are applied to study two modes of intraspecific competitions; scramble and contest. The study focuses on several comparisons of the dynamical outcomes of the two competitions. For a constant resource, it is shown, using analytical and numerical approaches, that solutions of the contest model monotonically equilibrate, while solutions of the scramble model oscillate and become chaotic. It is also shown that the inherent net reproductive number of each population affects the comparison of equilibrium points in the two populations. By considering cases on the resource and model parameters, the local as well as the global stability of nontrivial equilibrium points are studied. The impact of a contest and a scramble consumer on a time dependent resource is considered numerically. [ABSTRACT FROM AUTHOR]

Where Youden's extreme rank sum test is used, an early decision is often possible by performing the trials in sequence with the following method. After each trial is completed, the greatest lower bound and least upper bound of both extreme rank sums are calculated for all remaining trials. If the critical regions cover either or both intervals of the statistics or the intervals lie completely outside the critical regions, the round-robin test is terminated with the acceptance of the proper hypothesis. [ABSTRACT FROM AUTHOR]

Results are developed for play-the-winner sampling for choosing the better of two binomial populations when the maximum number of tests is specified. It is shown that for a fixed number of tests the probability of correct selection with alternating assignment exceeds that for play- the-winner sampling. However, when the probability of correct selection is fixed, neither sampling method is uniformly better than the other as measured by the expected number of tests on the poorer population, or on the total expected number of tests. [ABSTRACT FROM AUTHOR]