Search

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]

In the paper cited in the heading, we treated the problem of stabilizing a flexible cable with two rigid loads, which was described by two kinds of hyperbolic equations. To show the asymptotic stability of the closed-loop system with a controller derived there, we used the LaSalle's invariance principle. However, in that paper, we need to supplement the proof of Theorem 5.1 and to revise the proof of Theorem 5.2. Throughout this note, we use the same notation as in the paper cited in the heading. [ABSTRACT FROM PUBLISHER]

The purpose of the paper is to determine fine spectrum of newly introduced operator Δν on the sequence space c0. The operator Δν on c0 is defined by Δνχ = (νnχn - νn-1χn-1)n=0∞ with χ-1 = 0, where ν = (νk) is either constant or strictly decreasing sequence of positive real numbers such that lim νk = L > 0 and sup νk ≤ 2L. In this paper, it is shown that spectrum (These equations cannot be represented into ASCII text), the point spectrum σp(Δν,c0) = ϕ if ν is a constant and σp(Δν,c0) = {νn} if ν is a strictly decreasing sequence. We have also obtained the results on continuous spectrum σc(Δν,c0), residual spectrum σr(Δν,c0) and fine spectrum of the operator Δν on c0. [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]

This paper is concerned with refinement equations of the type [This symbol cannot be presented in ASCII format] where f is the unknown function defined on the s-dimensional Euclidean space Rs, a is a finitely supported sequence on Zs, and M is an s × s dilation matrix with m := | det M|. The solution of a refinement equation can be obtained by using the subdivision scheme associated with the mask. In this paper we give a characterization for the convergence of the subdivision scheme when the mask is nonnegative. Our method is to relate the problem of convergence to m column-stochastic matrices induced by the mask. In this way, the convergence of the subdivision scheme can be determined in a finite number of steps by checking whether each finite product of those column-stochastic matrices has a positive row. As a consequence of our characterization, we show that the convergence of the subdivision scheme with a nonnegative mask depends only on the location of its positive coefficients. Several examples are provided to demonstrate the power and applicability of our approach. [ABSTRACT FROM AUTHOR]

This paper studies the boundedness, global asymptotic stability, and periodicity of positive solutions of the equation xn = f(xn-2)/g(xn-1), n ϵ ℕ0, where f, g ϵ C[(0,∞), (0,∞)]. It is shown that if f and g are nondecreasing, then for every solution of the equation the subsequences {x2n} and {x2n-1} are eventually monotone. For the case when f(x) = a + βx and g satisfies the conditions g(0) = 1, g is nondecreasing, and x/g(x) is increasing, we prove that every prime periodic solution of the equation has period equal to one or two. We also investigate the global periodicity of the equation, showing that if all solutions of the equation are periodic with period three, then f(x) = c1/x and g(x) = c2x, for some positive c1 and c2. [ABSTRACT FROM AUTHOR]

The filter generator is an important building block in many stream ciphers. The generator consists of a linear feedback shift register of length n that generates an m-sequence of period 2′ - 1 filtered through a Boolean function of degree d that combines bits from the shift register and creates an output bit zt at any time t. The previous best attacks aimed at reconstructing the initial state from an observed keystream, have essentially reduced the problem to solving a nonlinear system of D = (Multiple line equation(s) cannot be represented in ASCII text) (i) equations in n unknowns using techniques based on linear algebra. This attack needs about D bits of keystream and the system can be solved in complexity O (Dω), where ω can be taken to be Strassen's reduction exponent ω = log2 (7) ≈ 2.807. This paper describes a new algorithm that recovers the initial state of most filter generators after observing O(D) keystream bits with complexity O((D - n)/2) ≈ O(D), after a pre-computation with complexity O(D(log2 D)³). [ABSTRACT FROM AUTHOR]

In this paper, it is proved that "Multiple line equation(s) cannot be represented in ASCII text." Where "Multiple line equation(s) cannot be represented in ASCII text." [ABSTRACT FROM AUTHOR]

In this correspondence, we point out an error in equation (4) in the above paper. We also correct equation (4) by a detailed mathematical deduction. [ABSTRACT FROM AUTHOR]

The notion of extended-well-posedness has been introduced by Zolezzi for scalar minimization problems and has been further generalized to vector minimization problems by Huang. In this paper, we study the extended well-posedness properties of vector minimization problems in which the objective function is C-quasiconvex. To achieve this task, we first study some stability properties of such problems. [ABSTRACT FROM AUTHOR]

This paper presents a new method to isolate all configurations that a multiloop linkage can adopt. The problem is tackled by means of formulation and resolution techniques that fit particularly well together. The adopted formulation yields a system of simple equations (only containing linear, bilinear, and quadratic monomials, and trivial trigonometric terms for the helical pair only) whose structure is later exploited by a branch-and-prune method based on linear relaxations. The method is general, as it can be applied to linkages with single or multiple loops with arbitrary topology, involving lower pairs of any kind, and complete, as all possible solutions get accurately bounded, irrespective of whether the linkage is rigid or mobile. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to determine fine spectrum of newly introduced operator Δν on the sequence space c0. The operator Δν on c0 is defined by Δνχ = (νnχn - νn-1χn-1)n=0∞ with χ-1 = 0, where ν = (νk) is either constant or strictly decreasing sequence of positive real numbers such that lim νk = L > 0 and sup νk ≤ 2L. In this paper, it is shown that spectrum (These equations cannot be represented into ASCII text), the point spectrum σp(Δν,c0) = ϕ if ν is a constant and σp(Δν,c0) = {νn} if ν is a strictly decreasing sequence. We have also obtained the results on continuous spectrum σc(Δν,c0), residual spectrum σr(Δν,c0) and fine spectrum of the operator Δν on c0. [ABSTRACT FROM AUTHOR]

Using the fixed point index, we study positive solutions of the multi-point fourth-order nonlinear boundary value problem u(4)(t) = λf (u(t)); 0 < t < 1, u(2i+1)(0) = 0, αiu(2i)(ηi) = u(2i)(1), i = 0,1, where (These equations cannot be represented into ASCII text). The intervals for the parameter λ are obtained for which the existence of at least one solution is guaranteed. [ABSTRACT FROM AUTHOR]

A recent study [D. N. Arnold, D. Boffi, and R. S. Falk, SIAM J. Numer. Anal., 42 (2005), pp. 2429-2451] reveals that convergence of finite element methods using H(div , O)-compatible finite element spaces deteriorates on nonaffine quadrilateral grids. This phenomena is particularly troublesome for the lowest-order Raviart-Thomas elements, because it implies loss of convergence in some norms for finite element solutions of mixed and least-squares methods. In this paper we propose reformulation of finite element methods, based on the natural mimetic divergence operator [M. Shashkov, Conservative Finite Difference Methods on General Grids, CRC Press, Boca Raton, FL, 1996], which restores the order of convergence. Reformulations of mixed Galerkin and leastsquares methods for the Darcy equation illustrate our approach. We prove that reformulated methods converge optimally with respect to a norm involving the mimetic divergence operator. Furthermore, we prove that standard and reformulated versions of the mixed Galerkin method lead to identical linear systems, but the two versions of the least-squares method are veritably different. The surprising conclusion is that the degradation of convergence in the mixed method on nonaffine quadrilateral grids is superficial, and that the lowest-order Raviart-Thomas elements are safe to use in this method. However, the breakdown in the least-squares method is real, and there one should use our proposed reformulation. [ABSTRACT FROM AUTHOR]

This paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions. We introduce what we call the geometric corrector, which is a sequence associated with every finite volume mesh in $\mathbf{R}^{nd}$ and every nonvanishing vector $\mathbf{a}$ of $\mathbf{R}^{nd}$. First we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size, then an order one error estimate for the finite volume scheme occurs. Afterwards we prove that this norm is indeed bounded by the mesh size in several cases, including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions. Computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases, such as the one with independent refined meshes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multilithology diffusion model used in the field of stratigraphic basin simulations to simulate large scale depositional transport processes of sediments described as a mixture of L lithologies. This model is a simplified one for which the surficial fluxes are proportional to the slope of the topography and to a lithology fraction with unitary diffusion coefficients. The main variables of the system are the sediment thickness h, the L surface concentrations cis in lithology i of the sediments at the top of the basin, and the L concentrations ci in lithology i in the sediments inside the basin. For this simplified model, the sediment thickness decouples from the other unknowns and satisfies a linear parabolic equation. The remaining equations account for the mass conservation of the lithologies, and couple, for each lithology, a first order linear equation for cis with a linear advection equation for ci for which cis appears as an input boundary condition. For this coupled system, a weak formulation is introduced. The system is discretized by an implicit time integration and a cell centered finite volume method. This numerical scheme is shown to satisfy stability estimates and to converge, up to a subsequence, to a weak solution of the problem. [ABSTRACT FROM AUTHOR]

In this paper we develop techniques for computing elementwise conservative approximations of the flux on element boundaries for the continuous Galerkin method. The technique is based on computing a correction of the average normal flux on an edge or face. The correction is a jump in a piecewise constant or linear function. We derive a basic algorithm which is based on solving a global system of equations and a parallel algorithm based on solving local problems on stars. The methods work on meshes with different element types and hanging nodes. We prove existence, uniqueness, and optimal order error estimates. Lastly, we illustrate our results by a few numerical examples. [ABSTRACT FROM AUTHOR]

We consider a parabolic equation with a drift term Δu+b∇u-ut=0. Under the condition divb=0, we prove that solutions possess dramatically better regularity than those provided by standard theory. For example, we prove continuity of solutions when not even boundedness is expected. [ABSTRACT FROM AUTHOR]

This paper is concerned with the ill-posed problems of identifying a parameter in an elliptic equation which appears in may applications in science and industry. Its solution is obtained by applying trust region method to a nonlinear least squares error problem. Trust region method has long been a popular method for well-posed problems. This paper indicates that it is also suitable for all-posed problems. Numerical experiment is given to compare the trust region method with the Tikhonov regularization method. It seems that the trust region method is more promising. [ABSTRACT FROM AUTHOR]

In this paper linear periodic systems of difference equations are classified with respect to periodic similarity and kinematic similarity. Complete sets of invariants of periodic difference equations relative to such similarity transformations are given, and corresponding canonical forms are described. Also the irreducible periodic difference equations, i.e., those that cannot be reduced by such similarities to a nontrivial direct sum, are identified. [ABSTRACT FROM AUTHOR]

This paper is an extension of previously published work by Bernard O. Koopman to the general case of any number of tasks and any effect function. A systematic method for the numerical calculation of the maximum effect and its corresponding optimum distribution is presented in a form which is also suitable for solution on an automatic digital computer. For the special case of additive returns which saturate, a special graphical or numerical method for any number of tasks is presented with an example worked out in detail A similar method for multiplicative effects is sketched. [ABSTRACT FROM AUTHOR]

Several corrections to the articles including "Finding Vintage Concentrations in a Sherry Solera," published in the 1990 issue, "Mathserve Lives On!," published in the 2004 issue, and "Author's Commentary: The Outstanding Kidney Exchange Papers," published in the 2007 issue are presented.

We prove the boundary Harnack principle for ratios of solutions u/ v of non-divergence second order elliptic equations Lu = a D u + b D u = 0 in a bounded domain Ω ⊂ $ {\mathbb R} $. We assume that b ∈ L(Ω) and Ω is a twisted Hölder domain of order α ∈ (1/2, 1]. Based on this result, we derive the Hölder regularity of u/ v for uniform domains. Bibliography: 27 titles. [ABSTRACT FROM AUTHOR]

In this paper, a new conjugate gradient formula βNewk is given to compute the search directions for unconstrained optimization problems. General convergence results for the proposed formula with some line searches such as the exact line search, the Grippo-Lucidi line search and the Wolfe-Powell line search are discussed. Under the above line searches and some assumptions, the global convergence properties of the given methods are discussed. The given formula βNewk ⩾ 0, and the search directions dk which are generated by the given βNewk under the strong Wolfe-Powell line search satisfy the sufficient descent condition. Preliminary numerical results show that the proposed methods are efficient. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence of solutions of a system of self-referred and hereditary differential equations. The initial data are assumed to be lower semi-continuous. We also formulate some open questions. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents new families of Newton-type iterative methods (Newton, Halley and Chebyshev methods) for finding simple zero of univariate non-linear equation, permitting in the vicinity of the root. Newton-type iterative methods have well-known geometric interpretation and admit their geometric derivation from a parabola. These algorithms are comparable to the well-known powerful classical methods of Newton, Halley and Chebyshev respectively, and these can be seen as special cases of these families. The efficiency of the presented methods is demonstrated by numerical examples. [Copyright &y& Elsevier]

Abstract: In this paper, we suggest and analyze a new three-step iterative method for solving nonlinear equations involving only first derive of the function using a new decomposition technique which is due to Noor [M. Aslam Noor, Numerical Analysis and Optimization, Lecture Notes, COMSATS Institute of Information Technology, Islamabad, Pakistan, 2006; M. Aslam Noor, New iterative schemes for nonlinear equations, Appl. Math. Comput., in press] and Noor and Noor [M. Aslam Noor, K. Inayat Noor, Some iterative schemes for nonlinear equations, Appl. Math. Comput., in press]. We show that this new iterative method has fourth-order convergence. Several numerical examples are given to illustrate the efficiency and performance of the new method. [Copyright &y& Elsevier]

Abstract: In the present paper, we introduce the generic extension graph G of a Dynkin or cyclic quiver Q and then compare this graph with the crystal graph C for the quantized enveloping algebra associated to Q via two maps ℘ Q , Q : Ω → ΛQ induced by generic extensions and Kashiwara operators, respectively, where Λ Q is the set of isoclasses of nilpotent representations of Q, and Ω is the set of all words on the alphabet I, the vertex set of Q. We prove that, if Q is a (finite or infinite) linear quiver, then the intersection of the fibres ℘ Q −1 (λ) and KQ −1 (λ) is non-empty for every λ ∈ Λ Q . We will also show that this non-emptyness property fails for cyclic quivers. [Copyright &y& Elsevier]

A cell-centered finite volume method is proposed to approximate numerically the solution to the steady convection-diffusion equation on unstructured meshes of $d$-simplexes, where $d\geq 2$ is the spatial dimension. The method is formally second-order accurate by means of a piecewise linear reconstruction within each cell and at mesh vertices. An algorithm is provided to calculate nonnegative and bounded weights. Face gradients, required to discretize the diffusive fluxes, are defined by a nonlinear strategy that allows us to demonstrate the existence of a maximum principle. Finally, a set of numerical results documents the performance of the method in treating problems with internal layers and solutions with strong gradients. [ABSTRACT FROM AUTHOR]

We obtain error bounds for monotone approximation schemes of Hamilton--Jacobi--Bellman equations. These bounds improve previous results of Krylov and the authors. The key step in the proof of these new estimates is the introduction of a switching system which allows the construction of approximate, (almost) smooth supersolutions for the Hamilton--Jacobi--Bellman equation. [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]

The paper presents a hybrid Trefftz (HT) element approach for the numerical solution of transient plate bending problems. In the proposed method, the dynamic plate equation is first discretized with respect to time and then the resulting set of elliptic equations is solved by the corresponding time independent hybrid Trefftz element approach. Two examples are considered to assess the effectiveness of the numerical method. [ABSTRACT FROM AUTHOR]

For transient thermal problems, discretizing in space by the finite element method yields a system of first-order ordinary differential equations which can be integrated in time by a finite difference based predictor-multicorrector scheme. At each time level the temperature field is obtained by solving a set of algebraic equations. In this paper the multigrid method is adopted for transient thermal problems. In order to model accurately the geometry of practical problems with various material properties, and to use the predictor-corrector scheme for the nonlinear transient analysis, the generalized multigrid approach, similar to that proposed in Reference 1, is employed. Some examples are included to illustrate the performance of this approach for nonlinear transient problems. [ABSTRACT FROM AUTHOR]

We show how the derivation of the Derrida–Lebowitz–Speer–Spohn equation can be prolonged to obtain a new equation, generalizing the models obtained in the paper by these authors. We then investigate its properties from both an analytical and numerical perspective. Specifically, a numerical method is presented to approximate solutions of the prolonged equation. Using this method, we investigate the relationship between the solutions of the prolonged equation and the Tracy–Widom GOE distribution. [ABSTRACT FROM AUTHOR]

We prove a general formula for a certain class of sums of tails. A new proof of some known identities is given, and a general identity that appears to be the first of its kind is established. [ABSTRACT FROM AUTHOR]

We consider the cubic defocusing nonlinear Schrödinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This behavior is quantified by the growth of higher Sobolev norms: given any δ≪1, K≫1, s>1, we construct smooth initial data u0 with $\|u_{0}\|_{{H}^{s}}<\delta$, so that the corresponding time evolution u satisfies $\|u(T)\|_{{H}^{s}}>K$ at some time T. This growth occurs despite the Hamiltonian’s bound on $\|u(t)\|_{\dot{H}^{1}}$ and despite the conservation of the quantity $\|u(t)\|_{L^{2}}$. The proof contains two arguments which may be of interest beyond the particular result described above. The first is a construction of the solution’s frequency support that simplifies the system of ODE’s describing each Fourier mode’s evolution. The second is a construction of solutions to these simpler systems of ODE’s which begin near one invariant manifold and ricochet from arbitrarily small neighborhoods of an arbitrarily large number of other invariant manifolds. The techniques used here are related to but are distinct from those traditionally used to prove Arnold Diffusion in perturbations of Hamiltonian systems. [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]

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]

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]

We show that under a suitable transversality condition, the intersection of two rational subtori in an algebraic torus (ℂ*)n is a finite group which can be determined using the torsion part of some associated lattice. We also give applications to the study of characteristic varieties of smooth complex algebraic varieties. As an example we discuss A. Suciu's line arrangement, the so-called deleted B3-arrangement. [ABSTRACT FROM AUTHOR]

We study the matrix equation C(BXC)†B = X†, where X† denotes the Moore-Penrose inverse. We derive conditions for the consistency of the equation and express all its solutions using singular vectors of B and C. Applications to compliance matrices in molecular dynamics, to mixed reverse-order laws of generalized inverses and to weighted Moore-Penrose inverses are given. [ABSTRACT FROM AUTHOR]

A hyperplane search tree is a binary tree used to store a set S of n d-dimensional data points. In a random hyperplane search tree for S, the root represents a hyperplane defined by d data points drawn uniformly at random from S. The remaining data points are split by the hyperplane, and the definition is used recursively on each subset. We assume that the data are points in general position in Rd. We show that, uniformly over all such data sets S, the expected height of the hyperplane tree is not worse than that of the k-d tree or the ordinary one-dimensional random binary search tree, and that, for any fixed d ≥ 3, the expected height improves over that of the standard random binary search tree by an asymptotic factor strictly greater than one. [ABSTRACT FROM AUTHOR]

A new discontinuous Galerkin finite element method for solving diffusion problems is introduced. Unlike the traditional local discontinuous Galerkin method, the scheme called the direct discontinuous Galerkin (DDG) method is based on the direct weak formulation for solutions of parabolic equations in each computational cell and lets cells communicate via the numerical flux ux only. We propose a general numerical flux formula for the solution derivative, which is consistent and conservative; and we then introduce a concept of admissibility to identify a class of numerical fluxes so that the nonlinear stability for both one-dimensional (1D) and multidimensional problems are ensured. Furthermore, when applying the DDG scheme with admissible numerical flux to the 1D linear case, kth order accuracy in an energy norm is proven when using kth degree polynomials. The DDG method has the advantage of easier formulation and implementation and efficient computation of the solution. A series of numerical examples are presented to demonstrate the high order accuracy of the method. In particular, we study the numerical performance of the scheme with different admissible numerical fluxes. [ABSTRACT FROM AUTHOR]

Thermomagnetic recording uses local heating of ferromagnetic media to locally decrease coercivity and change saturation magnetization of the material and alleviate magnetization reversal. A modified Landau-Lifshitz equation is proposed as a phenomenological model to allow for changes in magnetization magnitude and is studied both analytically and numerically. [ABSTRACT FROM AUTHOR]

A BDDC (balancing domain decomposition by constraints) method is developed for elliptic equations, with discontinuous coefficients, discretized by mortar finite element methods for geometrically nonconforming partitions in both two and three space dimensions. The coarse component of the preconditioner is defined in terms of one mortar constraint for each edge/face, which is the intersection of the boundaries of a pair of subdomains. A condition number bound of the form C maxi{(1 + log(Hi/hi))2} is established under certain assumptions on the geometrically nonconforming subdomain partition in the three-dimensional case. Here Hi and hi are the subdomain diameters and the mesh sizes, respectively. In the geometrically conforming case and the geometrically nonconforming cases in two dimensions, no assumptions on the subdomain partition are required. This BDDC preconditioner is also shown to be closely related to the Neumann-Dirichlet version of the FETI-DP algorithm. The results are illustrated by numerical experiments which confirm the theoretical results. [ABSTRACT FROM AUTHOR]

The article presents a study which examines the stability of a new pexider-type functional equation. This functional equation is f1(x+y+z)+f2(x-y)+f3(z-x)f4(x-y-z)-f5(x+y)-f6(x+z) = 0, a mixed of a quadratic and an additive functional equations. It established the generalized Hyers-Ulam-Rassias stability for new pexider-type functional equation on the punctured domain V\ {0} and obtain its general solution from the stability results. It also let V and X be normed space and Banach space, respectively. Results of the applications and stability of the equation are also given.

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]

Constitutive equations that describe the experimentally observed failure waves are proposed to model inelastic strains of brittle materials. The complete system of equations is hyperbolic, each equation of this system has divergent form. The model is based on the assumption that continual failure is the process of transition from an intact state to a “fully damaged” state described by the kinetics of the order parameter. The structure of stationary traveling compressive waves is analyzed using a simplified model. It is shown that in a certain range of amplitudes, the wave splits into an elastic precursor and a failure wave. [ABSTRACT FROM AUTHOR]

The article presents a discussion that focuses on the two methods of sharpening Jordan's inequality. It is mentioned that the first method conveys the possibility to come up with new strengthened Jordan's inequalities from old ones. On the other hand, it is also said that the second method shows that Jordan's inequality may be sharpened by choosing appropriate functions in the monotone form of L'Hopital's rule. A background of Jordan's inequality is also disclosed. Theorems that support the sharpening of Jordan's inequality are also provided.