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In this paper, we present a local Csörgö-Révész type functional limit theorem for increments of Brownian motion and give its convergence rate. The results also extend the functional forms of Lévy’s modulus of continuity for Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper, we unify previous definitions of weakened open subsets in a given topological space. We felt necessary to make this unification since we observed recently too many definitions, actually, more or less significant. We also show that our new framework is more general than the known supra-topological structure. Finally we show that, in our new framework, we can define well several standard topologies like concepts such as convergence, continuity, and compactness. We expect that our new framework could find application in images processing. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the strong convergence of an implicit iteration process with errors to a common fixed point for a finite family of asymptotically pseudocontractive mappings and nonexpansive mappings in normed linear spaces. The results in this paper improve and extend the corresponding results of Xu and Ori, Zhou and Chang, Sun, Yang and Yu in some aspects. [ABSTRACT FROM AUTHOR]

Let B = ( B( t), . . . , B( t)) be a d-dimensional fractional Brownian motion with Hurst index α < 1/4, or more generally a Gaussian process whose paths have the same local regularity. Defining properly iterated integrals of B is a difficult task because of the low Hölder regularity index of its paths. Yet rough path theory shows it is the key to the construction of a stochastic calculus with respect to B, or to solving differential equations driven by B. We intend to show in a series of papers how to desingularize iterated integrals by a weak, singular non-Gaussian perturbation of the Gaussian measure defined by a limit in law procedure. Convergence is proved by using 'standard' tools of constructive field theory, in particular cluster expansions and renormalization. These powerful tools allow optimal estimates and call for an extension of Gaussian tools such as, for instance, the Malliavin calculus. After a first introductory paper (Magnen and Unterberger in From constructive theory to fractional stochastic calculus. (I) An introduction: rough path theory and perturbative heuristics, ), this one concentrates on the details of the constructive proof of convergence for second-order iterated integrals, also known as Lévy area. A summary in French may be found in Unterberger (Mode d'emploi de la théorie constructive des champs bosoniques, avec une application aux chemins rugueux, ). [ABSTRACT FROM AUTHOR]

We extend some results due to Thanh-Hao (Acta Math. Vietnam. 31: 283-289, 2006) and Noor (J. Optim. Theory Appl. 115:447-452, 2002). The first paper established a convergence theorem for the Tikhonov regularization method (TRM) applied to finite-dimensional pseudomonotone variational inequalities (VIs), answering in the affirmative an open question stated by Facchinei and Pang (Finite- Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003). The second paper discussed the application of the proximal point algorithm (PPA) to pseudomonotone VIs. In this paper, new facts on the convergence of TRM and PPA (both the exact and inexact versions of PPA) for pseudomonotone VIs in Hilbert spaces are obtained and a partial answer to a question stated in (Acta Math. Vietnam. 31:283-289, 2006) is given. As a byproduct, we show that the convergence theorem for inexact PPA applied to infinite-dimensional monotone variational inequalities can be proved without using the theory of maximal monotone operators. [ABSTRACT FROM AUTHOR]

This paper proposes several globally convergent geometric optimization algorithms on Riemannian manifolds, which extend some existing geometric optimization techniques. Since any set of smooth constraints in the Euclidean space Rn (corresponding to constrained optimization) and the Rn space itself (corresponding to unconstrained optimization) are both special Riemannian manifolds, and since these algorithms are developed on general Riemannian manifolds, the techniques discussed in this paper provide a uniform framework for constrained and unconstrained optimization problems. Unlike some earlier works, the new algorithms have less restrictions in both convergence results and in practice. For example, global minimization in the one-dimensional search is not required. All the algorithms addressed in this paper are globally convergent. For some special Riemannian manifold other than Rn, the newalgorithms are very efficient. Convergence rates are obtained. Applications are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some existence results for a class of singular semilinear elliptic problems where we improve some earlier results of Zhijun Zhang. We show the existence of entire positive solutions without the monotonic condition imposed in Zhang’s paper. The main point of our technique is to choose an approximating sequence and prove its convergence. The desired compactness can be obtained by the Sobolev embedding theorems. [ABSTRACT FROM AUTHOR]

In this paper, a new sequential penalty algorithm, based on the L[sub ∞] exact penalty function, is proposed for a general nonlinear constrained optimization problem. The algorithm has the following characteristics: it can start from an arbitrary initial point; the feasibility of the subproblem is guaranteed; the penalty parameter is adjusted automatically; global convergence without any regularity assumption is proved. The update formula of the penalty parameter is new. It is proved that the algorithm proposed in this paper behaves equivalently to the standard SQP method after sufficiently many iterations. Hence, the local convergence results of the standard SQP method can be applied to this algorithm. Preliminary numerical experiments show the efficiency and stability of the algorithm. [ABSTRACT FROM AUTHOR]

The backward heat equation is a typical ill-posed problem. In this paper, we shall apply a dual least squares method connecting Shannon wavelet to the following equation Motivated by Regińska's work, we shall give two nonlinear approximate methods to regularize the approximate solutions for high-dimensional backward heat equation, and prove that our methods are convergent. [ABSTRACT FROM AUTHOR]

The Letac principle says that if the backward iterations of a random function converge almost surely, then the forward iterations converge in distribution. In this paper, we find conditions for the inverse implication to hold. [ABSTRACT FROM AUTHOR]

It is well known that the convergence of multivariate subdivision schemes with finite masks can be characterized via joint spectral radius. For nonnegative masks, we will present in this paper some computable simply sufficient conditions for the convergence, which will cover a substantially large class of schemes. [ABSTRACT FROM AUTHOR]

Quite recently, Savaş (Appl Math Lett 21:134-141, ), defined the lacunary statistical analogue for double sequence $$X=\{X_{k,l}\}$$ of fuzzy numbers as follows: a double sequence $$X=\{X_{k,l}\}$$ is said to be lacunary P-statistically convergent to $$X_{0}$$ provided that for each $$\epsilon >0$$In this paper we introduce and study double lacunary $$\sigma$$-statistical convergence for sequence of fuzzy numbers and also we get some inclusion theorems. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new kind of Lax-Oleinik type operator with parameters associated with positive definite Lagrangian systems for both the time-periodic case and the time-independent case. On one hand, the family of new Lax-Oleinik type operators with an arbitrary $${u \in C(M, \mathbb{R}^1)}$$ as initial condition converges to a backward weak KAM solution in the time-periodic case, while it was shown by Fathi and Mather that there is no such convergence of the Lax-Oleinik semigroup. On the other hand, the family of new Lax-Oleinik type operators with an arbitrary $${u \in C(M, \mathbb{R}^1)}$$ as initial condition converges to a backward weak KAM solution faster than the Lax-Oleinik semigroup in the time-independent case. [ABSTRACT FROM AUTHOR]

Let f∈ L be a real-valued even function with its Fourier series $\frac {a_{0}}{2}+\sum\limits_{n=1}^{\infty}{a_{n}}\cos nx$ , and let S( f, x) be the nth partial sum of the Fourier series, n≧1. The classical result says that if the nonnegative sequence { a} is decreasing and $\lim\limits_{n\to\infty} a_{n} =0$ , then $\lim\limits_{n\to\infty} \|{f-S_{n}(f)}\|_{L}=0$ if and only if $\lim\limits_{n\to\infty} a_{n}\log n=0$ . Later, the monotonicity condition set on { a} is essentially generalized to MVBV (Mean Value Bounded Variation) condition. Very recently, Kórus further generalized the condition in the classical result to the so-called GM condition in real space. In this paper, we give a complete generalization to the complex space. [ABSTRACT FROM AUTHOR]

In this paper, we considered a power of a non-Hermitian random matrix and prove that the empirical distribution of its singular values converges to the Fuss-Catalan distribution almost surely. Bibliography: 3 titles. [ABSTRACT FROM AUTHOR]

The matrix rank minimization problem has applications in many fields, such as system identification, optimal control, low-dimensional embedding, etc. As this problem is NP-hard in general, its convex relaxation, the nuclear norm minimization problem, is often solved instead. Recently, Ma, Goldfarb and Chen proposed a fixed-point continuation algorithm for solving the nuclear norm minimization problem ( Math. Program., doi:, ). By incorporating an approximate singular value decomposition technique in this algorithm, the solution to the matrix rank minimization problem is usually obtained. In this paper, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving affinely constrained matrix rank minimization problems are reported. [ABSTRACT FROM AUTHOR]

Nonlinear optimization algorithms could be divided into local exploitation methods such as Nelder-Mead (NM) algorithm and global exploration ones, such as differential evolution (DE). The former searches fast yet could be easily trapped by local optimum, whereas the latter possesses better convergence quality. This paper proposes hybrid differential evolution and NM algorithm with re-optimization, called as DE-NMR. At first a modified NM, called NMR is presented. It re-optimizes from the optimum point at the first time and thus being able to jump out of local optimum, exhibits better properties than NM. Then, NMR is combined with DE. To deal with equal constraints, adaptive penalty function method is adopted in DE-NMR, which relaxes equal constraints into unequal constrained functions with an adaptive relaxation parameter that varies with iteration. Benchmark optimization problems as well as engineering design problems are used to experiment the performance of DE-NMR, with the number of function evaluation times being employed as the main index of measuring convergence speed, and objective function values as the main index of optimum's quality. Non-parametric tests are employed in comparing results with other global optimization algorithms. Results illustrate the fast convergence speed of DE-NMR. [ABSTRACT FROM AUTHOR]

The PDE Δ u = u| u|, with $${m\in\mathbb{N}}$$ and $${p\in(1,\infty)}$$, serves as a paradigm for a large class of higher-order elliptic equations with power-like nonlinearities. In studying radially symmetric solutions of this PDE and their asymptotic behavior, it is of critical importance to understand the dynamics of the associated ODE, u = u| u|, which is the subject of the present paper. Most solutions of the ODE blow up in finite time, diverging to either ∞ or −∞. In the phase space $${\mathbb{R}^{2m}}$$, the orbits of these two types of solutions are separated by a (2 m−1)-dimensional manifold $${\fancyscript{M}}$$, which is unordered and homeomorphic to each of the coordinate hyperplanes under orthogonal projection. Solutions with orbits on $${\fancyscript{M}}$$ do not blow up or, else, are unbounded from above and from below (oscillatory blow-up). In the second-order case, $${\fancyscript{M}}$$ coincides with the stable manifold of the trivial equilibrium. In the fourth-order case, which is our main focus, $${\fancyscript{M}}$$ contains a two-dimensional manifold $${\fancyscript{M}_0}$$ of periodic orbits. There exists a unique periodic solution $${\bar u}$$ with $${\bar u(0)=1}$$ and $${\bar u'(0)=0}$$, which shares most of the symmetry properties of the common cosine; all nontrivial periodic solutions are obtained from $${\bar u}$$ via scaling and phase-shifting. Every solution with orbit on $${\fancyscript{M}\setminus\fancyscript{M}_0}$$ converges to either the trivial equilibrium or one of the nontrivial periodic orbits; oscillatory blow-up does not occur. In higher-order cases, the structure of the manifold $${\fancyscript{M}}$$ remains largely open. [ABSTRACT FROM AUTHOR]

Linear Programming, LP, problems with finite optimal value have a zero duality gap and a primal–dual strictly complementary optimal solution pair. On the other hand, there exist Semidefinite Programming, SDP, problems which have a nonzero duality gap (different primal and dual optimal values; not both infinite). The duality gap is assured to be zero if a constraint qualification, e.g., Slater’s condition (strict feasibility) holds. Measures of strict feasibility, also called distance to infeasibility, have been used in complexity analysis, and, it is known that (near) loss of strict feasibility results in numerical difficulties. In addition, there exist SDP problems which have a zero duality gap but no strict complementary primal–dual optimal solution. We refer to these problems as hard instances of SDP. The assumption of strict complementarity is essential for asymptotic superlinear and quadratic rate convergence proofs. In this paper, we introduce a procedure for generating hard instances of SDP with a specified complementarity nullity (the dimension of the common nullspace of primal–dual optimal pairs). We then show, empirically, that the complementarity nullity correlates well with the observed local convergence rate and the number of iterations required to obtain optimal solutions to a specified accuracy, i.e., we show, even when Slater’s condition holds, that the loss of strict complementarity results in numerical difficulties. We include two new measures of hardness that correlate well with the complementarity nullity. [ABSTRACT FROM AUTHOR]

This paper proposes a generalization of column generation, reformulating the master problem with fewer variables at the expense of adding more constraints; the sub-problem structure does not change. It shows both analytically and computationally that the reformulation promotes faster convergence to an optimal solution in application to a linear program and to the relaxation of an integer program at each node in the branch-and-bound tree. Further, it shows that this reformulation subsumes and generalizes prior approaches that have been shown to improve the rate of convergence in special cases. [ABSTRACT FROM AUTHOR]

A VU-decomposition method for solving a second-order cone problem is presented in this paper. It is first transformed into a nonlinear programming problem. Then, the structure of the Clarke subdifferential corresponding to the penalty function and some results of its VU-decomposition are given. Under a certain condition, a twice continuously differentiable trajectory is computed to produce a second-order expansion of the objective function. A conceptual algorithm for solving this problem with a superlinear convergence rate is given. [ABSTRACT FROM AUTHOR]

This paper studies the normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically –1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature –1. A relative estimate of Green’s function is proved as a tool. [ABSTRACT FROM AUTHOR]

In this paper a new hybrid conjugate gradient algorithm is proposed and analyzed. The parameter β k is computed as a convex combination of the Polak-Ribière-Polyak and the Dai-Yuan conjugate gradient algorithms, i.e. β=(1− θ k) β+ θ k β. The parameter θ k in the convex combination is computed in such a way that the conjugacy condition is satisfied, independently of the line search. The line search uses the standard Wolfe conditions. The algorithm generates descent directions and when the iterates jam the directions satisfy the sufficient descent condition. Numerical comparisons with conjugate gradient algorithms using a set of 750 unconstrained optimization problems, some of them from the CUTE library, show that this hybrid computational scheme outperforms the known hybrid conjugate gradient algorithms. [ABSTRACT FROM AUTHOR]

Let C be a nonempty closed convex subset of a Banach space E with the dual E*, let T: C→ E* be a Lipschitz continuous mapping and let S: C→ C be a relatively nonexpansive mapping. In this paper, by employing the notion of generalized projection operator, we study the following variational inequality (for short, VI( T− f, C)): find x∈ C such that where f∈ E* is a given element. Utilizing the modified Ishikawa iteration and the modified Halpern iteration for relatively nonexpansive mappings, we propose two modified versions of J.L. Li’s (J. Math. Anal. Appl. 295:115–126, ) iterative algorithm for finding approximate solutions of VI( T− f, C). Moreover, it is proven that these iterative algorithms converge strongly to the same solution of VI( T− f, C), which is also a fixed point of S. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in some convergent formulations for the unsymmetric collocation method or the so-called Kansa’s method. We review some newly developed theories on solvability and convergence. The rates of convergence of these variations of Kansa’s method are examined and verified in arbitrary–precision computations. Numerical examples confirm with the theories that the modified Kansa’s method converges faster than the interpolant to the solution; that is, exponential convergence for the multiquadric and Gaussian radial basis functions (RBFs). Some numerical algorithms are proposed for efficiency and accuracy in practical applications of Kansa’s method. In double–precision, even for very large RBF shape parameters, we show that the modified Kansa’s method, through a subspace selection using a greedy algorithm, can produce acceptable approximate solutions. A benchmark algorithm is used to verify the optimality of the selection process. [ABSTRACT FROM AUTHOR]

This paper provides a detailed analysis of a primal-dual interior-point method for PDE-constrained optimization. Considered are optimal control problems with control constraints in L p . It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier L ∞-setting is analyzed, but also a more involved L q -analysis, q < ∞, is presented. In L ∞, the set of feasible controls contains interior points and the Fréchet differentiability of the perturbed optimality system can be shown. In the L q -setting, which is highly relevant for PDE-constrained optimization, these nice properties are no longer available. Nevertheless, a convergence analysis is developed using refined techniques. In parti- cular, two-norm techniques and a smoothing step are required. The L q -analysis with smoothing step yields global linear and local superlinear convergence, whereas the L ∞-analysis without smoothing step yields only global linear convergence. [ABSTRACT FROM AUTHOR]

In this paper, we give an analysis and a general procedure for 4D variational data assimilation (4D-Var). In functional partial differential equation setting, the adjoint equation method, sensitivity analysis, and multicomponent operator splitting are discussed. Nonlinear optimization methods and convergence analysis are also investigated for 4D-Var. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the following implicit iteration scheme recently introduced by Xu and Ori [ Numer. Funct. Anal. Optim., 22, (2001) 767–773]: and to prove several strongly and weakly convergent theorems of the iteration for a finite family of pseudocontractive mappings under condition α n ∈ (0, b] ⊂ (0, 1). [ABSTRACT FROM AUTHOR]

An estimate for the Riemann localization principle for trigonometric series is established. At the same time, there is an error in treating this problem in the paper [3] which is now being corrected. [ABSTRACT FROM AUTHOR]

This paper considers the problems of determining center or focus and isochronous centers for the planar quasi-analytic systems. Two recursive formulas to determine the focal values and period constants are given. The convergence of first integral near the center is proved. Using the general results to quasi-quadratic systems, the problem of the isochronous center of the origin is completely solved. [ABSTRACT FROM AUTHOR]

In this paper we establish convergence to equilibrium of all global and bounded solutions of a gradient like system of second order with nonlinear dissipation and analytic nonlinearity. We estimate also the rate of convergence. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide an error analysis for the multicategory support vector machine (MSVM) classificaton problems. We establish the uniform convergency approach for MSVMs and estimate the misclassification error. The main difficulty we overcome here is to bound the offset vector. As a result, we confirm that the MSVM classification algorithm with polynomial kernels is always efficient when the degree of the kernel polynomial is large enough. Finally the rate of convergence and examples are given to demonstrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider algorithms for constructing n-terms approximations with nonnegative coefficients. The convergence theorem is proved for a “positive” analog of the Pure Greedy Algorithm. We establish a condition on the sequence of weakness coefficients which is sufficient for the convergence of the Positive Weak Greedy Algorithm. This condition is also necessary for the class of monotone sequences. [ABSTRACT FROM AUTHOR]

In the present paper, we investigate the asymptotic behavior of the solution of a variational inequality with one-sided constraints on ɛ-periodically located subsets G ε belonging to the boundary ∂Ω of the domain Ω ⊂ ℝ3. We construct a limit (homogenized) problem and prove the strong (in H 1(Ω)) convergence of the solutions of the original inequality to the solution the limit nonlinear boundary-value problem as ɛ → 0 in the so-called critical case. [ABSTRACT FROM AUTHOR]

For single splittings of Hermitian positive definite matrices, there are well-known convergence and comparison theorems. This paper gives new convergence and comparison results for double splittings of Hermitian positive definite matrices. Keywords: Hermitian positive definite matrix; convergence theorem; comparison theorem; double splitting Mathematics Subject Classification (2000): 65F10 [ABSTRACT FROM AUTHOR]

In this paper, the existence and smoothness of the collision local time are proved for two independent fractional Brownian motions, through L 2 convergence and Chaos expansion. Furthermore, the regularity of the collision local time process is studied. [ABSTRACT FROM AUTHOR]

In this paper we study inexact inverse iteration for solving the generalised eigenvalue problemAx=?Mx. We show that inexact inverse iteration is a modified Newton method and hence obtain convergence rates for various versions of inexact inverse iteration for the calculation of an algebraically simple eigenvalue. In particular, if the inexact solves are carried out with a tolerance chosen proportional to the eigenvalue residual then quadratic convergence is achieved. We also show how modifying the right hand side in inverse iteration still provides a convergent method, but the rate of convergence will be quadratic only under certain conditions on the right hand side. We discuss the implications of this for the preconditioned iterative solution of the linear systems. Finally we introduce a new ILU preconditioner which is a simple modification to the usual preconditioner, but which has advantages both for the standard form of inverse iteration and for the version with a modified right hand side. Numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

To verify the universal validity of the “two-sided” monotonicity condition introduced in [4], we will apply it to include more classical examples. The present paper selects the L p convergence case for this purpose. Furthermore, Theorem 3 shows that our improvements are not trivial. [ABSTRACT FROM AUTHOR]

In this paper, we discuss here-and-now type stochastic programs with equilibrium constraints. We give a general formulation of such problems and study their basic properties such as measurability and continuity of the corresponding integrand functions. We discuss also the consistency and rate of convergence of sample average approximations of such stochastic problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a monotone, Lipschitz-continuous mapping. The iterative process is based on the so-called extragradient method. We obtain a weak convergence theorem for two sequences generated by this process. [ABSTRACT FROM AUTHOR]

Let $\sigma_k^\delta$ denote the Cesaro means of order $\delta > -1$ of the spherical harmonic expansions on the unit sphere $S^{d-1}$, and let $E_j(f, H^1)$ denote the best approximation of $f$ in the Hardy space $H^1(S^{d-1})$ by spherical polynomials of degree at most $j$. It is known that $\lambda:= (d-2)/2$ is the critical index for the summability of the Cesaro means on $H^1(S^{d-1})$. The main result of this paper states that, for $ f\in H^1(S^{d-1})$, $$\sum_{j=0}^N \f 1{j+1} \|\sigma_j^\lambda (f) -f\|_{H^1}\approx \sum_{j=0}^N \f 1{j+1} E_j(f, H^1),$$ where “$\approx$” means that the ratio of both sides lies between two positive constants independent of $f$ and $N$. [ABSTRACT FROM AUTHOR]

The notion of weak sharp minima is an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems. It has been studied extensively by several authors. This paper is the second of a series on this subject where the basic results on weak sharp minima in Part I are applied to a number of important problems in convex programming. In Part II we study applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming. We obtain both new results and reproduce several existing results from a fresh perspective. [ABSTRACT FROM AUTHOR]

The paper is concerned with the Nitsche mortaring in the framework of domain decomposition where non-matching meshes and weak continuity of the finite element approximation at the interface are admitted. The approach is applied to singularly perturbed reaction-diffusion problems in 2D. Non-matching meshes of triangles being anisotropic in the boundary layers are applied. Some properties as well as error estimates of the Nitsche mortar finite element schemes are proved. In particular, using a suitable degree of anisotropy of triangles in the boundary layers of a rectangle, convergence rates as known for the conforming finite element method are derived. Numerical examples illustrate the approach and the results. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the steady-state acoustic wave equation in the space ℝ3 diffracted by an obstacle made by an inhomogeneous medium and located in a bounded domain. The inhomogeneity of the medium depends on a parameter ε > 0. If the solution uε converges to a solution u0 of the limit problem as ε → 0, as in the homogenization process, then we can use the two-scale convergence method to study the convergence of the gradient. [ABSTRACT FROM AUTHOR]

The present paper proposes a new condition to replace both the ( O-regularly varying) quasimonotone condition and a certain type of bounded variation condition, and shows the same conclusion for the uniform convergence of certain trigonometric series still holds. [ABSTRACT FROM AUTHOR]

In this paper, we consider weak efficiency in vector optimization problems with equilibrium constraints. We obtain results on the convergence of the marginal map, the value, and the solution sets. [ABSTRACT FROM AUTHOR]

We consider the problem of finding solutions of systems of monotone equations. The Newton-type algorithm proposed in Ref. 1 has a very nice global convergence property in that the whole sequence of iterates generated by this algorithm converges to a solution, if it exists. Superlinear convergence of this algorithm is obtained under a standard nonsingularity assumption. The nonsingularity condition implies that the problem has a unique solution; thus, for a problem with more than one solution, such a nonsingularity condition cannot hold. In this paper, we show that the superlinear convergence of this algorithm still holds under a local error-bound assumption that is weaker than the standard nonsingularity condition. The local error-bound condition may hold even for problems with nonunique solutions. As an application, we obtain a Newton algorithm with very nice global and superlinear convergence for the minimum norm solution of linear programs. [ABSTRACT FROM AUTHOR]

In this paper, we consider special classes of strongly convex functionals in Sobolev spaces. It is proved that functionals from such classes have the so-called H-property: weak convergence of sequences of arguments and convergence of such sequences with respect to a given functional imply strong convergence. [ABSTRACT FROM AUTHOR]

In this paper we establish the laws of large numbers for blockwise martingale differences and for blockwise adapted sequences which are stochastically dominated by a random variable. Some well-known results from the literature are extended. [ABSTRACT FROM AUTHOR]

There are many studies on the asymptotic behavior of solutions of differential equations. In the present paper, we consider another aspect of this problem, namely, the rate of the asymptotic convergence of solutions. Let ϕ ( t) be a scalar continuous monotonically increasing positive function tending to ∞ as t → ∞. It is established that if all solutions of a differential system satisfy the inequality then the solution x( t; t0, x0) of this differential system tends to 0 faster than $$M\frac{{\varphi (t_0 )}}{{\varphi (t)}}$$ . [ABSTRACT FROM AUTHOR]