The Thresholding Greedy Algorithm, Greedy Bases, and Duality

Some new conditions that arise naturally in the study of the Thresholding Greedy Algorithm are introduced for bases of Banach spaces. We relate these conditions to best n-term approximation and we study their duality theory. In particular, we obtain a complete duality theory for greedy bases.     

Greedy Algorithm for General Biorthogonal Systems

We consider biorthogonal systems in quasi-Banach spaces such that the greedy algorithm converges for each xX (quasi-greedy systems). We construct quasi-greedy conditional bases in a wide range of Banach spaces. We also compare the greedy algorithm for the multidimensional Haar system with the optimal m-term approximation for this system. This substantiates a conjecture by Temlyakov.     

A Greedy Algorithm for Capacitated Lot-Sizing Problems

We show that the convex hull of the set of feasible solutions of single-item capacitated lot-sizing problem (CLSP) is a base polyhedron of a polymatroid. We present a greedy algorithm to solve CLSP with linear objective function. The proposed algorithm is an effective implementation of the classical Edmonds' algorithm for maximizing linear function over a polymatroid. We consider some special cases of CLSP with nonlinear objective function that can be solved by the proposed greedy algorithm in O ( n ) time.     

The Weak Chebyshev X-Greedy Algorithm in the unweighted Bergman space

We study a greedy algorithm called the Weak Chebyshev X-Greedy Algorithm (WCXGA) and investigate its application to unweighted Bergman spaces. We first show that the WCXGA converges for a wide class of real and complex Banach spaces and dictionaries. We then prove that certain Bergman spaces and their holomorphic monomial dictionaries belong to the class of Banach spaces for which the WCXGA converges.     

On some characterizations of greedy-type bases

In 1999, S. V. Konyagin and V. N. Temlyakov introduced the so-called Thresholding Greedy Algorithm. Since then, there have been many interesting and useful characterizations of greedy-type bases in Banach spaces. In this article, we study and extend several characterizations of greedy and almost greedy bases in the literature. Along the way, we give various examples to complement our main results. Furthermore, we propose a new version of the so-called Weak Thresholding Greedy Algorithm (WTGA) and show that the convergence of this new algorithm is equivalent to the convergence of the WTGA.     

A step toward probabilistic analysis of simplex method convergence

Probability distributions are assumed for the coefficients in simplex tableaus. A probability of success in one simplex iteration is then derived; for example, the probability of a tableau which satisfies the criteria for optimality except in one row becoming fully optimal in one iteration. Such results are expressed in terms of tableau size parameters.     

Weak convergence of Markov-modulated diffusion processes with rapid switching

In this paper, we study the weak convergence of a sequence of Markov-modulated diffusion processes when the modulating Markov chain is ergodic and rapidly switching. We prove, in particular, its tightness property based on Aldous’ tightness criterion.     

On the Convergence of a Weak Greedy Algorithm for the Multivariate Haar Basis

We define a family of weak thresholding greedy algorithms for the multivariate Haar basis for L ₁[0,1] ^d (d≥1). We prove convergence and uniform boundedness of the weak greedy approximants for all f∈L ₁[0,1] ^d .     

Weak convergence of diffusion processes on Wiener space

Let γ be a Gaussian measure on a Suslin space X, H be the corresponding Cameron–Martin space and {e _i } ⊂ H be an orthonormal basis of H. Suppose that μ _n = ρ _n · γ is a sequence of probability measures which converges weakly to a probability measure μ = ρ · γ Consider a sequence of Dirichlet forms , where and . We prove some sufficient conditions for Mosco convergence where . In particular, if X is a Hilbert space, and can be uniformly approximated by finite dimensional conditional expectations for every fixed e _i , then under broad assumptions Mosco and the distributions of the associated stochastic processes converge weakly.     

Approximate solutions of partial differential equations by some Meshfree Greedy Algorithms

In this article, we use some greedy algorithms to avoid the ill‐conditioning of the final linear system in unsymmetric Kansa collocation method. The greedy schemes have the same background, but we use them in different settings. In the first algorithm, the optimal trial points for interpolation obtained among a huge set of initial points are used for numerical solution of partial differential equations (PDEs). In the second algorithm, based on the Kansa's method, the PDE is discretized to a finite number of test functional equations, and a greedy sparse discretization is applied for approximating the linear functionals. Each functional is stably approximated by some few trial points with an acceptable accuracy. The third greedy algorithm is used to generate the test points. This paper shows that the greedily selection of nodes yields a better conditioning in contrast with usual full meshless methods. Some well‐known PDE examples are solved and compared with the full unsymmetric Kansa's technique. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1884–1899, 2017     

On convergence of greedy algorithm by Walsh system in the space <Emphasis Type="Italic">C</Emphasis>(0, 1)

The paper studies convergence of the greedy algorithm by the Walsh system in the space C(0, 1). Some sufficient conditions for uniform convergence are given. It is proved that there exists a function satisfying more restrictive conditions, for which the sequence of the partial sums of the Fourier-Walsh series diverges at the point 0.     

On convergence of moment generating functions

Mukherjea et al. [Mukherjea, A., Rao, M., Suen, S., 2006. A note on moment generating functions. Statist. Probab. Lett. 76, 1185-1189] proved that if a sequence of moment generating functions M_n(t) converges pointwise to a moment generating function M(t) for all t in some open interval of the real line, not necessarily containing the origin, then the distribution functions F_n (corresponding to M_n) converge weakly to the distribution function F (corresponding to M). In this note, we improve this result and obtain conditions of the convergence which seem to be sharp: F_n converge weakly to F if M_n(t_k) converge to M(t_k), k=1,2,…, for some sequence {t₁,t₂,…} having the minimal and the maximal points. A similar result holds for characteristic functions.     

Weak convergence of the scaled median of independent Brownian motions

We consider the median of n independent Brownian motions, denoted by M _n (t), and show that $\sqrt{n}\,M_nWe consider the median of n independent Brownian motions, denoted by M _n (t), and show that converges weakly to a centered Gaussian process. The chief difficulty is establishing tightness, which is proved through direct estimates on the increments of the median process. An explicit formula is given for the covariance function of the limit process. The limit process is also shown to be H?lder continuous with exponent γ for all γ < 1/4.      

Weak convergence theorems for a countable family of Lipschitzian mappings

This paper is concerned with convergence of an approximating common fixed point sequence of countable Lipschitzian mappings in a uniformly convex Banach space. We also establish weak convergence theorems for finding a common element of the set of fixed points, the set of solutions of an equilibrium problem, and the set of solutions of a variational inequality. With an appropriate setting, we obtain and improve the corresponding results recently proved by Moudafi [A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems. J. Nonlinear Convex Anal. 9 (2008) 37–43], Tada–Takahashi [A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133 (2007) 359–370], and Plubtieng–Kumam [S. Plubtieng and P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings. J. Comput. Appl. Math. (2008) doi:10.1016/j.cam.2008.05.045]. Some of our results are established with weaker assumptions.     

On the Weak Order of Orthogonal Groups

A complete lattice structure is defined on the underlying set of the orthogonal group of a real Euclidean space, by a construction analogous to that of the weak order of Coxeter systems in terms of root systems. This produces a complete rootoid in the sense of Dyer, with the orthogonal group as underlying group. It is shown that this complete lattice has a saturation property which is used along with other properties of the lattice to characterize the maximal totally ordered subsets of the lattice as collections of initial sections with respect to a total ordering on the positive roots.     

Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces

Let C be a closed convex subset of a real Hilbert space H and assume that T is a κ-strict pseudo-contraction on C with a fixed point, for some 0?κ<1. Given an initial guess x₀∈C and given also a real sequence {αn} in (0,1). The Mann's algorithm generates a sequence {xn} by the formula: xn+1=αnxn+(1−αn)Txn, n?0. It is proved that if the control sequence {αn} is chosen so that κ<αn<1 and , then {xn} converges weakly to a fixed point of T. However this convergence is in general not strong. We then modify Mann's algorithm by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strong convergent sequence. This result extends a recent result of Nakajo and Takahashi [K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372-379] from nonexpansive mappings to strict pseudo-contractions.     

A convergence result in the study of bone remodeling contact problems

We consider the approximation of a bone remodeling model with the Signorini contact conditions by a contact problem with normal compliant obstacle, when the obstacle's deformability coefficient converges to zero (that is, the obstacle's stiffness tends to infinity). The variational problem is a coupled system composed of a nonlinear variational equation (in the case of normal compliance contact conditions) or a variational inequality (for the case of Signorini's contact conditions), for the mechanical displacement field, and a first-order ordinary differential equation for the bone remodeling function. A theoretical result, which states the convergence of the contact problem with normal compliance contact law to the Signorini problem, is then proved. Finally, some numerical simulations, involving examples in one and two dimensions, are reported to show this convergence behaviour.     

On the 2-Categories of Weak Distributive Laws

ABSTRACT

Let G be a torsion-free group with all subgroups subnormal of defect at most 4. We show that G is nilpotent of class at most 4.