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In this paper we are concerned with absolute, relative and Tate Tor modules. In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander—Buchweitz approximation theory, and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules. In the second part of the paper we consider a depth equality, called the depth formula, which has been initially introduced by Auslander and developed further by Huneke and Wiegand. As an application of our main result, we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension. [ABSTRACT FROM AUTHOR]

This paper discusses the asymptotic behaviors of the longest run on a countable state Markov chain. Let $$\left\{ {X_a } \right\}_{a \in Z_ + }$$ be a stationary strongly ergodic reversible Markov chain on countablestate space S = {1, 2, ...}. Let T ⊂ S be an arbitrary finite subset of S. Denote by Ln the length of the longest run of consecutive i’s for i ∈ T, that occurs in the sequence X1, ..., Xn. In this paper, we obtain a limit law and a week version of an Erdös-Rényi type law for Ln. A large deviation result of Ln is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we consider the strong approximation for locally square-integrable martingales. In our results, the limit process may be a process with jumps. This is an extension of the former results. [ABSTRACT FROM AUTHOR]

A highly celebrated problem in dyadic harmonic analysis is the pointwise convergence of the Fejér (or ( C, 1)) means of functions on unbounded Vilenkin groups. There are several papers of the author of this paper concerning this. That is, we know the a.e. convergence σ n f → f ( n → ∞) for functions f ∈ L p , where p > 1 ( Journal of Approximation Theory, 101(1), 1–36, (1999)) and also the a.e. convergence σ M n f → f ( n → ∞) for functions f ∈ L 1 ( Journal of Approximation Theory, 124(1), 25–43, (2003)). The aim of this paper is to prove the a.e. relation lim n → σ n f = f for each integrable function f on any rarely unbounded Vilenkin group. The concept of the rarely unbounded Vilenkin group is discussed in the paper. Basically, it means that the generating sequence m may be an unbounded one, but its "big elements" are not "too dense". [ABSTRACT FROM AUTHOR]

In this paper, we study radial operators in Toeplitz algebra on the weighted Bergman spaces over the polydisk by the (m, λ)-Berezin transform and find that a radial operator can be approximated in norm by Toeplitz operators without any conditions. We prove that the compactness of a radial operator is equivalent to the property of vanishing of its (0, λ)-Berezin transform on the boundary. In addition, we show that an operator S is radial if and only if its (m, λ)-Berezin transform is a separately radial function. [ABSTRACT FROM AUTHOR]

Empirical likelihood inference for parametric and nonparametric parts in functional coefficient ARCH-M models is investigated in this paper. Firstly, the kernel smoothing technique is used to estimate coefficient function δ(x). In this way we obtain an estimated function with parameter β. Secondly, the empirical likelihood method is developed to estimate the parameter β. An estimated empirical log-likelohood ratio is proved to be asymptotically standard chi-squred, and the maximum empirical likelihood estimation (MELE) for β is shown to be asymptotically normal. Finally, based on the MELE of β, the empirical likelihood approach is again applied to reestimate the nonparametric part δ(x). The empirical log-likelohood ratio for δ(x) is proved to be also asymptotically standard chi-squred. Simulation study shows that the proposed method works better than the normal approximation method in terms of average areas of confidence regions for β, and the empirical likelihood confidence belt for δ(x) performs well. [ABSTRACT FROM AUTHOR]

This paper gives the concepts of finite dimensional irreducible operators ((FDI) operators) and infinite dimensional irreducible operators ((IDI) operators). Discusses the relationships of (FDI) operators, (IDI) operators and strongly irreducible operators ((SI) operators) and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an (FDI) operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in (ΣFDI)( X) with respect to the strong-operator topology and every compact operator K can be approximated by operators in (ΣFDI)( X) with respect to the norm topology on a Banach space X with a Schauder basis, where (ΣFDI)( X):= { T ∈ B( X): T = Σ ⊕ T, T ∈ (FDI), k ∈ ℕ}. [ABSTRACT FROM AUTHOR]

The stability is an expected property for functions, which is widely considered in the study of approximation theory and wavelet analysis. In this paper, we consider the Lp,q-stability of the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1). We first show that the shifts ϕ(· − k) (k ∈ ℤd+1) are Lp,q-stable if and only if for any ξ ∈ ℝd+1, ∑k∈Zd+1|ϕ^(ξ+2πk)|2>0. Then we give a necessary and sufficient condition for the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1) to be Lp,q-stable which improves some known results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove the existence of a spatially periodic weak solution to the steady compressible isentropic MHD equations in ℝ for any specific heat ratio γ > 1. The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density, and the method of weak convergence. According to the author's knowledge, it is the first result that treats in three dimensions the existence of weak solutions to the steady compressible MHD equations with Γ > 1. [ABSTRACT FROM AUTHOR]

In this paper, we present a variable selection procedure by combining basis function approximations with penalized estimating equations for semiparametric varying-coefficient partially linear models with missing response at random. The proposed procedure simultaneously selects significant variables in parametric components and nonparametric components. With appropriate selection of the tuning parameters, we establish the consistency of the variable selection procedure and the convergence rate of the regularized estimators. A simulation study is undertaken to assess the finite sample performance of the proposed variable selection procedure. [ABSTRACT FROM AUTHOR]

Based on a new efficient identification technique of active constraints introduced in this paper, a new sequential systems of linear equations (SSLE) algorithm generating feasible iterates is proposed for solving nonlinear optimization problems with inequality constraints. In this paper, we introduce a new technique for constructing the system of linear equations, which recurs to a perturbation for the gradients of the constraint functions. At each iteration of the new algorithm, a feasible descent direction is obtained by solving only one system of linear equations without doing convex combination. To ensure the global convergence and avoid the Maratos effect, the algorithm needs to solve two additional reduced systems of linear equations with the same coefficient matrix after finite iterations. The proposed algorithm is proved to be globally and superlinearly convergent under some mild conditions. What distinguishes this algorithm from the previous feasible SSLE algorithms is that an improving direction is obtained easily and the computation cost of generating a new iterate is reduced. Finally, a preliminary implementation has been tested. [ABSTRACT FROM AUTHOR]

In this paper, we continue studying the so-called non-linear best m-term one-sided approximation problems and obtain the asymptotic estimations of non-linear best m-term one-sided trigonometric approximation under the norm L p (1 ≤ p ≤ ∞) of multiplier function classes and the corresponding m-term Greedy-liked one-sided trigonometric approximation results. [ABSTRACT FROM AUTHOR]

In this paper we study zero-sum stochastic games. The optimality criterion is the long-run expected average criterion, and the payoff function may have neither upper nor lower bounds. We give a new set of conditions for the existence of a value and a pair of optimal stationary strategies. Our conditions are slightly weaker than those in the previous literature, and some new sufficient conditions for the existence of a pair of optimal stationary strategies are imposed on the primitive data of the model. Our results are illustrated with a queueing system, for which our conditions are satisfied but some of the conditions in some previous literatures fail to hold. [ABSTRACT FROM AUTHOR]

In this paper, the non-linear approximation on the class of multivariate functions with bounded mixed derivatives is investigated, and the asymptotic degree of the non-linear width on this class is determined. [ABSTRACT FROM AUTHOR]

In this paper, we first consider the least-squares solution of the matrix inverse problem as follows: Find a hermitian anti–reflexive matrix corresponding to a given generalized reflection matrix J such that for given matrices X,B we have min A ‖ AX − B‖. The existence theorems are obtained, and a general representation of such a matrix is presented. We denote the set of such matrices by S E . Then the matrix nearness problem for the matrix inverse problem is discussed. That is: Given an arbitrary A*, find a matrix  ∈ S E which is nearest to A* in Frobenius norm. We show that the nearest matrix is unique and provide an expression for this nearest matrix. [ABSTRACT FROM AUTHOR]

Letbe a unit sphere of thed-dimensional Euclidean space Rd and let(0 < p= 1) denote the real Hardy space onFor 0 < p= 1 andletEj(f,Hp) (j= 0, 1, ...) be the best approximation offby spherical polynomials of degree less than or equal toj, in the spaceGiven a distributionfonits Cesàro mean of order d>-1 is denoted byFor 0p. In this paper, the following result is proved:TheoremLet0N(f)˜BN(f)means that there’s a positive constant C, independent of N and f, such thatIn the cased= 2,this result was proved by Belinskii in 1996. [ABSTRACT FROM AUTHOR]

This paper deals with the L-consistency of wavelet estimators for a density function based on size-biased random samples. More precisely, we firstly show the L-consistency of wavelet estimators for independent and identically distributed random vectors in R. Then a similar result is obtained for negatively associated samples under the additional assumptions d = 1 and the monotonicity of the weight function. [ABSTRACT FROM AUTHOR]

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property (OFDDP) if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces. This property has an influence in the non-Archimedean Grothendieck's approximation theory, where an open problem is the following: Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E. Does D have the OFDDP? In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP. Next we prove that, however, for certain classes of Banach spaces of countable type, the OFDDP is preserved by taking finite-codimensional subspaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the sharp Jackson inequality for the best approximation of f ∈ L(ℝ) by a subspace E( σ) ( SE( σ)), which is a subspace of entire functions of exponential type (spherical exponential type) at most σ. Here L(ℝ) denotes the space of all d-variate functions f endowed with the L-norm with the weight $v_\kappa (x) = \prod\nolimits_{\xi \in R_ + } {|(\xi ,x)|^{2\kappa (\xi )} } $, which is defined by a positive subsystem R of a finite root system R ⊂ ℝ and a function κ( ξ): R → ℝ invariant under the reflection group G( R) generated by R. In the case G( R) = ℤ, we get some exact results. Moreover, the deviation of best approximation by the subspace E( σ) ( SE( σ)) of some class of the smooth functions in the space L(ℝ) is obtained. [ABSTRACT FROM AUTHOR]

We consider a Markov chain X = { Xi, i = 1, 2, ...} with the state space {0, 1}, and define W = Σ i=1nXiXi+1, which is the number of 2-runs in X before time n+1. In this paper, we prove that the negative binomial distribution is an appropriate approximation for $$\mathcal{L}W$$ when Var W is greater than $$\mathbb{E}W$$ . The error estimate obtained herein improves the corresponding result in previous literatures. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a type of approximation operators of neural networks with sigmodal functions on compact intervals, and obtain the pointwise and uniform estimates of the approximation. To improve the approximation rate, we further introduce a type of combinations of neural networks. Moreover, we show that the derivatives of functions can also be simultaneously approximated by the derivatives of the combinations. We also apply our method to construct approximation operators of neural networks with sigmodal functions on infinite intervals. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of homoclinic solutions in periodic nonlinear difference equations with superlinear nonlinearity. The classical Ambrosetti-Rabinowitz superlinear condition is improved by a general superlinear one. The proof is based on the critical point theory in combination with periodic approximations of solutions. [ABSTRACT FROM AUTHOR]

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle (in short EVP), where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space (which is a uniform space) has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi's fixed point theorem and a general vectorial Takahashi's nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results. [ABSTRACT FROM AUTHOR]

In this paper, we shall give an elementary proof of a Bishop-Stone-Weierstrass theorem for ( M(ℂ)) with respect to its pure states. To be more precise, we shall show that the pure-state Bishop hull of a unital subalgebra (not necessarily self-adjoint) of ( M(ℂ)) is equal to itself. [ABSTRACT FROM AUTHOR]

In this paper, the concepts of approximate character amenability (contractibility), uniform approximate character amenability (contractibility) and w*-approximate character amenability are introduced. We are concerned with the relations among the generalized concepts of character amenability for Banach algebras. We show that approximate character amenability, w*-approximate character amenability and approximate character contractibility are the same properties, as uniform approximate character amenability and character amenability, as uniform approximate character contractibility and character contractibility. The general theory for these concepts is also developed. Moreover, approximate character amenability of several concrete classes of Banach algebras related to locally compact groups and also some discrete semigroups is considered. [ABSTRACT FROM AUTHOR]

Recently Takahashi established a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies the spherical approximation theorem due to Auslander and Bridger and the Cohen-Macaulay approximation theorem due to Auslander and Buchweitz. In this paper we generalize these results to much more general case over non-commutative rings. As an application, we establish a relation between the injective dimension of a generalized tilting module ω and the finitistic dimension with respect to ω. [ABSTRACT FROM AUTHOR]

In this paper, we study the approximation of identity operator and the convolution integral operator B by Fourier partial sum operators, Fejér operators, Vallée-Poussin operators, Cesáro operators and Abel mean operators, respectively, on the periodic Wiener space ( C(ℝ), W) and obtain the average error estimations. [ABSTRACT FROM AUTHOR]

The homogeneous approximation property (HAP) states that the number of building blocks involved in a reconstruction of a function up to some error is essentially invariant under time-scale shifts. In this paper, we show that every wavelet frame with nice wavelet function and arbitrary expansive dilation matrix possesses the HAP. Our results improve some known ones. [ABSTRACT FROM AUTHOR]

In this paper, we present a decomposition method of multivariate functions. This method shows that any multivariate function f on [0, 1] is a finite sum of the form Σ ϕψ, where each ϕ can be extended to a smooth periodic function, each ψ is an algebraic polynomial, and each φ ψ is a product of separated variable type and its smoothness is same as f. Since any smooth periodic function can be approximated well by trigonometric polynomials, using our decomposition method, we find that any smooth multivariate function on [0, 1] can be approximated well by a combination of algebraic polynomials and trigonometric polynomials. Meanwhile, we give a precise estimate of the approximation error. [ABSTRACT FROM AUTHOR]

In this paper, we establish a general solution and the generalized Hyers-Ulam-Rassias stability of the following general mixed additive-cubic functional equation in the quasi-Banach spaces. [ABSTRACT FROM AUTHOR]

A multi-class single server queue under non-preemptive static buffer priority (SBP) service discipline is considered in this paper. Using a bounding technique, we obtain the fluid approximation for the queue length and busy time processes. Furthermore, we prove that the convergence rate of the fluid approximation for the queue length and busy time processes is exponential for large N. Additionally, a sufficient condition for stability is obtained. [ABSTRACT FROM AUTHOR]

The first goal of this paper is to establish some properties of the ridge function representation for multivariate polynomials, and the second one is to apply these results to the problem of approximation by neural networks. We find that for continuous functions, the rate of approximation obtained by a neural network with one hidden layer is no slower than that of an algebraic polynomial. [ABSTRACT FROM AUTHOR]

This paper concerns the study of the numerical approximation for the following initialboundary value problem where f( s) is a positive, increasing, C convex function for the nonnegative values of s, f(0) > 0, $$ \frac{{ds}} {{f\left( s \right)}} $$ < ∞, u ∈ C([0, 1]), u(0) = 0, u′(1) = 0. We find some conditions under which the solution of a semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also prove the convergence of the semidiscrete blow-up time to the theoretical one. A similar study has been also undertaken for a discrete form of the above problem. Finally, we give some numerical results to illustrate our analysis. [ABSTRACT FROM AUTHOR]

In this paper, we consider the relations among L-fuzzy sets, rough sets and n-ary polygroup theory. Some properties of (normal) TL-fuzzy n-ary subpolygroups of an n-ary polygroup are first obtained. Using the concept of L-fuzzy sets, the notion of ϑ-lower and T-upper L-fuzzy rough approximation operators with respect to an L-fuzzy set is introduced and some related properties are presented. Then a new algebraic structure called (normal) TL-fuzzy rough n-ary polygroup is defined and investigated. Also, the (strong) homomorphism of ϑ-lower and T-upper L-fuzzy rough approximation operators is studied. [ABSTRACT FROM AUTHOR]

In this paper, we give some equivalent conditions on a Banach frame for a Banach space by using the pseudoinverse operator. We also consider the stability of a Banach frame for a Banach space X with respect to X or an X-frame for a Banach space X under perturbation. These results generalize and improve the related works of Balan, Casazza, Christensen, Stoeva and Jian et al. [ABSTRACT FROM AUTHOR]

Hudzik and Kurc discussed some best approximation problems in Banach lattices by means of monotonicities. This paper deals with more general best approximation problems in Banach lattices. Existence, uniqueness, stability and continuity for such best approximation problems are discussed. [ABSTRACT FROM AUTHOR]

Borwein and Choi conjectured that a polynomial P( x) with coefficients ±1 of degree N − 1 is cyclotomic iff , where N = p 1 p 2 ... p r and the p i are primes, not necessarily distinct. Here Φ p ( x):= ( x p − 1)/( x − 1) is the p-th cyclotomic polynomial. They also proved the conjecture for N odd or a power of 2. In this paper we introduce a so-called E-transformation, by which we prove the conjecture for a wider variety of cases and present the key as well as a new approach to investigate the conjecture. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problem of best weighted simultaneous approximations to totally bounded sequences in Banach spaces. Characterization results from convex sets in Banach spaces are established under the assumption that the Banach space is uniformly smooth. [ABSTRACT FROM AUTHOR]

The stationary and nonstationary rotating Navier-Stokes equations with mixed boundary conditions are investigated in this paper. The existence and uniqueness of the solutions are obtained by the Galerkin approximation method. Next, θ-scheme of operator splitting algorithm is applied to rotating Navier-Stokes equations and two subproblems are derived. Finally, the computational algorithms for these subproblems are provided. [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]

This paper deals with realizable adaptive algorithms of the nonlinear approximation with finite terms based on wavelets. We present a concrete algorithm by which we may find the required index set Λ m for the greedy algorithm $$ G^{p}_{m} {\left( { \cdot ,\psi } \right)}. $$ This makes the greedy algorithm realize the near best approximation in practice. Moreover, we study the efficiency of the finite-term approximation of another algorithm introduced by Birge and Massart. [ABSTRACT FROM AUTHOR]

In this paper we study the connection between the metric projection operator P K : B → K, where B is a reflexive Banach space with dual space B* and K is a non–empty closed convex subset of B, and the generalized projection operators ∏ K : B → K and π K : B* → K. We also present some results in non–reflexive Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated. We give some direct and inverse results on pointwise simultaneous approximation by the combinations of Baskakov operators. We also give a new equivalent result on pointwise approximation by these operators. [ABSTRACT FROM AUTHOR]

The study of zeros of orthogonal functions is an important topic. In this paper, by improving the middle variable x(t), we've got a new form of asymptotic approximation, completed with error bounds, it is constructed for the Jacobi functions ϕμ(αβ) (t) (α > -1) as μ → + ∞. Besides, an accurate approximation with error bounds is also constructed correspondingly for the zeros tμ,8 of ϕμ(αβ) (t) (α ≥ -0) as μ → + ∞, uniformly with respect to s = 1,2,.... [ABSTRACT FROM AUTHOR]

This paper deals with the cyclicity of a kind of degenerate planar polycycles through a saddle–node and two hyperbolic saddles, where the hyperbolicity ratio of the saddle (which connects the saddle–node with hp–connection) is equal to 1 and that of the other saddle is irrational. It is obtained that the cyclicity of this kind of polycycle is no more than 5 if the hp–connection keeps unbroken under the C∞ perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of optimization of adaptive direct methods of operator equations. Adaptivity of a direct method is understood in the sense that the subspace on the basis of which it is constructed is chosen depending on the operator of the concrete equation (otherwise, nonadaptive direct method is then concerned), which would essentially let us increase the precision. For some classes of the second kind of Fredhlom integral equations with anisotropic smooth kernels we determine the exact order of the error of adaptive direct methods, and we also give an optimal algorithm. [ABSTRACT FROM AUTHOR]

The stability is an expected property for functions, which is widely considered in the study of approximation theory and wavelet analysis. In this paper, we consider the Lp,q-stability of the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1). We first show that the shifts ϕ(· − k) (k ∈ ℤd+1) are Lp,q-stable if and only if for any ξ ∈ ℝd+1, ∑k∈Zd+1|ϕ^(ξ+2πk)|2>0. Then we give a necessary and sufficient condition for the shifts of finitely many functions in mixed Lebesgue spaces Lp,q(ℝd+1) to be Lp,q-stable which improves some known results. [ABSTRACT FROM AUTHOR]

To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined. [ABSTRACT FROM AUTHOR]

There have been many studies on the dense theorem of approximation by radial basis feedforword neural networks, and some approximation problems by Gaussian radial basis feedforward neural networks (GRBFNs) in some special function space have also been investigated. This paper considers the approximation by the GRBFNs in continuous function space. It is proved that the rate of approximation by GRNFNs with n neurons to any continuous function f defined on a compact subset K ⊂ ℝ can be controlled by ω( f, n), where ω( f, t) is the modulus of continuity of the function f. [ABSTRACT FROM AUTHOR]

We analyze the learning rates for the least square regression with data dependent hypothesis spaces and coefficient regularization algorithms based on general kernels. Under a very mild regularity condition on the regression function, we obtain a bound for the approximation error by estimating the corresponding K-functional. Combining this estimate with the previous result of the sample error, we derive a dimensional free learning rate by the proper choice of the regularization parameter. [ABSTRACT FROM AUTHOR]