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Abstract Let n ≥ 2 and let A ∈ L (C n) be such that m (A) > 0. In this paper, we use a variational result for A -normalized univalent subordination chains, to deduce that every normalized univalent mapping which has A -parametric representation on B n can be approximated locally uniformly on B n by mappings which have A -parametric representation on B n and admit extensions, on one hand, to automorphisms of C n and, on the other hand, to quasiconformal diffeomorphisms of class C ∞ from C n onto C n , which are equal to the identity mapping in a neighborhood of ∞. Finally, we obtain approximation results of A -spirallike, starlike, and convex mappings by automorphisms of C n and quasiconformal diffeomorphisms of class C ∞ from C n onto itself, which are equal to the identity mapping in a neighborhood of ∞ , whose restrictions to B n have the same geometric property. We remark that even though these extensions exist for the same mapping on B n , they are not necessarily equal on C n ∖ B n ¯. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Hankel determinants associated with the singularly perturbed Laguerre weight w ( x ) = x α e − x − t / x , x ∈ ( 0 , ∞ ) , t > 0 and α > 0 . When the matrix size n → ∞ , we obtain an asymptotic formula for the Hankel determinants, valid uniformly for t ∈ ( 0 , d ] , d > 0 fixed. A particular Painlevé III transcendent is involved in the approximation, as well as in the large- n asymptotics of the leading coefficients and recurrence coefficients for the corresponding perturbed Laguerre polynomials. The derivation is based on the asymptotic results in an earlier paper of the authors, obtained by using the Deift–Zhou nonlinear steepest descent method. [ABSTRACT FROM AUTHOR]

In this paper we investigate a relationship between fully k -rotundity properties, uniform K -monotonicity properties, reflexivity and K -order continuity in symmetric spaces E . We also answer a crucial question whether fully k -rotundity properties might be restricted in definition to E d the positive cone of all nonnegative and decreasing elements of E . We present a complete characterization of decreasing uniform K -monotonicity and K -order continuity in E . It is worth mentioning that we also establish several auxiliary results describing reflexivity in Lorentz spaces Γ p , w and K -order continuity in Orlicz spaces L ψ . Finally, we show an application of discussed geometric properties to the approximation theory. [ABSTRACT FROM AUTHOR]

In this paper, we construct a piecewise smooth solution for a 2-D supersonic potential flow past a curved wedge, whose boundary is assumed to have large variation. In fact, we originally provide a background solution, which is constructed by assuming the coming flow has limit speed. Its shock coincides with the curved wedge and the flow behind the shock is defined only along the wedge. Then our problem can be treated as a perturbation of this background solution and the solution is obtained by showing specific estimates. [ABSTRACT FROM AUTHOR]

This paper investigates asymptotical distributions of extremes on different grids and continuous time of stationary processes. The findings show that these joint extremes over certain threshold dependent grids are asymptotically completely dependent, dependent and independent respectively for dense, Pickands and sparse grids. Furthermore, an integral representation of the asymptotic function involved in the approximation is obtained. [ABSTRACT FROM AUTHOR]

Abstract: Text: The aim of this paper is to establish some sharp bounds for the Glaisher–Kinkelin and Bendersky–Adamchik constants. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=LmxI_0AW1vk. [Copyright &y& Elsevier]

For integers 1 ≤ l < k with k ≥ 3 and a real number ξ such that 1 , ξ l , ξ k are linearly independent over Q , we obtain two results on upper bound of the uniform exponent of simultaneous approximation to ( 1 , ξ l , ξ k ) by rational numbers, which improve the results in our previous paper. Our first result treats the case of arbitrary k , and our second and the main result, which gives a better bound, treats the case k = 5 , 7 , 9 . [ABSTRACT FROM AUTHOR]

Abstract: The context of the paper is: a locally compact Hausdorff space ; the space , equipped with the uniform norm, of real continuous functions on which vanish at infinity; a linear subspace , of finite dimension , of the space ; and the set-valued metric projection of into the family of non-empty compact convex subsets of , defined by  the set of best uniform approximations to from . Those which are Chebyshev (that is the metric projection is single point valued) were characterised by Haar (1918). Those for which the metric projection is lower semi-continuous have been characterised by Wu Li (1989) and A.L. Brown (2005); the paper interprets and exploits the characterisation. A ‘Generalised Haar Condition’ which is necessary for to be lower semi-continuous is identified; in some circumstances it is also sufficient. The condition has a calculable determinantal form. The results of the paper include an essentially complete determination, in case is a compact space in which no net of components is convergent to a single point set, of those subspaces of for which is lower semi-continuous. Simple examples show that if the space is compact and totally disconnected, but not finite, the situation is essentially different. [Copyright &y& Elsevier]

Let H be a Hilbert space, L ( H ) the algebra of bounded linear operators on H and W ∈ L ( H ) a positive operator such that W 1 / 2 is in the p-Schatten class, for some 1 ≤ p < ∞ . Given A ∈ L ( H ) with closed range and B ∈ L ( H ) , we study the following weighted approximation problem: analyze the existence of m i n X ∈ L ( H ) ‖ A X − B ‖ p , W , where ‖ X ‖ p , W = ‖ W 1 / 2 X ‖ p . In this paper we prove that the existence of this minimum is equivalent to a compatibility condition between R ( B ) and R ( A ) involving the weight W , and we characterize the operators which minimize this problem as W -inverses of A in R ( B ) . [ABSTRACT FROM AUTHOR]

In this paper, we derive a generalized multiplicative Hardy–Littlewood–Polya type inequality, as well as several related additive inequalities, for functions of operators in Hilbert spaces. In addition, we find the modulus of continuity of a function of an operator on a class of elements defined with the help of another function of the operator. We then apply the results to solve the following problems: (i) the problem of approximating a function of an unbounded self-adjoint operator by bounded operators, (ii) the problem of best approximation of a certain class of elements from a Hilbert space by another class, and (iii) the problem of optimal recovery of an operator on a class of elements given with an error. [ABSTRACT FROM AUTHOR]

We consider a two-dimensional motion of a thin film flowing down an inclined plane under the influence of the gravity and the surface tension. In order to investigate the stability of such flow, we often use a thin film approximation, which is an approximation obtained by the perturbation expansion with respect to the aspect ratio of the film. The famous examples of the approximate equations are the Burgers equation, Kuramoto–Sivashinsky equation, KdV–Burgers equation, KdV–Kuramoto–Sivashinsky equation, and so on. In this paper, we give a mathematically rigorous justification of a thin film approximation by establishing an error estimate between the solution of the Navier–Stokes equations and those of approximate equations. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper, we study conditions under which the metric projection of a polyhedral Banach space onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if satisfies (a geometric property stronger than polyhedrality) and is any proximinal subspace, then the metric projection is Hausdorff continuous and is strongly proximinal (i.e., if , and , then ). One of the main results of a different nature is the following: if satisfies and is a closed subspace of finite codimension, then the following conditions are equivalent: (a) is strongly proximinal; (b) is proximinal; (c) each element of attains its norm. Moreover, in this case the quotient is polyhedral. The final part of the paper contains examples illustrating the importance of some hypotheses in our main results. [Copyright &y& Elsevier]

Abstract: Recovering missing data from its partial samples is a fundamental problem in mathematics and it has wide range of applications in image and signal processing. While many such algorithms have been developed recently, there are very few papers available on their error estimations. This paper is to analyze the error of a frame based data recovery approach from random samples. In particular, we estimate the error between the underlying original data and the approximate solution that interpolates (or approximates with an error bound depending on the noise level) the given data that has the minimal norm of the canonical frame coefficients among all the possible solutions. [Copyright &y& Elsevier]

Abstract: The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets. In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well. [Copyright &y& Elsevier]

Abstract: In this paper we present a survey of rates of pointwise approximation of modified Gamma operators for locally bounded functions and absolutely continuous functions by using some inequalities and results of probability theory with the method of Bojanic–Cheng. In the paper a kind of locally bounded functions is introduced with different growth conditions in the fields of both ends of interval , and it is found out that the operators have different properties compared to the Gamma operators discussed in [X.M. Zeng, Approximation properties of Gamma operators, J. Math. Anal. Appl. 311 (2005) 389–401]. And we obtain two main theorems. Theorem 1 gives an estimate for locally bounded functions which subsumes the approximation of functions of bounded variation as a special case. Theorem 2 gives an estimate for absolutely continuous functions which is best possible in the asymptotical sense. [Copyright &y& Elsevier]

In this paper, we study the critical Sobolev embeddings W 1 , p ( ⋅ ) ( Ω ) ⊂ L p ⁎ ( ⋅ ) ( Ω ) for variable exponent Sobolev spaces from the point of view of the Γ-convergence. More precisely we determine the Γ-limit of subcritical approximation of the best constant associated with this embedding. As an application we provide a sufficient condition for the existence of extremals for the best constant. [ABSTRACT FROM AUTHOR]

In this paper, we investigate some approximations to inverse moments of double-indexed weighted sums of random variables. Let { Z n } be a sequence of nonnegative independent random variables and { w n i } be a triangular array of nonnegative non-random weights. With the finite first moment, we establish that E ( a + ∑ i = 1 n w n i Z i ) − α ∼ ( a + ∑ i = 1 n w n i E Z i ) − α holds for all a > 0 and α > 0 . By the finite r -th moment ( r > 2 ), the convergence rate of approximation between E ( a + ∑ i = 1 n w n i Z i ) − α and ( a + ∑ i = 1 n w n i E Z i ) − α is presented. On the other hand, we obtain some approximations for the estimators of E ( ∑ i = 1 n w n i Z i a + ∑ i = 1 n Z i ) , E ( ∑ i = 1 n w n i Z i a + ∑ i = 1 n Z i ) 2 and Var ( ∑ i = 1 n w n i Z i a + ∑ i = 1 n Z i ) for all a > 0 . Finally, some examples and simulations are illustrated. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the notion of linkage by perfect modules, which we call perfect linkage, for Cohen–Macaulay modules over Cohen–Macaulay local rings. We explore perfect linkage in connection with syzygies, maximal Cohen–Macaulay approximations and Yoshino–Isogawa linkage. We recover a theorem of Yoshino and Isogawa, and analyze the structure of double perfect linkage. Moreover, we establish a criterion for two Cohen–Macaulay modules of codimension one to be perfectly linked, and apply it to the classical linkage theory for ideals. We also construct various examples of linkage of modules and ideals. [ABSTRACT FROM AUTHOR]

In this paper, we establish a double inequality for the gamma function, from which we deduce the following approximation formula: Γ ( x + 1 ) ∼ 2 π x ( x e ) x ( 1 + 1 12 x 3 + 24 7 x − 1 2 ) x 2 + 53 210 , x → ∞ , which is more accurate than the Burnside, Gosper, Ramanujan, Windschitl, and Nemes formulas. We develop the previous approximation formula to produce an asymptotic expansion. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the double inequality λ S 11 / 4 , 7 / 4 ( 1 , r ′ ) < E ( r ) < μ S 11 / 4 , 7 / 4 ( 1 , r ′ ) holds for all r ∈ ( 0 , 1 ) if and only if λ ≤ π / 2 = 1.570796 ⋯ and μ ≥ 11 / 7 = 1.571428 ⋯ , where r ′ = ( 1 − r 2 ) 1 / 2 , E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 ⁡ ( t ) d t is the complete elliptic integral of the second kind, and S p , q ( a , b ) = [ q ( a p − b p ) / ( p ( a q − b q ) ) ] 1 / ( p − q ) is the Stolarsky mean of a and b . [ABSTRACT FROM AUTHOR]

In this paper, we give sharp Rusak- and Markov-type inequalities for rational functions on several intervals when the system of intervals is a “rational function inverse image” of an interval and those functions are large in gaps. [ABSTRACT FROM AUTHOR]

In 1987 the author gave an example of a non convex Chebyshev set S in the incomplete inner product space E consisting of the vectors in l 2 which have at most a finite number of non zero terms. In this paper, we show that the closure of S in the Hilbert space completion l 2 of E is not Chebyshev in l 2 . [ABSTRACT FROM AUTHOR]

Computing the roots of a univariate polynomial is a fundamental and long-studied problem of computational algebra with applications in mathematics, engineering, computer science, and the natural sciences. For isolating as well as for approximating all complex roots, the best algorithm known is based on an almost optimal method for approximate polynomial factorization, introduced by Pan in 2002. Pan's factorization algorithm goes back to the splitting circle method from Schönhage in 1982. The main drawbacks of Pan's method are that it is quite involved 2 2 In Victor Pan's own words: “Our algorithms are quite involved, and their implementation would require a non-trivial work, incorporating numerous known implementation techniques and tricks”. In fact, we are not aware of any implementation of Pan's method. and that all roots have to be computed at the same time. For the important special case, where only the real roots have to be computed, much simpler methods are used in practice; however, they considerably lag behind Pan's method with respect to complexity. In this paper, we resolve this discrepancy by introducing a hybrid of the Descartes method and Newton iteration, denoted A New Dsc , which is simpler than Pan's method, but achieves a run-time comparable to it. Our algorithm computes isolating intervals for the real roots of any real square-free polynomial, given by an oracle that provides arbitrary good approximations of the polynomial's coefficients. A New Dsc can also be used to only isolate the roots in a given interval and to refine the isolating intervals to an arbitrary small size; it achieves near optimal complexity for the latter task. [ABSTRACT FROM AUTHOR]

This paper studies the stability of periodic solutions of distributed parameters biochemical system with periodic input S in ( t ) . We prove that if S in ( t ) is periodic then the system has a periodic solution that is input to state stable when small perturbations are acting on the input concentration S in ( t ) . [ABSTRACT FROM AUTHOR]

The celebrated Hajnal–Szemerédi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K k -packing. Fischer's conjecture states that the analogous result holds for all multipartite graphs except for those formed by a single construction. Recently, we deduced an approximate version of this conjecture from new results on perfect matchings in hypergraphs. In this paper, we apply a stability analysis to the extremal cases of this argument, thus showing that the exact conjecture holds for any sufficiently large graph. [ABSTRACT FROM AUTHOR]

It is well-known from the recent literature that nonlocal integral models are suitable to approximate integral functionals or partial differential equations. In the present work, a nonlocal optimal design model has been considered as approximation of the corresponding classical or local optimal control problem. The new model is driven by a nonlocal elliptic equation and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove existence of optimal design for the new model. This work is complemented by showing that the limit of the nonlocal problem is the local one when the cost to minimize is the compliance functional (see [14] ). [ABSTRACT FROM AUTHOR]

We prove a few interesting inequalities for Lorentz polynomials. A highlight of this paper states that the Markov-type inequality max x ∈ [ − 1 , 1 ] | f ′ ( x ) | ≤ n max x ∈ [ − 1 , 1 ] | f ( x ) | holds for all polynomials f of degree at most n with real coefficients for which f ′ has all its zeros outside the open unit disk. Equality holds only for f ( x ) : = c ( ( 1 ± x ) n − 2 n − 1 ) with a constant 0 ≠ c ∈ R . This should be compared with Erdős’s classical result stating that max x ∈ [ − 1 , 1 ] | f ′ ( x ) | ≤ n 2 ( n n − 1 ) n − 1 max x ∈ [ − 1 , 1 ] | f ( x ) | for all polynomials f of degree at most n having all their zeros in R ∖ ( − 1 , 1 ) . [ABSTRACT FROM AUTHOR]

Padé approximation has two natural extensions to vector rational approximation through the so-called type I and type II Hermite–Padé approximants. The convergence properties of type II Hermite–Padé approximants have been studied. For such approximants Markov and Stieltjes type theorems are available. To the present, such results have not been obtained for type I approximants. In this paper, we provide Markov and Stieltjes type theorems on the convergence of type I Hermite–Padé approximants for Nikishin systems of functions. [ABSTRACT FROM AUTHOR]

In this paper, we will show that any domain D in C n which is spirallike with respect to a linear operator A , where m ( A ) > 0 , is Runge. We also show the local uniform approximation of biholomorphic mappings on a spirallike domain D with respect to A , where k + ( A ) < 2 m ( A ) , by automorphisms of C n . Finally, as an application of the above result, we will show that any Loewner PDE in a complete hyperbolic spirallike domain D with respect to A , where k + ( A ) < 2 m ( A ) , of C n admits an essentially unique univalent solution with values in C n . [ABSTRACT FROM AUTHOR]

This paper addresses two different but related questions regarding an unbounded symmetric tridiagonal operator: its self-adjointness and the approximation of its spectrum by the eigenvalues of its finite truncations. The sufficient conditions given in both cases improve and generalize previously known results. It turns out that, not only self-adjointness helps to study limit points of eigenvalues of truncated operators, but the analysis of such limit points is a key help to prove self-adjointness. Several examples show the advantages of these new results compared with previous ones. Besides, an application to the theory of continued fractions is pointed out. [ABSTRACT FROM AUTHOR]

Text: We consider the analogue, when Z is replaced with Fq[T] of the elementary cyclotomic polynomials and prove an analogue of Suzuki's Theorem. Video: For a video summary of this paper, please visit http://youtu.be/mVBw82N34NE. [ABSTRACT FROM AUTHOR]

The Douglas-Rachford splitting algorithm is a classical optimization method that has found many applications. When specialized to two normal cone operators, it yields an algorithm for finding a point in the intersection of two convex sets. This method for solving feasibility problems has attracted a lot of attention due to its good performance even in nonconvex settings. In this paper, we consider the Douglas-Rachford algorithm for finding a point in the intersection of two subspaces. We prove that the method converges strongly to the projection of the starting point onto the intersection. Moreover, if the sum of the two subspaces is closed, then the convergence is linear with the rate being the cosine of the Friedrichs angle between the subspaces. Our results improve upon existing results in three ways: First, we identify the location of the limit and thus reveal the method as a best approximation algorithm; second, we quantify the rate of convergence, and third, we carry out our analysis in general (possibly infinite-dimensional) Hilbert space. We also provide various examples as well as a comparison with the classical method of alternating projections. [ABSTRACT FROM AUTHOR]

Abstract: Let be a Hilbert -module over a -algebra , and . In this paper we study a problem of finding such that for all . We show that such an operator exists if and only if the range of is contained in the range of , and in this case it can be chosen to belong to . We also consider Hilbert -modules in which for every X and Y there is (a unique) A with the above property. [Copyright &y& Elsevier]

Abstract: In recent years many new definitions of shadowing, using a notion of ergodic (or average) pseudo-orbit were introduced. While, under the assumption of chain mixing, average pseudo-orbit usually can be equivalently replaced by pseudo-orbit in these definitions of shadowing, it is not completely clear when these shadowing properties (i.e. approximation of a pseudo-orbit by a real orbit on a sufficiently large set of indices) can or cannot occur. In this paper we analyze necessary and sufficient conditions for shadowing over a set with positive density (or syndetic). While we do not provide full characterization, a few relations to standard notions from topological dynamics are obtained. [Copyright &y& Elsevier]

Abstract: In this paper is shown that solutions associated with an extended affine control system can be obtained as a limit process using solutions for a parameterized affine control system and weak small controls. [Copyright &y& Elsevier]

Abstract: In both Lyubarskii and Madych (1994) [7], and Schlumprecht and Sivakumar (2009) [9], it was shown that Paley–Wiener functions may be recovered from their values on a complete interpolating sequence. This paper explores the same phenomenon, and gives a sufficient condition on a function , called an interpolator, so that scattered translates of this function may be used to interpolate and recover any given Paley–Wiener function. [Copyright &y& Elsevier]

Abstract: In this paper our aim is to deduce some Turán type inequalities for -hypergeometric and -confluent hypergeometric functions. In order to obtain the main results we apply the methods developed in the case of classical Kummer and Gauss hypergeometric functions. [Copyright &y& Elsevier]

Abstract: Bernstein inequalities and inverse theorems are a recent development in the theory of radial basis function (RBF) approximation. The purpose of this paper is to extend what is known by deriving Bernstein inequalities for RBF networks on . These inequalities involve bounding a Bessel-potential norm of an RBF network by its corresponding norm in terms of the separation radius associated with the network. The Bernstein inequalities will then be used to prove the corresponding inverse theorem. [Copyright &y& Elsevier]

Abstract: Compressive sensing (CS) has recently emerged as a framework for efficiently capturing signals that are sparse or compressible in an appropriate basis. While often motivated as an alternative to Nyquist-rate sampling, there remains a gap between the discrete, finite-dimensional CS framework and the problem of acquiring a continuous-time signal. In this paper, we attempt to bridge this gap by exploiting the Discrete Prolate Spheroidal Sequences (DPSSʼs), a collection of functions that trace back to the seminal work by Slepian, Landau, and Pollack on the effects of time-limiting and bandlimiting operations. DPSSʼs form a highly efficient basis for sampled bandlimited functions; by modulating and merging DPSS bases, we obtain a dictionary that offers high-quality sparse approximations for most sampled multiband signals. This multiband modulated DPSS dictionary can be readily incorporated into the CS framework. We provide theoretical guarantees and practical insight into the use of this dictionary for recovery of sampled multiband signals from compressive measurements. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce and solve of the radical quadratic and radical quartic functional equations: We also establish some stability results in 2-normed spaces and then the stability by using subadditive and subquadratic functions in -2-normed spaces for these functional equations. [Copyright &y& Elsevier]

Abstract: In this paper we explain how to characterize the best approximation to any in a Hilbert space from the set in the face of data uncertainty in the convex constraints, , where is a closed convex subset of . Following the robust optimization approach, we establish Lagrange multiplier characterizations of the robust constrained best approximation that is immunized against data uncertainty. This is done by characterizing the best approximation to any from the robust counterpart of the constraints where the constraints are satisfied for all possible uncertainties within the prescribed uncertainty sets. Unlike the traditional Lagrange multiplier characterizations without data uncertainty, for constrained best approximation problems in the face uncertainty, we show that the strong conical hull intersection property (strong CHIP) alone is not sufficient to guarantee the Lagrange multiplier characterizations. We present conditions which guarantee that the strong CHIP is necessary and sufficient for the multiplier characterization. We also establish that the strong CHIP is automatically satisfied for the cases of polyhedral constraints with polytope uncertainty, and linear constraints with interval uncertainty. As an application, we show how robust solutions of shape preserving interpolation problems under ellipsoidal and box uncertainty cases can be obtained in terms of Lagrange multipliers under strict robust feasibility conditions. [Copyright &y& Elsevier]

Abstract: We give necessary and sufficient conditions on the Fourier transform of a generator function of a principal shift-invariant subspace of providing density order . Our starting point is a paper by de Boor, DeVore, and Ron [C. de Boor, R.A. DeVore, A. Ron, Approximation from shift-invariant subspaces of , Trans. Amer. Math. Soc. 341 (2) (1994) 787–806] and the new conditions that we present here involve the classical notion of the point of approximate continuity. In addition, we study properties of the approximation of a shift-invariant subspace of when it is dilated by a diagonalizable expansive linear map . Indeed, we present a necessary and sufficient condition on a principal shift-invariant subspace such that its union with itself dilated by integer powers of is dense in . [Copyright &y& Elsevier]

Abstract: We consider two types of Schrödinger operators and defined on , where q is an even potential that is bounded from below, A is a constant, and is a parameter. We assume that has at least two eigenvalues below its essential spectrum; and we denote by and the lowest eigenvalue and the second one, respectively. The purpose of this paper is to study the asymptotics of the gap in the limit as . [Copyright &y& Elsevier]

Abstract: In this paper, the Berry–Esséen bound of sample quantiles for φ-mixing random variables is investigated under some weak conditions, for example, the condition of mixing coefficients only needs that for some . The rate of normal approximation is shown as . [Copyright &y& Elsevier]

Abstract: In the previous paper (Almira and Oikhberg, 2010 ), the authors investigated the existence of an element x of a quasi-Banach space X whose errors of best approximation by a given approximation scheme (defined by ) decay arbitrarily slowly. In this work, we consider the question of whether x witnessing the slowness rate of approximation can be selected in a prescribed subspace of X. In many particular cases, the answer turns out to be positive. [Copyright &y& Elsevier]

Abstract: In recent years, many approaches have been utilized for finding the exact solutions of nonlinear systems of partial differential equations. In this paper, the first integral method introduced by Feng is adopted for solving some important nonlinear systems of partial differential equations, including, KdV, Kaup–Boussinesq and Wu–Zhang systems, analytically. By means of this method, some exact solutions for these systems of equations are formally obtained. The results obtained confirm that the proposed method is an efficient technique for analytic treatment of a wide variety of nonlinear systems of partial differential equations. [Copyright &y& Elsevier]

Abstract: In this paper, we establish approximation formulas for evaluating Landauʼs constants. [Copyright &y& Elsevier]

Abstract: Our paper is devoted to a study of an eddy current problem with a nonlinear evolution degenerate boundary condition of the type with a power-law nonlinearity. We have designed a nonlinear numerical scheme for approximation in suitable function spaces. The well-posedness of the problem is addressed and the error estimates are derived. Monotonicity methods and the Minty–Browder argument are used in the proofs. [Copyright &y& Elsevier]

Abstract: In this paper, we consider partial regularity for weak solutions of second-order nonlinear sub-elliptic systems in Carnot groups. By the method of -harmonic approximation, we establish optimal interior partial regularity of weak solutions to systems under natural growth conditions with subquadratic growth in Carnot groups. [Copyright &y& Elsevier]