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Recently, Alaca et al. [Alaca et al., Chaos, Solitons & Fractals 2006;29:1073–9] proved some fixed point theorems in intuitionistic fuzzy metric spaces by a strong definition of Cauchy sequence (see [George and Veeramani, Fuzzy Sets Syst 1994;64:395–9] and [Veeramani and Vasuki, Fuzzy Sets Syst 2003;135:409–13]), also the intuitionistic fuzzy metric space has extra conditions (see [Gregori et al., Chaos, Solitons & Fractals, 2006;28:902–5]). In this paper, we consider generalized intuitionistic fuzzy metric spaces i.e., L -fuzzy metric spaces and prove the fuzzy version of Banach and Edelstein contraction theorems in these spaces for modified definition of Cauchy sequence.

It is pointed out that the proof of the basic result of Rahmanov's paper has a serious gap. It is documented by original sources that a statement he relied on in the proof contains a misprint, and it is shown by a counterexample that this statement (with the misprint) is, in fact, false. A somewhat weaker statement is proved true.

The converse of the Borel–Cantelli Lemma states that if { A i } i = 1 ∞ is a sequence of independent events such that ∑ P ( A i ) = ∞ , then almost surely infinitely many of these events will occur. Erdős and Renyi proved that it is sufficient to weaken the condition of independence to pairwise independence. Later, several other weakenings of the condition appeared in the literature. The aim of this paper is to provide a collection of conditions, all of which imply that almost surely infinitely many of the events occur, and determine the complete implicational relationship between them. Many of these results are known, or follow from known results, however, they are not widely known among non-specialists. Yet, the results can be extremely useful for areas outside of probability theory, as evidenced by the original motivation of this paper emerging from infinite combinatorics. Our proofs are aimed to be accessible to a general mathematical audience.

The norm of a partition is defined as the product of its parts. This paper aims to conduct a thorough study of norms of prime partitions which are partitions with all prime parts. The analysis begins by defining r p ( i , n ) , called the primal norm counting function, which refers to number of times i appears as a norm to the prime partitions of n. The generating function for this function is deduced, using which a wealth of intriguing relations are proved like series, product, integral representations, and recursive relations, etc. The special cases of these results bear resemblance to other results known in classical partition theory. An analogue of the Goldbach conjecture in the theory of norms is presented, and it is stressed that an explicit formula for r p ( i , n ) must be extracted. Two approaches are discussed for this purpose, and the paper is concluded with a potential scope for future work.

We present a covering conjecture that we expect to be true below superstrong cardinals. We then show that the conjecture is true in hod mice. This work is a continuation of the work that started in Covering with Universally Baire Functions Advances in Mathematics, and the main conjecture of the current paper is a revision of the UB Covering Conjecture of the aforementioned paper.

In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is “the generalized Neumann lemma” which is quite different from the method in [12] where “the generalized Banach lemma” was used. By the method of the perturbation analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.

In this paper we compute topological entropy with prescribed accuracy for different economic models, showing the existence of a topologically chaotic regime for them. In order to make the paper self-contained, a general overview on the topological entropy of continuous interval maps is given. More precisely, we focus on piecewise monotone maps which often appear as dynamical models in economy, but also in population growth and physics. Our main aim is to show that when topological entropy can be approximated up to a given error, it is a useful tool which helps to analyze the chaotic dynamics in one dimensional models.

It is well-known that dyadic martingale transforms are a good model for Calderon–Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that if W is an A 2 matrix weight, then the weighted L 2 -norm of a Calderon–Zygmund operator with cancellation has the same dependence on the A 2 characteristic of W as the weighted L 2 -norm of an appropriate matrix martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderon–Zygmund operators on the A 2 characteristic of the weight is reduced to the case of dyadic martingales and paraproducts. We also show a slightly different proof for the special case of Calderon–Zygmund operators with even kernel, where only scalar martingale transforms are required. We conclude the paper by proving a version of the matrix-weighted Carleson Embedding Theorem. Our method uses a Bellman function technique introduced by S. Treil to obtain the right estimates for the norm of dyadic Haar shift operators. We then apply the representation theorem of T. Hytonen to extend the result to general Calderon–Zygmund operators.

We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an e -approximation for the d -variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on e − 1 . The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on e − 1 , we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential ( s , t ) -weak tractability with max ( s , t ) ≥ 1 . For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential ( s , t ) -weak tractability with max ( s , t ) 1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woźniakowski (2019).

Let E be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. Let τ be a translation automorphism of E that is not of order 2. In a previous paper we studied an algebra A = A ( E , τ ) that depends on this data: A ( E , τ ) = ( S ( E , τ ) ⊗ M 2 ( k ) ) Γ where S ( E , τ ) is the 4-dimensional Sklyanin algebra associated to ( E , τ ) , M 2 ( k ) is the ring of 2 × 2 matrices over k, and Γ is ( Z / 2 ) × ( Z / 2 ) acting in a particular way as automorphisms of S and M 2 ( k ) . The action of Γ on S is compatible with the translation action of the 2-torsion subgroup E [ 2 ] on E. Following the ideas and results in papers of Artin–Tate–Van den Bergh, Smith–Stafford, and Levasseur–Smith, this paper examines the line modules, point modules, and fat point modules, over A, and their incidence relations. The right context for the results is non-commutative algebraic geometry: we view A as a homogeneous coordinate ring of a non-commutative analogue of P 3 that we denote by Proj n c ( A ) . Point modules and fat point modules determine “points” in Proj n c ( A ) . Line modules determine “lines” in Proj n c ( A ) . Line modules for A are in bijection with certain lines in P ( A 1 ⁎ ) ≅ P 3 and therefore correspond to the closed points of a certain subscheme L of the Grassmannian G ( 1 , 3 ) . Shelton–Vancliff call L the line scheme for A. We show that L is the union of 7 reduced and irreducible components, 3 quartic elliptic space curves and 4 plane conics in the ambient Plucker P 5 , and that deg ⁡ ( L ) = 20 . The union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in P ( A 1 ⁎ ) . Thus, the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for A.

The focus of this paper commences with an examination of three (not obviously related) pages in Ramanujan's lost notebook, pages 336, 335, and 332, in decreasing order of attention. On page 336, Ramanujan proposes two identities, but the formulas are wrong – each is vitiated by divergent series. We concentrate on only one of the two incorrect “identities,” which may have been devised to attack the extended divisor problem. We prove here a corrected version of Ramanujan's claim, which contains the convergent series appearing in it. The convergent series in Ramanujan's faulty claim is similar to one used by G.F. Voronoi, G.H. Hardy, and others in their study of the classical Dirichlet divisor problem. This now brings us to page 335, which comprises two formulas featuring doubly infinite series of Bessel functions, the first being conjoined with the classical circle problem initiated by Gauss, and the second being associated with the Dirichlet divisor problem. The first and fourth authors, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this monograph, we return to these two formulas and examine them in more general settings. The aforementioned convergent series in Ramanujan's “identity” is also similar to one that appears in a curious identity found in Chapter 15 in Ramanujan's second notebook, written in a more elegant, equivalent formulation on page 332 in the lost notebook. This formula may be regarded as a formula for ζ ( 1 2 ) , and in 1925, S. Wigert obtained a generalization giving a formula for ζ ( 1 k ) for any even integer k ≥ 2 . We extend the work of Ramanujan and Wigert in this paper. The Voronoi summation formula appears prominently in our study. In particular, we generalize work of J.R. Wilton and derive an analogue involving the sum of divisors function σ s ( n ) . The modified Bessel functions K s ( x ) arise in several contexts, as do Lommel functions. We establish here new series and integral identities involving modified Bessel functions and modified Lommel functions. Among other results, we establish a modular transformation for an infinite series involving σ s ( n ) and modified Lommel functions. We also discuss certain obscure related work of N.S. Koshliakov. We define and discuss two new related classes of integral transforms, which we call Koshliakov transforms, because he first found elegant special cases of each.

In this paper we study d -variate approximation for weighted Korobov spaces in the worst-case setting. The considered algorithms use finitely many evaluations of arbitrary linear functionals. We give matching necessary and sufficient conditions for some notions of tractability in terms of two weight parameters. Our result is an affirmative answer to a problem which is left open in a recent paper of Kritzer, Pillichshammer and Wo?niakowski.

In this paper we consider the following critical nonlocal problem { − L K u = λ u + | u | 2 ⁎ − 2 u in Ω u = 0 in R n ∖ Ω , where s ∈ ( 0 , 1 ) , Ω is an open bounded subset of R n , n > 2 s , with continuous boundary, λ is a positive real parameter, 2 ⁎ : = 2 n / ( n − 2 s ) is the fractional critical Sobolev exponent, while L K is the nonlocal integrodifferential operator L K u ( x ) : = ∫ R n ( u ( x + y ) + u ( x − y ) − 2 u ( x ) ) K ( y ) d y , x ∈ R n , whose model is given by the fractional Laplacian − ( − Δ ) s . Along the paper, we prove a multiplicity and bifurcation result for this problem, using a classical theorem in critical points theory. Precisely, we show that in a suitable left neighborhood of any eigenvalue of − L K (with Dirichlet boundary data) the number of nontrivial solutions for the problem under consideration is at least twice the multiplicity of the eigenvalue. Hence, we extend the result got by Cerami, Fortunato and Struwe in [14] for classical elliptic equations, to the case of nonlocal fractional operators.

In this paper we report for the first time on the binary generalized synchronization, when for the certain values of the coupling strength two unidirectionally coupled dynamical systems generating the aperiodic binary sequences are in the generalized synchronization regime. The presence of the binary generalized synchronization has been revealed with the help of both the auxiliary system approach and the largest conditional Lyapunov exponent calculation. The mechanism resulting in the binary generalized synchronization has been explained. The finding discussed in this paper gives a strong potential for new applications under many relevant circumstances.

In this paper, we investigate the factor properties and gap sequence of the Tribonacci sequence, the fixed point of the substitution σ( a , b , c ) = ( ab , ac , a ). Let ω p be the p-th occurrence of ω and G p ( ω ) be the gap between ω p and ω p +1 . We introduce a notion of kernel for each factor ω , and then give the decomposition of the factor ω with respect to its kernel. Using the kernel and the decomposition, we prove the main result of this paper: for each factor ω , the gap sequence { G p ( ω )} p ≤1 is the Tribonacci sequence over the alphabet { G 1 ( ω ), G 2 ( ω ), G 4 ( ω )}, and the expressions of gaps are determined completely. As an application, for each factor ω and p ∈ ℕ , we determine the position of ω p . Finally we introduce a notion of spectrum for studying some typical combinatorial properties, such as power, overlap and separate of factors.

In this paper we investigate quasi-Monte Carlo (QMC) integration using digital nets over Z b in reproducing kernel Hilbert spaces. The tent transformation (previously called baker’s transform) was originally used for lattice rules by Hickernell (2002) to achieve higher order convergence of the integration error for smooth non-periodic integrands, and later, has been successfully applied to digital nets over Z 2 by Cristea et al. (2007) and Goda (2015). The aim of this paper is to generalize the latter two results to digital nets over Z b for an arbitrary prime b . For this purpose, we introduce the b -adic tent transformation for an arbitrary positive integer b greater than 1, which is a generalization of the original (dyadic) tent transformation. Further, again for an arbitrary positive integer b greater than 1, we analyze the mean square worst-case error of QMC rules using digital nets over Z b which are randomly digitally shifted and then folded using the b -adic tent transformation in reproducing kernel Hilbert spaces. Using this result, for a prime b , we prove the existence of good higher order polynomial lattice rules over Z b among a smaller number of candidates as compared to the result by Dick and Pillichshammer (2007), which achieve almost the optimal convergence rate of the mean square worst-case error in unanchored Sobolev spaces of smoothness of arbitrary high order.

Let C be the complex Levi-Civita field and let c 0 ( C ) or, simply, c 0 denote the space of all null sequences z = ( z n ) n ∈ N of elements of C . The natural inner product on c 0 induces the sup-norm of c 0 . In a previous paper Aguayo et al. (2013), we presented characterizations of normal projections, adjoint operators and compact operators on c 0 . In this paper, we work on some B ∗ -algebras of operators, including those mentioned above; then we define an inner product on such algebras and prove that this inner product induces the usual norm of operators. We finish the paper with a characterization of closed subspaces of the B ∗ -algebra of all adjoint and compact operators on c 0 which admit normal complements.

We introduce a covering conjecture and show that it holds below A D R + “ Θ is regular” . We then use it to show that in the presence of mild large cardinal axioms, PFA implies that there is a transitive model containing the reals and ordinals and satisfying A D R + “ Θ is regular” . The method used to prove the Main Theorem of this paper is the core model induction. The paper contains the first application of the core model induction that goes significantly beyond the region of A D + + θ 0 Θ .

The Calkin theorem provides a one-to-one correspondence between all operator ideals A(H) over the separable infinite-dimensional Hilbert space H and all symmetric sequence ideals a(N) over the index set N≔{1,2,…}. The main idea of the present paper is to replace a(N) by the ideal z(N0) that consists of all sequences (αh) indexed by N0≔{0,1,2,…} for which (α0,α1,α1,…,αh,…,αh︷2hterms,…)∈a(N). This new kind of sequence ideals is characterized by two properties: (1) For (αh)∈z(N0) there is a non-increasing (βh)∈z(N0) such that ∣αh∣≤βh. (2) z(N0) is invariant under the operator S+:(α0,α1,α2,…)↦(0,α0,α1,…). Using this modification of the Calkin theorem, we simplify, unify, and complete earlier results of [4,5,7–9,13,14,19–21,25] The central theorem says that there are canonical isomorphisms between the linear spaces of all traces on A(H), all symmetric linear forms on a(N), and all 12S+-invariant linear forms on z(N0). In this way, the theory of linear forms on ideals of a non-commutative algebra that are invariant under the members of a non-commutative group is reduced to the theory of linear forms on ideals of a commutative algebra that are invariant under a single operator. It is hoped that the present approach deserves the rating “streamlined”. Our main objects are linear forms in the purely algebraic sense. Only at the end of this paper continuity comes into play, when the case of quasi-normed ideals is considered. We also sketch a classification of operator ideals according to the existence of various kinds of traces. Details will be discussed in a subsequent publication.

In this paper, we introduce and study an implicit iterative method to approximate a common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup in real Hilbert spaces. Further, we prove that the nets generated by the implicit iterative method converge strongly to the common solution of split equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. Furthermore, we justify our main result through a numerical example. The results presented in this paper extend and generalize the corresponding results given by Plubtieng and Punpaeng [S. Plubtieng, R. Punpaeng, Fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Model. 48 (2008) 279–286] and Cianciaruso et al. [F. Cianciaruso, G. Marino, L. Muglia, Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert space, J. Optim. Theory Appl. 146 (2010) 491–509].

In this paper, we investigate the bounds of the coefficients of several classes of bi-univalent functions. The results presented in this paper improve or generalize the recent works of other authors.

An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^{m}, which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S_n on these spaces is conjectured to be closely related to the natural representation of S_n on (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S_n acts on labelled intervals of T_n^{m} by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S_n. In particular, the dimension of the representation, that is, the number of labelled m-Tamari intervals of size n, is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables., Comment: 29 pages --- This paper subsumes the research report arXiv:1109.2398, which will not be submitted to any journal

The goal of the present paper is to investigate some new HUR-stability results by applying the alternative fixed point of generalized quartic functional equation ∑ k = 2 n ( ∑ i 1 = 2 k ∑ i 2 = i 1 + 1 k + 1 … ∑ i n − k + 1 = i n − k + 1 n ) f ( ∑ i = 1 , i ≠ i 1 , … , i n − k + 1 n x i − ∑ r = 1 n − k + 1 x i r ) + f ( ∑ i = 1 n x i ) = 2 n − 2 ∑ 1 ≤ i j ≤ n ( f ( x i + x j ) + f ( x i − x j ) ) − 2 n − 5 ( n − 2 ) ∑ i = 1 n f ( 2 x i ) ( n ∈ ℕ , n ≥ 3 ) in β-Banach modules on Banach algebras. The concept of Ulam-Hyers-Rassias stability (briefly, HUR-stability) originated from Th. M. Rassias stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300.

The subject of this paper is to give the description, up to topological conjugacy, of possible minimal sets of triangular maps of the square of type 2 ∞ . In [4] , we give a general method allowing to embed any zero-dimensional almost 1–1 extension of the dyadic odometer (in particular any dyadic Toeplitz system) as a minimal set of a triangular map of this type. In this paper we present a method (a combination of that described in [4] with one introduced in [1] ) of similarly embedding a special class of zero-dimensional almost 2–1 extensions of the odometer. We conjecture that these two embedding theorems exhaust all possibilities for nonperiodic minimal sets. The paper was inspired by the last unsolved problem in the Sharkovski classification program of triangular maps: does there exist a triangular map with positive entropy attained on the set of uniformly recurrent points but with entropy zero on the set of regularly recurrent points. The paper answers this question positively, concluding the program.

In this paper, we introduce the new notion of g-monotone mapping and prove some fixed point theorems for multi-valued and single-valued g-increasing mappings in partially ordered metric spaces. The mappings considered in this paper are assumed to satisfy certain metric inequalities which are established by an altering distance function. The presented results extend and improve the main results of Choudhury and Metiya [B.S. Choudhury, N. Metiya, Multi-valued and single-valued fixed point results in partially ordered metric spaces, Arab J. Math. Sci. 17 (2011) 135–151].

In an earlier paper (arxiv:1108.4292) we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on $K$, namely $\sup{\dim_{H}f^{-1}(y) : y \in \mathbb{R}} = \dim_{tH} K - 1$ for the generic $f \in C(K)$, provided that $K$ is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if $K$ is not totally disconnected and sufficiently homogeneous then $\dim_{H}f^{-1}(y) = \dim_{tH} K - 1$ for the generic $f \in C(K)$ and the generic $y \in f(K)$. The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic $f\in C(K)$ and the generic $y\in f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. We also generalize a result of B. Kirchheim by showing that if $K$ is self-similar then for the generic $f\in C(K)$ for every $y\in \inter f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. Finally, we prove that the graph of the generic $f\in C(K)$ has the same Hausdorff and topological Hausdorff dimension as $K$., 20 pages

As known, the method to obtain a sequence space by using convergence field of an infinite matrix is an old method in the theory of sequence spaces. However, the study of convergence field of an infinite matrix in the space of almost convergent sequences is so new (see [15]). The purpose of this paper is to introduce the new spaces and consisting of all sequences whose Cesaro transforms of order one are in the spaces f and f0, respectively. Also, in this paper, we show that and are linearly isomorphic to the spaces f and f0, respectively. The β- and γ-duals of the spaces and are computed. Furthermore, the classes ( : μ) and (μ: ) of infinite matrices are characterized for any given sequence space μ, and determined the necessary and sufficient conditions on a matrix A to satisfy BC-core(Ax) ⊆ K-core(x), K-core(Ax) ⊆ BC-core(x), BC-core(Ax) ⊆ BC-core(x), BC-core(Ax) ⊆ st-core(x) for all x ∈ l∈.

Let a and b be two even integers with 2 ⩽ a < b, and let k be a nonnegative integer. Let G be a graph of order n. Its binding number bind(G) is defined as follows, bind(G)=min|NG(X)||X|:∅≠X⊆V(G),NG(X)≠V(G). In this paper, it is proved that G is an (a, b, k)-critical graph if bind(G)>(a+b-1)(n-1)bn-(a+b)-bk+3 and n⩾(a+b)(a+b-3)b+bkb-1. Furthermore, it is shown that the result in this paper is best possible in some sense.

This is a follow-up paper of “Liberating the dimension for function approximation”, where we studied approximation of infinitely variate functions by algorithms that use linear information consisting of finitely many linear functionals. In this paper, we study similar approximation problems, however, now the algorithms can only use standard information consisting of finitely many function values. We assume that the cost of one function value depends on the number of active variables. We focus on polynomial tractability, and occasionally also study weak tractability. We present non-constructive and constructive results. Non-constructive results are based on known relations between linear and standard information for finitely variate functions, whereas constructive results are based on Smolyak’s construction generalized to the case of infinitely variate functions. Surprisingly, for many cases, the results for standard information are roughly the same as for linear information.

The purpose of this paper is to study necessary and sufficient condition for the strong convergence of a new parallel iterative algorithm with errors for two finite families of uniformly L -Lipschitzian mappings in Banach spaces. The results presented in this paper improve and extend the recent ones announced by [2–7].

This paper is another case study in the program of logically analyzing proofs to extract new (typically effective) information (‘proof mining’). We extract explicit uniform rates of metastability (in the sense of T. Tao) from two ineffective proofs of a classical theorem of F.E. Browder on the convergence of approximants to fixed points of nonexpansive mappings as well as from a proof of a theorem of R. Wittmann which can be viewed as a nonlinear extension of the mean ergodic theorem. The first rate is extracted from Browder's original proof that is based on an application of weak sequential compactness (in addition to a projection argument). Wittmann's proof follows a similar line of reasoning and we adapt our analysis of Browder's proof to get a quantitative version of Wittmann's theorem as well. In both cases one also obtains totally elementary proofs (even for the strengthened quantitative forms) of these theorems that neither use weak compactness nor the existence of projections anymore. In this way, the present article also discusses general features of extracting effective information from proofs based on weak compactness. We then extract another rate of metastability (of similar nature) from an alternative proof of Browder's theorem essentially due to Halpern that already avoids any use of weak compactness. The paper is concluded by general remarks concerning the logical analysis of proofs based on weak compactness as well as a quantitative form of the so-called demiclosedness principle. In a subsequent paper these results will be utilized in a quantitative analysis of Baillon's nonlinear ergodic theorem.

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ _S= +\lambda , $$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $\lambda$ is a positive constant. In this paper we get algebraic and differential/difference properties for such polynomials as well as algebraic relations between them and the polynomial sequence orthogonal with respect to the semiclassical functional $\bf u$. The main goal of this article is to give a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator ${\mathscr D}$ considered. Finally, we illustrate our results by applying them to some known families of Sobolev orthogonal polynomials as well as to some new ones introduced in this paper for the first time., Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez lagomasino on the occasion of his 60th birthday, accepted in Journal of Approximation Theory

We study the optimal approximation of the solution of an operator equation A(u)=f by linear and different types of nonlinear mappings. In our earlier papers we only considered the error with respect to a certain H^s-norm where s was given by the operator since we assumed that A:H"0^s(@W)->H^-^s(@W) is an isomorphism. The most typical case here is s=1. It is well known that for certain regular problems the order of convergence is improved if one takes the L"2-norm. In this paper we study error bounds with respect to such a weaker norm, i.e., we assume that H"0^s(@W) is continuously embedded into a space X and we measure the error in the norm of X. A major example is X=L"2(@W) or X=H^r(@W) with r

This paper is largely concerned with constructing coarse moduli spaces for Artin stacks. The main purpose of this paper is to introduce the notion of stability on an arbitrary Artin stack and construct a coarse moduli space for the open substack of stable points. Also, we present an application to coherent cohomology of Artin stacks.

In this paper, we have tried to apply the concepts of fuzzy set to Lie groups and fuzzy differential invariant (FDI) in order to provide suitable conditions for applying Lie symmetry method in solving fuzzy differential equations (FDEs). For this, we define a C 1 -fuzzy submanifold and fuzzy immersion with some examples. In main section, we defined the fuzzy Lie group and some its relative concepts such as fuzzy transformation group and fuzzy G-invariant. The goal of this paper is to introduce and study new defining for fuzzy Lie group and fuzzy differential invariant (FDI). Also, some illustrative examples are given.

In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F1, F2 such that the arithmetic difference set F2 − F1 does not contain any intervals but ℒeb(F2 − F1)> 0 almost surely, conditioned on non-extinction.

We study multivariate generalisations of the classical Wiener--Hopf algebra, which is the C$^*$-algebra generated by the Wiener--Hopf operators, given by the convolutions restricted to convex cones. By the work of Muhly and Renault, this C$^*$-algebra is known to be isomorphic to the reduced C$^*$-algebra of a certain restricted action groupoid. In a previous paper, we have determined a composition series of this C$^*$-algebra, and compute the $K$-theory homomorphisms induced by the `symbol' maps given by the subquotients of the composition series in terms of the analytical index of a continuous family of Fredholm operators. In this paper, we obtain a topological expression for these index maps in terms of geometric-topological data naturally associated to the underlying convex cone. The resulting index formula is expressed in the framework of Kasparov's bivariant $KK$-theory. Our proof relies heavily on groupoid methods., Comment: 46 pages, 1 figure; last version prior to publication, journal reference added

Many papers study polynomial tractability for multivariate problems. Let n(@?,d) be the minimal number of information evaluations needed to reduce the initial error by a factor of @? for a multivariate problem defined on a space of d-variate functions. Here, the initial error is the minimal error that can be achieved without sampling the function. Polynomial tractability means that n(@?,d) is bounded by a polynomial in @?^-^1 and d and this holds for all (@?^-^1,d)@?[1,~)xN. In this paper we study generalized tractability by verifying when n(@?,d) can be bounded by a power of T(@?^-^1,d) for all (@?^-^1,d)@[email protected], where @W can be a proper subset of [1,~)xN. Here T is a tractability function, which is non-decreasing in both variables and grows slower than exponentially to infinity. In this article we consider the set @W=[1,~)x{1,2,...,d^*}@?[1,@?"0^-^1)xN for some d^*>=1 and @?"[email protected]?(0,1). We study linear tensor product problems for which we can compute arbitrary linear functionals as information evaluations. We present necessary and sufficient conditions on T such that generalized tractability holds for linear tensor product problems. We show a number of examples for which polynomial tractability does not hold but generalized tractability does.

In this paper, we develop the theory of the necklace ring and the logarithmic function. Regarding the necklace ring, we introduce the necklace ring functor Nr from the category of special λ -rings into the category of special λ -rings and then study the associated Adams operators. As far as the logarithmic function is concerned, we generalize the results in Bryant's paper [Free Lie algebras and formal power series, J. Algebra 253(1) (2002) 167–188] to the case of graded Lie (super)algebras with a group action by applying the Euler–Poincare principle.

In the paper [Y. Xu, Lagrange interpolation on Chebyshev points of two variables, J. Approx. Theory 87 (1996) 220–238], the author introduced a set of Chebyshev-like points for polynomial interpolation (by a certain subspace of polynomials) in the square [-1,1]2, and derived a compact form of the corresponding Lagrange interpolation formula. In [L. Bos, M. Caliari, S. De Marchi, M. Vianello, A numerical study of the Xu polynomial interpolation formula in two variables, Computing 76(3–4) (2005) 311–324], we gave an efficient implementation of the Xu interpolation formula and we studied numerically its Lebesgue constant, giving evidence that it grows like O((logn)2), n being the degree. The aim of the present paper is to provide an analytic proof to show that the Lebesgue constant does have this order of growth.

Let φ be an analytic self-map of the complex unit disk and X a Banach space. This paper studies the action of composition operator C φ : f → f ˆ φ on the vector-valued Nevanlinna classes N(X) and Na(X). Certain criteria for such operators to be weakly compact are given. As a consequence, this paper shows that the composition operator Cφ is weakly compact on N(X) and Na(X) if and only if it is weakly compact on the vector-valued Hardy space H1(X) and Bergman space B1(X) respectively.

In this paper the authors investigate the representation type of the blocks of the relative (parabolic) category O S for complex semisimple Lie algebras. A complete classification of the blocks corresponding to regular weights is given. The main results of the paper provide a classification of the blocks in the “mixed” case when the simple roots corresponding to the singular set and S do not meet.

In this paper we exhibit a triangular map F of the square with the following properties: (i) F is of type 2∞ but has positive topological entropy; we recall that similar example was given by Kolyada in 1992, but our argument is much simpler. (ii) F is distributionally chaotic in the wider sense, but not distributionally chaotic in the sense introduced by Schweizer and Smital [Trans. Amer. Math. Soc. 344 (1994) 737]. In other words, there are lower and upper distribution functions Φxy and Φ xy ∗ generated by F such that Φ xy ∗ ≡1 and Φxy(0+) Φ uv ∗ such that Φ uv ∗ ≡1 and Φuv(t)=0 whenever 0 0. We also show that the two notions of distributional chaos used in the paper, for continuous maps of a compact metric space, are invariants of topological conjugacy.

One of the most celebrated problems in dyadic harmonic analysis is the pointwise convergence of the Fejer (or (C, 1)) means of functions on unbounded Vilenkin groups. There was no known positive result before the author's paper appeared in 1999 (J. Approx. Theory 101(1) (1999) 1) with respect to the a.e. convergence of the one-dimensional (C, 1) means of Lp (p > 1) functions. This paper is concerned with the almost everywhere convergence of a subsequence of the two-dimensional Fejer means of functions in L log- L. Namely, we prove the a.e. relation limn,k → ∞ σMnċM-k f = f (for the indices the condition |n - k| > α is provided, where α > 0 is some constant).

Let (\Pi,\Sigma) be a Coxeter system. An ordered list of elements in \Sigma and an element in \Pi determine a {\em subword complex}, as introduced in our paper on Gr\"obner geometry of Schubert polynomials (math.AG/0110058). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which is due to Fomin and Kirillov, are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented., Comment: 14 pages. Final version, to appear in Advances in Mathematics. This paper was split off from math.AG/0110058v2, whose version 3 is now shorter

Let Q: R → R be even, nonnegative and continuous, Q' be continuous, Q' > 0 in (0, ∞), and let Q'' be continuous in (0, ∞). Furthermore, Q satisfies further conditions. We consider a certain generalized Freud-type weight WrQ2(x) = |x|2r exp(-2Q(x)). In previous paper (J. Approx. Theory 121 (2003) 13) we studied the properties of orthonormal polynomials {Pn(WrQ2; x)}n=0x with the generalized Freud-type weight WrQ2(x) on R. In this paper we treat three themes. Firstly, we give an estimate of Pn(WrQ2; x) in the Lp-space, 0 < p ≤ ∞. Secondly, we obtain the Markov inequalities, and third we study the higher-order Hermite Fejer interpolation polynomials based at the zeros {xkn}k=1n of Pn(WrQ2; x). In Section 5 we show that our results are applicable to the study of approximation for continuous functions by the higher-order Hermite-Fejer interpolation polynomials.

The paper considers properties of compactly supported, locally linearly independent refinable function vectors Φ = (φ 1 , ..., φ r ) T , r ∈ N. In the first part of the paper, we show that the interval endpoints of the global support of φ v , v = 1,..., r , are special rational numbers. Moreover, in contrast with the scalar case r = 1. we show that components φ v of a locally linearly independent refinable function vector Φ can have holes. In the second part of the paper we investigate the problem whether any shift-invariant space generated by a refinable function vector Φ possesses a basis which is linearly independent over (0, 1). We show that this is not the case. Hence the result of Jia, that each finitely generated shift-invariant space possesses a globally linearly independent basis, is in a certain sense the strongest result which can be obtained.