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In this short paper, generally, we define a family of functions fk depends on the Eisenstein series with weight 2k, for k ( N. More detail, by considering the function fk, we define a derivative formula for generating weakly modular forms with weight 12. As a result for this, we claim that this formula gives an advantage to find the special solutions of some differential equations.

In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research.

In the authors' first paper, Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. Current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator., We deleted a proposition and a corollary from section 4, and simplified the proof of the main theorem. **The article has been published in Archiv der Mathematik**

This paper suggests new types of weak nonexpansive mappings defined from normed space X into its Cartesian product X × X, studies the main features of the fixed points for those mappings and extends the concept of (C)-contractivity condition introduced in some previous research papers. On other side, it introduces new types of contraction mappings with a mixed monotone property; the {a, b, c} M-first type and the {a, b, c} M-second type contractions, these types are defined from the Cartesian product space X × X into X, where X is a sequentially ordered Banach space, proves the existence of first-anti-second and second-anti-first couple fixed points of such types and generalizes some of the results given before.

In the paper under review, we analyze various types of degenerate abstract Volterra integrodifferential equations in sequentially complete locally convex spaces. From the theory of non-degenerate equations, it is well known that the class of (a,k)-regularized C-resolvent families provides an efficient tool for dealing with abstract Volterra integro-differential equations of scalar type. Following the approach of T.-J. Xiao and J. Liang [41]-[43], we introduce the class of degenerate exponentially equicontinuous (a,k)- regularized C-resolvent families and discuss its basic structural properties. In the final section of paper, we will look at generation of degenerate fractional resolvent operator families associated with abstract differential operators.

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 view of the fact that there are not appropriate frame operators of frames in Banach spaces, this paper considers a class of sequences satisfying certain conditions in Banach spaces which is called as the $\sigma$-frame, and the corresponding concept of frame operators is given. The $\sigma$-frames and $\sigma$-frame operators are natural generalizations of frames and frame operators in Hilbert spaces. This paper illustrates that $\sigma$-frame operators are positive, self-adjoint and they can be decomposed through $l_2$. The perturbation result under operators of $\sigma$-frame is obtained. This paper also shows that the kind of $\sigma$-frames contains two other kinds of frames in Banach space---$p$-frames ($1＜p\leq 2$) and $\sigma {\rm HS}$ frames which are a kind of frames according to the definition of the Hilbert-Schauder frames. The perturbation results under operators of $p$-frames ($1＜p\leq 2$) and $\sigma {\rm HS}$ frames are obtained.

In the 1960s Bers showed how to uniformize simultaneously two Riemann surfaces of the same finite analytic type by using a single quasi-Fuchsian group of the first kind. In this paper, we show how to uniformize simultaneously two uniformly quasisymmetric circle endomorphisms of the same degree by a unique normalized branched covering of the Riemann sphere of the same degree such that this branched covering has a unique normalized quasicircle as an invariant limit set. We use this simultaneous uniformization to define a transformation between their spaces of probability invariant measures and formulate several equivalent conjectures to the uniqueness conjecture for symmetric invariant probability measures. In a subsequent paper, we study these conjectures.

In the paper [Differ. Uravn., 2007, vol. 43, no. 2, pp. 191–202], we defined the noncoinciding irreducibility sets N 2(a, σ) and N 3(a, σ), σ ∈ (0, 2a], of all n-dimensional linear differential systems with piecewise continuous coefficient matrices A(t) bounded on the half-line [0,+∞) with norms ||A(t)|| ≤ a < +∞ for each of which there exists a linear differential system that cannot be reduced to it by Lyapunov transformations and whose coefficient matrix B(t) satisfies the condition ||B(t) - A(t)|| ≤ const × e −σt , t ≥ 0, or the more general condition that the Lyapunov exponent of the difference B(t) - A(t) does not exceed -σ, respectively. In the present paper, we study the properties of irreducibility sets treated as functions of the parameters σ and a.

The present paper gives several subordination results involving a generalized Srivastava–Attiya operator (defined below). Among the results presented in this paper include also a sufficiency condition for the convexity of the convolution of certain functions and a sharp result relating to the convolution structure. We also mention various useful special cases of the main results including those which are related to the Zeta function.

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.

In this paper we introduce generalized or vector-valued Orlicz-Lorentz sequence spaces l p,q,M (X) on Banach space X with the help of an Orlicz function M and for different positive indices p and q. We study their structural properties and investigate cross and topological duals of these spaces. Moreover these spaces are generalizations of vector-valued Orlicz sequence spaces l M (X) for p = q and also Lorentz sequence spaces for M(x) = x q for q ≥ 1. Lastly we prove that the operator ideals defined with the help of scalar valued sequence spaces l p,q,M and additive s-numbers are quasi-Banach operator ideals for p < q and Banach operator ideals for p ≥ q. The results of this paper are more general than the work of earlier mathematicians, say A. Pietsch, M. Kato, L. R. Acharya, etc.

The present paper deals mainly with seven fundamental theorems of mathematical analysis, numerical analysis, and number theory, namely the generalized Parseval decomposition formula (GPDF), introduced 15 years ago, the well-known approximate sampling theorem (ASF), the new approximate reproducing kernel theorem, the basic Poisson summation formula, already known to Gaus, a newer version of the GPDF having a structure similar to that of the Poisson summation formula, namely, the Parseval decomposition–Poisson summation formula, the functional equation of Riemann’s zeta function, as well as the Euler–Maclaurin summation formula. It will in fact be shown that these seven theorems are all equivalent to one another, in the sense that each is a corollary of the others. Since these theorems can all be deduced from each other, one of them has to be proven independently in order to verify all. It is convenient to choose the ASF, introduced in 1963. The epilogue treats possible extensions to the more general contexts of reproducing kernel theory and of abstract harmonic analysis, using locally compact abelian groups. This paper is expository in the sense that it treats a number of mathematical theorems, their interconnections, their equivalence to one another. On the other hand, the proofs of the many intricate interconnections among these theorems are new in their essential steps and conclusions.

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.

This paper is devoted to the study of so-called finitely bi-Lipschitz mappings, which are a far-reaching generalization of isometries and quasi-isometries. We obtain several criteria for the homeomorphic extension to the boundary of finitely bi-Lipschitz homeomorphisms f between domains in ℝn, n ≥ 2, whose outer dilatations KO(x, f) satisfy the integral constraints $$\int {\Phi (K_O^{n - 1} (x,f))dm(x) < \infty } $$ with an increasing convex function Φ: [0,∞] → [0,∞]. Note that the integral conditions on the function Φ (obtained in the paper) are not only sufficient, but also necessary for the continuous extension of f to the boundary.

Recently, q-Dedekind-type sums related to q-zeta function and basic L-series are studied by Simsek in [13] (Y. Simsek, q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), 333–351) and Dedekind-type sums related to Euler numbers and polynomials are introduced in the previous paper [11] (T. Kim, Note on Dedekind type DC sums, Adv. Stud. Contem. Math. 18 (2009), 249–260). It is the purpose of this paper to construct a p-adic continuous function for an odd prime to contain a p-adic q-analogue of the higher order Dedekind the type sums related to q-Euler polynomials and numbers by using an invariant p-adic q-integrals.

This paper is concerned with the distribution of normalized zero-sets of random meromorphic functions. The normalization of the zero-set plays the same role as the counting function for a meromorphic function in Nevanlinna theory. The results generalize the theory of Shiffman and Zelditch on the distribution of the zeroes of random holomorphic sections of powers of positive Hermitian holomorphic line bundles. As in a very special case, our paper resembles a form of First Main Theorem in classical Nevanlinna Theory.

In this paper we classify, up to analytic isomorphism, the family of almost stretched Artinian complete intersection A=R/I with a given Hilbert function, in the case R is a power series ring with an arbitrary number of variables., 20 pages. This paper generalizes a previous version where the result was proven for a power series ring in two variables

This paper complements a recent result of Dilworth, Ferenczi, Kutzarova and Odell regarding the ergodicity of strongly asymptotic ℓ p \ell _p spaces. We state this result in a more general form, involving domination relations, and we show that every asymptotically Hilbertian space which is not isomorphic to ℓ 2 \ell _2 is ergodic. In particular, every weak Hilbert space which is not isomorphic to ℓ 2 \ell _2 must be ergodic. Throughout the paper we construct explicitly the maps which establish the fact that the relation E 0 E_0 is Borel reducible to isomorphism between subspaces of the Banach spaces involved.

Let H be a Hilbert space which is the direct sum of five closed subspaces X0, X1, X2, X3 and X4 with X1, X2, X3 of finite dimension. Let J be a C 1,1 functional defined on H with J(0) = 0. We show the existence of at least four nontrivial critical points when the sublevels of J (the torus with three holes and sphere) link and the functional J satisfies sup-inf variational inequality on the linking subspaces, and the functional J satisfies (P.S.)⁄ condition and f|X0'X4 has no critical point with level c. For the proof of main theorem we use the nonsmooth version of the classical deformation lemma and the limit relative category theory. 1. Introduction and statement of main result Let H be a Hilbert space which is a direct sum of five closed subspaces X0, X1, X2, X3 and X4 with X1, X2, X3 of finite dimension. Let J be a C 1,1 functional defined on H with J(0) = 0. In this paper we investigate the number of nontrivial critical points of the C 1,1 functional J under some conditions on the sublevels of J and the shape of J. We show the existence of at least four nontrivial critical points when the sublevels of J (the torus with three holes and sphere) link and the functional J satisfies the sup-inf variational inequality on the linking subspaces, and the functional J satisfies (P.S.) ⁄ condition and J|X0'X4 has no critical point with level c. Micheletti and Saccon prove in (13) that the functional J has at least two nontrivial critical points under the same conditions on J except the condition that the sublevel sets are the torus with one hole and the sphere. In this paper we improve this result to the case that the sublevel sets are the torus with three holes and sphere. Now, we state the main result

Regular polynomials with quaternionic coefficients admit only isolated zeroes and spherical zeroes. In this paper we prove a factorization theorem for such polynomials. Specifically, we show that every regular polynomial can be written as a product of degree one binomials and special second degree polynomials with real coefficients. The degree one binomials are determined (but not uniquely) by the knowledge of the isolated zeroes of the original polynomial, while the second degree factors are uniquely determined by the spherical zeroes. We also show that the number of zeroes of a polynomial, counted with their multiplicity as defined in this paper, equals the degree of the polynomial. While some of these results are known in the general setting of an arbitrary division ring, our proofs are based on the theory of regular functions of a quaternionic variable, and as such they are elementary in nature and offer explicit constructions in the quaternionic setting.

application/pdf, Synthetic di¤erential geometry occupies a unique position in topos-theoretic physics. Nevertheless it has appeared somewhat too conceptual to physicists in general, partly because it has appeared to lack computational aspects. Its computational facets are really concerned with computation of the quasi-colimit of a …nite diagram of in…nitesimal spaces, or equivalently, with computation of the limit of a …nite diagram of Weil algebras. Indeed we have been forced to do a highly invovled computation of the above kind by hand in our previous papers ([International Journal of Theoretical Physics, 36 (1997) , 1099-1131] and [International Journal of Theoretical Physics, 38 (1999) , 2163-2174]). The principal objective in this paper is to show that Gröbner bases techniques pro- vide us with means that relegate such computations to computers.

We consider partially hyperbolic maps which are close to the direct product of a hyperbolic map and an identity map and prove that their central leaves depend Holder continuously on the base point in the -metric. We use this result to construct an open set of diffeomorphisms with rather unusual properties (they have transitive sets with periodic points of different indices and orbits with zero Lyapunov exponent). This paper concludes a series of joint papers with Yu. S. Ilyashenko.

In a previous paper, Auslander–Reiten triangles and quivers were introduced into algebraic topology. This paper shows that over a Poincare duality space, each component of the Auslander–Reiten quiver is isomorphic to \(\mathbb{Z}A_{\infty }\).

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.

Let (G, ρ, X) be a prehomogeneous vector space with singular set S over an algebraic number field F. The main result of this paper is a proof for the convergence of the zeta fucntions Z(Φ, s) associated with (G, ρ, X) for large Re s under the assumption that S is a hypersurface. This condition is satisfied if G is reductive and (G, ρ, X) is regular. When the connected component of the stabilizer of a generic point x is semisimple and the group Πx of connected components of Gx is abelian, a clear estimate of the domain of convergence is given.Moreover when S is a hypersurface and the Hasse principle holds for G, it is shown that the zeta fucntions are sums of (usually infinite) Euler products, the local components of which are orbital local zeta functions. This result has been proved in a previous paper by the author under the more restrictive condition that (G, ρ, X) is irreducible, regular, and reduced, and the zeta function is absolutely convergent.

In a recent paper on semidefinite linear complementarity problems, Gowda and Song (2000) introduced and studied the P-property, P2-property, GUS-property, and strong monotonicity property for linear transformation L: Sn → Sn, where Sn is the space of all symmetric and real n × n matrices. In an attempt to characterize the P2-property, they raised the following two questions: (i) Does the strong monotonicity imply the P2-property? (ii) Does the GUS-property imply the P2-property? In this paper, we show that the strong monotonicity property implies the P2-property for any linear transformation and describe an equivalence between these two properties for Lyapunov and other transformations. We show by means of an example that the GUS-property need not imply the P2-property, even for Lyapunov transformations.

The Rees algebra is the homogeneous coordinate ring of a blowing-up. The present paper gives a necessary and sufficient condition for a Noetherian local ring to have a Cohen-Macaulay Rees algebra: A Noetherian local ring has a Cohen-Macaulay Rees algebra if and only if it is unmixed and all the formal fibers of it are Cohen-Macaulay. As a consequence of it, we characterize a homomorphic image of a Cohen-Macaulay local ring. For non-local rings, this paper gives only a sufficient condition. By using it, however, we obtain the affirmative answer to Sharp’s conjecture. That is, a Noetherian ring having a dualizing complex is a homomorphic image of a finite-dimensional Gorenstein ring.

We determine the periodic cyclic homology of the Iwahori-Hecke algebras $\Hecke_q$, for $q \in \CC^*$ not a ``proper root of unity.'' (In this paper, by a {\em proper root of unity} we shall mean a root of unity other than 1.) Our method is based on a general result on periodic cyclic homology, which states that a ``weakly spectrum preserving'' morphism of finite type algebras induces an isomorphism in periodic cyclic homology. The concept of a weakly spectrum preserving morphism is defined in this paper, and most of our work is devoted to understanding this class of morphisms. Results of Kazhdan--Lusztig and Lusztig show that, for the indicated values of $q$, there exists a weakly spectrum preserving morphism $\phi_q : \Hecke_q \to J$, to a fixed finite type algebra $J$. This proves that $\phi_q$ induces an isomorphism in periodic cyclic homology and, in particular, that all algebras $\Hecke_q$ have the same periodic cyclic homology, for the indicated values of $q$. The periodic cyclic homology groups of the algebra $\Hecke_1$ can then be determined directly, using results of Karoubi and Burghelea, because it is the group algebra of an extended affine Weyl group.

“Weyl’s theorem” for an operator on a Hilbert space is a statement that the complement in the spectrum of the “Weyl spectrum” coincides with the isolated eigenvalues of finite multiplicity. In this paper we consider how Weyl’s theorem survives for polynomials of operators and under quasinilpotent or compact perturbations. First, we show that if T is reduced by each of its finite-dimensional eigenspaces then the weyl spectrum obeys the spectral mapping theorem, and further if T is reduction-isoloid then for every polynomial p, Weyl’s theorem holds for p(T ). The results on perturbations are as follows. If T is a “finite-isoloid” operator and if K commutes with T and is either compact or quasinilpotent then Weyl’s theorem is transmitted from T to T +K. As a non-commutative perturbation theorem, we also show that if the spectrum of T has no holes and at most finitely many isolated points, and if K is a compact operator then Weyl’s theorem holds for T +K when it holds for T . Introduction. H. Weyl [22] examined the spectra of all compact perturbations T +K of a hermitian operator T and discovered that λ ∈ σ(T + K) for every compact operator K if and only if λ is not an isolated eigenvalue of finite multiplicity in σ(T ). Today this result is known as Weyl’s theorem, and it has been extended from hermitian operators to hyponormal operators and to Toeplitz operators by L. Coburn [7], to several classes of operators including seminormal operators by S. Berberian [2],[3], and to a few classes of Banach space operators [15],[17]. Weyl’s theorem may fail for even the square of T when it holds for T (see [18, Example 1]). In [14], it was shown that Weyl’s theorem holds for polynomials of hyponormal operators. The first aim of this paper is to extend this result via “Berberian” spectra. On the other hand, Weyl’s theorem is liable to fail under “small” perturbations if “small” is interpreted in the sense of compact or quasinilpotent. Recently Weyl’s theorem under small perturbations has been considered in [11],[12],[13], and [18]. The second aim of this paper is to explore how Weyl’s theorem survives under quasinilpotent or compact perturbations. Throughout this paper let H denote an infinite dimensional separable Hilbert space. Let L(H) denote the algebra of bounded linear operators on H and let K(H) denote the closed ideal of compact operators on H. If T ∈ L(H) write ρ(T ) for the resolvent set of T ; σ(T ) for the spectrum of T ; π0(T ) for the set of eigenvalues of T ; π0f (T ) for the eigenvalues of finite multiplicity; π0i(T ) for the eigenvalues of infinite multiplicity. An operator T ∈ L(H) is said to be Fredholm if T−1(0) and T (H)⊥ are both finite-dimensional. The index of a Fredholm operator T ∈ L(H), denoted ind (T ), is given by ind (T ) = dimT−1(0)− dimT (H)⊥ (= dimT−1(0)− dimT ∗−1(0)). 2000 Mathematics Subject Classification. Primary 47A10,47A53,47A55

In this paper, a new method is presented for proving boundedness of singular Radon transforms on L p for 1 < p < ∞. Two closely related results generalizing the corresponding theorem of Christ, Nagel, Stein, and Wainger are proven. 1. Introduction. In this paper, a new method is presented for proving boundedness of singular Radon transforms on L p for 1 < p < ∞. Two closely related results generalizing the corresponding theorem of Christ, Nagel, Stein, and Wainger in (3) are proved. Singular Radon transforms are a type of singular integral operator, a prototype for which is the Hilbert transform along curves. Let γ :( − 1, 1) → R n be a curve with γ(0) = 0, and let f be a Schwarz function on R n+1 .

Let X and Y be finite connected CW complexes with base points, and let K denote the C*algebra of compact operators on a separable infinite dimensional complex Hilbert space. The purpose of this paper is to study the question of when C0(X)⊗K is homotopy equivalent to C0(Y )⊗K; here C0(X) is, as usual, the C*-algebra of continuous complex-valued functions which vanish at the base point. Recall that two C-algebras A and B are said to be homotopy equivalent, written A ' B, if there are ∗-homomorphisms φ : A → B and ψ : B → A for which ψ ◦ φ and φ ◦ ψ may be deformed by a path of endomorphisms to the identity maps idA : A → A and idB : B → B, respectively. Two C*-algebras A and B are called ‘stably’ homotopy equivalent if A⊗K ' B⊗K (this should not be confused with the notion of stable homotopy of spaces used in topology), so another way of stating the problem we consider is: when are C0(X) and C0(Y ) ‘stably’ homotopy equivalent? To begin with we should remark that the analogous question for homotopy equivalence has a simple answer: C0(X) and C0(Y ) are homotopy equivalent if and only if X and Y are based homotopy equivalent. The situation for ‘stable’ homotopy equivalence is more complicated and is closely related to K-theory: for example, if two C*-algebras are ‘stably’ homotopy equivalent then they have the same K-theoretic invariants. The main purpose of this paper is to show that the converse is not true: we give an example of two spaces X and Y which cannot be distinguished by

The paper is devoted to the Jacobian Conjecture: a polynomial mappingf∶ℂ2→ℂ2 with a constant nonzero Jacobian is polynomially invertible. The main result of the paper is as follows. There is no four-sheeted polynomial mapping whose Jacobian is a nonzero constant such that after the resolution of the indeterminacy points at infinity there is only one added curve whose image is not a point and does not belong to infinity.

In this paper we give two counterexamples to the closedness of the sum of two sectorial operators with commuting resolvents. In the first example the operators are defined on an L p-space, with \(1 \le p \neq 2 \le \infty \), and one of them admits bounded imaginary powers. The second example is concerned with operators defined on a Hilbert valued L p-space; one acts on L p and admits bounded imaginary powers as the other acts on the Hilbert space. In the last section of the paper we show that the two partial derivations on \(L^2 ({\Bbb R}^2;X)\) admit a so-called bounded joint functional calculus if and only if X is a UMD Banach space with property \((\alpha )\) (geometric property introduced by G. Pisier).

This paper concerns a Markov operator T on a space L I , and a Markov process P, which defines a Markov operator on a space M of finite signed measures. For T, the paper presents necessary and suf ic ient conditions for: (a) the existence of invariant probability densities (IPDs) (b) existence of strictly positive IPDs, and (c) existence and uniqueness of IPDs. Similar results on invariant probability measures for P are presented. The basic approach is to pose a fixed-point problem as the problem of solving a certain linear equation in a suitable Banach space, and then obtain necessary and sufficient conditions for this equation to have a solution.

In a 1988 paper, Cowen found a formula expressing the adjoint of any linear fractional composition operator on the Hardy space as a product of Toeplitz operators and another linear fractional composition operator. In this paper, we use Cowen's adjoint formula to give a unitary equivalence relating composition operators on different weighted Hardy spaces. This result is then applied to some composition operators on the Sa spaces. We find the spectrum of any linear fractional composition operator whose symbol has exactly one fixed point of multiplicity one on the unit circle.

This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras A ( + ) A(+) , A ( − ) A(-) , and A ( a ) A({\mathbf {a}}) , where a ∈ P 2 {\mathbf {a}}\in \mathbb {P}^{2} , which were first defined in an earlier paper. We omit a set S ⊂ P 2 S\subset \mathbb {P}^2 consisting of 11 specified points where the algebras A ( a ) A({\mathbf {a}}) become too degenerate to be regular. Theorem. Let A A represent A ( + ) A(+) , A ( − ) A(-) or A ( a ) A({\mathbf {a}}) , where a ∈ P 2 ∖ S {\mathbf {a}} \in \mathbb {P}^2\setminus S . Then A A is an Artin-Schelter regular algebra of global dimension three. Moreover, A A is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. This, combined with our earlier results, completes the classification.

In his paper [K], Kashiwara introduced the notion of characteristic cycle for complexes of constructible sheaves on manifolds: let X be a real analytic manifold, and F a complex of sheaves of C-vector spaces on X , whose cohomology is constructible with respect to a subanalytic strati cation; the characteristic cycle CC(F) is a subanalytic, Lagrangian cycle (with in nite support, and with values in the orientation sheaf of X ) in the cotangent bundle T ∗X . The de nition of CC(F) is Morse-theoretic. Heuristically, CC(F) encodes the in nitesimal change of the Euler characteristic of the stalks of F along the various directions in X . It tends to be di cult in practice to calculate CC(F) explicitly for all but the simplest complexes F; on the other hand, the characteristic cycle construction has good functorial properties. The behavior of CC(F) with respect to the operations of proper direct image, Verdier duality, and non-characteristic inverse image of F is well understood [KS]. In this paper, we describe the e ect of the operation of direct image by an open embedding. Combining our result with those that were previously known, we obtain descriptions of CC(Rf∗F) and CC(f∗F) – analogous to those in [KS] – for arbitrary morphisms f : X → Y in the semi-algebraic category, and complexes F with semi-algebraically constructible cohomology. In e ect, this provides an axiomatic characterization of the functor CC, at least in the semi-algebraic context. Our arguments do apply more generally in the subanalytic case, but because statements become quite convoluted, we shall not strive for the greatest degree of generality. As a concrete application, we consider the case of the ag manifold X of a complex semisimple Lie algebra g. Here the Weyl group W of g operates

By using Deodhar's combinatorial setting and Bernstein-Gelfand projective functors, this paper provides some necessary and sufficient conditions for a highest weight category to have a Kazhdan-Lusztig theory. A consequence of these conditions is that in the semisimple Lie algebra case, the Kazhdan-Lusztig conjecture on the multiplicities of a Verma module implies the nonnegativity conjecture on the coefficients of Kazhdan-Lusztig polynomials. One of the central topics in representation theory in recent years is the socalled Kazhdan-Lusztig theory. The Kazhdan-Lusztig polynomials play a key role in this theory. These polynomials can be defined by using a distinguished basis of the Hecke algebra associated to a Coxeter group. In [KL1], there are two conjectures about these polynomials: (a) For any Coxeter group, the coefficients of these polynomials are nonnegative integers; (b) If the Coxeter group is the Weyl group of a complex semisimple Lie algebra, then the multiplicities of the composition series of a Verma module are given by the values of these polynomials at 1. Conjecture (b) is usually referred to as the Kazhdan-Lusztig conjecture and was proven in [BB] and [BK] shortly thereafter. Conjecture (a) is now known to be true for all crystallographic Coxeter groups (for a more upto-date reference on recent developments of Kazhdan-Lusztig theory, we refer to [DS]). It was shown in [D] that if the coefficients of the Kazhdan-Lusztig polynomials of a Coxeter group are nonnegative, then these polynomials can be defined by using certain sets derived from the elements of the Coxeter group. In fact, these sets give a closed formula for the Kazhdan-Lusztig polynomials under the nonnegativity assumption (see [D]). Since the Kazhdan-Lusztig polynomials are not easy to get at in general, the results in [D] give strong evidence for the importance of the nonnegativitiness. In an attempt to understand the results of [D], we observed that in the semisimple Lie algebra case, conjecture (b) implies conjecture (a). The connection is provided by some tensor functors called projective functors defined in [BG]. In this paper, we will give some necessary and sufficient conditions for the validity of the Kazhdan-Lusztig conjecture in certain special cases of the highest weight categories defined by CPS (see [CPS1] Received by the editors June 2, 1993; the contents of this paper have been presented to the Nineteenth Holiday Symposium held in December 1992 at New Mexico State University. 1991 Mathematics Subject Classification. Primary 22E47, 17B10; Secondary 22E46, 17B35.

In a recent paper Lal and Yadav [1] obtained a theorem on the degree of approximation for a function belonging to a Lipschitz class using a triangular matrix transform of the Fourier series representation of the function. The matrix involved was the product of $ (C, 1) $, the Cesaro matrix of order one, with $ (E, 1) $, the Euler matrix of order one. In this paper we extend this result to a much wider class of Hausdorff matrices.

In this paper, by means of a combinatorial-geometric method, we give a new proof of Knopp's formula for Dedekind sums and its generalizations to multiple Dedekind sums attached to Dirichlet characters. The combinatorial-geometric method for studying Dedekind sums were introduced by Beck, who proved the well-known reciprocity formula for Dedekind sums and some of its generalizations by the method. The motive of this paper is to find a similar approch to Knopp's formula .

It is well known that a generic compact Riemann surface of genus greater than two admits only the identity automorphism; however, examples of such Riemann surfaces with their defining algebraic equations have not appeared in the literature. In this paper we give the defining equations of a doubly infinite, two-parameter family of projective curves (Riemann surfaces if defined over the complex numbers), whose members admit only the identity automorphism. It is well known that a generic curve of genus greater than two admits only the identity automorphism. Although this result was probably known by the turn of the century, the first published proof was given by Bailey in 1961 [1]. To obtain the strongest results, Bailey's method is necessarily nonconstructive; it does not yield an example of a defining algebraic equation for a curve with no nontrivial automorphisms. Similarly a proof by Greenberg [4], using techniques of Teichmuller theory, does not yield an explicit example of a Riemann surface with a trivial automorphism group. Much of the subsequent work on automorphisms of Riemann surfaces, including the author's, has relied on the representation of a given Riemann surface as the upper half plane under the action of a Fuchsian group. This again has the disadvantage of rarely yielding a defining algebraic equation for the given Riemann surface. Indeed, in the preface to his book, The complex analytic theory of Teichmiiller spaces, Subhashis Nag exclaimed, "Almost every compact Riemann surface of genus g > 3 allows only the identity automorphism. (I don't know, though, of even a single explicit such algebraic curve whose automorphism group is demonstrably trivial!)." The author believes that examples pertinent to famous theorems should be readily at hand. Therefore, in this paper we give the defining equations of a doubly infinite, two-parameter family of curves which admit only the identity automorphism (see equation (1) below). The curves in this family have genus (n 1)(m 1)/2 for relatively prime integers m and n which satisfy n > m+ 1 > 3. Let C be a curve defined by (1) and let C' be a nonsingular projective model for C. The proof that C' admits only the trivial automorphism depends on the Received by the editors February 5, 1993. 1991 Mathematics Subject Classification. Primary 14E09, 14H55, 30F99. @ 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page

We answer a 1975 question of G. R. Gordh by showing that if X is a homogeneous compactum which is the continuous image of a compact ordered space then at least one of the following holds: (i) X is metrizable, (ii) dimX = 0 or (iii) X is a union of finitely many pairwise disjoint generalized simple closed curves. We begin to examine the structure of homogeneous 0-dimensional spaces which are continuous images of ordered compacta. 1. Introduction. The aim of this paper is to investigate homogeneous spaces which are continuous images of ordered compacta. In 1975, G. R. Gordh proved that if a homo­ geneous and hereditarily unicoherent continuum is the continuous image of an ordered compactum, then it is metrizable, and so indecomposable (7, Theorem 3). Further, he asked if, in general, every homogeneous continuum which is the continuous image of an ordered compactum must be either metrizable or a generalized simple closed curve. Our Theorem 1 provides an affirmative answer to Gordh's question. Moreover, in Theorem 2, we prove that a homogeneous space which is not 0-dimensional and which is the continuous image of an ordered compactum is either metrizable or a union of finitely many pairwise disjoint generalized simple closed curves. Our methods of proof involve characterizations of continuous images of arcs obtained in ( 16) in terms of cyclic elements and T-sets. When dealing with the class A of all homogeneous and 0-dimensional spaces which are the continuous images of ordered compacta, the situation becomes less clear. By a recent theorem of M. Bell, each member of A is first countable. Moreover, by a result of (18), each member of A can be embedded into a dendron. We give a rather simple construction leading to a wide subclass of A. In particular, we show that not all members of A are orderable, and that there exists a strongly homogeneous space X which is the continuous image of an ordered compactum and which is not first countable. It follows that X $ A. Our investigations of the class A led to some natural questions which are stated at the end of the paper. All spaces considered in this paper are Hausdorff.

Introduction In his paper [10] the author investigated the structure of m-full ideals by analysing their syzygies and, as one special case, showed how the Betti numbers of Borel stable ideals over polynomial rings can be computed. The same result, among other things, was also obtained by Eliahou and Kervaire[l] by a different method. Let a be a Borel stable ideal in a polynomial ring R and let V{ be the ideal generated by i generic linear forms and let tn_i_l be the type of the ideal a+ VJVt over R/Vt. Then the Betti numbers of a are linear combinations of to,...,tn_l with certain binomial numbers as coefficients, and conversely from the numbers t0, ...,tn_1 the Betti numbers can be recovered (see [10], corollary 9 and proposition 4). The present paper grew out of the question of deciding what sequences t0, ...,tn_1 of integers can arise from a Borel stable ideal as above. Our goal of this paper is to prove Theorems 4l and 4-2 of Section 4, in which we show that if we make the restriction on the ideal that it be generated by monomials of a fixed degree then the sequence t0,..., tn_^ is precisely the same as what has been known as the Hilbert series of a graded Artin algebra. In an earlier paper [11] the author showed some properties of m-primary m-full ideals. In this present paper we need to generalize these results to ideals which are not necessarily m-primary. Most results of [11] can be generalized to general m-full ideals. The Oth local cohomology module Ua:mYa of R/a plays an important role. These are the contents of Section 1 and have independent interest. In Section 2 we summarize some combinatorial formulae derived from Macaulay's theorem which characterizes the Hilbert function of homogeneous algebras (Theorem 2-2). In Section 3 we consider m-full ideals o in regular local rings which satisfy the condition m (lo = ma. The result (Theorem 3-1) will be applied, in Section 4, to Borel stable ideals generated by monomials all of degree d to obtain the theorems mentioned above.

The main purpose of this paper is to establish a Galois correspondence for a given covariant group system, its associated C*-algebra and Hopf C*-algebra. On the way to this, we first study covariance group C*-algebras and their representations, and prove a result which is simpler but yet very similar to the C*-algebra case in the main body of the paper. We then show that there is a Galois correspondence between the lattice of normal subgroups of the given covariant group system and a corresponding lattice of certain invariant *-subalgebras of the covariant group C*-algebra; in particular, there is a natural Galois correspondence for the group C*-algebra. We further study this Galois correspondence for the Hopf C*-algebras associated with covariant group systems.

The geometry of the orbits of a minimal parabolick-subgroup acting on a symmetrick-variety is essential in several areas, but its main importance is in the study of the representations associated with these symmetrick-varieties (see for example [5, 6, 20, and 31]). Up to an action of the restricted Weyl group ofG, these orbits can be characterized by theHk-conjugacy classes of maximalk-split tori, which are stable underk-involutionθassociated with the symmetrick-variety. HereHis a openk-subgroup of the fixed point group ofθ. This is the second in a series of papers in which we characterize and classify theHk-conjugacy classes of maximalk-split tori. The first paper in this series dealt with the case of algebraically closed fields. In this paper we lay the foundation for a characterization and classification for the case of nonalgebraically closed fields. This includes a partial classification in the cases, where the base field is the real numbers, p -adic numbers, finite fields, and number fields.

The authors study some properties of the ring of abstract semialgebraic functions over a constructible subset of the real spectrum of an excellent ring. To be more precise, let X be a constructible subset of the real spectrum of a ring A. The ring S(X) of abstract semialgebraic functions over X was introduced bz N. Schwartz [see Mem. Am. Math. Soc. 397 (1989; Zbl 0697.14015)], as a generalization of continuous functions with semialgebraic graph to the context of real spectra. Unfortunately the utility of this functions is not yet quite established. The main result of the paper states that if A is excellent, the Krull dimension of S(X) equals the dimension of X (defined as the maximum of the heights of the supports of points lying in X), which in turn, as J. M. Ruiz showed in C. R. Acad. Sci. Paris, S´er. I 302, 67-69 (1986; Zbl 0591.13017) coincides with its topological dimension. This was first shown by M. Carral and M. Coste [J. Pure Appl. Algebra 30, 227-235 (1983; Zbl 0525.14015)] for the particular case of X being a ‘true’ semialgebraic subset which is locally closed, and the result extends readily to abstract locally closed constructible sets. Then the authors use the compactness of the constructible topology of real spectra and the properties of excellent rings to reduce the general case to the locally closed one. The paper finishes by characterizing the finitely generated prime ideals of S(X), namely they are the ideals of the open constructible points of X whose closure in X is open of dimension 6= 1.

International audience; In the context of Mather's theory of Lagrangian systems, we study the decomposition in chain-transitive classes of the Mather invariant sets. As an application, we prove, under appropriate hypotheses, the semi-continuity of the so-called Aubry set as a function of the Lagrangian. In the study of Lagrangian systems, John Mather introduced several invariant sets composed of globally minimizing extremals. He developed methods to construct several orbits undergoing interesting behaviors in phase space under some assumptions on these invariant sets, see [14]. In order to pursue this theory and to apply it on examples, it is necessary to have tools to describe precisely the invariant sets. At least two points of view can be adopted. One can study the invariant set from a purely topological point of view in the style of Conley as compact metric spaces with flows, and study their transitive components. One can also study these set from the point of view of action minimization, and decompose them in invariant subsets that have been called static classes. These points of view are very closely related, but each of them has specific features. For example, understanding the decomposition in static classes is necessary for the variational construction of interesting orbits, while the topological decomposition behaves well under perturbations. Our goal in the present paper is to explicit the links between these two decompositions. We explain that the topological decomposition is finer than the variational one, and that they coincide for most (but not all) systems. As an application, we prove a result of semi-continuity of the so-called Aubry set as a function of the Lagrangian, under certain non-degeneracy hypotheses. The semi-continuity of the Aubry set is a subtle problem, which has remained open for several years, until John Mather gave a counter example, see §18 in [16]. In the same paper, he also states without proof that semi-continuity holds under appropriate hypotheses. Our result extends the one of Mather. The methods we use are inspired from the recent work of Fathi, Figalli and Rifford, [9].

In this paper we give sucient conditions for weak isomorphism of Lebesgue measure-preserving expanding endomorphisms of S 1 : rst author gave necessary and sucient conditions for two real an- alytic Lebesgue measure-preserving expanding endomorphisms of the circle to be isomorphic upto a phase factor. This was a partial answer to the problem of nding complete measure theoretic invariants for isomorphisms posed by Shub and Sullivan in (5). In this paper it is shown that the condition given in (2) is sucient for weak-isomorphism. For i = 1; 2 let fi be endomorphisms of the Lebesgue spaces (Xi;Bi; i): We say that the two systems (X1;B1; 1;f1) and (X2;B2; 2;f2) are isomorphic if there are sets of measure zero A1 X1;A2 X2 and a one-to-one onto map : X1nA1! X2nA2 such that f 1 = f2 on X1nA1 and 1( 1 E) = 2(E) for all measurable E X2nA2: The classication