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M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.

In this short paper, we prove a Hitchin-Thorpe type inequality for closed 4-manifolds with non-positive Yamabe invariant, and admitting long time solutions of the normalized Ricci flow equation with bounded scalar curvature.

In this short paper we prove that, for3≤N≤93 \le N \le 9, the problem−Δu=eu-\Delta u = e^uon the entire Euclidean spaceRN\mathbb {R}^Ndoes not admit any solution stable outside a compact set ofRN\mathbb {R}^N. This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems.

In this paper, we study boundedness questions for (simply-connected) smooth Calabi-Yau threefolds. The diffeomorphism class of such a threefold is known to be determined up to finitely many possibilities by the integral middle cohomology and two integral forms on the integral second cohomology, namely the cubic cup-product form and the linear form given by cup-product with the second Chern class. The motivating question for this paper is whether knowledge of these cubic and linear forms determines the threefold up to finitely many families, that is the moduli of such threefolds is bounded. If this is true, then in particular the middle integral cohomology would be bounded by knowledge of these two forms on the second cohomology. Crucial to this question is the study of rigid rational surfaces on the threefold, which are the irreducible surfaces for which no multiple moves and which do not disappear under a general deformation of the complex structure of the threefold. If for instance there are no such surfaces, then the answer to the motivating question is yes (Theorem 0.1). In particular, for given cubic and linear forms on the second cohomology, there must exist such surfaces for large enough third Betti number (Corollary 0.2). The paper starts by proving general results on these rigid rational surfaces and boundedness of the family of threefolds. The basic principle is that if the cohomology classes of these surfaces are also known, then boundedness should hold (Theorem 4.5). The second half of the paper restricts to the case of Picard number 2, where it is shown that knowledge of the cubic and linear forms does indeed bound the family of Calabi-Yau threefolds (Theorem 0.3). As far as the author is aware, this is the first non-trivial case where a general boundedness result for Calabi-Yau threefolds has been proved (without the assumption of a special structure).

In this paper, we study a problem posed by Furstenberg on intersections between × 2 , × 3 \times 2, \times 3 invariant sets. We present here a direct geometrical counting argument to revisit a theorem of Wu and Shmerkin. This argument can be used to obtain further improvements. For example, we show that if A 2 , A 3 ⊂ [ 0 , 1 ] A_2,A_3\subset [0,1] are closed and × 2 , × 3 \times 2, \times 3 invariant respectively, assuming that dim ⁡ A 2 + dim ⁡ A 3 > 1 \dim A_2+\dim A_3>1 then A 2 ∩ ( u A 3 + v ) A_2\cap (uA_3+v) is sparse (defined in this paper) and has box dimension zero uniformly with respect to the real parameters u , v u,v such that u u and u − 1 u^{-1} are both bounded away from 0 0 .

In this paper we characterize the Nazarov-Sklyanin hierarchy for the classical periodic Benjamin-Ono equation in two complementary degenerations: for the multi-phase initial data (the periodic multi-solitons) at fixed dispersion and for bounded initial data in the limit of small dispersion. First, we express this hierarchy in terms of a piecewise-linear function of an auxiliary real variable which we call a dispersive action profile and whose regions of slope $\pm 1$ we call gaps and bands, respectively. Our expression uses Kerov's theory of profiles and Kre\u{\i}n's spectral shift functions. Next, for multi-phase initial data, we identify Baker-Akhiezer functions in Dobrokhotov-Krichever and Nazarov-Sklyanin and prove that multi-phase dispersive action profiles have finitely-many gaps determined by the singularities of their Dobrokhotov-Krichever spectral curves. Finally, for bounded initial data independent of the coefficient of dispersion, we show that in the small dispersion limit, the dispersive action profile concentrates weakly on a convex profile which encodes the conserved quantities of the dispersionless equation. To establish the weak limit, we reformulate Szeg\H{o}'s first theorem for Toeplitz operators using spectral shift functions. To illustrate our results, we identify the dispersive action profile of sinusoidal initial data with a profile found by Nekrasov-Pestun-Shatashvili and its small dispersion limit with the convex profile found by Vershik-Kerov and Logan-Shepp., Comment: This is a revised version of the paper submitted on 13 January 2019 to appear in the Quarterly of Applied Mathematics; 28 pages, 8 figures

In the present paper, we define a notion of good coverings of Alexandrov spaces with curvature bounded below, and prove that every Alexandrov space admits such a good covering and that it has the same homotopy type as the nerve of the good covering. We also prove the stability of the isomorphism classes of the nerves of good coverings in the non-collapsing case. In the proof, we need a version of Perelman's fibration theorem, which is also proved in this paper., Minor change basically on the proof of Theorem 1.2 in Section 5

Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert's results are not uniform in n: unsurprisingly, polynomials in n variables will exhibit greater complexity as n increases. However, an array of recent work has shown that in a certain regime---namely, that where the number of polynomials and their degrees are fixed---the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's Conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including: the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables., This expository paper was written in conjunction with Craig Huneke's talk on Stillman's Conjecture at the 2018 JMM Current Events Bulletin

In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $\mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.

This is the third in a series of papers in which we prove a conjecture of Boston and Shalev that the proportion of derangements (fixed point free elements) is bounded away from zero for transitive actions of finite simple groups on a set of size greater than one. This paper treats the case of primitive subspace actions. It is also shown that if the dimension and codimension of the subspace go to infinity, then the proportion of derangements goes to one. Similar results are proved for elements in finite classical groups in cosets of the simple group. The results in this paper have applications to probabilistic generation of finite simple groups and maps between varieties over finite fields.

This paper is about two arrangements of hyperplanes. The first --- the Shi arrangement --- was introduced by Jian-Yi Shi to describe the Kazhdan-Lusztig cells in the affine Weyl group of type $A$. The second --- the Ish arrangement --- was recently defined by the first author who used the two arrangements together to give a new interpretation of the $q,t$-Catalan numbers of Garsia and Haiman. In the present paper we will define a mysterious "combinatorial symmetry" between the two arrangements and show that this symmetry preserves a great deal of information. For example, the Shi and Ish arrangements share the same characteristic polynomial, the same numbers of regions, bounded regions, dominant regions, regions with $c$ "ceilings" and $d$ "degrees of freedom", etc. Moreover, all of these results hold in the greater generality of "deleted" Shi and Ish arrangements corresponding to an arbitrary subgraph of the complete graph. Our proofs are based on nice combinatorial labelings of Shi and Ish regions and a new set partition-valued statistic on these regions.

In this paper we give simple, sufficient conditions for the existence of the stochastic integral for vector-valued processes X with values in a Banach space E; namely, X is of class (LD), and the stochastic measure I X is bounded and strongly additive in L p E (in particular, if I X is bounded in L p E and c 0 ⊈ E) and has bounded semivariation. The result is then applied to martingales and processes with integrable variation or semivariation. For martingales the condition of being of class (LD) is superfluous. For a square-integrable martingale with values in a Hilbert space, all the conditions are superfluous. For processes with p-integrable semivariation or p-integrable variation, the conditions of I X to be bounded and have bounded semivariation are superfluous. For processes with 1-integrable variation, all conditions are superfluous. In a forthcoming paper, we shall extend these results to local summability. The extension needs additional nontrivial work.

In 1945 Duffin and Schaeffer proved that a power series that is bounded in a sector and has coefficients from a finite subset of C is already a rational function. Their proof is relatively indirect. It is one purpose of this paper to give a shorter direct proof of this beautiful and surprising theorem. This will allow us to give an easy proof of a recent result of two of the authors stating that a sequence of polynomials with coefficients from a finite subset of C cannot tend to zero uniformly on an arc of the unit circle. Another main result of this paper gives explicit estimates for the number and location of zeros of polynomials with bounded coefficients. Let n be so large that v satisfies δn≤ 1. We show that any polynomial in has at least zeros in any disk with center on the unit circle and radius αn.

The aims of this paper are to discuss local existence and uniqueness of solutions for a class of non-Newtonian fluids with vacuum in one-dimensional bounded intervals. The important point in this paper is that we allow the initial vacuum.

This paper is concerned with the behaviour of the period function of the quadratic reversible centers. In this context the interesting stratum is the family of the so-called Loud's dehomogenized systems, namely { x = -y + xy, {y = x+Dx 2 +Fy 2 . In this paper we show that the bifurcation set of the period function of these centers is contained in the rectangle K = (-7, 2) x (0,4). More concretely, we prove that if (D, F) ∉ K, then the period function of the center is monotonically increasing.

This paper deepens the connections between critically finite rational maps and finite subdivision rules. The main theorem is that if f is a critically finite rational map with no periodic critical points, then for any sufficiently large integer n the iterate f is the subdivision map of a finite subdivision rule. We are interested here in connections between finite subdivision rules and rational maps. Finite subdivision rules arose out of our attempt to resolve Cannon’s Conjecture: If G is a Gromov-hyperbolic group whose space at infinity is a 2-sphere, then G has a cocompact, properly discontinuous, isometric action on hyperbolic 3-space. Cannon’s Conjecture can be reduced (see, for example, the Cannon-Swenson paper [5]) to a conjecture about (combinatorial) conformality for the action of such a group G on its space at infinity, and finite subdivision rules were developed to give models for the action of a Gromov-hyperbolic group on the 2-sphere at infinity. There is also a connection between finite subdivision rules and rational maps. If R is an orientation-preserving finite subdivision rule such that the subdivision complex SR is a 2-sphere, then the subdivision map σR is a critically finite branched map of this 2-sphere. In joint work [3] with Kenyon we consider these subdivision maps under the additional hypotheses that R has bounded valence (this is equivalent to its not having periodic critical points) and mesh approaching 0. In [3, Theorem 3.1] we show that if R is conformal (in the combinatorial sense) then the subdivision map σR is equivalent to a rational map. The converse follows from [4, Theorem 4.7]. In this paper we consider the problem of when a rational map f can be equivalent to the subdivision map of a finite subdivision rule. Since a subdivision complex has only finitely many vertices, such a rational map must be critically finite. We specialize here to the case that f has no periodic critical points. Our main theorem, which has also been proved by Bonk and Meyer [1] when f has a hyperbolic orbifold (which includes all but some well-understood examples), is the following: Date: March 15, 2007. 2000 Mathematics Subject Classification. Primary 37F10, 52C20; Secondary 57M12.

We study a stochastically perturbed reaction-diffusion equation by using the methods of the theory of stochastic attractors. It is proved that solutions of the equation form a multivalued random dynamic system for which there exists a random attractor in the phase space. Introduction The theory of random dynamic systems [1] provides convenient tools to study many random and stochastic equations. Evolution systems described by these equations have, as a rule, unbounded trajectories in ordinary time. The theory of random attractors (i.e., measurable invariant sets attracting trajectories in inverse time) is developed in the papers [5]–[7] to obtain a qualitative description of such systems and to study finitedimensional systems of stochastically perturbed differential equations that have unique solutions. A generalization of the theory of random attractors is proposed in the paper [8] for the case of infinite-dimensional systems that do not necessarily have unique solutions. However an extra assumption is made in [8] that the corresponding multivalued random dynamic system is dissipative with probability one. In this paper, we develop the theory of random attractors for a system that is dissipative in probability. Based on this extension we study the qualitative behavior of solutions of the reaction-diffusion equation perturbed by a cadlag process. The setting of the problem Consider the problem (1) { ∂u(t,x) ∂t = a u(t, x)− f(u(t, x)) + h(x) + g(u(t, x))ξ(t, w), u ∣∣ ∂Q = 0, u ∣∣ t=0 = u0(x), where a > 0, Q ⊂ R is a bounded domain with a smooth boundary, f, g ∈ C(R), h ∈ L2(Q), uf(u) ≥ α|u| − C, |f(u)| ≤ C(1 + |u|p−1), p ≥ 2, α > 0, and |g(u)| ≤ C1|u|+ C2. We assume that ξ(t, w) is a stochastic cadlag process; that is, its trajectories have no discontinuities of the second kind. In view of the canonical representation of such processes 2000 Mathematics Subject Classification. Primary 34F05, 60H10. c ©2007 American Mathematical Society 57 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 58 O. V. KAPUSTYAN, J. VALERO, AND O. V. PEREGUDA described in [1, 2], we assume in what follows that ξ(t, w) : R+ × Ω → R, where Ω = D(R) = { w(·) : R → R ∣∣∣ for all t ∈ R there exist lim s→t− w(s) =: w(t−) and lim s→t+ w(s) = w(t) } is the Skorokhod metric space [2] with the Borel σ-algebra Φ, the transformation θtw(·) = w(t+ ·), and with a θt-invariant probabilistic measure P. In this case ξ(t, w) = w(t) = π(θtw), where π : Ω → R and π(w) = w(0). The metric ρ is defined by

In 1918 Pólya and Vinogradov gave an upper bound for the maximal size of character sums, which still remains the best known general estimate. One of the main results of this paper provides a substantial improvement of the Pólya-Vinogradov bound for characters of odd, bounded order. In 1977 Montgomery and Vaughan showed how the Pólya-Vinogradov inequality may be sharpened assuming the Generalized Riemann Hypothesis. We give a simple proof of their estimate and provide an improvement for characters of odd, bounded order. The paper also gives characterizations of the characters for which the maximal character sum is large, and it finds a hidden structure among these characters.

This paper describes the existence and representation of certain finite energy ( L 2 (L^2 -)solutions of weighted div-curl systems on bounded 3D regions with C 2 C^2 -boundaries and mixed boundary data. Necessary compatibility conditions on the data for the existence of solutions are described. Subject to natural integrability assumptions on the data, it is then shown that there exist L 2 L^2 -solutions whenever these compatibility conditions hold. The existence results are proved by using a weighted orthogonal decomposition theorem for L 2 L^2 -vector fields in terms of scalar and vector potentials. This representation theorem generalizes the classical Hodge-Weyl decomposition. With this special choice of the potentials, the mixed div-curl problem decouples into separate problems for the scalar and vector potentials. Variational principles for the solutions of these problems are described. Existence theorems, and some estimates, for the solutions of these variational principles are obtained. The unique solution of the mixed system that is orthogonal to the null space of the problem is found and the space of all solutions is described. The second part of the paper treats issues concerning the non-uniqueness of solutions of this problem. Under additional assumptions, this space is shown to be finite dimensional and a lower bound on the dimension is described. Criteria that prescribe the harmonic component of the solution are investigated. Extra conditions that determine a well-posed problem for this system on a simply connected region are given. A number of conjectures regarding the results for bounded regions with handles are stated.

Let q 1 > q 2 > q 3 q_1>q_2>q_3 be odd primes and N = q 1 q 2 q 3 N=q_1q_2q_3 . Put \[ d = gcd ( q 1 − 1 , q 2 − 1 , q 3 − 1 ) and h i = q i − 1 d , i = 1 , 2 , 3. d=\gcd (q_1-1,q_2-1,q_3-1) \text { and } h_i=\tfrac {q_i-1}d, \;i=1,2,3. \] Then we call d d the kernel, the triple ( h 1 , h 2 , h 3 ) (h_1,h_2,h_3) the signature, and H = h 1 h 2 h 3 H=h_1h_2h_3 the height of N N , respectively. We call N N a C 3 C_3 -number if it is a Carmichael number with each prime factor q i ≡ 3 mod 4 q_i\equiv 3\mod 4 . If N N is a C 3 C_3 -number and a strong pseudoprime to the t t bases b i b_i for 1 ≤ i ≤ t 1\leq i\leq t , we call N N a C 3 C_3 -spsp ( b 1 , b 2 , … , b t ) (b_1, b_2,\dots ,b_t) . Since C 3 C_3 -numbers have probability of error 1 / 4 1/4 (the upper bound of that for the Rabin-Miller test), they often serve as the exact values or upper bounds of ψ m \psi _m (the smallest strong pseudoprime to all the first m m prime bases). If we know the exact value of ψ m \psi _m , we will have, for integers n > ψ m n>\psi _m , a deterministic efficient primality testing algorithm which is easy to implement. In this paper, we first describe an algorithm for finding C 3 C_3 -spsp(2)’s, to a given limit, with heights bounded. There are in total 21978 21978 C 3 C_3 -spsp ( 2 ) (2) ’s > 10 24 >10^{24} with heights > 10 9 >10^9 . We then give an overview of the 21978 C 3 C_3 - spsp(2)’s and tabulate 54 54 of them, which are C 3 C_3 -spsp’s to the first 8 8 prime bases up to 19 19 ; three numbers are spsp’s to the first 11 prime bases up to 31. No C 3 C_3 -spsp’s > 10 24 >10^{24} to the first 12 12 prime bases with heights > 10 9 >10^9 were found. We conjecture that there exist no C 3 C_3 -spsp’s > 10 24 >10^{24} to the first 12 12 prime bases with heights ≥ 10 9 \geq 10^9 and so that ψ 12 a m p ; = 3186 65857 83403 11511 67461 (24 digits) a m p ; = 399165290221 ⋅ 798330580441 , \begin{equation*} \begin {split} \psi _{12}&= 3186\; 65857\; 83403\; 11511\; 67461\; \text {(24 digits)} \\ &=399165290221\cdot 798330580441, \end{split}\end{equation*} which was found by the author in an earlier paper. We give reasons to support the conjecture. The main idea of our method for finding those 21978 21978 C 3 C_3 -spsp ( 2 ) (2) ’s is that we loop on candidates of signatures and kernels with heights bounded, subject those candidates N = q 1 q 2 q 3 N=q_1q_2q_3 of C 3 C_3 -spsp ( 2 ) (2) ’s and their prime factors q 1 , q 2 , q 3 q_1,q_2,q_3 to Miller’s tests, and obtain the desired numbers. At last we speed our algorithm for finding larger C 3 C_3 -spsp’s, say up to 10 50 10^{50} , with a given signature to more prime bases. Comparisons of effectiveness with Arnault’s and our previous methods for finding C 3 C_3 -strong pseudoprimes to the first several prime bases are given.

In [Y], Yau proposed to study the uniformization of complete Kahler manifolds with nonnegative curvature. In particular, one wishes to determine whether or not a complete Kahler manifold M with positive bisectional curvature is biholomorphic to Cm. See also [GW], [Si]. For this sake, it was further asked in [Y] whether or not the ring of the holomorphic functions with polynomial growth, which we denote by Op(M), is finitely generated, and whether or not the dimension of the spaces of holomorphic functions of polynomial growth is bounded from above by the dimension of the corresponding spaces of polynomials on Cm. This paper addresses the latter questions. We denote by 0?(M) the space of holomorphic functions of polynomial growth with degree d. (See Section 3 for the precise definition.) Then Op(M) = \Jd>0 Od(M). In this paper, we show that

The infinitesimal earthquake theorem gives a one-to-one correspondence between Thurston bounded earthquake measures and normalized Zygmund bounded functions. In this paper, we provide an intrinsic proof of a theorem given in an earlier paper by the author; that is, we show that the cross-ratio norm of a Zygmund bounded function is equivalent to the Thurston norm of the earthquake measure in the correspondence.

x0. Introduction In this paper we construct a new linearly similar functional model for linear operators and study its elementary properties. This model generalizes the Sz.-Nagy{Foia s model for C0-contractions and also forC0-dissipative operators. We shall not restrict ourselves to the disk or the half-plane: the model will be constructed in a fairly arbitrary domain. The reduction of an operator to an \almost diagonal" model form will be written directly via the resolvent of the operator. Attention will be focused on the case of C00-operators. The main results of this paper were announced in [76]. Let be a union of nitely many piecewise-smooth contours in the complex plane C. Suppose that the sets int ,ext are open and have empty intersection,C =int[ [ext, and = @int = @ext (here we omit a certain technical condition on ). The main objects of this paper are model spacesH() that will be associated with operator-valued bounded analytic functions dened on int and having some special properties. Consider the special case where 2 H 1 int;L(R;R) , 1 is a meromorphic function on int ,a ndk 1 () kC a.e. on . Here R and R are Hilbert spaces, and L(R;R) denotes the space of all bounded linear operators from R to R .W e always consider only separable Hilbert spaces. Let E 2 (ext;R) denote the Smirnov class of functions on ext whose values are vectors in the space R (seex2). Suppose that int and ext are unbounded. In this special case the model spaceH() consists of all functions f meromorphic in int, holomorphic in ext, and satisfying fjext2 E 2 (ext;R) ;fjint2 E 2 (int;R) ;f int = fext a.e. on : This is a Hilbert space. The general denition ofH() will be given inx2. Ap air (A;J) of linear operators (possibly, unbounded) will be called a 2-system if 1) A is a closed operator in a Hilbert space H with nonempty eld of regularity (A )= Cn(A) and with domainD(A )( here(A) is the spectrum of the operator A); 2) J : D(J)! R ,w hereD(J )= D(A) H and J is bounded in the graph norm kxkG def = kxk 2 +kAxk 2 1=2 inD(A). Here R is a Hilbert space.

This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green’s functions for the equations are then random variables. Regularity properties for expectation values of Green’s functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green’s function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

The first section of this paper provides an improvement upon known finiteness theorems for Riemannian submersions; that is, theorems which conclude that there are only finitely many isomorphism types of fiber bundles among Riemannian submersions whose total spaces and base spaces both satisfy certain geometric bounds. The second section of this paper provides a sharpening of some recent theorems which conclude that, for an open manifold of nonnegative curvature satisfying certain geometric bounds near its soul, there are only finitely many possibilities for the isomorphism class of a normal bundle of the soul. A common theme to both sections is a reliance on basic facts about Riemannian submersions whose A A and T T tensors are both bounded in norm.

In this paper it is shown that if MC = ( iS a 2 x 2 upper triangular operator matrix acting on the Hilbert space 7H E IC and if W(.) denotes the "Weyl spectrum", then the passage from w(A) U w(B) to w(MC) is accomplished by removing certain open subsets of w(A) n w(B) from the former, that is, there is equality w(A) U w(B) = w(MC) U G, where G is the union of certain of the holes in w(Mc) which happen to be subsets of w (A) n w (B). Let At and IC be Hilbert spaces, let 1C(7-, 1C) denote the set of bounded linear operators from 7t to IC, and abbreviate LC(QH,7) to C(H). When A E LC(t) and B E L(/C) are given we denote by Mc an operator acting on 1t E 1C of the form M A C) Mc:=( 0 ), where C E iC(!, 7-). The invertibility and spectra of Mc were considered by Du and Jin [5]. In this paper we give some conditions for operators A and B to exist an operator C such that Mc is Weyl, and describe the Weyl spectra of Mc. Recall ([7], [8]) that an operator A E 12(X, Y) for Banach spaces X and Y is called regular if there is an operator A' E L(Y, X) for which A = AA'A; then A' is called a generalized inverse for A. In this case, X and Y can be decomposed as follows (cf. [8, Theorem 3.8.2]): A-1(0) e A'A(X) = X and A(X) e (AA')-'(0) = Y. It is familiar ([6], [8]) that A E LC((, IC) is regular if and only if A has closed range. An operator A E LC(7-, IC) is called relatively Weyl if there is an invertible operator A' E 12(IC,7t) for which A = AA'A. It is known ([8, Theorem 3.8.6]) that A is relatively Weyl if and only if A is regular and A-1(0) A(H)', where means a topological isomorphism between spaces. An operator A E LQ(H, IC) is called left-Fredholm if it is regular with finite dimensional null space and rightFredholm if it is regular with its range of finite co-dimension. If A is both leftand right-Fredholm, we call it Fredholm. The index, ind A, of a leftor right-Fredholm Received by the editors November 21, 1997 and, in revised form, May 1, 1998 and March 10, 1999. 1991 Mathematics Subject Classification. Primary 47A53, 47A55.

In reply to a problem posed by Jean Leray in 1950, a nontrivial example of application of the Nielsen fixed-point theory to differential systems is given. So the existence of two entirely bounded solutions or three periodic (harmonic) solutions of a planar system of ODEs is proved by means of the Nielsen number. Subsequently, in view of T. Matsuoka's results in Invent. Math. (70 (1983), 319-340) and Japan J. Appl. Math. (1 (1984), no. 2, 417434), infinitely many subharmonics can be generically deduced for a smooth system. Unlike in other papers on this topic, no parameters are involved and no simple alternative approach can be used for the same goal. 1. HISTORICAL REMARKS Unlike any standard fixed-point principle, Nielsen theory gives us additional information about the lower estimate of the fixed points number. It was originated by the Danish mathematician Jakob Nielsen in 1927 [N], who studied with this respect self-maps of compact surfaces. Later on, finite polyhedra were systematically treated by FRanz Wecken in 1941-42 [W], who also gave an alternative definition of the Nielsen relations which we use below. The crucial step in this development was also accomplished by Andrzej Granas who defined a strictly related fixed-point index for arbitrary (i.e. also infinite-dimensional) ANRs in 1972 [G]. Thus, a sufficiently general fixed-point principle, preserving the number of essential classes of fixed points under homotopy, could be formulated by U. K. Scholz [S], Boju Jiang [Bol], R. F. Brown [Brl], and some others (see the references in [AGJ1], [Bol], [Brl]). By sufficiency, we mean the appropriate form which was suitable for answering the question posed already by Jean Leray [L] at the first International Congress of Mathematicians held after World War II in Cambridge, Mass., in 1950. Namely, he suggested the problem of adapting the Nielsen theory to the needs of nonlinear analysis and, in particular, of its application to differential systems for obtaining multiplicity results (for more details see e.g. [Br3], [Br4]). Since that time only several papers have been devoted to this problem (see e.g. [BKM], [Br2], [Br3], [Br4], [Fl], [F2]), but either additional parameters had to be Received by the editors May 4, 1998 and, in revised form, November 6, 1998. 1991 Mathematics Subject Classification. Primary 34B15, 47H10.

In this paper we construct examples of uniformly quasiregular (uqr) mappings. These provide counterexamples for a rigidity conjecture in quasiregular dynamics. It states that a closed manifold of dimension at least three, which admits a branched uqr mapping, is quasiconformally equivalent to an ordinary sphere Sn. 1. Statements and results In this paper we construct examples of a certain type of quasiregular mappings. Topologically, quasiregular mappings are branched coverings, that is, covering mappings without the local injectivity requirement. This is a consequence of the completely analytic definition demanding that the mapping distorts the metric only by a bounded amount. More precisely: Definition 1. A continuous mapping f : G → R of a domain G ⊂ R is quasiregular, if it belongs to W 1 n,loc(G) and if there exists a number K, 1 ≤ K

We prove new results on the stability of the absolutely cont;inuous spectrum for perturbed Stark operators with decaying or satisfying certain smoothness assumption perturbation. We show that the absolutely continuous spectrum of the Stark operator is stable if the perturbing potential decays at the rate (1 + x) 3 -e or if it is continuously differentiable with derivative from the H6lder space C, (R), with any ae > 0. 0. INTRODUCTION In this paper, we study the stability of the absolutely continuous spect;rum of one-dimensional Stark operators under various classes of perturbations. Stark Schr6dinger operators describe behavior of the charged particle in the constant electric field. The absolutely continuous spectrum is a manifestation of the fact that the particle described by the operator propagates to infinity at a rather fast rate (see, e.g. [2], [12]). It is therefore interesting to describe the classes of perturbations which preserve the absolutely continuous spectrum of the Stark operators. In the first part of this work, we study perturbations of Stark operators by decaying potentials. This part is inspired by the recent work of Naboko and Pushnitski [14]. The general picture that we prove is very similar to the case of perturbatiorns of free Schr6dinger operators [9]. In accordance with physical intuition, however, the absolutely continuous spectrum is stable under stronger perturbations than in the free case. If in the free case the short range potentials preserving purely absolutely continuous spectrum of the free operator are given by condition (on the power scale) lq(x)l < C(1 + xKl)'E, in the Stark operator case the corresponding condition reads lq(x)l < C(1 + IXD)-2-E. If c is allowed to be zero in the above bounds, imbedded eigenvalues may occur in both cases (see, e.g. [14], [151). Moreover, in both cases if we allow potential to decay slower by an arbitrary function growing to infinity, very rich singular spectrum, such as a dense set of eigenvalues, -may occur (see [13] for the free case and [141 for the Stark case for precise formulation and proofs of these results). The first part of this work draws the parallel further, showing that the absolutely continuous spectrum of Stark operators is preserved under perturbations satisfying lq(x)l < C(1 + Jxl)-3-E, in particular even in the regimes where a dense set of eigenvalues occurs; hence in such cases these eigenvalues are genuinely imbedded. Similar results for the free case were proven in [9], Received by the editors April 14, 1997. 1991 Mathematics Subject Classification. Primary 34L40, 81Q10. ?)1999 American Mathematical Society 243 This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:57:39 UTC All use subject to http://about.jstor.org/terms 244 ALEXANDER KISELEV [10]. Our main strategy of the proof here is similar to that in [9] and [10]: we study the asymptotics of the generalized eigenfunctions and then apply Gilbert-Pearson theory [7] to derive spectral consequences. While the main new tool we introduce in our treatment of Stark operators is the same as in the free case, namely the a.e. convergence of the Fourier-type integral operators, there are some major differences. First of all, the spectral parameter enters the final equations that we study in a different way and this makes analysis more complicated. Secondly, we employ a different method to analyze the asymptotics. Instead of Harris-Lutz asymptotic method we study appropriate Prilfer transform variables, simplifying the overall consideration. In the second part of the work we discuss perturbations by potentials having some additional smoothness properties, but without decay. It turns out that for Stark operators the effects of decay or of additional smoothness of potential on the spectral properties are somewhat similar. It was known for a long time that if a potential perturbing Stark operator has two bounded derivatives the spectrum remains purely absolutely continuous (actually, certain growth of derivatives is also allowed, see Section 2 for details or Walter [21] for the original result). We note that the results similar to Walter's on the preservation of absolutely continuous spectrum were also obtained in [4] by applying different types of technique (Mourre method instead of studying asymptotics of solutions). On the other hand, if the perturbing potential is a sequence of derivatives of 6 functions in integer points on R with certain couplings, the spectrum may turn pure point [3], [5], [1]. In some sense, the 6' interaction is the most singular and least "differentiable" among all available natural perturbations of one-dimensional Schrbdinger operators [11]. Hence we have very different spectral properties on the very opposite sides of the smoothness scale. This work closes part of the gap. We improve the well-known results of Walter [21] concerning the minimal smoothness required for the preservation of the absolutely continuous spectrum and show that in fact existence and minimal smoothness of the first derivative is sufficient to imply absolute continuity of the spectrum. After submitting this paper, the author learned about the work of J. Sahbani [17], where, in particular, the results related to Theorem 2.1 of the present work are proven. Sahbani's results are slightly stronger than Theorem 2.1: the derivative of potential V'(x) is required to be bounded and Dini continuous in order for the absolutely continuous spectrum to be preserved. In addition, he shows that the imbedded singular spectrum in this case may only consist of isolated eigenvalues. The approach employed in [17] is an extension of conjugate operator method. 1. DECAYING PERTURBATIONS Consider a self-adjoint operator Hq defined by the differential expression Hqu =-u" -xu + q(x)u on the L2(-oo, oo). Let us introduce some notation. For the function f C L2 we denote by bf its Fourier transform

In this paper we prove the following result: Let A A be a nonunital Banach algebra with a bounded approximate identity. Then A A cannot be both Arens regular and weakly sequentially complete. The paper also contains some applications of this result.

In this paper we prove the algorithmic solvability of finite systems of equations over the Q-completion of a torsion-free hyperbolic group. It was recently proved in [1] that finite systems of equations over the Q–completion of a finitely generated free group are algorithmically solvable. In this paper we generalize the results of [1] to the case of the Q–completion G of an arbitrary torsion–free hyperbolic group G with generators d1, . . . , dN . (A detailed definition of the Q–completion of a group can be found in [1].) Main Theorem. Let G be a torsion–free hyperbolic group. Then there exists an algorithm that decides if a given finite system of equations over the Q–completion of G has a solution, and if it does, finds a solution. By a triangular equation we mean an equation with at most three terms. Given a system S of equations in G, we may assume that all equations in S are triangular. Indeed, an arbitrary equation x1 1 x e2 2 . . . x en n = 1 (where xi stands either for a variable or for a constant, ei = ±1) is equivalent to a finite system of triangular equations: x1 1 x e2 2 y −1 1 = 1, y1x e3 3 y −1 2 = 1, . . . , yn−2x en−1 n−1 x en n = 1. Adding a finite number of new variables and equations, we may also assume that S contains only equations with coefficients in G. To achieve this, we replace every constant of the form d m n by a new variable z satisfying the equation z = d. From now on, we fix a finite system S of triangular equations over G with coefficients in G. We will reduce the system S to a finite set of systems in a specific hyperbolic group G ∗ 〈t1, . . . , tψ(m)〉, where the number ψ(m) can be determined effectively given the number m of equations in the original system S. The resulting systems are accompanied by the restriction that some of the variables belong to a certain subgroup G ∗ 〈t1, . . . , ti〉 of G ∗ 〈t1, . . . , tψ(m)〉, i.e. do not contain certain t’s. To each of the systems we can apply the (slightly modified) method of Rips and Sela [4] to see if they are decidable. If none of them is consistent, then the original system S has no solution; if at least one has a solution, then it is possible to find a corresponding solution to the system S. Suppose that the system has a solution in G; let {X1, . . . , XL} be a solution with the minimal possible number of roots. Since the solution belongs to G, it is contained in a certain group K, obtained from G by adding finitely many roots. It is the union of a chain of subgroups Hi defined as follows. Let G = H0. Received by the editors August 14, 1996. 1991 Mathematics Subject Classification. Primary 20E05, 20F10. The first author was supported by grants from NSERC and FCAR. The third author was supported by the NSF grant DMS-9103098. c ©1999 American Mathematical Society 2961 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 2962 O. KHARLAMPOVICH, E. LIOUTIKOVA, AND A. MYASNIKOV Step 1. Consider pairwise nonconjugated cyclically minimal primitive elements u1, . . . , uk1 ∈ G, |u1| ≤ . . . ≤ |uk1 | (here |u| denote the length of u in G), and add roots t1, . . . , tk1 , such that uj = t sj j . (Notice that ui+1 does not become a proper power after we add roots t1, . . . , ti.) The corresponding groups are denoted by H1, . . . , Hk1 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Step 2. Consider pairwise nonconjugated primitive elements uk1+1, . . . , uk2 ∈ H1, cyclically reduced in the amalgamated product, each having the reduced form u = t1 1 c1 . . . t αk 1 ck, where αi 6= 0, αi ∈ Z, ci ∈ G, |uk1+1|H1 ≤ . . . ≤ |uk2 |H1 ; and add roots tk1+1, . . . , tk2 , such that uj = t sj j , to the group Hk1 . The corresponding groups are denoted by Hk1+1, . . . , Hk2 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Step i+1. Suppose that H1, . . . , Hki have been constructed. Consider pairwise nonconjugated primitive elements uki+1, . . . , uki+1 ∈ Hi, cyclically reduced in the amalgamated product, each having the reduced form t1 i c1 . . . tk i ck, where αi 6= 0, αi ∈ Z, ci ∈ Hi−1 ( ck is not a power of ui, because the elements are cyclically reduced), |uki+1|Hi ≤ . . . ≤ |uki+1 |Hi , and add roots tki+1, . . . , tki+1 , such that uj = t sj j , to the group Hki . The corresponding groups are denoted by Hki+1, . . . , Hki+1 , where Hj+1 = Hj ∗uj+1=tsj+1 j+1 〈tj+1〉. Finally, for some number i one has K = Hki+1 . The group Hki+1 is called the group at level i, corresponding to the sequence u1, . . . , uki+1 . The group Hi will be called the group of rank i. We also order the set of tj ’s: tk < tl if k < l. Let Fi be the free group with the same generating set as Hi, i = 1, . . . , k1, and let β : Hi → Fi be a section, i.e. a mapping of sets such that π ◦ β = id. For our purposes, β has to be of specific form; we will define it explicitly in section 2. For X ∈ Hi we will call β(X) the canonical representative of X . Definition 1. By an equational diagram over a group Hi we mean an equation X1 . . . Xn = 1 together with a solution A1, . . . , An and a diagram over Hi having β(A1) . . . β(An) as its boundary label. An equational triangle is an equational diagram corresponding to a triangular equation. A system of equational diagrams is a system of equations together with the system of diagrams such that the solution associated to each equational diagram is a solution of the whole system. A free equational diagram is an equational diagram with no cells; the corresponding equations are called free equations. If the level i of the group Hki+1 is greater than zero, we can use the method of [1] to reduce each equation of the system S to a bounded number of free equations and at most one equation over the group in the previous rank. We can proceed in the same way until we reach level 0 (i.e. the group Hk1), for which we need to develop a separate method. In what follows we will denote Hk1 by H . 1. Some properties of the Cayley graph of H We will frequently use the following lemma: Lemma 1. ([3, Lemma 1.5]) For each geodesic triangle [x1, x2, x3] in a δ-space, there are points yi ∈ [xi−1, xi+1] (indices are considered modulo 3) such that d(xi, yi−1) = d(xi, yi+1) = (xi−1 · xi+1)xi , License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use THE Q–COMPLETION OF A TORSION–FREE HYPERBOLIC GROUP 2963

It is shown that if the Banach space operator T has regular normsequence, its vector orbits are asymptotically non-vanishing and there exists a complete vector orbit satisfying the growth condition of non-quasianalycity, then T has infinitely many disjoint hyperinvariant subspaces. In the recent paper [7] we initiated the study of operators possessing regular norm-sequences. The definition was based on the concept of almost convergence. We recall that a bounded, complex sequence S E '??(N,C) almost converges to the complex number c if limk,oo supneN Ic k-l Z+k 1 (i)I = 0. (The reader is referred to [9] for fundamental properties of this type of convergence.) We say that the sequence J almost converges to c in the strong sense if | cll almost converges to zero, where 1 stands for the constant 1 sequence. A gauge function is a mapping p : N -* (0, oo) with the property that {p(n + 1 )/p(n)}nEN almost converges in the strong sense to a positive number c. The set of all gauge functions is denoted by 7P. Let us consider now a complex Banach space X and let L(X) denote the set of bounded, linear operators acting on X. We say that the norm-sequence of an operator T E L(X) is compatible with the gauge function p E P, if ITnll < p(n) holds for every n E N and the sequence {lTnll/p(n)},EN does not almost converge to zero. The set of all such operators is denoted by L(p, X), and T E X is said to have a regular norm-sequence if T belongs to L(p, X) for some p E P. It was shown in [7] that {p(n + l)/p(n)}nEN almost converges to the spectral radius r(T) for every T E ?(p, X), and that a large class of operators with positive spectral radius possesses regular norm-sequence, including power bounded operators of spectral radius one. With regard to the asymptotic behaviour of vector orbits the following subclasses can be introduced. The operator T E ?(p, X) belongs to the class Ci.(p, X) (C*.(p, X)) if {llTnsll/p(n)}nEN does not almost converge to zero for all non-zero vectors x E X (for some non-zero vector x E X, respectively). In the present paper we are concerned with the hyperinvariant subspace lattice Hlat(T) of operators T of class C1. (p, X) and C*. (p, X). We remind the reader that a (closed) subspace M is hyperinvariant for T, and so it belongs to Hlat(T), if CM C M holds for every operator C commuting with T. In [7] we proved among other things the existence of a proper hyperinvariant subspace under different conditions, Received by the editors August 6, 1997. 1991 Mathematics Subject Classification. Primary 47A15, 47A60. Research partially supported by Hungarian NFS Research grant no. T 022920. (1999 American Mathematical Society

In this paper we develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. As an application we show that the logarithmic densities of any automatic sequence along squares $(n^2)_{n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, we give for these subsequences a criterion to decide whether the densities exist, in which case they are also computable. In particular in the prime case these densities are all rational. We also deduce from a recent result of the third author and Lema\'nczyk that all subshifts generated by automatic sequences are orthogonal to any bounded multiplicative aperiodic function., Comment: 35 pages. We added an Appendix concerning upper densities of subsequences of automatic sequences

In this paper, the authors establish a multiplier theorem for Herz type Hardy spaces. Let Tm be a multiplier operator defined in terms of Fourier transforms by Tmf Q) = M(W for suitable functions f. It is well-known that there is a multiplier theorem for H1 (Rn) (see [FS]): if a > n/2 and (1) J ID3m(e)12d d 0, let us denote m6QW) = M(6077(0) It is easy to check that (1) is equivalent to (2) sup |m6||Ka 2(Rn) < 0, 6 2 where K2'2 2(R) is a non-homogeneous Herz space (see [BS]). By using some embedding relations on Herz spaces, A. Baernstein II and E. T. Sawyer [BS] weakened (2) into (3) sup iM6i||Kf1l (Rn) < 0, 6 where 0 < E < an2. In fact, this is just a special case of their theorem. In [BS], Baernstein and Sawyer showed that m is a bounded multiplier of H1 (Rn) under an even weaker condition than (3); see Theorem 3b in [BS, page 21]. By using the technique of Herz type Hardy spaces developed by the authors in [LY1]-[LY3] and [Y], in this paper, we shall first establish a multiplier theorem for the homogeneous Herz type Hardy space Hk n(1-1/q), 1(IRn) which is introduced by y ~~~q the authors of this paper in [LY1]. Then as simple consequences of this theorem, a multiplier theorem for the corresponding non-homogeneous version of the space Received by the editors April 13, 1995 and, in revised form, April 5, 1997. 1991 Mathematics Subject Classification. Primary 42B15; Secondary 42B30.

The purpose of the present paper is to prove that the topological degree of the subdifferential of a coercive lower semi-continuous function on a sufficiently large ball in a reflexive Banach space is equal to one. A well known result of Krasnoselskii [7] says that the degree of the gradient of a coercive potential mapping on a sufficiently large ball in RN is equal to one. As Nirenberg pointed out in [8], this result seems intuitively clear, but the proof is not. Many authors had new views of this theorem. See [1], [9]-[11]. The aim of this note is to give a generalization of the degree of a maximal monotone mapping and prove that the topological degree of the subdifferential of a coercive convex function is equal to one on a sufficiently large ball. The main result of this paper is Theorem 5. Its proof is very simple and elementary. Let us recall the properties of the degree of mappings of class (S+), as introduced by Browder [4]. See also [3], [6]. Let E be a real reflexive Banach space of dual E* . We can always assume that both E and E* are locally uniformly convex [12]. The duality pairing between E* and E is denoted by (.,.). Let 0* be the zero element of E*, J : E -E* the duality mapping, Q C E an open bounded subset, and T: Q -E* a demi-continuous operator of class (S+) (see [4] for a definition); if 0* 0 T(Q9), then there is a topological degree deg(T, Q, 0*) satisfying Theorem 1. (a) deg(J, Q, 0*) = { 0 ifo* E J(Q). (b) If deg(T, Q, 0*) $7 0, then Tx = 0* has a solution in Q. (c) If Q1, Q2 are two disjoint open subsets of Q, then deg(T, Q, 0*) + deg(T,Q2,0*) = deg(T, Q, U Q2, 0*). (d) If {Tt} o,1] is a homotopy of class (S+), and 0* f TtQ9Q), then deg(T, Q, 0*) doesn't depend on t E [0, 1]. Now, suppose that A: D(A) C E -2 is a maximal monotone operator. Let AA = (AJ-1+A-')-', RA = I-AJ-1AA be its Yosida approximation and resolvent Received by the editors February 25, 1997. 1991 Mathematics Subject Classification. Primary 47H10, 47H15; Secondary 55M25.

The purpose of this paper is to develop a new approach to certain approximation and factorization problems for matrix-valued functions. The approach is based on studying properties of maximizing vectors of Hankel operators with matrix-valued symbols and on the solution of the so-called recovery problem for unitary-valued matrix functions. In the case of scalar functions such problems were studied in detail in [PK]. It turns out however, that the case of matrix functions is considerably more complicated than the scalar case. In this paper we solve the recovery problem with the help of Hankel operators. This solution also allows us to give a new approach to Wiener-Hopf factorization problems and obtain some new results. Another application we consider here is the problem of finding a superoptimal analytic approximation to a matrix function on the unit circle. We improve earlier results on hereditary properties of such approximations (see [PY1]) and on continuity properties of the operator of superoptimal approximation (see [PY3]). Recall that for a bounded m x n matrix-function b on the unit circle Tf the Hankel operator Hp is defined on the Hardy class H2 (Cn) of Cn-valued functions by HDf = PADf, f E H (Cn)

Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain tree-like obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. We prove that if H has tree duality then the H-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the X-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the H-colouring problem is NP -complete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial H-colouring problems. If P 6= NP , then no oriented triad H with an NP -complete H-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NP -complete Hcolouring problems given in the companion paper has tree duality.

A transcendental entire function is called criniferous if every point in its escaping set can eventually be connected to infinity by a curve of escaping points. Many transcendental entire functions with bounded singular set have this property, and this class has recently attracted much attention in complex dynamics. In the presence of escaping critical values, these curves break or split at (preimages of) critical points. In this paper, we develop combinatorial tools that allow us to provide a complete description of the escaping set of any criniferous function without asymptotic values on its Julia set. In particular, our description precisely reflects the splitting phenomenon. This combinatorial structure provides the foundation for further study of this class of functions. For example, we use these results in [arXiv:1905.03778] to give the first full description of the topological dynamics of a class of transcendental entire maps with unbounded postsingular set., Comment: 28 pages, 3 figures. Refined results of those in sections 2,3 and 4 of the second version of arXiv:1905.03778

We characterize the generalized inner derivations on a unital Banach algebra which are spectrally bounded. In particular, a simplified argument for the recent result due to Bresar that every spectrally bounded inner derivation maps into the radical is given. In [3], a linear mapping T on a complex unital Banach algebra A was defined to be spectrally bounded if r(Tx) 0O where r(.) denotes the spectral radius. This concept arises in connection with the non-commutative Singer-Wermer conjecture stating that the image 3A of any derivation 3 on A is contained in the Jacobson radical rad(A) if and only if x6x (3x)x E rad(A) for all x E A. Although some progress on this conjecture has been made lately (see [6] and the references in [3]), it is still open and seems hard to verify. A related, also open, conjecture is the following: The image 3A of a derivation 3 on A is contained in rad(A) if and only if 3 is spectrally bounded. This was recently affirmed for inner derivations by Bresar [2]. His argument rests in an essential way on two results due to Ptak [5], in particular on [5, Proposition 2.1] stating that a spectrally bounded inner derivation 3 has the property that 652A C d(A), the set of quasinilpotent elements of A. In an earlier paper, however, Ptak obtained a number of equivalent conditions for the image of an inner derivation to lie in the radical. For our purposes, the following suffices. Theorem A [4, Proposition 2.1]. Let A be a unital Banach algebra and a e A. The following conditions are equivalent. (a) ax xa E rad(A) for all x E A. (b) r(ax) 0. While (a) => (b) is immediate, the implication (b) => (a) is a consequence of the subharmonicity of the spectral radius. (But see the note at the end of this paper.) Let La and Ra denote the left and right multiplications by a E A, respectively. Combining Theorem A with Bresar's result we have that La Ra is Received by the editors July 8, 1993 and, in revised form, November 28, 1993; presented by the first-named author at the Special Session on Nonselfadjoint Operator Algebras, 886th Regional Meeting of the AMS, College Station, Texas, October 22, 1993. 1991 Mathematics Subject Classification. Primary 46H99; Secondary 47B47. The first-named author's research was partially supported by an NSF grant. ? 1995 American Mathematical Society

The purpose of this paper is to obtain the extensions of Borsuk's antipodal and fixed-point theorems for set-valued maps. Borsuk [3] related the following relative results: Borsuk's antipodal theorem. A single-valued antipodal-preserving continuous map f: Sn -> En has a zero value. Borsuk's fixed-point theorem. Let U be a bounded symmetric convex open neighborhood of the origin in En, and let f: U -> En be antipodal-preserving on a U, i.e., -f(a) = f(-a) for each a E a U. Then f has a fixed point. In this note we consider the set-valued maps. With a proper definition of antipodality, we prove that Borsuk's antipodal theorem can be extended for setvalued maps on the boundary of a symmetric bounded balanced neighborhood of the origin. Such boundaries are more general than the homeomorphic image of spheres. At the same time we also prove Borsuk's fixed-point theorem on a symmetric balanced set of a locally convex space. The main results are summarized in Theorems 4, 5, 6, and 7. Lemmas 1 and 2 are two approximation properties which play a crucial role in this paper. Let us first recall some definitions. Let X and Y be two topological spaces; capital letters F: X -> Y denote nonempty set-valued maps while noncapital letters f: X -Y will denote single-valued functions. F is called upper semicontinuous (u.s.c.) if for each open set G of Y the set {x E XIF(x) c G} is open in X. F is said to have open lower sections if, for each y E Y, F-1(y) = {x E Xly E F(x)} is open in X. The set GrF is the graph of F which is the set {(x, Y)ly E F(x), x E X}I. When Y is a topological vector space and S c Y, co S denotes the convex hull of S. A set B c Y is said to be balanced if rB c B for every real number r with Irn n} is denoted by En; the unit n-sphere Sn = {x E En+1I lxll = 1}. Let X and Y be subsets of E, S c X be a Received by the editors February 14, 1991 and, in revised form, August 3, 1992. 1991 Mathematics Subject Classification. Primary 54H25, 55M20, 47H 10.

A necessary and sufficient condition is given to show that the indefinite integral of a vector-valued pseudo-almost periodic function is again pseudo-almost periodic. Then we use this result to answer a question about weakly almost periodic functions. Throughout this paper, X denotes a Banach space and Ja stands for [a, ox) when a E R and for R when a = -xc; W(Ja, X) denotes the space of all bounded continuous functions from Ja to X. Also, m denotes Lebesgue measure on R. Let f E (R, X). The translate of f by s E R is the function R,f(t) = f(t + s), t E R. f is called (weakly) almost periodic if the set {RRf} = {RJf s E R} is (weakly) relatively compact in W(R, X) [3]. Denote by ( 'YV (R, X)) ,V (R, X) all such functions. In case X = C, we will omit X from our notation and write, for example, '(Ja) for iF(Ja, X) . For a function f E s 9s(R), the classical Bohl-Bohr integration theorem (cf. [2]) asserts that the indefinite integral F(t) = fJ f(u) du will also be almost periodic on R whenever F is bounded. For f E ? 9 (R, X), there is Kadets's generalized Bohl-Bohr theorem (see [4]). A question arises naturally in 71W(R, X); that is, if f E 7?9s(R, X), what is a necessary and sufficient condition for F to be again in WVY(D(R, X)? Let f E 7AW,9 (R, X). Then f -g + (a, where g E W?9 (R, X) and o E 71Q 9(R,X), the subspace of ZYY(R, X) whose members have the zero function in the weak closure of the set of translates [3, Theorem 4.11]. The difficult part of answering the question is to handle the function ~o. For this purpose we introduce a new generalization of almost periodic functions, which we call pseudo-almost periodic functions. In this paper, we first set up some theorems of the indefinite integral of a pseudo-almost periodic function from Ja to X. Then we use these theorems to answer the question. Received by the editors June 30, 1992 and, in revised form, August 20, 1992. 1991 Mathematics Subject Classification. Primary 43A60; Secondary 28B05.

We show a relation between O'Hara's energy of knots and the Douglas functional. In his paper [4], O'Hara introduced the following energy of knots 2 X JJslxSl {I?p(X) (y)12 sin27r(X y) d for any C2-embedding (o: SI = R/Z 1&3 such that I ('I = 1. This quantity is admitted to be negative, but it is bounded from below (Eknot((o) > -2) and blows up if a knot has a self-intersection. O'Hara showed that there exist only finitely many ambient isotopy classes of knots whose energy and bending energy (see [4] for the definition) are bounded from above by any given constant. In his prize winning paper [3] for the Plateau problem, Douglas employed the Douglas functional D(f) 1 J27t 27 1 ^(e v a) (el-1)12 D ___ k -(evN)-P(V )2da d/i 16Dt = ? sin2(a ,6/2) 7f kop(X) (p(y)12 4 JSXsl sin% t(x-y) dx where (e\v'ia) = (o(a/27r). (See also Struwe [7].) The Douglas functional is closely related to the Dirichlet integral (or more generally the energy functional) of a C1-map f: B2 -+ R3 E(f)=IJIIdf II =I Jj( f 2 of 2)dx dy, 2 jJ2 11tl 2 tJ2 O11x 1 1ay 1 where B2 {z = x+ /Ty E C; lzl < 1} . Indeed D(p) is equal to the energy E(fq,) of the harmonic extension fv, of (0, i.e., Afv, = 0 in B2, f, = ^ on 0B2. (See Douglas [3], Ahlfors [1, Theorems 2-5].) Courant [2] minimized the Dirichlet functional to solve the Plateau problem. The Douglas functional of a curve (0 is an energy of the harmonic surface fv, with the boundary ( . In other Received by the editors July 24, 1991 and, in revised form, August 23, 1991. 1991 Mathematics Subject Classification. Primary 58E12; Secondary 57M25.

In previous papers the author characterized free arrangements of hyperplanes by the vanishing of cohomology of the intersection lattice with coefficients in a certain sheaf of graded modules over a polynomial ring. The main result of this paper is that for a locally free arrangement the degrees of nonzero homogeneous components of the cohomology modules are bounded by a number depending only on the intersection lattice. In particular, the Hilbert coefficients of the module of derivations of a locally free arrangement are combinatorial invariants. Another result of the paper asserts that the set of free arrangements is Zariski open in the set of all arrangements with a given intersection lattice.

Let A A be a densely defined closed (linear) operator, and { A α } \{ {A_\alpha }\} , { B α } \{ {B_\alpha }\} be two nets of bounded operators on a Banach space X X such that | | A α | | = O ( 1 ) , A α A ⊂ A A α , | | A A α | | = o ( 1 ) ||{A_\alpha }|| = O(1),{A_\alpha }A \subset A{A_\alpha },||A{A_\alpha }|| = o(1) , and B α A ⊂ A B α = I − A α {B_\alpha }A \subset A{B_\alpha } = I - {A_\alpha } . Denote the domain, range, and null space of an operator T T by D ( T ) D(T) , R ( T ) R(T) , and N ( T ) N(T) , respectively, and let P ( resp . B ) P(\operatorname {resp} .B) be the operator defined by P x = lim α A α x ( r e s p . B y = lim α B α y ) Px = {\lim _\alpha }{A_\alpha }x(resp. By = {\lim _\alpha }{B_\alpha }y) for all those x ∈ X ( resp . y ∈ R ( A ) ¯ ) x \in X(\operatorname {resp} .y \in \overline {R(A)} ) for which the limit exists. It is shown in a previous paper that D ( P ) = N ( A ) ⊕ R ( A ) ¯ , R ( P ) = N ( A ) , D ( B ) = A ( D ( A ) ∩ R ( A ) ¯ ) , R ( B ) = D ( A ) ∩ R ( A ) ¯ D(P) = N(A) \oplus \overline {R(A)} ,R(P) = N(A),D(B) = A(D(A) \cap \overline {R(A)} ),R(B) = D(A) \cap \overline {R(A)} , and that B B sends each y ∈ D ( B ) y \in D(B) to the unique solution of A x = y in R ( A ) ¯ Ax = y{\text { in }}\overline {R(A)} . In this paper, we prove that D ( P ) = X D(P) = X and | | A α − P | | → 0 ||{A_\alpha } - P|| \to 0 if and only if | | B α | D ( B ) − B | | → 0 ||{B_\alpha }|D(B) - B|| \to 0 , if and only if | | B α | D ( B ) | | = O ( 1 ) ||{B_\alpha }|D(B)|| = O(1) , if and only if R ( A ) R(A) is closed. Moreover, when X X is a Grothendieck space with the Dunford-Pettis property, all these conditions are equivalent to the mere condition that D ( P ) = X D(P) = X . The general result is then used to deduce uniform ergodic theorems for n n -times integrated semigroups, ( Y ) (Y) -semigroups, and cosine operator functions.

We discuss the term "thick" set. With the help of this term we deduce a strong Uniform Boundedness Principle valid for all Banach spaces. As an application we give a new proof of Seever's theorem. 1. DEFINITIONS AND PRELIMINARY RESULTS In this paper X is a Banach space. In [3] the notion of a "thin" set was introduced. It turns out that if a bounded set A C X is not thin, then for any Banach space Y, any T E L(Y, X) onto A has to be onto X. Further, "not thin" is the weakest general condition such that "onto A" implies "onto X". It is natural to ask for the weakest general condition to be put on a bounded set A to assure that pointwise boundedness of a family of operators in L(X, Y) implies uniform boundedness. The Banach-Steinhaus theorem tells us that "second category" is a sufficient condition, but the Nikodrm boundedness theorem shows that a uniform boundedness principle is true under weaker conditions, in particular spaces at least. Our main result in this paper is that "not thin" is the condition sought for this problem as well. Since, for our purposes at least, "not thin" seems to be more important than "thin", we prefer to use the terms "thin" and "thick" instead of "thin" and "not thin". Definition 1.1. A bounded set B C X is called non-norming for X* if inf sup If(x) I 0. fCSx* xCzB A bounded set C C X is called thin if it is the countable union of an increasing family of sets which are non-norming for X*. If a bounded set C is not thin, we will call it thick. In [3] Kadets and Fonf showed that there is an "open mapping principle" associated to thickness: Theorem 1.1. Suppose B C X is bounded. Then the following are equivalent statements: 1) For all Banach spaces Y and all T E L2(Y, X) with TY D B, we have TY = X. Received by the editors October 9, 1997 and, in revised form, May 13, 1998 and June 1, 1999. 2000 Mathematics Subject Classification. Primary 46B20; Secondary 28A33.

Let F F be an analytic function on the bounded domain R R , a multiply-connected region of the complex plane. The Bloch constant of F F is defined by \[ β F = sup | z | > 1 ( 1 − | z | 2 ) | ( F ∘ p ) ′ ( z ) | , \beta _F = \sup _{|z| > 1} (1 - |z|^2)|(F \circ p)’(z)|, \] where p p is a conformal universal cover of R R with domain Δ \Delta , the open unit disk. If F F is bounded, then β F ≤ | | F | | ∞ {\beta _F} \leq ||F|{|_\infty } , the sup-norm of F F . In this paper we characterize those functions F F for which β F = | | F | | ∞ {\beta _F} = ||F|{|_\infty } in terms of the zeros of F F when the boundary of R R is the union of finitely many curves. We conclude this paper by showing the existence of extremal functions, and generalizing the results to bounded harmonic mappings on these domains.

The aim of this paper is to present an example of a bounded balanced domain of holomorphy in C" (n > 3) with continuous Minkowski function that is not Kobayashi-finitely-compact. INTRODUCTION It is known [6] that if G c C" is a bounded Reinhardt-domain of holomorphy with 0 C G then G is finitely-compact with respect to (w.r.t.) the Carath6odorydistance cG, i.e., all cG-balls are relatively compact subsets of G w.r.t. the usual topology. In a more general case, if G = Gh = {z C Cn: h(z) < 1} is a bounded balanced domain of holomorphy with continuous Minkowski function h, then G is finitely-compact w.r.t. the Bergman-distance bG [4]. On the other hand, the continuity of h is a necessary condition for G = Gh to be finitely-compact w.r.t. the Kobayashi-distance kG [5, 1]. In this paper we give an example of a bounded balanced domain of holomorphy G = Gh C C3 with continuous h that is not kG-finitely compact and therefore, not cG-finitely compact. This answers a question formulated by J. Siciak in [7]. In particular, the example shows that, in general, there is no comparison of type bG < CkG for bounded balanced domains of holomorphy with continuous Minkowski function. DEFINITIONS AND STATEMENT We repeat some of the notions that will be needed in the sequel. Definition. A domain G c Cn is called balanced' iff whenever z C G and E C, 11?

Consider a semisimple connected Lie group G with an aiSine symmetric space X. We study abstractly the intertwining operators from the discrete series of X into representations with reproducing kernel and in particular, into the discrete series of G; each such is given by a convolution with an analytic function. For X of Hermitian type, we consider the holomorphic discrete series of X and here derive very explicit formulas for the intertwining operators. As a corollary we get a multiplicity one result for the series in question. 0. INTRODUCTION The purpose of this work is twofold. The first is to study abstractly the intertwining operators between discrete series representations of an affine symmetric space X and discrete series of the corresponding semisimple group. The second is to obtain very explicit formulas for certain discrete representations on X in effect giving the analogue of Harish-Chandra's original construction of the holomorphic discrete series in the group case. Not surprisingly, these two problems are highly connected. Thus Theorem 5.4 below gives the formula for the lowest K-type (the Flensted-Jensen function) on X, whereas in Theorem 7.3 essentially the same function gives the intertwining operator. Much of our analysis builds on the framework in [14], but otherwise we have tried to make the paper reasonably self-contained. This is, in particular, the case in §1, §2, and §4, where we abstract the properties of our intertwining operators with an eye towards later applications. We shall always let G be a connected, noncompact semisimple Lie group such that the associated Riemannian symmetric space is a bounded symmetric domain D c Cn . The best known and most interesting discrete series representations of G are the holomorphic discrete series representations. As explained above, one of our aims in this paper is to generalize this construction to affine symmetric spaces of Hermitian type and relate those with other well-known representations. Received by the editors September 23, 1988 and, in revised form, June 27, 1989. 1980 Mathematics Subject Classiffcation (1985 Rerision). Primary 22E46; Secondary 43A85. (D 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

In this paper the Kobayashi metric on a domain in Cn is used to define a new function space. Elements of this space belong to a nonisotropic Lipschitz class. It is proved that if f is holomorphic on the domain and in the classical Lipschitz space A. then in fact f is in the new function space. The result contains the original result of Stein on this subject and provides the optimal result adapted to any domain. In particular, it recovers the Hartogs extension phenomenon in the category of Lipschitz spaces. This paper is part of a program to work out the function theory-especially the harmonic analysis-of domains in Cn in terms of invariant metrics and related constructs. The point is that, whereas in one complex dimension the principal objects of study are functions, it turns out that in several complex dimensions the principal objects of study should be domains. The correct context in which to perceive the differences between domains seems to be that of metric geometry. The present paper is concerned with Lipschitz spaces, and draws its inspiration from [ST]. In [ST], Stein announced that holomorphic Lipschitz a functions on bounded C2 domains in Cn, n > 1, are in fact Lipschitz 2a in complex tangential directions; a detailed proof appears, for instance, in [KR3]. It turns out that this result is optimal only for strongly pseudoconvex domains (see [KR6] for improved results on some other domains), and our purpose here is to find a language in which to formulate the best result for any domain. The Eisenman-Kobayashi metric and volume construction provides the most natural language. For related work using the Eisenman-Kobayashi ideas, see [KR1, KR2, KM, GK]. A second point of the present work is to eliminate the arbitrary distinction between normal and tangential directions. All directions should be treated equally, and the metric should read off the variation in smoothness. An interesting by-product of the work is that it rediscovers the Hartogs extension phenomenon (or "Kugelsatz"-see [KR3]) in the category of Lipschitz functions. This feature is explored in ??2 and 4. ? 1 contains basic definitions and terminology, including the definition of a new Lipschitz class on domains in Cn. It also contains the statement of our Received by the editors October 12, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 32A10; Secondary 32A30. Work supported in part by the National Science Foundation. ? 1990 American Mathematical Society 0002-9947/90 $1.00 + $.25 per page

In the present paper the reflexivity of a WOT-closed algebra generated by certain pairs of commuting shifts, which are not necessarily doubly commuting, is proved. In what follows B(H) denotes the algebra of all (linear bounded) operators on a separable complex Hilbert space H. Throughout the paper we simply write a subspace for a closed subspace of H and an-algebra (of operators) on H for a subalgebra of B(H) with the unit (the identity on H). If 5 is a subset of B(H) then W(() stands for the WOT (Weak Operator Topology)closed algebra generated by 5 while Lat5 denotes the lattice of all invariant subspaces for all operators from Y. For operators T1,..., Tn on H we write in the sequel I(T1, ..., Tn), Lat(T1, ..., T.), respectively, instead of ({ T,, . T.., T}) and Lat({T, ..., Tn}). AlgLat Y stands for the algebra of all operators on H which leave invariant all subspaces from Lat 59. An algebra v is called reflexive if and only if _v = AlgLat? . A family 5 c B(H) is called reflexive if the algebra 41(Y) is reflexive. An operator T E B(H) is called reflexive if 5 = { T} . Deddens [1] proved the reflexivity of an isometry using the Wold decomposition and a model for a shift. If a pair of commuting isometries is considered then a Wold-type decomposition need not exist [7]. Another difficulty is a lack of any model for a pair of commuting shifts in a general situation. In [3] the reflexivity of a pair { VI, V2J} of doubly commuting isometries (i.e. VJ/, V2 commute and so do Vi, J2*') was proved. Thus a pair of doubly commuting shifts is also reflexive. In this paper we prove, using a model from [3], the reflexivity of certain pairs of shifts, which are not necessarily doubly commuting. Let S denote a set of all pairs of nonnegative integers and G be a set of all pairs of integers. Following [3], we introduce some notations and definitions. Namely, a subset X c G is called a diagram if E E X, s E S imply q+s E X. The set of all diagrams is denoted by X. For 4 E G we define E.= {X c X: 4 E X}I. It is obvious that E. c E,+, (q E G, s E S). Let XF be a c-algebra Received by the editors July 31, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 47D25; Secondary 47B35.