Critical points of essential norms of singular integral operators in weighted spaces

We show that for any simple piecewise Ljapunov contour there exists a power weight such that the essential norm |S | in the spaceL ₂(, ) does not depend on the angles of the contour and it is given by formula (2). All such weights are described. For the union =₁₂ of two simple piecewise Lyapunov curves we prove that the essential norm |S | inL ₂() is minimal if both ₁ and ₂ are smooth in some neighborhoods of the common points. It is the case when the norm |S | in the spaceL ₂() as well as inL ₂(, ) does not depend on the values of the angles and it can be calculated by formula (5).     

A remark on Nehari's problem

An example of a strickly contractive Hankel matrix is given such that the central contractive extensionf _c of satisfies f _c =1. This way we answer a problem raised by Ciprian Foias.Partially supported by a Georgia State University Research Grant.     

A note on Brown—McCoy radicals ofΓ-rings

In [1] we defined the Brown—McCoy radical,B(M), of a-ringM. In this note we show thatB is a special radical. The simplicial radical, defined by Kyuno [4] for-rings with left and right unities, is extended to arbitrary-rings. The simplicial radicalS is shown to be a generalization of the Brown—McCoy radical of a ring. In general,B(M) S(M). This work is supported financially by the Technikon Natal.     

1-Homogeneous Graphs with Cocktail Party μ-Graphs

Let be a graph with diameter d 2. Recall is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of the distance partition{{z V() | (z, y) = i, (x, z) = j} | 0 i, j d}is equitable and its parameters do not depend on the edge xy. Let be 1-homogeneous. Then is distance-regular and also locally strongly regular with parameters (v,k,,), where v = k, k = a ₁, (v – k – 1) = k(k – 1 – ) and c ₂ + 1, since a -graph is a regular graph with valency . If c ₂ = + 1 and c ₂ 1, then is a Terwilliger graph, i.e., all the -graphs of are complete. In [11] we classified the Terwilliger 1-homogeneous graphs with c ₂ 2 and obtained that there are only three such examples. In this article we consider the case c ₂ = + 2 3, i.e., the case when the -graphs of are the Cocktail Party graphs, and obtain that either = 0, = 2 or is one of the following graphs: (i) a Johnson graph J(2m, m) with m 2, (ii) a folded Johnson graph J¯(4m, 2m) with m 3, (iii) a halved m-cube with m 4, (iv) a folded halved (2m)-cube with m 5, (v) a Cocktail Party graph K _{m × 2} with m 3, (vi) the Schläfli graph, (vii) the Gosset graph.     

Special radicals of near-rings and Γ-near-rings

It was previously shown that every special radical classR of rings induces a special radical class R of -rings. Amongst the special radical classes of near-rings, there are some, called the -special radical classes, which induce, special radical classes of -near-rings by the same procedure as used in the ring case. The -special radicals of near-rings possess very strong hereditary properties. In particular, this leads to some new results for the equiprime andI ₃ radicals.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

Splitting with Continuous Control in Algebraic K-Theory

In this work, the continuously controlled assembly map in algebraic K-theory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups that satisfy certain geometric conditions. The group is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, K₀(k) is proved to be isomorphic to the colimit of K₀(kH) over the finite subgroups H of , when is a virtually polycyclic group and k is a field of characteristic zero.     

Free Algebras of Automorphic Forms on the Upper Half-Plane

We find all pairs (,a) consisting of a cocompact Fuchsian group of genus zero and an automorphy factor a of for which the graded algebra of a--automorphic forms is free.     

The Spectral Theory of Amenable Actions and Invariants of Discrete Groups

Let G denote a semisimple group, a discrete subgroup, B=G/P the Poisson boundary. Regarding invariants of discrete subgroups we prove, in particular, the following:(1) For any -quasi-invariant measure on B, and any probablity measure on , the norm of the operator () on L ²(B,) is equal to (), where is the unitary representation in L ²(X,), and is the regular representation of .(2) In particular this estimate holds when is Lebesgue measure on B, a Patterson–Sullivan measure, or a -stationary measure, and implies explicit lower bounds for the displacement and Margulis number of (w.r.t. a finite generating set), the dimension of the conformal density, the -entropy of the measure, and Lyapunov exponents of .(3) In particular, when G=PSL₂() and is free, the new lower bound of the displacement is somewhat smaller than the Culler–Shalen bound (which requires an additional assumption) and is greater than the standard ball-packing bound.We also prove that ()=_G() for any amenable action of G and L ¹(G), and conversely, give a spectral criterion for amenability of an action of G under certain natural dynamical conditions. In addition, we establish a uniform lower bound for the -entropy of any measure quasi-invariant under the action of a group with property T, and use this fact to construct an interesting class of actions of such groups, related to 'virtual' maximal parabolic subgroups. Most of the results hold in fact in greater generality, and apply for instance when G is any semi-simple algebraic group, or when is any word-hyperbolic group, acting on their Poisson boundary, for example.     

Solvable Lie Algebras, Lie Groups andPolynomial Structures

In this paper, we study polynomial structures by starting on the Lie algebra level, thenpassing to Lie groups to finally arrive at the polycyclic-by-finite group level. To be more precise,we first show how a general solvable Lie algebra can be decomposed into a sum of two nilpotentsubalgebras. Using this result, we construct, for any simply connected, connected solvable Lie groupG of dim n, a simply transitive action on R ⁿ which is polynomial and of degree n³. Finally, we show the existence of a polynomial structure on any polycyclic-by-finite group , which is of degree h()³ on almost the entire group (h () being the Hirsch length of ).     

Linear Versus Nonlinear Rules for Mixture Normal Priors

The problem under consideration is the -minimax estimation, under L₂loss, of a multivariate normal mean when the covariance matrix is known. The family of priors is induced by mixing zero mean multivariate normals with covariance matrix I by nonnegative random variables , whose distributions belong to a suitable family G. For a fixed family G, the linear (affine) -minimax rule is compared with the usual -minimax rule in terms of corresponding -minimax risks. It is shown that the linear rule is "good", i.e., the ratio of the risks is close to 1, irrespective of the dimension of the model. We also generalize the above model to the case of nonidentity covariance matrices and show that independence of the dimensionality is lost in this case. Several examples illustrate the behavior of the linear -minimax rule.     

Twisted derivations and distinct sets of lines that cover the same pairs of points

A partial projective plane of ordern consists of lines andn ² +n + 1 points such that every line hasn+1 points and distinct lines meet in a unique point. Suppose that two essentially different partial projective planes and of ordern, n a perfect square, that are defined on the same set of points cover the same pairs of points. For sufficiently largen we show that this implies that and have at leastn(n+1) lines. This bound is sharp and there exist essentially two different types of examples meeting the bound.As an application, we can show that derived planes provide an example for a pair of projective planes of square order with as much structure as possible in common, that is, as many lines as possible in common. Furthermore, we present a new method (twisted derivations) to obtain planes from one another by replacing the same number of lines as in a derivation.     

Boundary control of parabolic systems: Regularity of optimal solutions

Boundary control problems for the linear, parabolic equations and a quadratic performance index are considered. The controls are allowed to be in the spaceL ²[OT;L²()], where is a boundary. Exploiting the semigroup approach, it is shown that optimal control belongs toL ²[OT;H^1/2()] and, as a consequence, optimal trajectory belongs toL ¹[OT;H¹()]. This result is obtained for two kinds of domains. The first are the domains withC -boundary and the second are the domains being the parallelepipeds.     

Convergence proofs and error estimates for an iterative method for conformal mapping

Summary The iterative method as introduced in [8] and [9] for the determination of the conformal mapping of the unit disc onto a domainG is here described explicitly in terms of the operatorK, which assigns to a periodic functionu its periodic conjugate functionK u. It is shown that whenever the boundary curve ofG is parametrized by a function with Lipschitz continuous derivative then the method converges locally in the Sobolev spaceW of 2-periodic absolutely continuous functions with square integrable derivative. If is in a Hölder classC ²⁺, the order of convergence is at least 1+. If is inC ^l+1+ withl1, 0<<1, then the iteration converges inC ^l+. For analytic boundary curves the convergence takes place in a space of analytic functions.For the numerical implementation of the method the operatorK can be approximated by Wittich's method, which can be applied very effectively using fast Fourier transform. The Sobolev norm of the numerical error can be estimated in terms of the numberN of grid points. It isO(N ^1–l–) if is inC ^l+1+, andO (exp (–N/2)) if is an analytic curve. The number in the latter formula is bounded by logR, whereR is the radius of the largest circle into which can be extended analytically such that'(z)0 for |z|<R. The results of some test calculations are reported.     

Zur Beziehung zwischen Basen und Vektoren kürzester Länge in Gittern

LetR be an order of an algebraic number field of degreen over ,V ann-dimensional real vector space and the class of lattices inV which are free rank 1 modules overR. For certain ordersR and distance functionsd onV a method of computingd-minimal vectors of is described; further it is shown how to constructs anR-basis for by comparing thed-length of vectors of . An application to the computation of fundamental units and class numbers of real abelian number fields is mentioned.     

Terminal 3-fold Divisorial Contractions of a Surface to a Curve I

Let X be a smooth curve on a 3-fold which has only index 1 terminal singularities along . In this paper we investigate the existence of extremal terminal divisorial contractions E Y X, contracting an irreducible surface E to . We consider cases with respect to the singularities of the general hypersurface section S of X through . We completely classify the cases when S is A _i , i 3, and D _2n for any n.     

Lefschetz type theorem

Summary Let X=Cⁿ / be a toroidal group of rank =n+m. If X is compact, then it is a complex torus. In the compact case, we have the theorem of Lefschetz which asserts that if L is a positive line bundle over a complex torus X, then gives an embedding of X for any integer l3. This theorem is generalized to noncompact toroidal groups in this paper. In fact, we prove the following: (I) In the case of rank =n+1, H⁰(X, (Ll)) gives an embedding of a toroidal group X for l3, if L is positive. (II) In the case of rank =n+m, 2m相似文献   

Continuous Quotients for Lattice Actions on Compact Spaces

Let < SL _n ( ) be a subgroup of finite index, where n 5. Suppose acts continuously on a manifold M, where ₁(M) = ⁿ , preserving a measure that is positive on open sets. Further assume that the induced action on H ¹(M) is non-trivial. We show there exists a finite index subgroup < and a equivariant continuous map : M ⁿ that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where ₁(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with actions.     

K-theory of fine topological algebras, Chern character, and assembly

TheK-theory of the group algebra [] for a countable, discrete group is defined in terms of the simplicial ring of smooth simplices on [], where [] is given the fine topology with respect to its finite-dimensional, linear subspaces. The assembly map for this theory :K _* B K _* [] is studied and shown to be a rational injection. The proof uses the Connes-Karoubi Chern character fromK-theory of Banach algebras to cyclic homology, here generalized to any fine topological algebra, and proved to be multiplicative.