A homotopy principle for maps with prescribed Thom-Boardman singularities

Let and be smooth manifolds of dimensions and ( ) respectively. Let denote an open subspace of which consists of all Boardman submanifolds of symbols with . An -regular map refers to a smooth map such that . We will prove what is called the homotopy principle for -regular maps on the existence level. Namely, a continuous section of over has an -regular map such that and are homotopic as sections.

     

Characterizations of function spaces on the sphere using frames

In this paper we introduce a polynomial frame on the unit sphere of , for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere , such as , and Besov spaces, can be characterized in terms of the coefficients in the wavelet decompositions, as in the usual Euclidean case . We also study a related nonlinear -term approximation problem on . In particular, we prove both a Jackson-type inequality and a Bernstein-type inequality associated to wavelet decompositions, which extend the corresponding results obtained by R. A. DeVore, B. Jawerth and V. Popov (``Compression of wavelet decompositions', Amer. J. Math. 114 (1992), no. 4, 737-785).

     

Generalized Ahlfors functions

Let be a bordered Riemann surface with genus and boundary components. Let be a smooth family of smooth Jordan curves in which all contain the point 0 in their interior. Let and let be the family of all bounded holomorphic functions on such that and for almost every . Then there exists a smooth up to the boundary holomorphic function with at most zeros on so that for every and such that for every . If, in addition, all the curves are strictly convex, then is unique among all the functions from the family .

     

Scrollar syzygies of general canonical curves with genus

We prove that for a general canonical curve of genus , the space of th (last) scrollar syzygies is isomorphic to the Brill-Noether locus . Schreyer has conjectured that these scrollar syzygies span the space of all th (last) syzygies of . Using Mukai varieties we prove this conjecture for genus , and .

     

Generating continuous mappings with Lipschitz mappings

If is a metric space, then and denote the semigroups of continuous and Lipschitz mappings, respectively, from to itself. The relative rank of modulo is the least cardinality of any set where generates . For a large class of separable metric spaces we prove that the relative rank of modulo is uncountable. When is the Baire space , this rank is . A large part of the paper emerged from discussions about the necessity of the assumptions imposed on the class of spaces from the aforementioned results.

     

Tangentially positive isometric actions and conjugate points

Let be a complete Riemannian manifold with no conjugate points and a principal -bundle, where is a Lie group acting by isometries and the smooth quotient with the Riemannian submersion metric.

We obtain a characterization of conjugate point-free quotients in terms of symplectic reduction and a canonical pseudo-Riemannian metric on the tangent bundle , from which we then derive necessary conditions, involving and , for the quotient metric to be conjugate point-free, particularly for a reducible Riemannian manifold.

Let , with the Lie Algebra of , be the moment map of the tangential -action on and let be the canonical pseudo-Riemannian metric on defined by the symplectic form and the map , . First we prove a theorem, stating that if is not positive definite on the action vector fields for the tangential action along then acquires conjugate points. (We proved the converse result in 2005.) Then, we characterize self-parallel vector fields on in terms of the positivity of the -length of their tangential lifts along certain canonical subsets of . We use this to derive some necessary conditions, on and , for actions to be tangentially positive on relevant subsets of , which we then apply to isometric actions on complete conjugate point-free reducible Riemannian manifolds when one of the irreducible factors satisfies certain curvature conditions.

     

Deformed preprojective algebras of generalized Dynkin type

We introduce the class of deformed preprojective algebras of generalized Dynkin graphs (), (), , , and () and prove that it coincides with the class of all basic connected finite-dimensional self-injective algebras for which the inverse Nakayama shift of every non-projective simple module is isomorphic to its third syzygy .

     

Minimal polynomials and radii of elements in finite-dimensional power-associative algebras

In the first section of this paper we revisit the definition and some of the properties of the minimal polynomial of an element of a finite-dimensional power-associative algebra over an arbitrary field . Our main observation is that , the minimal polynomial of , may depend not only on , but also on the underlying algebra. More precisely, if is a subalgebra of , and if is the minimal polynomial of in , then may differ from , in which case we have .

In the second section we restrict attention to the case where is either the real or the complex numbers, and define , the radius of an element in , to be the largest root in absolute value of the minimal polynomial of . We show that possesses some of the familiar properties of the classical spectral radius. In particular, we prove that is a continuous function on .

In the third and last section, we deal with stability of subnorms acting on subsets of finite-dimensional power-associative algebras. Following a brief survey, we enhance our understanding of the subject with the help of our findings of the previous section. Our main new result states that if , a subset of an algebra , satisfies certain assumptions, and is a continuous subnorm on , then is stable on if and only if majorizes the radius defined above.

     

The average norm of polynomials of fixed height

Let be any integer and let

be the set of all polynomials of height 1 and degree . Let

Here is the power of the norm on the boundary of the unit disc. So is the average of the power of the norm over

In this paper we give exact formulae for for various values of . We also give a variety of related results for different classes of polynomials including polynomials of fixed height H, polynomials with coefficients and reciprocal polynomials. The results are surprisingly precise. Typical of the results we get is the following.

Theorem 0.1. For , we have

and

     

Rotation topological factors of minimal -actions on the Cantor set

In this paper we study conditions under which a free minimal -action on the Cantor set is a topological extension of the action of rotations, either on the product of -tori or on a single -torus . We extend the notion of linearly recurrent systems defined for -actions on the Cantor set to -actions, and we derive in this more general setting a necessary and sufficient condition, which involves a natural combinatorial data associated with the action, allowing the existence of a rotation topological factor of one of these two types.

     

Counting integral Lamé equations by means of dessins d'enfants

We obtain an explicit formula for the number of Lamé equations (modulo linear changes of variable) with index and projective monodromy group of order , for given and . This is done by performing the combinatorics of the `dessins d'enfants' associated to the Belyi covers which transform hypergeometric equations into Lamé equations by pull-back.

     

Lusternik-Schnirelmann category of

First we give an upper bound of , the L-S category of a principal -bundle for a connected compact group with a characteristic map . Assume that there is a cone-decomposition of in the sense of Ganea that is compatible with multiplication. Then we have for , if is compressible into with trivial higher Hopf invariant . Second, we introduce a new computable lower bound, for . The two new estimates imply , where is a category weight due to Rudyak and Strom.

     

Rigidity of smooth Schubert varieties in Hermitian symmetric spaces

In this paper we study the space of effective -cycles in with the homology class equal to an integral multiple of the homology class of Schubert variety of type . When is a proper linear subspace of a linear space in , we know that is already complicated. We will show that for a smooth Schubert variety in a Hermitian symmetric space, any irreducible subvariety with the homology class , , is again a Schubert variety of type , unless is a non-maximal linear space. In particular, any local deformation of such a smooth Schubert variety in Hermitian symmetric space is obtained by the action of the Lie group .

     

Bochner-Riesz means with respect to a rough distance function

The generalized Bochner-Riesz operator may be defined as

where is an appropriate distance function and is the inverse Fourier transform. The behavior of on is described for , a rough distance function. We conjecture that this operator is bounded on when and , and unbounded when . This conjecture is verified for large ranges of .

     

Classification of homomorphisms and dynamical systems

Let be a unital simple -algebra, with tracial rank zero and let be a compact metric space. Suppose that are two unital monomorphisms. We show that and are approximately unitarily equivalent if and only if

for every and every trace of Inspired by a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let be a compact metric space and let be two minimal homeomorphisms. Using the above-mentioned result, we show that two dynamical systems are approximately conjugate in that sense if and only if a -theoretical condition is satisfied. In the case that is the Cantor set, this notion coincides with the strong orbit equivalence of Giordano, Putnam and Skau, and the -theoretical condition is equivalent to saying that the associate crossed product -algebras are isomorphic.

Another application of the above-mentioned result is given for -dynamical systems related to a problem of Kishimoto. Let be a unital simple AH-algebra with no dimension growth and with real rank zero, and let We prove that if fixes a large subgroup of and has the tracial Rokhlin property, then is again a unital simple AH-algebra with no dimension growth and with real rank zero.

     

A generalization of half-plane mappings to the ball in

Let be a normalized (, ) biholomorphic mapping of the unit ball onto a convex domain that is the union of lines parallel to some unit vector . We consider the situation in which there is one infinite singularity of on . In one case with a simple change-of-variables, we classify all convex mappings of that are half-plane mappings in the first coordinate. In the more complicated case, when is not in the span of the infinite singularity, we derive a form of the mappings in dimension .

     

Analytic models for commuting operator tuples on bounded symmetric domains

For a domain in and a Hilbert space of analytic functions on which satisfies certain conditions, we characterize the commuting -tuples of operators on a separable Hilbert space  such that is unitarily equivalent to the restriction of to an invariant subspace, where is the operator -tuple on the Hilbert space tensor product  . For the unit disc and the Hardy space , this reduces to a well-known theorem of Sz.-Nagy and Foias; for a reproducing kernel Hilbert space on such that the reciprocal of its reproducing kernel is a polynomial in and  , this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces for which ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) on a Cartan domain corresponding to the parameter in the continuous Wallach set, and reproducing kernel Hilbert spaces for which is a rational function. Further, we treat also the more general problem when the operator is replaced by ,  being a certain generalization of a unitary operator tuple. For the case of the spaces on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on , which seems to be of an independent interest.

     

Topological dynamics on moduli spaces II

Let be an orientable genus 0$"> surface with boundary . Let be the mapping class group of fixing . The group acts on the space of -gauge equivalence classes of flat -connections on with fixed holonomy on . We study the topological dynamics of the -action and give conditions for the individual -orbits to be dense in .