Scrambled sets of continuous maps of 1-dimensional polyhedra

Let be a 1-dimensional simplicial complex in without isolated vertexes, be the polyhedron of with the metric induced by , and be a continuous map. In this paper we prove that if is finite, then the interior of every scrambled set of in is empty. We also show that if is an infinite complex, then there exist continuous maps from to itself having scrambled sets with nonempty interiors, and if or , then there exist maps of with the whole space being a scrambled set.

     

Galois coverings of selfinjective algebras by repetitive algebras

In the representation theory of selfinjective artin algebras an important role is played by selfinjective algebras of the form where is the repetitive algebra of an artin algebra and is an admissible group of automorphisms of . If is of finite global dimension, then the stable module category of finitely generated -modules is equivalent to the derived category of bounded complexes of finitely generated -modules. For a selfinjective artin algebra , an ideal and , we establish a criterion for to admit a Galois covering with an infinite cyclic Galois group . As an application we prove that all selfinjective artin algebras whose Auslander-Reiten quiver has a non-periodic generalized standard translation subquiver closed under successors in are socle equivalent to the algebras , where is a representation-infinite tilted algebra and is an infinite cyclic group of automorphisms of .

     

Overgroups of irreducible linear groups, II

Determining the subgroup structure of algebraic groups (over an algebraically closed field of arbitrary characteristic) often requires an understanding of those instances when a group and a closed subgroup both act irreducibly on some module , which is rational for and . In this paper and in Overgroups of irreducible linear groups, I (J. Algebra 181 (1996), 26-69), we give a classification of all such triples when is a non-connected algebraic group with simple identity component , is an irreducible -module with restricted -high weight(s), and is a simple algebraic group of classical type over sitting strictly between and .

     

Embeddings of open manifolds

Let be the simplicial group of homeomorphisms of . The following theorems are proved.

Theorem A. Let be a topological manifold of dim 5 with a finite number of tame ends , . Let be the simplicial group of end preserving homeomorphisms of . Let be a periodic neighborhood of each end in , and let be manifold approximate fibrations. Then there exists a map such that the homotopy fiber of is equivalent to , the simplicial group of homeomorphisms of which have compact support.

Theorem B. Let be a compact topological manifold of dim 5, with connected boundary , and denote the interior of by . Let be the restriction map and let be the homotopy fiber of over . Then is isomorphic to for , where is the concordance space of .

Theorem C. Let be a manifold approximate fibration with dim 5. Then there exist maps and for , such that , where is a compact and connected manifold and is the infinite cyclic cover of .

     

On the depth of the tangent cone and the growth of the Hilbert function

For a dimensional Cohen-Macaulay local ring we study the depth of the associated graded ring of with respect to an -primary ideal in terms of the Vallabrega-Valla conditions and the length of , where is a minimal reduction of and . As a corollary we generalize Sally's conjecture on the depth of the associated graded ring with respect to a maximal ideal to -primary ideals. We also study the growth of the Hilbert function.

     

Isoperimetric Estimates on Sierpinski Gasket Type Fractals

For a compact Hausdorff space that is pathwise connected, we can define the connectivity dimension to be the infimum of all such that all points in can be connected by a path of Hausdorff dimension at most . We show how to compute the connectivity dimension for a class of self-similar sets in that we call point connected, meaning roughly that is generated by an iterated function system acting on a polytope such that the images of intersect at single vertices. This class includes the polygaskets, which are obtained from a regular -gon in the plane by contracting equally to all vertices, provided is not divisible by 4. (The Sierpinski gasket corresponds to .) We also provide a separate computation for the octogasket (), which is not point connected. We also show, in these examples, that , where the infimum is taken over all paths connecting and , and denotes Hausdorff measure, is equivalent to the original metric on . Given a compact subset of the plane of Hausdorff dimension and connectivity dimension , we can define the isoperimetric profile function to be the supremum of , where is a region in the plane bounded by a Jordan curve (or union of Jordan curves) entirely contained in , with . The analog of the standard isperimetric estimate is . We are particularly interested in finding the best constant and identifying the extremal domains where we have equality. We solve this problem for polygaskets with . In addition, for we find an entirely different estimate for as , since the boundary of has infinite measure. We find that the isoperimetric profile function is discontinuous, and that the extremal domains have relatively simple polygonal boundaries. We discuss briefly the properties of minimal paths for the Sierpinski gasket, and the isodiametric problem in the intrinsic metric.

     

Extendability of Large-Scale Lipschitz Maps

Let be metric spaces, a subset of , and a large-scale lipschitz map. It is shown that possesses a large-scale lipschitz extension (with possibly larger constants) if is a Gromov hyperbolic geodesic space or the cartesian product of finitely many such spaces. No extension exists, in general, if is an infinite-dimensional Hilbert space. A necessary and sufficient condition for the extendability of a lipschitz map is given in the case when is separable and is a proper, convex geodesic space.

     

Conjugacy Classes of

Let be a quadratic extension of a global field , of characteristic not two, and the integral closure in of a Dedekind ring of -integers in . Then is isomorphic to the spinorial kernel for an indefinite quadratic -lattice of rank 4. The isomorphism is used to study the conjugacy classes of unitary groups of primitive odd binary hermitian matrices under the action of .

     

Minimal lattice-subspaces

In this paper the existence of minimal lattice-subspaces of a vector lattice containing a subset of is studied (a lattice-subspace of is a subspace of which is a vector lattice in the induced ordering). It is proved that if there exists a Lebesgue linear topology on and is -closed (especially if is a Banach lattice with order continuous norm), then minimal lattice-subspaces with -closed positive cone exist (Theorem 2.5).

In the sequel it is supposed that is a finite subset of , where is a compact, Hausdorff topological space, the functions are linearly independent and the existence of finite-dimensional minimal lattice-subspaces is studied. To this end we define the function where . If is the range of and the convex hull of the closure of , it is proved:

(i)

There exists an -dimensional minimal lattice-subspace containing if and only if is a polytope of with vertices (Theorem 3.20).

(ii)

The sublattice generated by is an -dimensional subspace if and only if the set contains exactly points (Theorem 3.7).

This study defines an algorithm which determines whether a finite-dimensional minimal lattice-subspace (sublattice) exists and also determines these subspaces.

     

Examples of Möbius-like groups which are not Möbius groups

In this paper we give two basic constructions of groups with the following properties:

(a)

, i.e., the group is acting by orientation preserving homeomorphisms on ;

(b)

every element of is Möbius-like;

(c)

, where denotes the limit set of ;

(d)

is discrete;

(e)

is not a conjugate of a Möbius group.

Both constructions have the same basic idea (inspired by Denjoy): we start with a Möbius group (of a certain type) and then we change the underlying circle upon which acts by inserting some closed intervals and then extending the group action over the new circle. We denote this new action by . Now we form a new group which is generated by all of and an additional element whose existence is enabled by the inserted intervals. This group has all the properties (a) through (e).

     

On the number of terms in the middle of almost split sequences over tame algebras

Let be a finite dimensional tame algebra over an algebraically closed field . It has been conjectured that any almost split sequence with indecomposable modules has and in case , then exactly one of the is a projective-injective module. In this work we show this conjecture in case all the are directing modules, that is, there are no cycles of non-zero, non-iso maps between indecomposable -modules. In case, and are isomorphic, we show that and give precise information on the structure of .

     

On the non-vanishing of cubic twists of automorphic -series

Let be a normalised new form of weight for over and , its base change lift to . A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the -function of . There is an algorithm to check the condition for any given form. The new form of level is used to illustrate our method.

     

Building blocks for Quadratic Julia sets

We obtain results on the structure of the Julia set of a quadratic polynomial with an irrationally indifferent fixed point in the iterative dynamics of . In the Cremer point case, under the assumption that the Julia set is a decomposable continuum, we obtain a building block structure theorem for the corresponding Julia set : there exists a nowhere dense subcontinuum such that , is the union of the impressions of a minimally invariant Cantor set of external rays, contains the critical point, and contains both the Cremer point and its preimage. In the Siegel disk case, under the assumption that no impression of an external ray contains the boundary of the Siegel disk, we obtain a similar result. In this case contains the boundary of the Siegel disk, properly if the critical point is not in the boundary, and contains no periodic points. In both cases, the Julia set is the closure of a skeleton which is the increasing union of countably many copies of the building block joined along preimages of copies of a critical continuum containing the critical point. In addition, we prove that if is any polynomial of degree with a Siegel disk which contains no critical point on its boundary, then the Julia set is not locally connected. We also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected Julia set have homeomorphic Julia sets.

     

Golubev series for solutions of elliptic equations

Let be an elliptic system with real analytic coefficients on an open set and let be a fundamental solution of Given a locally connected closed set we fix some massive measure on . Here, a non-negative measure is called massive, if the conditions and imply that We prove that, if is a solution of the equation in then for each relatively compact open subset of and every there exist a solution of the equation in and a sequence () in satisfying such that for This complements an earlier result of the second author on representation of solutions outside a compact subset of

     

Homogeneous projective varieties with degenerate secants

The secant variety of a projective variety in , denoted by , is defined to be the closure of the union of lines in passing through at least two points of , and the secant deficiency of is defined by . We list the homogeneous projective varieties with under the assumption that arise from irreducible representations of complex simple algebraic groups. It turns out that there is no homogeneous, non-degenerate, projective variety with and , and the -variety is the only homogeneous projective variety with largest secant deficiency . This gives a negative answer to a problem posed by R. Lazarsfeld and A. Van de Ven if we restrict ourselves to homogeneous projective varieties.

     

Witten-Helffer-Sjöstrand theory for -equivariant cohomology

Given an -invariant Morse function and an -invariant Riemannian metric , a family of finite dimensional subcomplexes , , of the Witten deformation of the -equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian as . In fact the spectrum of can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains as the complex of eigenforms corresponding to the small eigenvalues of . This permits us to verify the -equivariant Morse inequalities. Moreover suppose is self-indexing and satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as to a geometric complex which is induced by and calculates the -equivariant cohomology of .

     

An algorithm for calculating the Nielsen number on surfaces with boundary

Let be a self-map of a hyperbolic surface with boundary. The Nielsen number, , depends only on the induced map of the fundamental group, which can be viewed as a free group on generators, . We determine conditions for fixed points to be in the same fixed point class and if these conditions are enough to determine the fixed point classes, we say that is -characteristic. We define an algebraic condition on the and show that ``most' maps satisfy this condition and that all maps which satisfy this condition are -characteristic. If is -characteristic, we present an algorithm for calculating and prove that the inequality holds, where denotes the Lefschetz number of and the Euler characteristic of , thus answering in part a question of Jiang and Guo.

     

The problem on domains with piecewise smooth boundaries with applications

Let be a bounded domain in such that has piecewise smooth boudnary. We discuss the solvability of the Cauchy-Riemann equation

where is a smooth -closed form with coefficients up to the bundary of , and . In particular, Equation (0.1) is solvable with smooth up to the boundary (for appropriate degree if satisfies one of the following conditions:

i)

is the transversal intersection of bounded smooth pseudoconvex domains.

ii)

where is the union of bounded smooth pseudoconvex domains and is a pseudoconvex convex domain with a piecewise smooth boundary.

iii)

where is the intersection of bounded smooth pseudoconvex domains and is a pseudoconvex domain with a piecewise smooth boundary.

The solvability of Equation (0.1) with solutions smooth up to the boundary can be used to obtain the local solvability for on domains with piecewise smooth boundaries in a pseudoconvex manifold.

     

Capacity convergence results and applications to a Berstein-Markov inequality

Given a sequence of Borel subsets of a given non-pluripolar Borel set in the unit ball in with , we show that the relative capacities converge to if and only if the relative (global) extremal functions () converge pointwise to (). This is used to prove a sufficient mass-density condition on a finite positive Borel measure with compact support in guaranteeing that the pair satisfy a Bernstein-Markov inequality. This implies that the orthonormal polynomials associated to may be used to recover the global extremal function .

     

The maximality of the core model

Our main results are: 1) every countably certified extender that coheres with the core model is on the extender sequence of , 2) computes successors of weakly compact cardinals correctly, 3) every model on the maximal 1-small construction is an iterate of , 4) (joint with W. J. Mitchell) is universal for mice of height whenever , 5) if there is a such that is either a singular countably closed cardinal or a weakly compact cardinal, and fails, then there are inner models with Woodin cardinals, and 6) an -Erdös cardinal suffices to develop the basic theory of .