Finite-dimensional lattice-subspaces of

Let be linearly independent positive functions in , let be the vector subspace generated by the and let denote the curve of determined by the function , where . We establish that is a vector lattice under the induced ordering from if and only if there exists a convex polygon of with vertices containing the curve and having its vertices in the closure of the range of . We also present an algorithm which determines whether or not is a vector lattice and in case is a vector lattice it constructs a positive basis of . The results are also shown to be valid for general normed vector lattices.

     

Cohomological dimension and metrizable spaces. II

The main result of the first part of the paper is a generalization of the classical result of Menger-Urysohn : . Theorem. Suppose are subsets of a metrizable space and and are CW complexes. If is an absolute extensor for and is an absolute extensor for , then the join is an absolute extensor for .

As an application we prove the following analogue of the Menger-Urysohn Theorem for cohomological dimension: Theorem. Suppose are subsets of a metrizable space. Then

for any ring with unity and

for any abelian group .

The second part of the paper is devoted to the question of existence of universal spaces: Theorem. Suppose is a sequence of CW complexes homotopy dominated by finite CW complexes. Then

a.

Given a separable, metrizable space such that , , there exists a metrizable compactification of such that , .

b.

There is a universal space of the class of all compact metrizable spaces such that for all .

c.

There is a completely metrizable and separable space such that for all with the property that any completely metrizable and separable space with for all embeds in as a closed subset.

     

Topological entropy of standard type monotone twist maps

We study invariant measures of families of monotone twist maps with periodic Morse potential . We prove that there exist a constant such that the topological entropy satisfies . In particular, for . We show also that there exist arbitrary large such that has nonuniformly hyperbolic invariant measures with positive metric entropy. For large , the measures are hyperbolic and, for a class of potentials which includes , the Lyapunov exponent of the map with invariant measure grows monotonically with .

     

Relatively free invariant algebras of finite reflection groups

Let be a finite subgroup of is a field of characteristic and acting by linear substitution on a relatively free algebra of a variety of unitary associative algebras. The algebra of invariants is relatively free if and only if is a pseudo-reflection group and contains the polynomial

     

On Jacobian Ideals Invariant by a Reducible Action

This paper deals with a reducible action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau Conjecture: suppose that acts on the formal power series ring via . Then modulo some one dimensional representations where is an irreducible representation of dimension or empty set and . Unlike classical invariant theory which deals only with irreducible action and 1--dimensional representations, we treat the reducible action and higher dimensional representations succesively.

     

Extremal functions for Moser's inequality

Let be a bounded smooth domain in , and a function with compact support in . Moser's inequality states that there is a constant , depending only on the dimension , such that

where is the Lebesgue measure of , and the surface area of the unit ball in . We prove in this paper that there are extremal functions for this inequality. In other words, we show that the

is attained. Earlier results include Carleson-Chang (1986, is a ball in any dimension) and Flucher (1992, is any domain in 2-dimensions).

     

Noncomplete linear systems on abelian varieties

Let be a smooth projective variety. Every embedding is the linear projection of an embedding defined by a complete linear system. In this paper the geometry of such not necessarily complete embeddings is investigated in the special case of abelian varieites. To be more precise, the properties of complete embeddings are extended to arbitrary embeddings, and criteria for these properties to be satisfied are elaborated. These results are applied to abelian varieties. The main result is: Let be a general polarized abelian variety of type and , such that is even, and . The general subvector space of codimension satisfies the property .

     

Further Nice Equations for Nice Groups

Nice sextinomial equations are given for unramified coverings of the affine line in nonzero characteristic with P and as Galois groups where is any integer and is any power of .

     

Defect zero blocks for finite simple groups

We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a -block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero blocks remained unclassified were the alternating groups . Here we show that these all have a -block with defect 0 for every prime . This follows from proving the same result for every symmetric group , which in turn follows as a consequence of the -core partition conjecture, that every non-negative integer possesses at least one -core partition, for any . For , we reduce this problem to Lagrange's Theorem that every non-negative integer can be written as the sum of four squares. The only case with , that was not covered in previous work, was the case . This we prove with a very different argument, by interpreting the generating function for -core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne's Theorem (née the Weil Conjectures). We also consider congruences for the number of -blocks of , proving a conjecture of Garvan, that establishes certain multiplicative congruences when . By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime and positive integer , the number of blocks with defect 0 in is a multiple of for almost all . We also establish that any given prime divides the number of modularly irreducible representations of , for almost all .

     

Covering the integers by arithmetic sequences. II

Let ( be a system of arithmetic sequences where and . For system will be called an (exact) -cover of if every integer is covered by at least (exactly) times. In this paper we reveal further connections between the common differences in an (exact) -cover of and Egyptian fractions. Here are some typical results for those -covers of : (a) For any there are at least positive integers in the form where . (b) When (, either or , and for each positive integer the binomial coefficient can be written as the sum of some denominators of the rationals if forms an exact -cover of . (c) If is not an -cover of , then have at least distinct fractional parts and for each there exist such that (mod 1). If forms an exact -cover of with or () then for every and there is an such that (mod 1).

     

Immersed

We give two congruence formulas concerning the number of non-trivial double point circles and arcs of a smooth map with generic singularities --- the Whitney umbrellas --- of an -manifold into , which generalize the formulas by Szücs for an immersion with normal crossings. Then they are applied to give a new geometric proof of the congruence formula due to Mahowald and Lannes concerning the normal Euler number of an immersed -manifold in . We also study generic projections of an embedded -manifold in into and prove an elimination theorem of Whitney umbrella points of opposite signs, which is a direct generalization of a recent result of Carter and Saito concerning embedded surfaces in . The problem of lifting a map into to an embedding into is also studied.

     

Regularity and Algebras of Analytic Functions in Infinite Dimensions

A Banach space is known to be Arens regular if every continuous linear mapping from to is weakly compact. Let be an open subset of , and let denote the algebra of analytic functions on which are bounded on bounded subsets of lying at a positive distance from the boundary of We endow with the usual Fréchet topology. denotes the set of continuous homomorphisms . We study the relation between the Arens regularity of the space and the structure of .

     

Fine structure of the space of spherical minimal immersions

The space of congruence classes of full spherical minimal immersions of a given source dimension and algebraic degree is a compact convex body in a representation space of the special orthogonal group . In Ann. of Math. 93 (1971), 43--62 DoCarmo and Wallach gave a lower bound for and conjectured that the estimate was sharp. Toth resolved this ``exact dimension conjecture' positively so that all irreducible components of became known. The purpose of the present paper is to characterize each irreducible component of in terms of the spherical minimal immersions represented by the slice . Using this geometric insight, the recent examples of DeTurck and Ziller are located within .

     

Packing dimension and Cartesian products

We show that for any analytic set in , its packing dimension can be represented as , where the supremum is over all compact sets in , and denotes Hausdorff dimension. (The lower bound on packing dimension was proved by Tricot in 1982.) Moreover, the supremum above is attained, at least if . In contrast, we show that the dual quantity , is at least the ``lower packing dimension' of , but can be strictly greater. (The lower packing dimension is greater than or equal to the Hausdorff dimension.)

     

Lévy group action and invariant measures on

For let be defined by . We investigate permutations of , which satisfy as with for (i.e. is in the Lévy group , or for in the subspace of Cesàro-summable sequences. Our main interest are -invariant means on or equivalently -invariant probability measures on . We show that the adjoint of maps measures supported in onto a weak*-dense subset of the space of -invariant measures. We investigate the dynamical system and show that the support set of invariant measures on is the closure of the set of almost periodic points and the set of non-topologically transitive points in . Finally we consider measures which are invariant under .

     

The Morse spectrum of linear flows on vector bundles

For a linear flow on a vector bundle a spectrum can be defined in the following way: For a chain recurrent component on the projective bundle consider the exponential growth rates associated with (finite time) -chains in , and define the Morse spectrum over as the limits of these growth rates as and . The Morse spectrum of is then the union over all components . This spectrum is a synthesis of the topological approach of Selgrade and Salamon/Zehnder with the spectral concepts based on exponential growth rates, such as the Oseledec spectrum or the dichotomy spectrum of Sacker/Sell. It turns out that contains all Lyapunov exponents of for arbitrary initial values, and the are closed intervals, whose boundary points are actually Lyapunov exponents. Using the fact that is cohomologous to a subflow of a smooth linear flow on a trivial bundle, one can prove integral representations of all Morse and all Lyapunov exponents via smooth ergodic theory. A comparison with other spectral concepts shows that, in general, the Morse spectrum is contained in the topological spectrum and the dichotomy spectrum, but the spectral sets agree if the induced flow on the base space is chain recurrent. However, even in this case, the associated subbundle decompositions of may be finer for the Morse spectrum than for the dynamical spectrum. If one can show that the (closure of the) Floquet spectrum (i.e. the Lyapunov spectrum based on periodic trajectories in ) agrees with the Morse spectrum, then one obtains equality for the Floquet, the entire Oseledec, the Lyapunov, and the Morse spectrum. We present an example (flows induced by vector fields with hyperbolic chain recurrent components on the projective bundle) where this fact can be shown using a version of Bowen's Shadowing Lemma.

     

Package deal theorems and splitting orders in dimension 1

Let be a module-finite algebra over a commutative noetherian ring of Krull dimension 1. We determine when a collection of finitely generated modules over the localizations , at maximal ideals of , is the family of all localizations of a finitely generated -module . When is semilocal we also determine which finitely generated modules over the -adic completion of are completions of finitely generated -modules.

If is an -order in a semisimple artinian ring, but not contained in a maximal such order, several of the basic tools of integral representation theory behave differently than in the classical situation. The theme of this paper is to develop ways of dealing with this, as in the case of localizations and completions mentioned above. In addition, we introduce a type of order called a ``splitting order' of that can replace maximal orders in many situations in which maximal orders do not exist.

     

Abstract functions with continuous differences and Namioka spaces

Let be a semigroup and a topological space. Let be an Abelian topological group. The right differences of a function are defined by for . Let be continuous at the identity of for all in a neighbourhood of . We give conditions on or range under which is continuous for any topological space . We also seek conditions on under which we conclude that is continuous at for arbitrary . This led us to introduce new classes of semigroups containing all complete metric and locally countably compact quasitopological groups. In this paper we study these classes and explore their relation with Namioka spaces.

     

The fixed-point property for simply connected plane continua

We answer a question of R. Ma\'{n}ka by proving that every simply-connected plane continuum has the fixed-point property. It follows that an arcwise-connected plane continuum has the fixed-point property if and only if its fundamental group is trivial. Let be a plane continuum with the property that every simple closed curve in bounds a disk in . Then every map of that sends each arc component into itself has a fixed point. Hence every deformation of has a fixed point. These results are corollaries to the following general theorem. If is a plane continuum, is a decomposition of , and each element of is simply connected, then every map of that sends each element of into itself has a fixed point.

     

On polarized surfaces

Let be a smooth projective surface over and an ample Cartier divisor on . If the Kodaira dimension or , the author proved , where . If , then the author studied with . In this paper, we study the polarized surface with , , and .