Invertibility preserving linear maps and algebraic reflexivity of elementary operators of length one

Let and be real or complex Banach spaces. We show that a surjective linear map preserving invertibility in both directions is either of the form or the form , where , , , and are bounded invertible linear operators. As an application we improve a result of Larson and Sourour on algebraic reflexivity of elementary operators of length one.

     

Radicals and Plotkin's problem concerning geometrically equivalent groups

If and are groups and is a normal subgroup of , then the -closure of in is the normal subgroup of . In particular, is the -radical of . Plotkin calls two groups and geometrically equivalent, written , if for any free group of finite rank and any normal subgroup of the -closure and the -closure of in are the same. Quasi-identities are formulas of the form for any words in a free group. Generally geometrically equivalent groups satisfy the same quasi-identities. Plotkin showed that nilpotent groups and satisfy the same quasi-identities if and only if and are geometrically equivalent. Hence he conjectured that this might hold for any pair of groups. We provide a counterexample.

     

Van der Waerden spaces

A topological space is van der Waerden if for every sequence in there exists a converging subsequence so that contains arbitrarily long finite arithmetic progressions. Not every sequentially compact space is van der Waerden. The product of two van der Waerden spaces is van der Waerden.

The following condition on a Hausdorff space is sufficent for to be van der Waerden:

The closure of every countable set in is compact and first-countable.

A Hausdorff space that satisfies satisfies, in fact, a stronger property: for every sequence in :

There exists so that is converging, and contains arbitrarily long finite arithmetic progressions and sets of the form for arbitrarily large finite sets .

There are nonmetrizable and noncompact spaces which satisfy . In particular, every sequence of ordinal numbers and every bounded sequence of real monotone functions on satisfy .

     

On the absolute continuity of a class of invariant measures

Let be a compact connected subset of , let , be contractive self-conformal maps on a neighborhood of , and let be a family of positive continuous functions on . We consider the probability measure that satisfies the eigen-equation

for some 0$">. We prove that if the attractor is an -set and is absolutely continuous with respect to , the Hausdorff -dimensional measure restricted on the attractor , then is absolutely continuous with respect to (i.e., they are equivalent). A special case of the result was considered by Mauldin and Simon (1998). In another direction, we also consider the -property of the Radon-Nikodym derivative of and give a condition for which is unbounded.

     

On the maximal inequalities for martingales involving two functions

Let and be nonnegative convex functions, and let and be the right continuous derivatives of and respectively. In this paper, we prove the equivalence of the following three conditions: (i) (ii) and (iii) s_0,$">where and are the Orlicz martingale spaces. As a corollary, we get a sufficient and necessary condition under which the extension of Doob's inequality holds. We also discuss the converse inequalities.

     

-Weyl's theorem for operator matrices

If is a upper triangular matrix on the Hilbert space , then -Weyl's theorem for and need not imply -Weyl's theorem for , even when . In this note we explore how -Weyl's theorem and -Browder's theorem survive for operator matrices on the Hilbert space.

     

Subgroup growth in some pro- groups

For a group let be the number of subgroups of index and let be the number of normal subgroups of index . We show that for 2$">. If and does not divide or if and or , we show that for all sufficiently large . On the other hand if and divides , then is not even bounded as a function of .

     

Numerical bounds for critical exponents of crossing Brownian motion

We consider -dimensional crossing Brownian motion in a truncated Poissonian potential conditioned to reach a fixed hyperplane at distance from the starting point. The transverse fluctuation of the path is expected to be of order . We prove that for : . As a second critical exponent we introduce , which describes the fluctuations of naturally defined distance functions for crossing Brownian motion. The numerical bound we obtain is an improvement of Corollary 3.1 in Scaling identity for crossing Brownian motion in a Poissonian potential (Probab. Theory Related Fields 112 (1998), 299-319), resulting in if and if the killing rate is strictly positive (0$">).

     

Free products in linear groups

Let be a commutative integral domain of characteristic , and let be a finite subgroup of , the projective general linear group of degree over . In this note, we show that if , then also contains the free product , where is the infinite cyclic group generated by the image of a suitable transvection.

     

The globally irreducible representations of symmetric groups

Let be an algebraic number field and be the ring of integers of . Let be a finite group and be a finitely generated torsion free -module. We say that is a globally irreducible -module if, for every maximal ideal of , the -module is irreducible, where stands for the residue field .

Answering a question of Pham Huu Tiep, we prove that the symmetric group does not have non-trivial globally irreducible modules. More precisely we establish that if is a globally irreducible -module, then is an -module of rank with the trivial or sign action of .

     

Strongly representable atom structures of relation algebras

A relation algebra atom structure is said to be strongly representable if all atomic relation algebras with that atom structure are representable. This is equivalent to saying that the complex algebra is a representable relation algebra. We show that the class of all strongly representable relation algebra atom structures is not closed under ultraproducts and is therefore not elementary. This answers a question of Maddux (1982).

Our proof is based on the following construction. From an arbitrary undirected, loop-free graph , we construct a relation algebra atom structure and prove, for infinite , that is strongly representable if and only if the chromatic number of is infinite. A construction of Erdös shows that there are graphs () with infinite chromatic number, with a non-principal ultraproduct whose chromatic number is just two. It follows that is strongly representable (each ) but is not.

     

The homogeneous spectrum of a graded commutative ring

Suppose is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a -graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring have Noetherian spectrum. If , we show that has Noetherian spectrum, while for each we establish existence of an example where the homogeneous spectrum of is not Noetherian.

     

On the total curvature of convex hypersurfaces in hyperbolic spaces

Let be two convex compact subsets of the hyperbolic space with smooth boundary. It is shown that the total curvature of the hypersurface is larger than the total curvature of .

     

Composition of inner mappings on the ball

Suppose that is an inner map and that . We show that the identity

holds with an abstract boundary value . If the natural compatibility condition is satisfied, then . Here, denotes the image of the surface measure on under . In particular, is inner if and are inner and . Furthermore, we characterize the boundedness of composition operators on Hardy spaces in terms of the absolute continuity of .

     

A global pinching theorem for surfaces with constant mean curvature in

Let be a compact immersed surface in the unit sphere with constant mean curvature . Denote by the linear map from into , , where is the linear map associated to the second fundamental form and is the identity map. Let denote the square of the length of . We prove that if , then is either totally umbilical or an -torus, where is a constant depending only on the mean curvature .

     

Exactness of one relator groups

A discrete group is -exact if the reduced crossed product with converts a short exact sequence of --algebras into a short exact sequence of -algebras. A one relator group is a discrete group admitting a presentation where is a countable set and is a single word over . In this short paper we prove that all one relator discrete groups are -exact. Using the Bass-Serre theory we also prove that a countable discrete group acting without inversion on a tree is -exact if the vertex stabilizers of the action are -exact.

     

Lightface -indescribable cardinals

-absoluteness for forcing means that for any forcing , . `` inaccessible to reals' means that for any real , . To measure the exact consistency strength of `` -absoluteness for forcing and is inaccessible to reals', we introduce a weak version of a weakly compact cardinal, namely, a (lightface) -indescribable cardinal; has this property exactly if it is inaccessible and .

     

Julia sets converging to the unit disk

We consider the family of rational maps , where and is small. If is equal to 0, the limiting map is and the Julia set is the unit circle. We investigate the behavior of the Julia sets of when tends to 0, obtaining two very different cases depending on and . The first case occurs when ; here the Julia sets of converge as sets to the closed unit disk. In the second case, when one of or is larger than , there is always an annulus of some fixed size in the complement of the Julia set, no matter how small is.

     

A generalized Kolmogorov inequality for the Hilbert transform

If we can define the Hilbert transform almost everywhere (Lebesgue) and obtain an estimate for where is a suitable finite measure. The classical Kolmogorov inequality for the Lebesgue measure of is obtained by a scaling argument.

     

Isomorphisms of subalgebras of nest algebras

Let be a subalgebra of a nest algebra . If contains all rank one operators in , then is said to be large; if the set of rank one operators in coincides with that in the Jacobson radical of , is said to be radical-type. In this paper, algebraic isomorphisms of large subalgebras and of radical-type subalgebras are characterized. Let be a nest of subspaces of a Hilbert space and be a subalgebra of the nest algebra associated to (). Let be an algebraic isomorphism from onto . It is proved that is spatial if one of the following occurs: (1) () is large and contains a masa; (2) () is large and closed; (3) () is a closed radical-type subalgebra and ( is quasi-continuous (i.e. the trivial elements of are limit points); (4) () is large and one of and is not quasi-continuous.