Helly-type theorems for hollow axis-aligned boxes

A hollow axis-aligned box is the boundary of the cartesian product of compact intervals in . We show that for , if any of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has non-empty intersection; and if any of a collection of hollow axis-aligned rectangles in have non-empty intersection, then the whole collection has non-empty intersection. The values for and for are the best possible in general. We also characterize the collections of hollow boxes which would be counterexamples if were lowered to , and to , respectively.

     

On the excess of sets of complex exponentials

For let be complex numbers such that is bounded. For define , where . Then the excesses in the sense of Paley and Wiener satisfy .

     

A characterization of total reflection orders

Let be a Coxeter system with set of reflections . It is known that if is a total reflection order for , then, for each , and its complement are stable under conjugation by . Moreover the upper and lower -conjugates of are still total reflection orders. For any total order on , say that is stable if is stable under conjugation by for each . We prove that if and all orders obtained from by successive lower or upper -conjugations are stable, then is a total reflection order.

     

Remarkable asymmetric random walks

There exists an asymmetric probability measure on the real line with for every . can be chosen absolutely continuous and can be chosen to be concentrated on the integers. In both cases, can be chosen to have moments of every order, but cannot be determined by its moments.

     

New facts

We use ``iterated square sequences' to show that there is an -definable partition such that if is an inner model not containing :

(a)

For some is stationary.

(b)

For each there is a generic extension of in which does not exist and is non-stationary.

This result is then applied to show that if is an inner model without , then some sentence not true in can be forced over .

     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

On the irrationality of a certain series

We prove that if is an integer greater than one, and are any positive rationals such that for all integers , then

is irrational and is not a Liouville number.

     

Catenarity in module-finite algebras

The main theorem says that any module-finite (but not necessarily commutative) algebra over a commutative Noetherian universally catenary ring is catenary. Hence the ring is catenary if is Cohen-Macaulay. When is local and is a Cohen-Macaulay -module, we have that is a catenary ring, for any , and the equality holds true for any pair of prime ideals in and for any saturated chain of prime ideals between and .

     

Open covers and partition relations

An open cover of a topological space is said to be an -cover if there is for each finite subset of the space a member of the cover which contains the finite set, but the space itself is not a member of the cover. We prove theorems which imply that a set of real numbers has Rothberger's property if, and only if, for each positive integer , for each -cover of , and for each function from the two-element subsets of , there is a subset of such that is constant on , and each element of belongs to infinitely many elements of (Theorem 1). A similar characterization is given of Menger's property for sets of real numbers (Theorem 6).

     

The location of the zeros of the higher order derivatives of a polynomial

Let be a complex polynomial of degree having zeros in a disk . We deal with the problem of finding the smallest concentric disk containing zeros of . We obtain some estimates on the radius of this disk in general as well as in the special case, where zeros in are isolated from the other zeros of . We indicate an application to the root-finding algorithms.

     

Cyclic torsion of elliptic curves

Let be an elliptic curve over a number field such that
and let denote the number of roots of unity in . Ross proposed a question: Is isogenous over to an elliptic curve such that is cyclic of order dividing ? A counter-example of this question is given. We show that is isogenous to such that . In case has complex multiplication and , we obtain certain criteria whether or not is isogenous to such that .

     

Some Lie superalgebras associated to the Weyl algebras

Let be the Lie superalgebra . We show that there is a surjective homomorphism from to the Weyl algebra , and we use this to construct an analog of the Joseph ideal. We also obtain a decomposition of the adjoint representation of on and use this to show that if is made into a Lie superalgebra using its natural -grading, then . In addition, we show that if and are isomorphic as Lie superalgebras, then . This answers a question of S. Montgomery.

     

Extremal points of a functional on the set of convex functions

We investigate the extremal points of a functional , for a convex or concave function . The admissible functions are convex themselves and satisfy a condition . We show that the extremal points are exactly and if these functions are convex and coincide on the boundary . No explicit regularity condition is imposed on , , or . Subsequently we discuss a number of extensions, such as the case when or are non-convex or do not coincide on the boundary, when the function also depends on , etc.

     

Existence of unliftable modules

Let be a commutative Noetherian local ring, and let where is a non-zerodivisor of contained in . Then a finitely generated -module is said to lift to if there exists a finitely generated -module such that is -regular and . In this paper we give a general construction of finitely generated -modules of finite projective dimension over which fail to lift to provided and the depth of is at least 2.

     

On a theorem of Scott and Swarup

Let be an exact sequence of hyperbolic groups induced by an automorphism of the free group . Let be a finitely generated distorted subgroup of . Then there exist and a free factor of such that the conjugacy class of is preserved by and contains a finite index subgroup of a conjugate of . This is an analog of a theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.

     

A central limit theorem for Markov chains and applications to hypergroups

Let be a homogeneous Markov chain on an unbounded Borel subset of with a drift function which tends to a limit at infinity. Under a very simple hypothesis on the chain we prove that converges in distribution to a normal law where the variance depends on the asymptotic behaviour of . When goes to zero quickly enough and , the random centering may be replaced by These results are applied to the case of random walks on some hypergroups.

     

On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions

We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem on ; on , where is a bounded region in , is an indefinite weight function and may be positive, negative or zero.

     

The Weierstrass approximation theorem and a characterization of the unit circle

We study real algebraic morphisms from nonsingular real algebraic varieties with into nonsingular real algebraic curves . We show, among other things, that the set of real algebraic morphisms from into is never dense in the space of all maps from into , unless is biregularly isomorphic to a Zariski open subset of the unit circle.

     

Generating sets for compact semisimple Lie groups

Let be a compact connected semisimple Lie group. We prove that the subset of consisting of pairs which topologically generate is Zariski open.

     

Uniqueness of the least-energy solution for a semilinear Neumann problem

We prove that the least-energy solution of the problem

where is a ball, and if , if , is unique (up to rotation) if is small enough.