-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 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 .

     

The Furuta inequality with negative powers

Let be bounded linear operators on a Hilbert space satisfying . Furuta showed the operator inequality
as long as positive real numbers satisfy and . In this paper, we show this inequality is valid if negative real numbers satisfy a certain condition. Also, we investigate the optimality of that condition.

     

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.

     

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.

     

The distribution of solutions of the congruence

For a cube of size , we obtain a lower bound on so that is nonempty, where is the algebraic subset of defined by

a positive integer and an integer not divisible by . For we obtain that is nonempty if , for we obtain that is nonempty if , and for we obtain that is nonempty if . Using the assumption of the Grand Riemann Hypothesis we obtain is nonempty if .

     

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 .

     

Spectral multiplier theorem for spaces associated with some Schrödinger operators

Let be the semigroup of linear operators generated by a Schrödinger operator , where is a nonnegative polynomial. We say that is an element of if the maximal function belongs to . A criterion on functions which implies boundedness of the operators on is given.

     

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.

     

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 .

     

Behaviour of holomorphic automorphisms on equicontinuous subsets of the space

Consider a compact Hausdorff topological space , a -triple and , the -triple of all continuous -valued functions with the pointwise operations and the norm of the supremum. Let be the group of all holomorphic automorphisms of the unit ball of that map every equicontinuous subset lying strictly inside into another such a set. The real Banach-Lie group and its Lie algebra are investigated. The identity connected component of is identified when has the strong Banach-Stone property. This extends to the infinite dimensional setting a well known result concerning the case .

     

Smith equivalence of representations for finite perfect groups

Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group , we compute a certain subgroup of the representation ring . This allows us to prove that a finite perfect group has a smooth -proper action on a sphere with isolated fixed points at which the tangent representations of are mutually nonisomorphic if and only if contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer and primes , we prove similar results for the group , , or . In particular, has Smith equivalent representations that are not isomorphic if and only if , , .

     

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).

     

Finite time blow-up for the inhomogeneous equation

We consider the inhomogeneous equation

where , and , and give criteria on , and which determine whether for all and all the solution blows up in finite time or whether for and sufficiently small, the solution exists for all time.

     

Maximal estimates for the -dimensional Walsh-Fourier series

The -dimensional dyadic martingale Hardy spaces are introduced and it is proved that the maximal operator of the means of a Walsh-Fourier series is bounded from to and is of weak type , provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the means of a function converge a.e. to the function in question. Moreover, we prove that the means are uniformly bounded on whenever . Thus, in case , the means converge to in norm. The same results are proved for the conjugate means, too.

     

Common fixed points of commuting holomorphic maps in the unit ball of

Let be the unit ball of (). We prove that if are holomorphic self-maps of such that , then and have a common fixed point (possibly at the boundary, in the sense of -limits). Furthermore, if and have no fixed points in , then they have the same Wolff point, unless the restrictions of and to the one-dimensional complex affine subset of determined by the Wolff points of and are commuting hyperbolic automorphisms of that subset.

     

Weyl's construction and tensor power decomposition for

Let be the 7-dimensional irreducible representations of . We decompose the tensor power into irreducible representations of and obtain all irreducible representations of in the decomposition. This generalizes Weyl's work on the construction of irreducible representations and decomposition of tensor products for classical groups to the exceptional group .

     

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 an analogue of Selberg's eigenvalue conjecture for

Let be the homogeneous space associated to the group
. Let where and consider the first nontrivial eigenvalue of the Laplacian on . Using geometric considerations, we prove the inequality . Since the continuous spectrum is represented by the band , our bound on can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.

     

On smoothness of carrying simplices

We consider dissipative strongly competitive systems of ordinary differential equations. It is known that for a wide class of such systems there exists an invariant attracting hypersurface , called the carrying simplex. In this note we give an amenable condition for to be a submanifold-with-corners. We also provide conditions, based on a recent work of M. Benaïm (On invariant hypersurfaces of strongly monotone maps, J. Differential Equations 136 (1997), 302-319), guaranteeing that is of class .