Convex curves, Radon transforms and convolution operators defined by singular measures

Let be a convex curve in the plane and let be the arc-length measure of Let us rotate by an angle and let be the corresponding measure. Let . Then This is optimal for an arbitrary . Depending on the curvature of , this estimate can be improved by introducing mixed-norm estimates of the form where and are conjugate exponents.     

On the Lipschitz classification of normed spaces, unit balls, and spheres

For every normed space , we note its closed unit ball and unit sphere by and , respectively. Let and be normed spaces such that is Lipschitz homeomorphic to , and is Lipschitz homeomorphic to .

We prove that the following are equivalent:

1. is Lipschitz homeomorphic to .

2. is Lipschitz homeomorphic to .

3. is Lipschitz homeomorphic to .

This result holds also in the uniform category, except (2 or 3) 1 which is known to be false.

     

Restrictions of bounded linear operators: Closed range

Let be a bounded linear operator on a Banach space and let be a subspace of which is a Banach space and invariant. Denote by the restriction of to This paper explores the questions:

If the range of is closed, under what conditions is the range of closed?

If the range of is closed, under what conditions is the range of closed?

     

The Postnikov Tower and the Steenrod problem

The Steenrod problem asks: given a -module, when does there exist a Moore space realizing the module? By using the equivariant Postnikov Tower, it is shown that a -module is -realizable if and only if it is -realizable for all -Sylow subgroups , for all primes .

     

Equivariant crystalline cohomology and base change

Given a perfect field of characteristic , a smooth proper -scheme , a crystal on relative to and a finite group acting on and , we show that, viewed as a virtual -module, the reduction modulo of the crystalline cohomology of is the de Rham cohomology of modulo . On the way we prove a base change theorem for the virtual -representations associated with -equivariant objects in the derived category of -modules.

     

Linear maps preserving ideals of -algebras

We show that unital self-adjoint linear bijections of matrix algebras, type factors and abelian -algebras preserving maximal left ideals are isomorphisms and we show that a unital continuous linear map of a -algebra that maps the minimal left ideal into itself is the identity map.     

Irreducible restriction and zeros of characters

Let be a finite group, let be normal in and suppose that is an irreducible complex character of . Then is not irreducible if and only if vanishes on some coset of in .

     

Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian

We show how for every integer one can explicitly construct distinct plane quartics and one hyperelliptic curve over all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When we say that the curves can be constructed ``explicitly', we mean that the coefficients of the defining equations of the curves are simple rational expressions in algebraic numbers in whose minimal polynomials over can be given exactly and whose decimal approximations can be given to as many places as is necessary to distinguish them from their conjugates. We also prove a simply-stated theorem that allows one to decide whether or not two plane quartics over , each with a pair of commuting involutions, are isomorphic to one another.

     

Types in reductive -adic groups: The Hecke algebra of a cover

In this paper, is a non-Archimedean local field and is the group of -points of a connected reductive algebraic group defined over . Also, is an irreducible representation of a compact open subgroup of , the pair being a type in . The pair is assumed to be a cover of a type in a Levi subgroup of . We give conditions, generalizing those of earlier work, under which the Hecke algebra is the tensor product of a canonical image of and a sub-algebra , for a compact open subgroup of containing .

     

Proof of Wang's conjecture on subspaces of an inner product space

B.Y. Wang conjectured that if and are subspaces of an -dimensional complex inner product space , and their dimensions are and , respectively, where , then there exists a -dimensional subspace having two orthonormal bases and with and for all .

We prove this conjecture and its real counterpart. The proof is in essence an application of a real version of the Bézout Theorem for the product of several projective spaces.

     

The abelianization of almost free groups

We will construct an almost free non-free group of cardinality such that is a free abelian group.

     

On classification of polarized varieties with non-integral nef values

Let be an -dimensional normal projective variety with only Gorenstein, terminal, -factorial singularities. Let be an ample line bundle on . Let denote the nef value of . The classification of via the nef value morphism is given for the situations when satisfies or .

     

On characterization and perturbation of local -semigroups

Let be a -group with generator , and let be a local -semigroup commuting with . Then the operators , , form a local -semigroup. It is proved that if is injective and is the generator of , then is closable and is the generator of . Also proved are a characterization theorem for local -semigroups with not necessarily injective and a theorem about solvability of the abstract inhomogeneous Cauchy problem:

     

The asymptotic behavior of a class of nonlinear delay difference equations

The asymptotic behavior of difference equations of type 0, \end{equation*}">is studied, where and each are continuous real functions with decreasing and increasing. Results include sufficient conditions for permanence, oscillations and global attractivity.

     

A prediction problem in

For a nonnegative integrable weight function on the unit circle , we provide an expression for , in terms of the series coefficients of the outer function of , for the weighted distance , where is the normalized Lebesgue measure and ranges over trigonometric polynomials with frequencies in , , . The problem is open for .

     

Free summands of conormal modules and central elements in homotopy Lie algebras of local rings

If is a surjective local homomorphism with kernel , such that and the conormal module has a free summand of rank , then the degree central subspace of the homotopy Lie algebra of has dimension greater than or equal to . This is a corollary of the Main Theorem of this note. The techniques involved provide new proofs of some well known results concerning the conormal module.

     

Asymptotic behavior of Fourier transforms of self-similar measures

Let be a self-similar probability measure on satisfying where 0$"> and Let be the Fourier transform of A necessary and sufficient condition for to approach zero at infinity is given. In particular, if and for then 0$"> if and only if is a PV-number and is not a factor of . This generalizes the corresponding theorem of Erdös and Salem for the case

     

Simultaneous non-vanishing of twists

Let be a newform of even weight , level and character and let be a newform of even weight , level and character . We give a generalization of a theorem of Elliott, regarding the average values of Dirichlet -functions, in the context of twisted modular -functions associated to and . Using this result, we find a lower bound in terms of for the number of primitive Dirichlet characters modulo prime whose twisted product -functions are non-vanishing at a fixed point with .

     

The degree of the bicanonical map of a surface with

In this note it is shown that, given a smooth minimal complex surface of general type with , , for which the bicanonical map is a morphism, the degree of is not 3. This completes our earlier results, showing that if is a minimal surface of general type with , such that is free, then the bicanonical map of can have degree 1, 2 or 4.

     

Character degree graphs that are complete graphs

Let be a finite group, and write for the set of degrees of irreducible characters of . We define to be the graph whose vertex set is , and there is an edge between and if . We prove that if is a complete graph, then is a solvable group.