Noncoherence of some rings of functions

Let , denote the unit disc and unit circle, respectively, in , with center 0. If , then let denote the set of complex-valued functions defined on that are analytic in , and continuous and bounded on . Then is a ring with pointwise addition and multiplication. We prove that if the intersection of with the set of limit points of is not empty, then the ring is not coherent.

     

On zeros of Eisenstein series for genus zero Fuchsian groups

Let SL be a genus zero Fuchsian group of the first kind with as a cusp, and let be the holomorphic Eisenstein series of weight on that is nonvanishing at and vanishes at all the other cusps (provided that such an Eisenstein series exists). Under certain assumptions on and on a choice of a fundamental domain , we prove that all but possibly of the nontrivial zeros of lie on a certain subset of . Here is a constant that does not depend on the weight, is the upper half-plane, and is the canonical hauptmodul for

     

-isomorphisms, Jordan isomorphisms, and numerical range preserving maps

Let or , where is the algebra of a bounded linear operator acting on the Hilbert space , and is the set of self-adjoint operators in . Denote the numerical range of by It is shown that a surjective map satisfies

if and only if there is a unitary operator such that has the form

where is the transpose of with respect to a fixed orthonormal basis. In other words, the map or is a -isomorphism on and a Jordan isomorphism on . Moreover, if has finite dimension, then the surjective assumption on can be removed.

     

Exponential growth of Lie algebras of finite global dimension

Let be a connected finite type graded Lie algebra. If dim and gldim , then log index . If, moreover, , then for some ,    dim where log index as

     

A strong comparison principle for the -Laplacian

We consider weak solutions of the differential inequality of p-Laplacian type

such that on a smooth bounded domain in and either or is a weak solution of the corresponding Dirichlet problem with zero boundary condition. Assuming that on the boundary of the domain we prove that , and assuming that on the boundary of the domain we prove unless . The novelty is that the nonlinearity is allowed to change sign. In particular, the result holds for the model nonlinearity with .

     

Cofinality changes required for a large set of unapproachable ordinals below

In , assume that is a strong limit cardinal and . Let be the set of approachable ordinals less than . An open question of M. Foreman is whether can be non-stationary in some and preserving extension of . It is shown here that if is such an outer model, then is infinite, for each positive integer .

     

Upper bounds for the volume and diameter of -dimensional sections of convex bodies

In this paper some upper bounds for the volume and diameter of central sections of symmetric convex bodies are obtained in terms of the isotropy constant of the polar body. The main consequence is that every symmetric convex body in of volume one has a proportional section , dim ( ), of diameter bounded by

whenever the polar body is in isotropic position ( is some absolute constant).

     

On the index and spectrum of differential operators on

If is an system of differential operators on having continuous coefficients with vanishing oscillation at infinity, the Cordes-Illner theory ensures that is Fredholm from to for all or no value We prove that both the index (when defined) and the spectrum of are independent of

     

Operator theory on noncommutative varieties, II

An -tuple of operators on a Hilbert space is called a -constrained row contraction if and

where is a WOT-closed two-sided ideal of the noncommutative analytic Toeplitz algebra and is defined using the -functional calculus for row contractions.

We show that the constrained characteristic function associated with and is a complete unitary invariant for -constrained completely non-coisometric (c.n.c.) row contractions. We also provide a model for this class of row contractions in terms of the constrained characteristic functions. In particular, we obtain a model theory for -commuting c.n.c. row contractions.

     

Stability and exact multiplicity of periodic solutions of Duffing equations with cubic nonlinearities

We study the stability and exact multiplicity of periodic solutions of the Duffing equation with cubic nonlinearities,

where and are positive constants and is a positive -periodic function. We obtain sharp bounds for such that has exactly three ordered -periodic solutions. Moreover, when is within these bounds, one of the three solutions is negative, while the other two are positive. The middle solution is asymptotically stable, and the remaining two are unstable.

     

On support varieties for modules over complete intersections

Let be a complete intersection of codimension , and let be the algebraic closure of . We show that every homogeneous algebraic subset of is the cohomological support variety of an -module, and that the projective variety of a complete indecomposable maximal Cohen-Macaulay -module is connected.

     

Some exact sequences for Toeplitz algebras of spherical isometries

A family of commuting bounded operators on a Hilbert space is said to be a spherical isometry if in the weak operator topology. We show that every commuting family of spherical isometries is jointly subnormal, which means that it has a commuting normal extension on some Hilbert space Suppose now that the normal extension is minimal. Then we show that every bounded operator in the commutant of has a unique norm preserving extension to an operator in the commutant of Moreover, if is the commutator ideal in then is *-isomorphic to We also show that the commutant of the minimal normal extension is completely isometric, via the compression mapping, to the space of Toeplitz-type operators associated to We apply these results to construct exact sequences for Toeplitz algebras on generalized Hardy spaces associated to strictly pseudoconvex domains.

     

Multiple solutions for elliptic problems with singular and sublinear potentials

For certain positive numbers and we establish the multiplicity of solutions to the problem

where is a bounded open domain in containing the origin with smooth boundary while is continuous, superlinear at zero and sublinear at infinity.

     

Some Hopf Galois structures arising from elementary abelian -groups

Let be an odd prime, , the elementary abelian -group of rank , and let be the group of principal units of the ring . If is a Galois extension with Galois group , then we show that for , the number of Hopf Galois structures on afforded by -Hopf algebras with associated group is greater than , where .

     

Character sums over shifted smooth numbers

We give nontrivial bounds in various ranges for character sums of the form

where is a nontrivial multiplicative character modulo a prime and is the set of positive integers that are divisible only by primes .

     

Finite heat kernel expansions on the real line

Let be the one-dimensional Schrödinger operator and let be the corresponding heat kernel. We prove that the th Hadamard's coefficient is equal to 0 if and only if there exists a differential operator of order such that . Thus, the heat expansion is finite if and only if the potential is a rational solution of the KdV hierarchy decaying at infinity studied by Adler and Moser (1978) and Airault, McKean and Moser (1977). Equivalently, one can characterize the corresponding operators as the rank one bispectral family given by Duistermaat and Grünbaum (1986).

     

Unbounded solutions and periodic solutions for second order differential equations with asymmetric nonlinearity

In this paper we will prove the coexistence of unbounded solutions and periodic solutions for the asymmetric oscillator

where and are positive constants satisfying the nonresonant condition

and is periodic in the first variable and bounded.

     

Order-weakly compact operators from vector-valued function spaces to Banach spaces

Let be an ideal of over a  -finite measure space , and let stand for the order dual of . For a real Banach space let be a subspace of the space of -equivalence classes of strongly -measurable functions and consisting of all those for which the scalar function belongs to . For a real Banach space a linear operator is said to be order-weakly compact whenever for each the set is relatively weakly compact in . In this paper we examine order-weakly compact operators . We give a characterization of an order-weakly compact operator in terms of the continuity of the conjugate operator of with respect to some weak topologies. It is shown that if is an order continuous Banach function space, is a Banach space containing no isomorphic copy of and is a weakly sequentially complete Banach space, then every continuous linear operator is order-weakly compact. Moreover, it is proved that if is a Banach function space, then for every Banach space any continuous linear operator is order-weakly compact iff the norm is order continuous and is reflexive. In particular, for every Banach space any continuous linear operator is order-weakly compact iff is reflexive.

     

Polynomial maps and even dimensional spheres

We construct, for every even dimensional sphere , , and every odd integer , a homogeneous polynomial map of Brouwer degree and algebraic degree .

     

On a Littlewood-Paley type inequality

The following is proved: If is a function harmonic in the unit ball and if then the inequality

holds, where is the nontangential maximal function of This improves a recent result of Stoll. This inequality holds for polyharmonic and hyperbolically harmonic functions as well.