Subdirect products of hereditary congruence lattices

A congruence lattice L of an algebra A is called power-hereditary if every 0-1 sublattice of Lⁿ is the congruence lattice of an algebra on Aⁿ for all positive integers n. Let A and B be finite algebras. We prove

Dominated, diagonal polynomials on ℓ p spaces

We show that the r-dominated polynomials on _p(2 p ) are integral on ₁, and give examples proving that the converse is not true. We characterize when the 2-homogeneous, diagonal polynomials on _p(1 < p ) are r-dominated. We prove that, unlike the linear case, there are nuclear polynomials which are not 1-dominated.Received: 6 June 2004; revised: 28 September 2004     

Variants of recognition problems for modular forms

We consider the problem of distinguishing two modular forms, or two elliptic curves, by looking at the coefficients of their L-functions for small primes (compared to their conductor). Using analytic methods based on large-sieve type inequalities we give various upper bounds on the number of forms having the first few coefficients equal to those of a fixed one. In addition, we consider similar questions of recognizing symmetric squares and CM forms from the behavior of small primes.Received: 30 August 2004     

Notes on very ample vector bundles on 3-folds

Let be a very ample vector bundle of rank two on a smooth complex projective threefold X. An inequality about the third Segre class of is provided when is nef but not big, and when a suitable positive multiple of defines a morphism X → B with connected fibers onto a smooth projective curve B, where K_X is the canonical bundle of X. As an application, the case where the genus of B is positive and has a global section whose zero locus is a smooth hyperelliptic curve of genus ≧ 2 is investigated, and our previous result is improved for threefolds. Received: 27 January 2005; revised: 26 March 2005     

Some remarks on factoriality of certain hypersurfaces in \mathbb{P}^4

Let V be a reduced and irreducible hypersurface of degree k 3. In this paper we prove that if the singular locus of V consists of ₂ ordinary double points, ₃ ordinary triple points and if ₂ + 4₃ < (k – 1)², then any smooth surface contained in V is a complete intersection on V.Received: 7 January 2004     

On the number of nodal solutions to a singularly perturbed Neumann problem

We show that for ε small, there are arbitrarily many nodal solutions for the following nonlinear elliptic Neumann problem where Ω is a bounded and smooth domain in ℝ² and f grows superlinearly. (A typical f(u) is f(u)= a₁ u₊^p – a₁ u_-^p, a₁, a₂ >0, p, q>1.) More precisely, for any positive integer K, there exists ε_K>0 such that for 0<ε<ε_K, the above problem has a nodal solution with K positive local maximum points and K negative local minimum points. This solution has at least K+1 nodal domains. The locations of the maximum and minimum points are related to the mean curvature on ∂Ω. The solutions are constructed as critical points of some finite dimensional reduced energy functional. No assumption on the symmetry, nor the geometry, nor the topology of the domain is needed.     

A non-MRA C^r Frame Wavelet with Rapid Decay

A generalized filter construction is used to build an example of a non-MRA normalized tight frame wavelet for dilation by 2 in . This example has the same multiplicity function as the Journé wavelet, yet has a Fourier transform and can be made to be for any fixed postive integer . L. Baggett and P. Jorgensen were supported by a US–NSF Focused Research Group (FRG) grant.     

Linear Orders on General Algebras

We answer the question, when a partial order in a partially ordered algebraic structure has a compatible linear extension. The finite extension property enables us to show, that if there is no such extension, then it is caused by a certain finite subset in the direct square of the base set. As a consequence, we prove that a partial order can be linearly extended if and only if it can be linearly extended on every finitely generated subalgebra. Using a special equivalence relation on the above direct square, we obtain a further property of linearly extendible partial orders. Imposing conditions on the lattice of compatible quasi orders, the number of linear orders can be determined. Our general approach yields new results even in the case of semi-groups and groups.     

High Order Singular Rank One Perturbations of a Positive Operator

In this paper self-adjoint realizations in Hilbert and Pontryagin spaces of the formal expression are discussed and compared. Here L is a positive self-adjoint operator in a Hilbert space with inner product 〈·,·〉, α is a real parameter, and φ in the rank one perturbation is a singular element belonging to with n ≥ 3, where is the scale of Hilbert spaces associated with L in