On the first Hodge eigenvalue of isometric immersions

We give an extrinsic upper bound for the first positive eigenvalue of the Hodge Laplacian acting on -forms on a compact manifold without boundary isometrically immersed in or . The upper bound generalizes an estimate of Reilly for functions; it depends on the mean value of the squared norm of the mean curvature vector of the immersion and on the mean value of the scalar curvature. In particular, for minimal immersions into a sphere the upper bound depends only on the degree, the dimension and the mean value of the scalar curvature.

     

Asymptotics of ends of constant mean curvature surfaces with bubbletons

A constant mean curvature surface with bubbletons is defined by the loop group action on the set of extended framings for constant mean curvature surfaces by simple factors. Classically such surfaces were obtained by the transformation of tangential line congruences, the so-called Bianchi-Bäcklund transformations.

In this paper, we consider constant mean curvature surfaces with Delaunay ends in three-dimensional space forms , and and their surfaces with bubbletons for which the topology is preserved. We show that the ends of such surfaces are again asymptotic to Delaunay surfaces.

     

Rational Landen transformations on

The Landen transformation preserves the value of an elliptic integral, and its iteration produces the classical arithmetic-geometric mean AGM. We present analogous transformations for rational functions integrated over the whole real line.

     

A spectral characterization of the -torus by the first stability eigenvalue

Let be a compact hypersurface with constant mean curvature immersed into the unit Euclidean sphere . In this paper we derive a sharp upper bound for the first eigenvalue of the stability operator of in terms of the mean curvature and the length of the total umbilicity tensor of the hypersurface. Moreover, we prove that this bound is achieved only for the so-called -tori in , with . This extends to the case of constant mean curvature hypersurfaces previous results given by Wu (1993) and Perdomo (2002) for minimal hypersurfaces.

     

Means on chainable continua

By a mean on a space we understand a mapping such that and for . A chainable continuum is a metric compact connected space which admits an - mapping onto the interval for every number . We show that every chainable continuum that admits a mean is homeomorphic to the interval. In this way we answer a question by P. Bacon. We answer some other questions concerning means as well.

     

Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models

Let be a nonnegative, smooth function with , supported in , symmetric, , and strictly increasing in . We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation

We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as : they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments.

     

Quadratic extensions of totally real quintic fields

In this work, we establish lists for each signature of tenth degree number fields containing a totally real quintic subfield and of discriminant less than in absolute value. For each field in the list we give its discriminant, the discriminant of its subfield, a relative polynomial generating the field over one of its subfields, the corresponding polynomial over , and the Galois group of its Galois closure.

We have examined the existence of several non-isomorphic fields with the same discriminants, and also the existence of unramified extensions and cyclic extensions.

     

A note on extensions of asymptotic density

By a density we mean any extension of the asymptotic density to a finitely additive measure defined on all sets of natural numbers. We consider densities associated to ultrafilters on and investigate two additivity properties of such densities. In particular, we show that there is a density for which is complete.

     

Coarse classification of constant mean curvature cylinders

We give a coarse classification of constant mean curvature (CMC) immersions of cylinders into via the loop group method. Particularly for this purpose, we consider double loop groups and a new type of ``potentials' which are meromorphic 1-forms on Riemann surfaces.

     

On the stability index of hypersurfaces with constant mean curvature in spheres

Barbosa, do Carmo and Eschenburg characterized the totally umbilical spheres as the only weakly stable compact constant mean curvature hypersurfaces in the Euclidean sphere . In this paper we prove that the weak index of any other compact constant mean curvature hypersurface in n+1 which is not totally umbilical and has constant scalar curvature is greater than or equal to , with equality if and only if is a constant mean curvature Clifford torus with radius .

     

Biharmonic lifts by means of pseudo-Riemannian submersions in dimension three

We study the total lifts of curves by means of a submersion that satisfy the condition analyzing, in particular, the cases in which the submersion has totally geodesic fibres or integrable horizontal distribution. We also consider in detail the case (biharmonic lifts). Moreover, we obtain a biharmonic lift in by means of a Riemannian submersion that has non-constant mean curvature, getting so a counterexample to the Chen conjecture for with a non-flat Riemannian metric.

     

A point set whose space of triangulations is disconnected

By the ``space of triangulations" of a finite point configuration we mean either of the following two objects: the graph of triangulations of , whose vertices are the triangulations of and whose edges are the geometric bistellar operations between them or the partially ordered set (poset) of all polyhedral subdivisions of ordered by coherent refinement. The latter is a modification of the more usual Baues poset of . It is explicitly introduced here for the first time and is of special interest in the theory of toric varieties.

We construct an integer point configuration in dimension 6 and a triangulation of it which admits no geometric bistellar operations. This triangulation is an isolated point in both the graph and the poset, which proves for the first time that these two objects cannot be connected.

     

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.

     

On the existence of solutions to the Monge-Ampère equation with infinite boundary values

Given a positive and an increasing nonlinearity that satisfies an appropriate growth condition at infinity, we provide a condition on for which the Monge-Ampère equation admits a solution with infinite boundary value on a strictly convex domain . Sufficient conditions for the nonexistence of such solutions will also be given.

     

On the largest prime divisor of an odd harmonic number

A positive integer is called a (Ore's) harmonic number if its positive divisors have integral harmonic mean. Ore conjectured that every harmonic number greater than is even. If Ore's conjecture is true, there exist no odd perfect numbers. In this paper, we prove that every odd harmonic number greater than must be divisible by a prime greater than .

     

Computations of Eisenstein series on Fuchsian groups

We present numerical investigations of the value distribution and distribution of Fourier coefficients of the Eisenstein series on arithmetic and non-arithmetic Fuchsian groups. Our numerics indicate a Gaussian limit value distribution for a real-valued rotation of as , and also, on non-arithmetic groups, a complex Gaussian limit distribution for when near and , at least if we allow at some rate. Furthermore, on non-arithmetic groups and for fixed with near , our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.

     

A selection theorem in metric trees

In this paper, we show that nonempty closed convex subsets of a metric tree enjoy many properties shared by convex subsets of Hilbert spaces and admissible subsets of hyperconvex spaces. Furthermore, we prove that a set-valued mapping of a metric tree with convex values has a selection for which for each . Here by we mean the Hausdroff distance. Many applications of this result are given.

     

Quantitative estimates of unique continuation for parabolic equations and inverse initial-boundary value problems with unknown boundaries

In this paper we obtain quantitative estimates of strong unique continuation for solutions to parabolic equations. We apply these results to prove stability estimates of logarithmic type for an inverse problem consisting in the determination of unknown portions of the boundary of a domain in , from the knowledge of overdetermined boundary data for parabolic boundary value problems.

     

A Fejér type theorem to determine jumps in terms of the Abel-Poisson mean of double Fourier series

We extend from single to double Fourier series a theorem of Zygmund to determine the generalized jumps of a periodic integrable function at a simple discontinuity point. As a by-product of the proof, we obtain an estimate of the fourth mixed partial derivative of the Abel-Poisson mean of any integrable function at such a point where is smooth. We also consider the extension of the Zygmund classes and to the two-dimensional torus .

     

Algebraic polynomials with non-identical random coefficients

There are many known asymptotic estimates for the expected number of real zeros of a random algebraic polynomial The coefficients are mostly assumed to be independent identical normal random variables with mean zero and variance unity. In this case, for all sufficiently large, the above expected value is shown to be . Also, it is known that if the have non-identical variance , then the expected number of real zeros increases to . It is, therefore, natural to assume that for other classes of distributions of the coefficients in which the variance of the coefficients is picked at the middle term, we would also expect a greater number of zeros than . In this work for two different choices of variance for the coefficients we show that this is not the case. Although we show asymptotically that there is some increase in the number of real zeros, they still remain . In fact, so far the case of is the only case that can significantly increase the expected number of real zeros.