Ergodic theory and the duality principle on homogeneous spaces

We prove mean and pointwise ergodic theorems for the action of a lattice subgroup in a connected algebraic Lie group on infinite volume homogeneous algebraic varieties. Under suitable necessary conditions, our results are quantitative, namely we establish rates of convergence in the mean and pointwise ergodic theorems, which can be estimated explicitly. Our results give a precise and in most cases optimal quantitative form to the duality principle governing dynamics on homogeneous spaces. We illustrate their scope in a variety of equidistribution problems.     

Dynamics of digging in wet soil

Numerous animals live in, and locomote through, subsea soils. To move in a medium dominated by frictional interactions, many of these animals have adopted unique burrowing strategies. This paper presents a burrowing model inspired by the Atlantic razor clam (Ensis directus), which uses deformations of its body to cyclically loosen and re-pack the surrounding soil in order to locally manipulate burrowing drag. The model reveals how an anisotropic body - composed of a cylinder and sphere varying sinusoidally in size and relative displacement - achieves unidirectional motion through a medium with variable frictional properties. This net displacement is attained even though the body kinematics are reciprocal and inertia of both the model organism and the surrounding medium are negligible. Our results indicate that body aspect ratio has a strong effect on burrowing velocity and efficiency, with a well-defined maximum for given kinematics and soil material properties.     

Prime and almost prime integral points on principal homogeneous spaces

We develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space. In particular, we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers. When the orbit is the set of integral matrices of a fixed determinant, we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups, and sharp and uniform counting of points on such orbits when ordered by various norms.     

A Metric Version of Milyutin Theorem

Open covering and metric regularity are properties playing a crucial role in several topics of modern variational analysis. Here their stability behaviour in the presence of perturbations is investigated in a purely metric setting. Some results in this sense are obtained, which lead to extend the known Milyutin theorem, and then to expand the concept of radius of regularity, a quantitative measure of the open covering stability. Estimations for the latter in terms of covering moduli are provided.     

The weakly neighborly polyhedral maps on the 2-manifold with euler characteristic −1

A weakly neighborly polyhedral map (w.n.p. map) is a two-dimensional cell-complex which decomposes a closed 2-manifold without boundary, such that for every two vertices there is a 2-cell containing them. We prove that there are just four w.n.p. maps with Euler characteristic –1 and we describe them.     

The wavelet dimension function is the trace function of a shift-invariant system

In this note, we observe that the dimension function associated with a wavelet system is the trace of the Gramian fibers of the shift-invariant system generated by the negative dilations of the mother wavelets. When this shift-invariant system is a tight frame, each of the Gramian fibers is an orthogonal projector, and its trace, then, coincides with its rank. This connection leads to simple proofs of several results concerning the dimension function, and the arguments extend to the bi-frame case.

     

Application of a femtosecond self-sustaining mode-locked Ti:sapphire laser to the field of laser scanning confocal microscopy

Developments in ultrafast Ti:sapphire laser technology can be applied in the investigation of nonlinear optical processes. We describe the application of a self-sustaining femtosecond Ti:sapphire laser as an illumination source in the field of confocal laser scanning fluorescence microscopy (LSM). We present spectra for various fluorescent stains under two-photon excitation and present LSM images of stained samples under mode-locked illumination. The potential for such a system as a non-destructive technique for studying live cells in biomedical research is discussed.