Generalized incidence theorems, homogeneous forms and sum–product estimates in finite fields

In recent years, sum–product estimates in Euclidean space and finite fields have received great attention. They can often be interpreted in terms of Erdős type incidence problems involving the distribution of distances, dot products, areas, and so on, which have been studied quite extensively by way of combinatorial and Fourier analytic techniques. We use both kinds of techniques to obtain sharp or near-sharp results on the distribution of volumes (as examples of d-linear homogeneous forms) determined by sufficiently large subsets of vector spaces over finite fields and the associated arithmetic expressions. Arithmetic–combinatorial techniques turn out to be optimal for dimension d≥4 to this end, while for d=3 they have failed to provide us with a result that follows from the analysis of exponential sums. To obtain the latter result we prove a relatively straightforward function version of an incidence results for points and planes previously established in [D. Hart, A. Iosevich, Sums and products in finite fields: An integral geometric viewpoint, in: Radon Transforms, Geometry, and Wavelets, Contemp. Math. 464 (2008); D. Hart, A. Iosevich, D. Koh, M. Rudnev, Averages over hyperplanes, sum–product theory in vector spaces over finite fields and the Erdős–Falconer distance conjecture, arXiv:math/0711.4427, preprint 2007].More specifically, we prove that if E=A××A is a product set in , d≥4, the d-dimensional vector space over a finite field , such that the size |E| of E exceeds (i.e. the size of the generating set A exceeds ) then the set of volumes of d-dimensional parallelepipeds determined by E covers . This result is sharp as can be seen by taking , a prime sub-field of its quadratic extension , with q=p². For in three dimensions, however, we are able to establish the same result only if (i.e., , for some C; in fact, the bound can be justified for a slightly wider class of “Cartesian product-like” sets), and this uses Fourier methods. Yet we do prove a weaker near-optimal result in three dimensions: that the set of volumes generated by a product set E=A×A×A covers a positive proportion of if (so ). Besides, without any assumptions on the structure of E, we show that in three dimensions the set of volumes covers a positive proportion of if |E|≥Cq², which is again sharp up to the constant C, as taking E to be a 2-plane through the origin shows.     

Are Corals Colorful?

Using in situ spectrometry data and visual system modeling, we investigate whether the colors conferred to the reef-building corals by GFP-like proteins would look colorful not only to humans, but also to fish occupying different ecological niches on the reef. Some GFP-like proteins, most notably fluorescent greens and nonfluorescent chromoproteins, indeed generate intense color signals. An unexpected finding was that fluorescent proteins might also make corals appear less colorful to fish, counterbalancing the effect of absorption by the photosynthetic pigments of the endosymbiotic algae, which might be a form of protection against herbivores. We conclude that GFP-determined coloration of corals may be an important factor in visual ecology of the reef fishes.     

Tunnel ionization of molecules and orbital imaging

We study whether tunnel ionization of aligned molecules can be used to map out the electronic structure of the ionizing orbitals. We show that the common view, which associates tunnel ionization rates with the electronic density profile of the ionizing orbital, is not always correct. Using the example of tunnel ionization from the CO(2) molecule, we show how and why the angular structure of the alignment-dependent ionization rate moves with increasing the strength of the electric field. These modifications reflect a general trend for molecules.     

Wirtinger numbers and holomorphic symplectic immersions

For any subvariety of a compact holomorphic symplectic K?hler manifold, we define the symplectic Wirtinger number W(X). We show that W(X) \leqslant 1,W(X) \leqslant 1, and the equality is reached if and only if the subvariety X ì MX \subset M is trianalytic, i.e. compatible with the hyperk?hler structure on M. For a sequence X₁ ? X₂ ? ?X_n ? MX_1 \to X_2 \to \ldots X_n \to M of immersions of simple holomorphic symplectic manifolds, we show that W( X₁ ) \leqslant W( X₂ ) \leqslant ?\leqslant W( X_n ).W\left( {X_1 } \right) \leqslant W\left( {X_2 } \right) \leqslant \ldots \leqslant W\left( {X_n } \right).     

On the Mattila Integral Associated with Sign Indefinite Measures

In order to quantitatively illustrate the rôle of positivity in the Falconer distance problem, we construct a family of sign indefinite, compactly supported measures in \({\Bbb R}^d\), such that their Fourier transform and Fourier energy of dimension \(s \in (0, d)\) are uniformly bounded. However, the Mattila integral, associated with the Falconer distance problem for these measures is unbounded in the range \(0 < s < \frac{d^2}{2d-1}\).     

Vanishing Viscosity Solutions of the Compressible Euler Equations with Spherical Symmetry and Large Initial Data

We are concerned with spherically symmetric solutions of the Euler equations for multidimensional compressible fluids, which are motivated by many important physical situations. Various evidences indicate that spherically symmetric solutions of the compressible Euler equations may blow up near the origin at a certain time under some circumstance. The central feature is the strengthening of waves as they move radially inward. A longstanding open, fundamental problem is whether concentration could be formed at the origin. In this paper, we develop a method of vanishing viscosity and related estimate techniques for viscosity approximate solutions, and establish the convergence of the approximate solutions to a global finite-energy entropy solution of the isentropic Euler equations with spherical symmetry and large initial data. This indicates that concentration is not formed in the vanishing viscosity limit, even though the density may blow up at a certain time. To achieve this, we first construct global smooth solutions of appropriate initial-boundary value problems for the Euler equations with designed viscosity terms, approximate pressure function, and boundary conditions, and then we establish the strong convergence of the viscosity approximate solutions to a finite-energy entropy solution of the Euler equations.     

Colouring triangle-free intersection graphs of boxes on the plane

We prove that intersection graphs of boxes on the plane with girth 6 and 8 are 3- and 2-degenerate, respectively. This implies that these graphs are 4- and 3-list-colourable, respectively.     

Incompressible Flows of an Ideal Fluid¶with Unbounded Vorticity

In this paper we study solutions to the Euler equations of an ideal incompressible fluid in R ⁿ singular at the origin with a finite symmetry group. For an “admissible” class of finite groups we prove a local existence and uniqueness theorem. In even dimensions this theorem covers some symmetric flows with essentially unbounded vorticity. In arbitrary dimension (including n=3) we construct local in time solutions with vorticity that behaves, e.g., like a function of homogeneous degree zero near the origin. The symmetry condition provides necessary additional cancellations and is preserved by the evolution due to uniqueness. Received: 31 March 1999 / Accepted: 10 July 2000     

Plateau-Stein manifolds

We study/construct (proper and non-proper) Morse functions f on complete Riemannian manifolds X such that the hypersurfaces f(x) = t for all ?∞ < t < +∞ have positive mean curvatures at all non-critical points x ∈ X of f. We show, for instance, that if X admits no such (not necessarily proper) function, then it contains a (possibly, singular) complete (possibly, compact) minimal hypersurface of finite volume.