Optimal quality of exceptional points for the Lebesgue density theorem

In spite of the Lebesgue density theorem, there is a positive $\delta$ such that, for every non-trivial measurable set $S$ of real numbers, there is a point at which both the lower densities of $S$ and of the complement of $S$ are at least $\delta$. The problem of determining the supremum of possible values of this $\delta$ was studied in a paper of V. I. Kolyada, as well as in some recent papers. We solve this problem in the present work.Comment: 45 page

Residue Formulas for the Large k Asymptotics of Witten's Invariants of Seifert Manifolds. The Case of SU(2)

We derive the large k asymptotics of the surgery formula for SU(2) Witten's invariants of general Seifert manifolds. The contributions of connected components of the moduli space of flat connections are identified. The contributions of irreducible connections are presented in a residue form. This form is similar to the one used by A. Szenes, L. Jeffrey and F. Kirwan. This similarity allows us to express the contributions of irreducible connections in terms of intersection numbers on their moduli spaces.Comment: 39 pages, no figures, LaTe

On Witten multiple zeta-functions associated with semisimple Lie algebras IV

In our previous work, we established the theory of multi-variable Witten zeta-functions, which are called the zeta-functions of root systems. We have already considered the cases of types $A_2$, $A_3$, $B_2$, $B_3$ and $C_3$. In this paper, we consider the case of $G_2$-type. We define certain analogues of Bernoulli polynomials of $G_2$-type and study the generating functions of them to determine the coefficients of Witten's volume formulas of $G_2$-type. Next we consider the meromorphic continuation of the zeta-function of $G_2$-type and determine its possible singularities. Finally, by using our previous method, we give explicit functional relations for them which include Witten's volume formulas.Comment: 22 pag

(0,2) Deformations of Linear Sigma Models

We study (0,2) deformations of a (2,2) supersymmetric gauged linear sigma model for a Calabi-Yau hypersurface in a Fano toric variety. In the non-linear sigma model these correspond to some of the holomorphic deformations of the tangent bundle on the hypersurface. Combinatorial formulas are given for the number of these deformations, and we show that these numbers are exchanged by mirror symmetry in a subclass of the models.Comment: 35 pages; uses xy-fig; typos fixed, acknowledgments adde

Comparative study on the uniform energy deposition achievable via optimized plasmonic nanoresonator distributions

Plasmonic nanoresonators of core-shell composition and nanorod shape were optimized to tune their absorption cross-section maximum to the central wavelength of a short pulse. Their distribution along a pulse-length scaled target was optimized to maximize the absorptance with the criterion of minimal absorption difference in between neighbouring layers. Successive approximation of layer distributions made it possible to ensure almost uniform deposited energy distribution up until the maximal overlap of two counter-propagating pulses. Based on the larger absorptance and smaller uncertainty in absorptance and energy distribution core-shell nanoresonators override the nanorods. However, optimization of both nanoresonator distributions has potential applications, where efficient and uniform energy deposition is crucial, including phase transitions and even fusion

Crater formation by fast ions: comparison of experiment with Molecular Dynamics simulations

An incident fast ion in the electronic stopping regime produces a track of excitations which can lead to particle ejection and cratering. Molecular Dynamics simulations of the evolution of the deposited energy were used to study the resulting crater morphology as a function of the excitation density in a cylindrical track for large angle of incidence with respect to the surface normal. Surprisingly, the overall behavior is shown to be similar to that seen in the experimental data for crater formation in polymers. However, the simulations give greater insight into the cratering process. The threshold for crater formation occurs when the excitation density approaches the cohesive energy density, and a crater rim is formed at about six times that energy density. The crater length scales roughly as the square root of the electronic stopping power, and the crater width and depth seem to saturate for the largest energy densities considered here. The number of ejected particles, the sputtering yield, is shown to be much smaller than simple estimates based on crater size unless the full crater morphology is considered. Therefore, crater size can not easily be used to estimate the sputtering yield.Comment: LaTeX, 7 pages, 5 EPS figures. For related figures/movies, see: http://dirac.ms.virginia.edu/~emb3t/craters/craters.html New version uploaded 5/16/01, with minor text changes + new figure

Parametric Polyhedra with at least kk Lattice Points: Their Semigroup Structure and the k-Frobenius Problem

Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup \mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\} can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x: Ax=b, x\geq0\}$. Such families of parametric polyhedra appear in many areas of combinatorics, convex geometry, algebra and number theory. The key themes of this paper are: (1) A structure theory that characterizes precisely the subset \mbox{ Sg}_{\geq k}(A) of all vectors b \in \mbox{ Sg}(A) such that $P_A(b) \cap {\mathbb Z}^n$ has at least $k$ solutions. We demonstrate that this set is finitely generated, it is a union of translated copies of a semigroup which can be computed explicitly via Hilbert bases computations. Related results can be derived for those right-hand-side vectors $b$ for which $P_A(b) \cap {\mathbb Z}^n$ has exactly $k$ solutions or fewer than $k$ solutions. (2) A computational complexity theory. We show that, when $n$, $k$ are fixed natural numbers, one can compute in polynomial time an encoding of \mbox{ Sg}_{\geq k}(A) as a multivariate generating function, using a short sum of rational functions. As a consequence, one can identify all right-hand-side vectors of bounded norm that have at least $k$ solutions. (3) Applications and computation for the $k$-Frobenius numbers. Using Generating functions we prove that for fixed $n,k$ the $k$-Frobenius number can be computed in polynomial time. This generalizes a well-known result for $k=1$ by R. Kannan. Using some adaptation of dynamic programming we show some practical computations of $k$-Frobenius numbers and their relatives

Creation of multiple nanodots by single ions

In the challenging search for tools that are able to modify surfaces on the nanometer scale, heavy ions with energies of several 10 MeV are becoming more and more attractive. In contrast to slow ions where nuclear stopping is important and the energy is dissipated into a large volume in the crystal, in the high energy regime the stopping is due to electronic excitations only. Because of the extremely local (< 1 nm) energy deposition with densities of up to 10E19 W/cm^2, nanoscaled hillocks can be created under normal incidence. Usually, each nanodot is due to the impact of a single ion and the dots are randomly distributed. We demonstrate that multiple periodically spaced dots separated by a few 10 nanometers can be created by a single ion if the sample is irradiated under grazing angles of incidence. By varying this angle the number of dots can be controlled.Comment: 12 pages, 6 figure

Laser Wake Field Collider

Recently NAno-Plasmonic, Laser Inertial Fusion Experiments (NAPLIFE) were proposed, as an improved way to achieve laser driven fusion. The improvement is the combination of two basic research discoveries: (i) the possibility of detonations on space-time hyper-surfaces with time-like normal (i.e. simultaneous detonation in a whole volume) and (ii) to increase this volume to the whole target, by regulating the laser light absorption using nanoshells or nanorods as antennas. These principles can be realized in a one dimensional configuration, in the simplest way with two opposing laser beams as in particle colliders. Such, opposing laser beam experiments were also performed recently. Here we study the consequences of the Laser Wake Field Acceleration (LWFA) if we experience it in a colliding laser beam set-up. These studies can be applied to laser driven fusion, but also to other rapid phase transition, combustion, or ignition studies in other materials.publishedVersio