Numerical study of the effect of structure and geometry on van der Waals forces

We use multipolar expansions to find the force on a gold coated sphere above a gold substrate; we study both an empty gold shell and a gold coated polystyrene sphere. We find four characteristic separation ranges. In the first region, which for the empty gold shell occurs for distances, d, smaller than the thickness of the coating, the result agrees with that on a solid gold sphere and varies as d^(-2); for larger separations there is a region where the force behaves as if the coating is strictly two dimensional and varies as d^(-5/2); in the third region the dependence is more unspecific; in the forth region when d is larger than the radius, the force varies as d^(-4). For homogeneous objects of more general shapes we introduce a numerical method based on the solution of an integral equation for the electric field over a system of objects with arbitrary shapes. We study the effect of shape and orientation on the van der Waals interaction between an object and a substrate and between two objects.Comment: 8 pages, presented in the QFEXT07 conference, submitted to Journal of Physics

On compact hyperbolic Coxeter d-polytopes with d+4 facets

We show that there is no compact hyperbolic Coxeter d-polytope with d+4 facets for d>7. This bound is sharp: examples of such polytopes up to dimension 7 were found by Bugaenko (1984). We also show that in dimension d=7 the polytope with 11 facets is unique.Comment: v2: the paper is rewritten. A new section added in which 7-dimensional polytopes are classified. 43 pages, a lot of figure

On prisms, M\"obius ladders and the cycle space of dense graphs

For a graph X, let f_0(X) denote its number of vertices, d(X) its minimum degree and Z_1(X;Z/2) its cycle space in the standard graph-theoretical sense (i.e. 1-dimensional cycle group in the sense of simplicial homology theory with Z/2-coefficients). Call a graph Hamilton-generated if and only if the set of all Hamilton circuits is a Z/2-generating system for Z_1(X;Z/2). The main purpose of this paper is to prove the following: for every s > 0 there exists n_0 such that for every graph X with f_0(X) >= n_0 vertices, (1) if d(X) >= (1/2 + s) f_0(X) and f_0(X) is odd, then X is Hamilton-generated, (2) if d(X) >= (1/2 + s) f_0(X) and f_0(X) is even, then the set of all Hamilton circuits of X generates a codimension-one subspace of Z_1(X;Z/2), and the set of all circuits of X having length either f_0(X)-1 or f_0(X) generates all of Z_1(X;Z/2), (3) if d(X) >= (1/4 + s) f_0(X) and X is square bipartite, then X is Hamilton-generated. All these degree-conditions are essentially best-possible. The implications in (1) and (2) give an asymptotic affirmative answer to a special case of an open conjecture which according to [European J. Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.Comment: 33 pages; 5 figure

Extremal properties for dissections of convex 3-polytopes

A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific non-simplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial d-cubes.Comment: 19 page

Self-calibrating d-scan: measuring ultrashort laser pulses on-target using an arbitrary pulse compressor

In most applications of ultrashort pulse lasers, temporal compressors are used to achieve a desired pulse duration in a target or sample, and precise temporal characterization is important. The dispersion-scan (d-scan) pulse characterization technique usually involves using glass wedges to impart variable, well-defined amounts of dispersion to the pulses, while measuring the spectrum of a nonlinear signal produced by those pulses. This works very well for broadband few-cycle pulses, but longer, narrower bandwidth pulses are much more difficult to measure this way. Here we demonstrate the concept of self-calibrating d-scan, which extends the applicability of the d-scan technique to pulses of arbitrary duration, enabling their complete measurement without prior knowledge of the introduced dispersion. In particular, we show that the pulse compressors already employed in chirped pulse amplification (CPA) systems can be used to simultaneously compress and measure the temporal profile of the output pulses on-target in a simple way, without the need of additional diagnostics or calibrations, while at the same time calibrating the often-unknown differential dispersion of the compressor itself. We demonstrate the technique through simulations and experiments under known conditions. Finally, we apply it to the measurement and compression of 27.5 fs pulses from a CPA laser.Comment: 11 pages, 5 figures, Scientific Reports, in pres