A nondifferentiable extension of a theorem of Pucci and Serrin and applications

We study the multiplicity of critical points for functionals which are only differentiable along some directions. We extend to this class of functionals the three critical point theorem of Pucci and Serrin and we apply it to a one-parameter family of functionals Jλ, λ∈I⊂R. Under suitable assumptions, we locate an open subinterval of values λ in I for which Jλ possesses at least three critical points. Applications to quasilinear boundary value problems are also given.     

An Enumerative Geometry for Magic and Magilatin Labellings

A magic labelling of a set system is a labelling of its points by distinct positive integers so that every set of the system has the same sum, the magic sum. Examples are magic squares (the sets are the rows, columns, and diagonals) and semimagic squares (the same, but without the diagonals). A magilatin labelling is like a magic labelling but the values need be distinct only within each set. We show that the number of n × n magic or magilatin labellings is a quasipolynomial function of the magic sum, and also of an upper bound on the entries in the square. Our results differ from previous ones because we require that the entries in the square all be different from each other, and because we derive our results not by ad hoc reasoning but from a general theory of counting lattice points in rational inside-out polytopes. We also generalize from set systems to rational linear forms. Dedicated to the memory of Claudia Zaslavsky, 1917–2006 Received August 10, 2005     

Effects of in situ thermal annealing on the structural, optical, and electrical properties in Hg0.7Cd0.3Te epilayers grown on CdTe buffer layers

Scanning electron microscopy (SEM), Fourier transform infrared (FTIR) transmission, and Hall effect measurements were performed to investigate the structural, optical, and electrical properties of as-grown and in situ-annealed Hg_0.7Cd_0.3Te epilayers grown on CdTe buffer layers by using molecular beam epitaxy. After the Hg_0.7Cd_0.3Te epilayers had been annealed in a Hg-cell flux atmosphere, the SEM images showed that the surface morphologies of the Hg_0.7Cd_0.3Te thin films were mirror-like with no indication of pinholes or defects, and the FTIR spectra showed that the transmission intensities had increased in comparison to that of the as-grown Hg_0.7Cd_0.3Te epilayer. Hall-effect measurements showed that n-Hg_0.7Cd_0.3Te epilayers were converted to p-Hg_0.7Cd_0.3Te epilayers. These results indicate that the surface, optical, and electrical properties of the Hg_1 − xCd_xTe epilayers are improved by annealing and that as-grown n-Hg_1 − xCd_xTe epilayers can be converted to p-Hg_1 − xCd_xTe epilayers by in situ annealing.     

Application of Electron-Shelving Detection via 423 nm Transition in Calcium-Beam Optical Frequency Standard

A new scheme of small compact optical frequency standard based on thermal calcium beam with application of 423 nm shelving detection and sharp-angle velocity selection detection is proposed. Combining these presented techniques, we conclude that a small compact optical frequency standard based on thermal calcium beam will outperform the commercial caesium-beam microwave dock, like the 5071 Cs clock （from Hp to Agilent, now Symmetricom company）, both in accuracy and stability.     

Molecular dynamics simulations on the structures and properties of ε-CL-20-based PBXs ——Primary theoretical studies on HEDM formulation design

Five polymer bonded explosives (PBXs) with the base explosive ε-CL-20 (hexanitrohexaazaisowurtzi- tane), the most important high energy density compound (HEDC), and five polymer binders (Estane 5703, GAP, HTPB, PEG, and F2314) were constructed. Molecular dynamics (MD) method was employed to investigate their binding energies (Ebind), compatibility, safety, mechanical properties, and energetic properties. The information and rules were reported for choosing better binders and guiding formula- tion design of high energy density material (HEDM). According to the calculated binding energies, the ordering of compatibility and stability of the five PBXs was predicted as ε-CL-20/PEG > ε-CL-20/ Estane5703 ≈ε-CL-20/GAP > ε-CL-20/HTPB > ε-CL-20/F2314. By pair correlation function g(r) analyses, hydrogen bonds and vdw are found to be the main interactions between the two components. The elasticity and isotropy of PBXs based ε-CL-20 can be obviously improved more than pure ε-CL-20 crystal. It is not by changing the molecular structures of ε-CL-20 for each binder to affect the sensitivity. The safety and energetic properties of these PBXs are mainly influenced by the thermal capability (C°p) and density (ρ) of binders, respectively.     

Euler configurations and quasi-polynomial systems

Consider the problem of three point vortices (also called Helmholtz’ vortices) on a plane, with arbitrarily given vorticities. The interaction between vortices is proportional to 1/r, where r is the distance between two vortices. The problem has 2 equilateral and at most 3 collinear normalized relative equilibria. This 3 is the optimal upper bound. Our main result is that the above standard statements remain unchanged if we consider an interaction proportional to r ^b, for any b < 0. For 0 < b < 1, the optimal upper bound becomes 5. For positive vorticities and any b < 1, there are exactly 3 collinear normalized relative equilibria. The case b = −2 of this last statement is the well-known theorem due to Euler: in the Newtonian 3-body problem, for any choice of the 3 masses, there are 3 Euler configurations (also known as the 3 Euler points). These small upper bounds strengthen the belief of Kushnirenko and Khovanskii [18]: real varieties defined by simple systems should have a simple topology. We indicate some hard conjectures about the configurations of relative equilibrium and suggest they could be attacked within the quasi-polynomial framework.     

Some remarks concerning C-spaces

We comment on the definition of C-spaces in [D.F. Addis, J.H. Gresham, A class of infinite-dimensional spaces. Part I: Dimension theory and Alexandroff's Problem, Fund. Math. 101 (1978) 195-205] and [W.E. Haver, A covering property for metric spaces, in: Topology Conference at Virginia Polytechnic Institute 1973, in: Lecture Notes in Math., vol. 375, 1974, pp. 108-113]. Furthermore we introduce two types of ‘finite’ C-spaces one of which gives an internal characterization of all spaces having a metrizable compactification satisfying property C. We also introduce a transfinite dimension function for those finite C-spaces. Several questions arise that are related to Alexandrov's problem.     

Almost sure estimates for the concentration neighborhood of Sinai’s walk

We consider Sinai’s random walk in random environment. We prove that infinitely often (i.o.) the size of the concentration neighborhood of this random walk is bounded almost surely. We also get that i.o. the maximal distance between two favorite sites is bounded almost surely.     

Bias of the structural quasi-score estimator of a measurement error model under misspecification of the regressor distribution

In a structural measurement error model the structural quasi-score (SQS) estimator is based on the distribution of the latent regressor variable. If this distribution is misspecified, the SQS estimator is (asymptotically) biased. Two types of misspecification are considered. Both assume that the statistician erroneously adopts a normal distribution as his model for the regressor distribution. In the first type of misspecification, the true model consists of a mixture of normal distributions which cluster around a single normal distribution, in the second type, the true distribution is a normal distribution admixed with a second normal distribution of low weight. In both cases of misspecification, the bias, of course, tends to zero when the size of misspecification tends to zero. However, in the first case the bias goes to zero in a flat way so that small deviations from the true model lead to a negligible bias, whereas in the second case the bias is noticeable even for small deviations from the true model.     

A stable approach to the equivariant Hopf theorem

Let G be a finite group. For semi-free G-manifolds which are oriented in the sense of Waner [S. Waner, Equivariant RO(G)-graded bordism theories, Topology and its Applications 17 (1984) 1-26], the homotopy classes of G-equivariant maps into a G-sphere are described in terms of their degrees, and the degrees occurring are characterised in terms of congruences. This is first shown to be a stable problem, and then solved using methods of equivariant stable homotopy theory with respect to a semi-free G-universe.