Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity

We present some open problems and obtain some partial results for spectral optimization problems involving measure, torsional rigidity and first Dirichlet eigenvalue.Comment: 18 pages, 4 figure

On the torsion function with Robin or Dirichlet boundary conditions

For $p\in (1,+\infty)$ and $b \in (0, +\infty]$ the $p$-torsion function with Robin boundary conditions associated to an arbitrary open set \Om \subset \R^m satisfies formally the equation $-\Delta_p =1$ in \Om and $|\nabla u|^{p-2} \frac{\partial u}{\partial n} + b|u|^{p-2} u =0$ on \partial \Om. We obtain bounds of the $L^\infty$ norm of $u$ {\it only} in terms of the bottom of the spectrum (of the Robin $p$-Laplacian), $b$ and the dimension of the space in the following two extremal cases: the linear framework (corresponding to $p=2$) and arbitrary $b>0$, and the non-linear framework (corresponding to arbitrary $p>1$) and Dirichlet boundary conditions ($b=+\infty$). In the general case, $p\not=2, p \in (1, +\infty)$ and $b>0$ our bounds involve also the Lebesgue measure of \Om.Comment: 19 page

High resolution angular sensor

Specifications for the pointing stabilization system of the large space telescope were used in an investigation of the feasibility of reducing ring laser gyro output quantization to the sub-arc-second level by the use of phase locked loops and associated electronics. Systems analysis procedures are discussed and a multioscillator laser gyro model is presented along with data on the oscillator noise. It is shown that a second order closed loop can meet the measurement noise requirements when the loop gain and time constant of the loop filter are appropriately chosen. The preliminary electrical design is discussed from the standpoint of circuit tradeoff considerations. Analog, digital, and hybrid designs are given and their applicability to the high resolution sensor is examined. the electrical design choice of a system configuration is detailed. The design and operation of the various modules is considered and system block diagrams are included. Phase 1 and 2 test results using the multioscillator laser gyro are included

Higgs diphoton rate enhancement from supersymmetric physics beyond the MSSM

We show that supersymmetric "new physics" beyond the MSSM can naturally accommodate a Higgs mass near 126 GeV and enhance the signal rate in the Higgs to diphoton channel, while the signal rates in all the other Higgs decay channels coincide with Standard Model expectations, except possibly the Higgs to Z-photon channel. The "new physics" that corrects the relevant Higgs couplings can be captured by two supersymmetric effective operators. We provide a simple example of an underlying model in which these operators are simultaneously generated. The scale of "new physics" that generates these operators can be around 5 TeV or larger, and outside the reach of the LHC.Comment: 24 pages, 4 figure

Wind speed statistics for Goldstone, California, anemometer sites

An exploratory wind survey at an antenna complex was summarized statistically for application to future windmill designs. Data were collected at six locations from a total of 10 anemometers. Statistics include means, standard deviations, cubes, pattern factors, correlation coefficients, and exponents for power law profile of wind speed. Curves presented include: mean monthly wind speeds, moving averages, and diurnal variation patterns. It is concluded that three of the locations have sufficiently strong winds to justify consideration for windmill sites

On the minimization of Dirichlet eigenvalues of the Laplace operator

We study the variational problem \inf \{\lambda_k(\Omega): \Omega\ \textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \}, where $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega)$, \h(\partial \Omega) is the $(m-1)$- dimensional Hausdorff measure of the boundary of $\Omega$, and $|\Omega|$ is the Lebesgue measure of $\Omega$. If $m=2$, and $k=2,3, \cdots$, then there exists a convex minimiser $\Omega_{2,k}$. If $m \ge 2$, and if $\Omega_{m,k}$ is a minimiser, then $\Omega_{m,k}^*:= \textup{int}(\overline{\Omega_{m,k}})$ is also a minimiser, and $\R^m\setminus \Omega_{m,k}^*$ is connected. Upper bounds are obtained for the number of components of $\Omega_{m,k}$. It is shown that if $m\ge 3$, and $k\le m+1$ then $\Omega_{m,k}$ has at most $4$ components. Furthermore $\Omega_{m,k}$ is connected in the following cases : (i) $m\ge 2, k=2,$ (ii) $m=3,4,5,$ and $k=3,4,$ (iii) $m=4,5,$ and $k=5,$ (iv) $m=5$ and $k=6$. Finally, upper bounds on the number of components are obtained for minimisers for other constraints such as the Lebesgue measure and the torsional rigidity.Comment: 16 page

Large deviations for ideal quantum systems

We consider a general d-dimensional quantum system of non-interacting particles, with suitable statistics, in a very large (formally infinite) container. We prove that, in equilibrium, the fluctuations in the density of particles in a subdomain of the container are described by a large deviation function related to the pressure of the system. That is, untypical densities occur with a probability exponentially small in the volume of the subdomain, with the coefficient in the exponent given by the appropriate thermodynamic potential. Furthermore, small fluctuations satisfy the central limit theorem.Comment: 28 pages, LaTeX 2

Bose-Einstein Condensation in Geometrically Deformed Tubes

We show that Bose-Einstein condensate can be created in quasi-one-dimensional systems in a purely geometrical way, namely by bending or other suitable deformation of a tube.Comment: RevTex, 4pages, no figure

Density of states and Fisher's zeros in compact U(1) pure gauge theory

We present high-accuracy calculations of the density of states using multicanonical methods for lattice gauge theory with a compact gauge group U(1) on 4^4, 6^4 and 8^4 lattices. We show that the results are consistent with weak and strong coupling expansions. We present methods based on Chebyshev interpolations and Cauchy theorem to find the (Fisher's) zeros of the partition function in the complex beta=1/g^2 plane. The results are consistent with reweighting methods whenever the latter are accurate. We discuss the volume dependence of the imaginary part of the Fisher's zeros, the width and depth of the plaquette distribution at the value of beta where the two peaks have equal height. We discuss strategies to discriminate between first and second order transitions and explore them with data at larger volume but lower statistics. Higher statistics and even larger lattices are necessary to draw strong conclusions regarding the order of the transition.Comment: 14 pages, 16 figure

String Loop Corrections to Kahler Potentials in Orientifolds

We determine one-loop string corrections to Kahler potentials in type IIB orientifold compactifications with either N=1 or N=2 supersymmetry, including D-brane moduli, by evaluating string scattering amplitudes.Comment: 80 pages, 4 figure