Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams

We present a complete solution to the problem of coherent-mode decomposition of the most general anisotropic Gaussian Schell-model (AGSM) beams, which constitute a ten-parameter family. Our approach is based on symmetry considerations. Concepts and techniques familiar from the context of quantum mechanics in the two-dimensional plane are used to exploit the Sp(4, R) dynamical symmetry underlying the AGSM problem. We take advantage of the fact that the symplectic group of first-order optical system acts unitarily through the metaplectic operators on the Hilbert space of wave amplitudes over the transverse plane, and, using the Iwasawa decomposition for the metaplectic operator and the classic theorem of Williamson on the normal forms of positive definite symmetric matrices under linear canonical transformations, we demonstrate the unitary equivalence of the AGSM problem to a separable problem earlier studied by Li and Wolf [Opt. Lett. 7, 256 (1982)] and Gori and Guattari [Opt. Commun. 48, 7 (1983)]. This conn ction enables one to write down, almost by inspection, the coherent-mode decomposition of the general AGSM beam. A universal feature of the eigenvalue spectrum of the AGSM family is noted.

Fluctuation-dissipation relation for nonlinear Langevin equations

It is shown that the fluctuation-dissipation theorem is satisfied by the solutions of a general set of nonlinear Langevin equations with a quadratic free-energy functional (constant susceptibility) and field-dependent kinetic coefficients, provided the kinetic coefficients satisfy the Onsager reciprocal relations for the irreversible terms and the antisymmetry relations for the reversible terms. The analysis employs a perturbation expansion of the nonlinear terms, and a functional integral calculation of the correlation and response functions, and it is shown that the fluctuation-dissipation relation is satisfied at each order in the expansion.

Work fluctuations in an elastic dumbbell model of polymers in planar elongational flow

We use a path-integral approach to calculate the distribution P(w, t) of the fluctuations in the work W at time t of a polymer molecule (modeled as an elastic dumbbell in a viscous solvent) that is acted on by an elongational flow field having a flow rate (gamma) over dot. We find that P(w, t) is non-Gaussian and that, at long times, the ratio P(w, t)/ P (-w, t) is equal to expw/(k(B)T)], independent of (gamma) over dot. On the basis of this finding, we suggest that polymers in elongational flows satisfy a fluctuation theorem.

Implications of the recent high statistics determination of the pion electromagnetic form factor in the timelike region

The recently evaluated two-pion contribution to the muon g - 2 and the phase of the pion electromagnetic form factor in the elastic region, known from pi pi scattering by Fermi-Watson theorem, are exploited by analytic techniques for finding correlations between the coefficients of the Taylor expansion at t = 0 and the values of the form factor at several points in the spacelike region. We do not use specific parametrizations, and the results are fully independent of the unknown phase in the inelastic region. Using for instance, from recent determinations, < r(pi)(2)> = (0.435 +/- 0.005) fm(2) and F(-1.6 GeV2) = 0.243(-0.014)(+0.022), we obtain the allowed ranges 3.75 GeV-4 less than or similar to c less than or similar to 3.98 GeV-4 and 9.91 GeV-6 less than or similar to d less than or similar to 10.46 GeV-6 for the curvature and the next Taylor coefficient, with a strong correlation between them. We also predict a large region in the complex plane where the form factor cannot have zeros.

Uniform algebras generated by holomorphic and close-to-harmonic functions

The initial motivation for this paper is to discuss a more concrete approach to an approximation theorem of Axler and Shields, which says that the uniform algebra on the closed unit disc (D) over bar generated by z and h, where h is a nowhere-holomorphic harmonic function on D that is continuous up to partial derivative D, equals C((D) over bar). The abstract tools used by Axler and Shields make harmonicity of h an essential condition for their result. We use the concepts of plurisubharmonicity and polynomial convexity to show that, in fact, the same conclusion is reached if h is replaced by h + R, where R is a non-harmonic perturbation whose Laplacian is ``small'' in a certain sense.

On Walkup's class K (d) and a minimal triangulation of (S-3 (sic) S-1)(#3)

For d >= 2, Walkup's class K (d) consists of the d-dimensional simplicial complexes all whose vertex-links are stacked (d - 1)-spheres. Kalai showed that for d >= 4, all connected members of K (d) are obtained from stacked d-spheres by finitely many elementary handle additions. According to a result of Walkup, the face vector of any triangulated 4-manifold X with Euler characteristic chi satisfies f(1) >= 5f(0) - 15/2 chi, with equality only for X is an element of K(4). Kuhnel observed that this implies f(0)(f(0) - 11) >= -15 chi, with equality only for 2-neighborly members of K(4). Kuhnel also asked if there is a triangulated 4-manifold with f(0) = 15, chi = -4 (attaining equality in his lower bound). In this paper, guided by Kalai's theorem, we show that indeed there is such a triangulation. It triangulates the connected sum of three copies of the twisted sphere product S-3 (sic) S-1. Because of Kuhnel's inequality, the given triangulation of this manifold is a vertex-minimal triangulation. By a recent result of Effenberger, the triangulation constructed here is tight. Apart from the neighborly 2-manifolds and the infinite family of (2d + 3)-vertex sphere products Sd-1 X S-1 (twisted for d odd), only fourteen tight triangulated manifolds were known so far. The present construction yields a new member of this sporadic family. We also present a self-contained proof of Kalai's result. (C) 2011 Elsevier B.V. All rights reserved.

Proof of a conjecture of Frankl and Furedi

We give a simple linear algebraic proof of the following conjecture of Frankl and Furedi [7, 9, 13]. (Frankl-Furedi Conjecture) if F is a hypergraph on X = {1, 2, 3,..., n} such that 1 less than or equal to /E boolean AND F/ less than or equal to k For All E, F is an element of F, E not equal F, then /F/ less than or equal to (i=0)Sigma(k) ((i) (n-1)). We generalise a method of Palisse and our proof-technique can be viewed as a variant of the technique used by Tverberg to prove a result of Graham and Pollak [10, 11, 14]. Our proof-technique is easily described. First, we derive an identity satisfied by a hypergraph F using its intersection properties. From this identity, we obtain a set of homogeneous linear equations. We then show that this defines the zero subspace of R-/F/. Finally, the desired bound on /F/ is obtained from the bound on the number of linearly independent equations. This proof-technique can also be used to prove a more general theorem (Theorem 2). We conclude by indicating how this technique can be generalised to uniform hypergraphs by proving the uniform Ray-Chaudhuri-Wilson theorem. (C) 1997 Academic Press.

Redistribution Mechanisms for Assignment of Heterogeneous Objects

There are p heterogeneous objects to be assigned to n competing agents (n > p) each with unit demand. It is required to design a Groves mechanism for this assignment problem satisfying weak budget balance, individual rationality, and minimizing the budget imbalance. This calls for designing an appropriate rebate function. When the objects are identical, this problem has been solved which we refer as WCO mechanism. We measure the performance of such mechanisms by the redistribution index. We first prove an impossibility theorem which rules out linear rebate functions with non-zero redistribution index in heterogeneous object assignment. Motivated by this theorem,we explore two approaches to get around this impossibility. In the first approach, we show that linear rebate functions with non-zero redistribution index are possible when the valuations for the objects have a certain type of relationship and we design a mechanism with linear rebate function that is worst case optimal. In the second approach, we show that rebate functions with non-zero efficiency are possible if linearity is relaxed. We extend the rebate functions of the WCO mechanism to heterogeneous objects assignment and conjecture them to be worst case optimal.

Irreversibility-to-reversibility crossover in transient response of an optically trapped particle

We study the transient response of a colloidal bead which is released from different heights and allowed to relax in the potential well of an optical trap. Depending on the initial potential energy, the system's time evolution shows dramatically different behaviors. Starting from the short-time reversible to long-time irreversible transition, a stationary reversible state with zero net dissipation can be achieved as the release point energy is decreased. If the system starts with even lower energy, it progressively extracts useful work from thermal noise and exhibits an anomalous irreversibility. In addition, we have verified the Transient Fluctuation Theorem and the Integrated Transient Fluctuation Theorem even for the non-ergodic descriptions of our system. Copyright (C) EPLA, 2011

Optimal control of a stochastic hybrid system with discounted cost

We address the optimal control problem of a very general stochastic hybrid system with both autonomous and impulsive jumps. The planning horizon is infinite and we use the discounted-cost criterion for performance evaluation. Under certain assumptions, we show the existence of an optimal control. We then derive the quasivariational inequalities satisfied by the value function and establish well-posedness. Finally, we prove the usual verification theorem of dynamic programming.

Tidally compressed gas in centers of early-type and ultraluminous galaxies

In this paper we propose that the compressive tidal held in the centers of flat-core early-type galaxies and ultraluminous galaxies compresses molecular clouds producing dense gas observed in the centers of these galaxies. The effect of galactic tidal fields is usually considered disruptive in the literature. However, for some galaxies, the mass profile flattens toward the center and the resulting galactic tidal field is not disruptive, but instead it is compressive within the flat-core region. We have used the virial theorem to determine the minimum density of a molecular cloud to be stable and gravitationally bound within the tidally compressive region of a galaxy. We have applied the mechanism to determine the mean molecular cloud densities in the centers of a sample of flat-core, early-type galaxies and ultraluminous galaxies. For early-type galaxies with a core-type luminosity profile, the tidal held of the galaxy is compressive within half the core radius. We have calculated the mean gas densities for molecular gas in a sample of early-type galaxies which have already been detected in CO emission, and we obtain mean densities of [n] similar to 10(3)-10(6) cm(-3) within the central 100 pc radius. We also use our model to calculate the molecular cloud densities in the inner few hundred parsecs of a sample of ultraluminous galaxies. From the observed rotation curves of these galaxies we show that they have a compressive core within their nuclear region. Our model predicts minimum molecular gas densities in the range 10(2)-10(4) cm(-3) in the nuclear gas disks; the smaller values are applicable typically for galaxies with larger core radii. The resulting density values agree well with the observed range. Also, for large core radii, even fairly low-density gas (similar to 10(2) cm(-3)) can remain bound and stable close to the galactic center.

Surfaces of Bounded Mean Curvature In Riemannian Manifolds

Consider a sequence of closed, orientable surfaces of fixed genus g in a Riemannian manifold M with uniform upper bounds on the norm of mean curvature and area. We show that on passing to a subsequence, we can choose parametrisations of the surfaces by inclusion maps from a fixed surface of the same genus so that the distance functions corresponding to the pullback metrics converge to a pseudo-metric and the inclusion maps converge to a Lipschitz map. We show further that the limiting pseudo-metric has fractal dimension two. As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface F subset of M, together with bounds on the geometry of M, give an upper bound on the diameter of F. Our proof is modelled on Gromov's compactness theorem for J-holomorphic curves.

Fast edge splitting and Edmonds' arborescence construction for unweighted graphs

Given an unweighted undirected or directed graph with n vertices, m edges and edge connectivity c, we present a new deterministic algorithm for edge splitting. Our algorithm splits-off any specified subset S of vertices satisfying standard conditions (even degree for the undirected case and in-degree ≥ out-degree for the directed case) while maintaining connectivity c for vertices outside S in Õ(m+nc2) time for an undirected graph and Õ(mc) time for a directed graph. This improves the current best deterministic time bounds due to Gabow [8], who splits-off a single vertex in Õ(nc2+m) time for an undirected graph and Õ(mc) time for a directed graph. Further, for appropriate ranges of n, c, |S| it improves the current best randomized bounds due to Benczúr and Karger [2], who split-off a single vertex in an undirected graph in Õ(n2) Monte Carlo time. We give two applications of our edge splitting algorithms. Our first application is a sub-quadratic (in n) algorithm to construct Edmonds' arborescences. A classical result of Edmonds [5] shows that an unweighted directed graph with c edge-disjoint paths from any particular vertex r to every other vertex has exactly c edge-disjoint arborescences rooted at r. For a c edge connected unweighted undirected graph, the same theorem holds on the digraph obtained by replacing each undirected edge by two directed edges, one in each direction. The current fastest construction of these arborescences by Gabow [7] takes Õ(n2c2) time. Our algorithm takes Õ(nc3+m) time for the undirected case and Õ(nc4+mc) time for the directed case. The second application of our splitting algorithm is a new Steiner edge connectivity algorithm for undirected graphs which matches the best known bound of Õ(nc2 + m) time due to Bhalgat et al [3]. Finally, our algorithm can also be viewed as an alternative proof for existential edge splitting theorems due to Lovász [9] and Mader [11].

{C}^0$ penalty methods for the fully nonlinear Monge-Ampère equation

In this paper, we develop and analyze C(0) penalty methods for the fully nonlinear Monge-Ampere equation det(D(2)u) = f in two dimensions. The key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric, stable, and consistent with the continuous linearization. We are then able to show the well-posedness of the penalty method as well as quasi-optimal error estimates using the Banach fixed-point theorem as our main tool. Numerical experiments are presented which support the theoretical results.

Static balancing of a four-bar linkage and its cognates

Motivated by the need to statically balance the inherent elastic forces in linkages, this paper presents three techniques to statically balance a four-bar linkage loaded by a zero-free-length spring attached between its coupler point and an anchor point on the ground. The number of auxiliary links and balancing springs required for the three techniques is less than or equal to that of the only technique currently in the literature. One of the three techniques does not require auxiliary links. In these techniques, the set of values for the spring constants and the ground-anchor point of the balancing springs can vary over a one-parameter family. Thrice as many balancing choices are available when the cognates are considered. The ensuing numerous options enable a user to choose the most practical solution. To facilitate the evaluation of the balancing choices for all the cognates, Roberts-Chebyshev cognate theorem is extended to statically balanced four-bar linkages. (C) 2011 Elsevier Ltd. All rights reserved.