Critical sets of nonlinear Sturm-Liouville operators of Ambrosetti-Prodi type

The critical set C of the operator F:H^2_D([0,pi]) -> L^2([0,pi]) defined by F(u)=-u''+f(u) is studied. Here X:=H^2_D([0,pi]) stands for the set of functions that satisfy the Dirichlet boundary conditions and whose derivatives are in L^2([0,pi]). For generic nonlinearities f, C=\cup C_k decomposes into manifolds of codimension 1 in X. If f''0, the set C_j is shown to be non-empty if, and only if, -j^2 (the j-th eigenvalue of u -> u'') is in the range of f'. The critical components C_k are (topological) hyperplanes.Comment: 6 pages, no figure

The fast sampling algorithm for Lie-Trotter products

A fast algorithm for path sampling in path integral Monte Carlo simulations is proposed. The algorithm utilizes the Levy-Ciesielski implementation of Lie-Trotter products to achieve a mathematically proven computational cost of n*log_2(n) with the number of time slices n, despite the fact that each path variable is updated separately, for reasons of optimality. In this respect, we demonstrate that updating a group of random variables simultaneously results in loss of efficiency.Comment: 4 pages, 1 figure; fast rejection from Phys. Rev. Letts; transfered to PRE as a Rapid Communication. Eq. 6 to 10 contained some inconsistencies that have been repaired in the present version; A sample code implementing the algorithm for LJ clusters is available from the author upon reques

A flux-ratio anomaly in the CO spectral line emission from gravitationally-lensed quasar MG J0414+0534

We present an analysis of archival observations with the Atacama Large (sub-)Millimetre Array (ALMA) of the gravitationally lensed quasar MG J0414+0534, which show four compact images of the quasar and an Einstein ring from the dust associated with the quasar host galaxy. We confirm that the flux-ratio anomalies observed in the mid-infrared and radio persists into the sub-mm for the continuum images of the quasar. We report the detection of CO (11-10) spectral line emission, which traces a region of compact gas around the quasar nucleus. This line emission also shows evidence of a flux-ratio anomaly between the merging lensed images that is consistent with those observed at other wavelengths, suggesting high-excitation CO can also provide a useful probe of substructures that is unaffected by microlensing or dust extinction. However, we do not detect the candidate dusty dwarf galaxy that was previously reported with this dataset, which we conclude is due to a noise artefact. Thus, the cause of the flux-ratio anomaly between the merging lensed images is still unknown. The composite compact and diffuse emission in this system suggest lensed quasar-starbursts will make excellent targets for detecting dark sub-haloes and testing models for dark matter.Comment: Accepted as MNRAS Lette

Volatility and dividend risk in perpetual American options

American options are financial instruments that can be exercised at any time before expiration. In this paper we study the problem of pricing this kind of derivatives within a framework in which some of the properties --volatility and dividend policy-- of the underlaying stock can change at a random instant of time, but in such a way that we can forecast their final values. Under this assumption we can model actual market conditions because some of the most relevant facts that may potentially affect a firm will entail sharp predictable effects. We will analyse the consequences of this potential risk on perpetual American derivatives, a topic connected with a wide class of recurrent problems in physics: holders of American options must look for the fair price and the optimal exercise strategy at once, a typical question of free absorbing boundaries. We present explicit solutions to the most common contract specifications and derive analytical expressions concerning the mean and higher moments of the exercise time.Comment: 21 pages, 5 figures, iopart, submitted for publication; deep revision, two new appendice

Persistence of Randomly Coupled Fluctuating Interfaces

We study the persistence properties in a simple model of two coupled interfaces characterized by heights h_1 and h_2 respectively, each growing over a d-dimensional substrate. The first interface evolves independently of the second and can correspond to any generic growing interface, e.g., of the Edwards-Wilkinson or of the Kardar-Parisi-Zhang variety. The evolution of h_2, however, is coupled to h_1 via a quenched random velocity field. In the limit d\to 0, our model reduces to the Matheron-de Marsily model in two dimensions. For d=1, our model describes a Rouse polymer chain in two dimensions advected by a transverse velocity field. We show analytically that after a long waiting time t_0\to \infty, the stochastic process h_2, at a fixed point in space but as a function of time, becomes a fractional Brownian motion with a Hurst exponent, H_2=1-\beta_1/2, where \beta_1 is the growth exponent characterizing the first interface. The associated persistence exponent is shown to be \theta_s^2=1-H_2=\beta_1/2. These analytical results are verified by numerical simulations.Comment: 15 pages, 3 .eps figures include

On the efficient Monte Carlo implementation of path integrals

We demonstrate that the Levy-Ciesielski implementation of Lie-Trotter products enjoys several properties that make it extremely suitable for path-integral Monte Carlo simulations: fast computation of paths, fast Monte Carlo sampling, and the ability to use different numbers of time slices for the different degrees of freedom, commensurate with the quantum effects. It is demonstrated that a Monte Carlo simulation for which particles or small groups of variables are updated in a sequential fashion has a statistical efficiency that is always comparable to or better than that of an all-particle or all-variable update sampler. The sequential sampler results in significant computational savings if updating a variable costs only a fraction of the cost for updating all variables simultaneously or if the variables are independent. In the Levy-Ciesielski representation, the path variables are grouped in a small number of layers, with the variables from the same layer being statistically independent. The superior performance of the fast sampling algorithm is shown to be a consequence of these observations. Both mathematical arguments and numerical simulations are employed in order to quantify the computational advantages of the sequential sampler, the Levy-Ciesielski implementation of path integrals, and the fast sampling algorithm.Comment: 14 pages, 3 figures; submitted to Phys. Rev.

2-D constrained Navier-Stokes equation and intermediate asymptotics

We introduce a modified version of the two-dimensional Navier-Stokes equation, preserving energy and momentum of inertia, which is motivated by the occurrence of different dissipation time scales and related to the gradient flow structure of the 2-D Navier-Stokes equation. The hope is to understand intermediate asymptotics. The analysis we present here is purely formal. A rigorous study of this equation will be done in a forthcoming paper

Quantum Effective Action in Spacetimes with Branes and Boundaries

We construct quantum effective action in spacetime with branes/boundaries. This construction is based on the reduction of the underlying Neumann type boundary value problem for the propagator of the theory to that of the much more manageable Dirichlet problem. In its turn, this reduction follows from the recently suggested Neumann-Dirichlet duality which we extend beyond the tree level approximation. In the one-loop approximation this duality suggests that the functional determinant of the differential operator subject to Neumann boundary conditions in the bulk factorizes into the product of its Dirichlet counterpart and the functional determinant of a special operator on the brane -- the inverse of the brane-to-brane propagator. As a byproduct of this relation we suggest a new method for surface terms of the heat kernel expansion. This method allows one to circumvent well-known difficulties in heat kernel theory on manifolds with boundaries for a wide class of generalized Neumann boundary conditions. In particular, we easily recover several lowest order surface terms in the case of Robin and oblique boundary conditions. We briefly discuss multi-loop applications of the suggested Dirichlet reduction and the prospects of constructing the universal background field method for systems with branes/boundaries, analogous to the Schwinger-DeWitt technique.Comment: LaTeX, 25 pages, final version, to appear in Phys. Rev.

The Casimir Effect for Parallel Plates Revisited

The Casimir effect for a massless scalar field with Dirichlet and periodic boundary conditions (b.c.) on infinite parallel plates is revisited in the local quantum field theory (lqft) framework introduced by B.Kay. The model displays a number of more realistic features than the ones he treated. In addition to local observables, as the energy density, we propose to consider intensive variables, such as the energy per unit area $\epsilon$, as fundamental observables. Adopting this view, lqft rejects Dirichlet (the same result may be proved for Neumann or mixed) b.c., and accepts periodic b.c.: in the former case $\epsilon$ diverges, in the latter it is finite, as is shown by an expression for the local energy density obtained from lqft through the use of the Poisson summation formula. Another way to see this uses methods from the Euler summation formula: in the proof of regularization independence of the energy per unit area, a regularization-dependent surface term arises upon use of Dirichlet b.c. but not periodic b.c.. For the conformally invariant scalar quantum field, this surface term is absent, due to the condition of zero trace of the energy momentum tensor, as remarked by B.De Witt. The latter property does not hold in tha application to the dark energy problem in Cosmology, in which we argue that periodic b.c. might play a distinguished role.Comment: 25 pages, no figures, late

The trace of the heat kernel on a compact hyperbolic 3-orbifold

The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold H^3/\Ga are evaluated in the case in which the discrete group \Ga contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expansion, but the Weyl term, which remains unchanged. Some physical consequences are briefly discussed in the examples.Comment: 11 page