Topological invariants of isolated complete intersection curve singularities

In this paper we present some formulae for topological invariants of projective complete intersection curves with isolated singularities in terms of the Milnor number, the Euler characteristic and the topological genus. We also present some conditions, involving the Milnor number and the degree of the curve, for the irreducibility of complete intersection curves.

Invariants of binary differential equations

In this paper, we study binary differential equations a(x, y)dy (2) + 2b(x, y) dx dy + c(x, y)dx (2) = 0, where a, b, and c are real analytic functions. Following the geometric approach of Bruce and Tari in their work on multiplicity of implicit differential equations, we introduce a definition of the index for this class of equations that coincides with the classical Hopf`s definition for positive binary differential equations. Our results also apply to implicit differential equations F(x, y, p) = 0, where F is an analytic function, p = dy/dx, F (p) = 0, and F (pp) not equal aEuro parts per thousand 0 at the singular point. For these equations, we relate the index of the equation at the singular point with the index of the gradient of F and index of the 1-form omega = dy -aEuro parts per thousand pdx defined on the singular surface F = 0.

Conformal deformation of equilibrium measures in normal random ensembles

We investigate the eigenvalue statistics of ensembles of normal random matrices when their order N tends to infinite. In the model, the eigenvalues have uniform density within a region determined by a simple analytic polynomial curve. We study the conformal deformations of equilibrium measures of normal random ensembles to the real line and give sufficient conditions for it to weakly converge to a Wigner measure.

A general treatment of geometric phases and dynamical invariants

Based only on the parallel-transport condition, we present a general method to compute Abelian or non-Abelian geometric phases acquired by the basis states of pure or mixed density operators, which also holds for nonadiabatic and noncyclic evolution. Two interesting features of the non-Abelian geometric phase obtained by our method stand out: i) it is a generalization of Wilczek and Zee`s non-Abelian holonomy, in that it describes nonadiabatic evolution where the basis states are parallelly transported between distinct degenerate subspaces, and ii) the non-Abelian character of our geometric phase relies on the transitional evolution of the basis states, even in the nondegenerate case. We apply our formalism to a two-level system evolving nonadiabatically under spontaneous decay to emphasize the non- Abelian nature of the geometric phase induced by the reservoir. We also show, through the generalized invariant theory, that our general approach encompasses previous results in the literature. Copyright (c) EPLA, 2008.

A conformal invariant growth model

We present a one-parameter extension of the raise and peel one-dimensional growth model. The model is defined in the configuration space of Dyck (RSOS) paths. Tiles from a rarefied gas hit the interface and change its shape. The adsorption rates are local but the desorption rates are non-local; they depend not only on the cluster hit by the tile but also on the total number of peaks (local maxima) belonging to all the clusters of the configuration. The domain of the parameter is determined by the condition that the rates are non-negative. In the finite-size scaling limit, the model is conformal invariant in the whole open domain. The parameter appears in the sound velocity only. At the boundary of the domain, the stationary state is an adsorbing state and conformal invariance is lost. The model allows us to check the universality of non-local observables in the raise and peel model. An example is given.

Quantized fields and gravitational particle creation in f (R) expanding universes

The problem of cosmological particle creation for a spatially flat, homogeneous and isotropic universes is discussed in the context of f (R) theories of gravity. Different from cosmological models based on general relativity theory, it is found that a conformal invariant metric does not forbid the creation of massless particles during the early stages (radiation era) of the universe. (C) 2010 Elsevier B.V. All rights reserved.

The analytic torsion of a cone over an odd dimensional manifold

We study the analytic torsion of a cone over an orientable odd dimensional compact connected Riemannian manifold W. We prove that the logarithm of the analytic torsion of the cone decomposes as the sum of the logarithm of the root of the analytic torsion of the boundary of the cone, plus a topological term, plus a further term that is a rational linear combination of local Riemannian invariants of the boundary. We show that this last term coincides with the anomaly boundary term appearing in the Cheeger Muller theorem [3, 2] for a manifold with boundary, according to Bruning and Ma (2006) [5]. We also prove Poincare duality for the analytic torsion of a cone. (C) 2010 Elsevier B.V. All rights reserved.

Invariant theory and reversible-equivariant vector fields

In this paper we present results for the systematic study of reversible-equivariant vector fields - namely, in the simultaneous presence of symmetries and reversing symmetries - by employing algebraic techniques from invariant theory for compact Lie groups. The Hilbert-Poincare series and their associated Molien formulae are introduced,and we prove the character formulae for the computation of dimensions of spaces of homogeneous anti-invariant polynomial functions and reversible-equivariant polynomial mappings. A symbolic algorithm is obtained for the computation of generators for the module of reversible-equivariant polynomial mappings over the ring of invariant polynomials. We show that this computation can be obtained directly from a well-known situation, namely from the generators of the ring of invariants and the module of the equivariants. (C) 2008 Elsevier B.V, All rights reserved.

Finite determinacy and Whitney equisingularity of map germs from C(n) to C(2n-1)

We show that a holomorphic map germ f : (C(n), 0) -> (C(2n-1), 0) is finitely determined if and only if the double point scheme D(f) is a reduced curve. If n >= 3, we have that mu(D(2)(f)) = 2 mu(D(2)(f)/S(2))+C(f)-1, where D(2)(f) is the lifting of the double point curve in (C(n) x C(n), 0), mu(X) denotes the Milnor number of X and C(f) is the number of cross-caps that appear in a stable deformation of f. Moreover, we consider an unfolding F(t, x) = (t, f(t)(x)) of f and show that if F is mu-constant, then it is excellent in the sense of Gaffney. Finally, we find a minimal set of invariants whose constancy in the family f(t) is equivalent to the Whitney equisingularity of F. We also give an example of an unfolding which is topologically trivial, but it is not Whitney equisingular.

Determination of Very Slow Diffusion of Paramagnetic Ions by NMR Spectroscopy

In this paper, we propose a new method of measuring the very slow paramagnetic ion diffusion coefficient using a commercial high-resolution spectrometer. If there are distinct paramagnetic ions influencing the hydrogen nuclear magnetic relaxation time differently, their diffusion coefficients can be measured separately. A cylindrical phantom filled with Fricke xylenol gel solution and irradiated with gamma rays was used to validate the method. The Fricke xylenol gel solution was prepared with 270 Bloom porcine gelatin, the phantom was irradiated with gamma rays originated from a (60)Co source and a high-resolution 200 MHz nuclear magnetic resonance (NMR) spectrometer was used to obtain the phantom (1)H profile in the presence of a linear magnetic field gradient. By observing the temporal evolution of the phantom NMR profile, an apparent ferric ion diffusion coefficient of 0.50 mu m(2)/ms due to ferric ions diffusion was obtained. In any medical process where the ionizing radiation is used, the dose planning and the dose delivery are the key elements for the patient safety and success of treatment. These points become even more important in modern conformal radio therapy techniques, such as stereotactic radiosurgery, where the delivered dose in a single session of treatment can be an order of magnitude higher than the regular doses of radiotherapy. Several methods have been proposed to obtain the three-dimensional (3-D) dose distribution. Recently, we proposed an alternative method for the 3-D radiation dose mapping, where the ionizing radiation modifies the local relative concentration of Fe(2+)/Fe(3+) in a phantom containing Fricke gel and this variation is associated to the MR image intensity. The smearing of the intensity gradient is proportional to the diffusion coefficient of the Fe(3+) and Fe(2+) in the phantom. There are several methods for measurement of the ionic diffusion using NMR, however, they are applicable when the diffusion is not very slow.

Fermion perturbations in string theory black holes

In this paper we study fermion perturbations in four-dimensional black holes of string theory, obtained either from a non-extreme configuration of three intersecting five-branes with a boost along the common string or from a non-extreme intersecting system of two two-branes and two five-branes. The Dirac equation for the massless neutrino field, after conformal re-scaling of the metric, is written as a wave equation suitable to study the time evolution of the perturbation. We perform a numerical integration of the evolution equation, and with the aid of Prony fitting of the time-domain profile, we calculate the complex frequencies that dominate the quasinormal ringing stage, and also determine these quantities by the semi-analytical sixth-order WKB method. We also find numerically the decay factor of fermion fields at very late times, and show that the falloff is identical to those showing for massless fields in other four-dimensional black hole spacetimes.

Quantum current algebra for the AdS(5) x S(5) superstring

The sigma model describing the dynamics of the superstring in the AdS(5) x S(5) background can be constructed using the coset PSU(2, 2 vertical bar 4)/SO(4, 1) x SO(5). A basic set of operators in this two dimensional conformal field theory is composed by the left invariant currents. Since these currents are not (anti) holomorphic, their OPE`s is not determined by symmetry principles and its computation should be performed perturbatively. Using the pure spinor sigma model for this background, we compute the one-loop correction to these OPE`s. We also compute the OPE`s of the left invariant currents with the energy momentum tensor at tree level and one loop.

Hierarchical spherical model from a geometric point of view

A continuous version of the hierarchical spherical model at dimension d=4 is investigated. Two limit distributions of the block spin variable X(gamma), normalized with exponents gamma = d + 2 and gamma=d at and above the critical temperature, are established. These results are proven by solving certain evolution equations corresponding to the renormalization group (RG) transformation of the O(N) hierarchical spin model of block size L(d) in the limit L down arrow 1 and N ->infinity. Starting far away from the stationary Gaussian fixed point the trajectories of these dynamical system pass through two different regimes with distinguishable crossover behavior. An interpretation of this trajectories is given by the geometric theory of functions which describe precisely the motion of the Lee-Yang zeroes. The large-N limit of RG transformation with L(d) fixed equal to 2, at the criticality, has recently been investigated in both weak and strong (coupling) regimes by Watanabe (J. Stat. Phys. 115:1669-1713, 2004) . Although our analysis deals only with N = infinity case, it complements various aspects of that work.

Conservation laws and tachyon potentials in the sliver frame

Conservation laws have provided an elegant and efficient tool to evaluate the open string field theory interaction vertex, they have been originally implemented in the case where the string field is expanded in the Virasoro basis. In this work we derive conservation laws in the case where the string field is expanded in the so-called sliver L(0)-basis. As an application of this new set of conservation laws, we compute the open string field action relevant to the tachyon condensation and in order to present not only an illustration but also an additional information, we evaluate the action without imposing a gauge choice.

Shared information in stationary states at criticality

We consider bipartitions of one-dimensional extended systems whose probability distribution functions describe stationary states of stochastic models. We define estimators of the information shared between the two subsystems. If the correlation length is finite, the estimators stay finite for large system sizes. If the correlation length diverges, so do the estimators. The definition of the estimators is inspired by information theory. We look at several models and compare the behaviors of the estimators in the finite-size scaling limit. Analytical and numerical methods as well as Monte Carlo simulations are used. We show how the finite-size scaling functions change for various phase transitions, including the case where one has conformal invariance.