Nonstationary Markov chains and convergence of the annealing algorithm

We study the asymptotic behavior as timet + of certain nonstationary Markov chains, and prove the convergence of the annealing algorithm in Monte Carlo simulations. We find that in the limitt + , a nonstationary Markov chain may exhibit phase transitions. Nonstationary Markov chains in general, and the annealing algorithm in particular, lead to biased estimators for the expectation values of the process. We compute the leading terms in the bias and the variance of the sample-means estimator. We find that the annealing algorithm converges if the temperatureT(t) goes to zero no faster thanC/log(t/t ₀) ast+, with a computable constantC andt ₀ the initial time. The bias and the variance of the sample-means estimator in the annealing algorithm go to zero likeO(t^–1+) for some 0<1, with =0 only in very special circumstances. Our results concerning the convergence of the annealing algorithm, and the rate of convergence to zero of the bias and the variance of the sample-means estimator, provide a rigorous procedure for choosing the optimal annealing schedule. This optimal choice reflects the competition between two physical effects: (a) The adiabatic effect, whereby if the temperature is loweredtoo abruptly the system may end up not in a ground state but in a nearby metastable state, and (b) the super-cooling effect, whereby if the temperature is loweredtoo slowly the system will indeed approach the ground state(s) but may do so extremely slowly.     

On defining OLAP formulations

^** Email: vutsinas{at}upatras.gr Recently there has been increasing interest in On Line AnalyticalProcessing (OLAP) to satisfy the organizational needs of high-levelinformation delivery and advanced data analysis. The actualapplication of OLAP tools involves the use of various functions,such as the common drilling down and slicing and dicing. Usuallyeach particular OLAP function is comprehensive and intuitive.However, sophisticated use of OLAP tools requires complicatedcombinations of different OLAP functions that are not straight-forwardfor end users or designers. In this paper we attempt to enumerateand formally define OLAP functions by defining a new OLAP modelthat provides a broader view of OLAP. We demonstrate the expressiveadequacy of the new OLAP model with application examples.     

Perturbations of spherically symmetric black holes

The linearized Einstein-Maxwell equations on a Reissner-Nördstrom background are reduced to two (decoupled) first order ordinary differential equations which determine the even and the odd parity perturbations via quadratures.     

Optimal transformations for prediction in continuous-time stochastic processes: finite past and future

In the classical Wiener-Kolmogorov linear prediction problem, one fixes a linear functional in the future of a stochastic process, and seeks its best predictor (in the L²-sense). In this paper we treat a variant of the prediction problem, whereby we seek the most predictable non-trivial functional of the future and its best predictor; we refer to such a pair (if it exists) as an optimal transformation for prediction. In contrast to the Wiener-Kolmogorov problem, an optimal transformation for prediction may not exist, and if it exists, it may not be unique. We prove the existence of optimal transformations for finite past and future intervals, under appropriate conditions on the spectral density of a weakly stationary, continuous-time stochastic process. For rational spectral densities, we provide an explicit construction of the transformations via differential equations with boundary conditions and an associated eigenvalue problem of a finite matrix.This research was partially supported by ARO (MURI grant) DAAH04-96-1-0445, NSF grant DMS-0074276, and CNPq grant 301179/00-0.     

Machine-part cell formation using biclustering

Cellular manufacturing is the cornerstone of many modern flexible manufacturing techniques, taking advantage of the similarities between parts in order to decrease the complexity of the design and manufacturing life cycle. Part-Machine Grouping (PMG) problem is the key step in cellular manufacturing aiming at grouping parts with similar processing requirements or similar design features into part families and by grouping machines into cells associated to these families. The PMG problem is NP-complete and the different proposed techniques for solving it are based on heuristics. In this paper, a new approach for solving the PMG problem is proposed which is based on biclustering. Biclustering is a methodology where rows and columns of an input data matrix are clustered simultaneously. A bicluster is defined as a submatrix spanned by both a subset of rows and a subset of columns. Although biclustering has been almost exclusively applied to DNA microarray analysis, we present that biclustering can be successfully applied to the PMG problem. We also present empirical results to demonstrate the efficiency and accuracy of the proposed technique with respect to related ones for various formations of the problem.     

A geometric notion of complete integrability

A system of partial differential equations which can be described as a harmonic mapping of riemannian manifolds is called completely integrable when the corresponding n-dimensional manifold of fields admits 2n?1 independent Killing vector fields. It is conjectured that, for systems of two independent variables, complete integrability in the present sense implies the existence of a Lax pair for the system, for which the theory of the inverse scattering method is applicable. The stationary axisymmetric Einstein and Einstein-Maxwell equations, the SU(n) self-dual Yang-Mills fields in 1+1 dimensions, and the two-dimensional non-linear σ-models are shown to satisfy the conjecture; the conjecture is also proved for any system of n = 2 and n = 3 partial differential equations for n unknown scalar fields.     

Sex pheromone biosynthesis in the female olive fruit-fly. Double labelling from [18O2]-dioxygen into 1,7-dioxaspiro[5.5]undecane

The demonstration that both oxygen atoms of 1,7-dioxaspiro[5.5]undecane (1), the sex-pheromone of the female olive fly, originate from dioxygen, strongly implicates monooxygenase mediated processes in assembly of (1), and reveals unexpected complexity in the formation of its nine-carbon precursor.     

The optical scalars in Kerr-Schild-type spacetimes

The optical scalars are defined and the optical equations are derived for a null, geodesic, but not necessarily affinely parametrized, vector field. The optical scalars are left invariant under the action of a Kerr-Schild-type transformation $\text{g}\text{?}{}_{\mathrm{ab}}\text{= g}{}_{\mathrm{ab}}\text{+ l}{}_{\mathrm{a}}\text{l}{}_{\mathrm{b}}$. The optical equations are integrated explicitly in Kerr-Schild-type spacetimes. As an application, the search for Kerr-Schild-type N spacetimes with twisting rays is formulated.