Constrained Molecular Dynamics Simulations of Atomic Ground-States

Constrained molecular dynamics(CoMD) model, previously introduced for nuclear dynamics, has been extended to the atomic structure and collision calculations. Quantum effects corresponding to the Pauli and Heisenberg principle are enforced by constraints, in a parameter-free way. Our calculations for small atomic system, H, He, Li, Be, F reproduce the ground-state binding energies within 3%, compared with the results of quantum mechanical Hartree-Fock calculations.Comment: 3 pages, 2 figure

Proof of some asymptotic results for a model equation for low Reynolds number flow

A two-point boundary value problem in the interval [ε, ∞], ε > 0 is studied. The problem contains additional parameters α ≥ 0, β ≥ 0, 0 ≤ U 0; for α = 0 an explicit construction shows that no solution exists unless k > 1. A special method is used to show uniqueness. For ε ↓ 0, k ≥ 1, various results had previously been obtained by the method of matched asymptotic expansions. Examples of these results are verified rigorously using the integral representation. For k < 1, the problem is shown not to be a layer-type problem, a fact previously demonstrated explicitly for k = 0. If k is an integer ≥ 0 the intuitive understanding of the problem is aided by regarding it as spherically symmetric in k + 1 dimensions. In the present study, however, k may be any real number, even negative

Viterbi Training for PCFGs: Hardness Results and Competitiveness of Uniform Initialization

We consider the search for a maximum likelihood assignment of hidden derivations and grammar weights for a probabilistic context-free grammar, the problem approximately solved by “Viterbi training.” We show that solving and even approximating Viterbi training for PCFGs is NP-hard. We motivate the use of uniformat-random initialization for Viterbi EM as an optimal initializer in absence of further information about the correct model parameters, providing an approximate bound on the log-likelihood.

Empirical Risk Minimization for Probabilistic Grammars: Sample Complexity and Hardness of Learning

Probabilistic grammars are generative statistical models that are useful for compositional and sequential structures. They are used ubiquitously in computational linguistics. We present a framework, reminiscent of structural risk minimization, for empirical risk minimization of probabilistic grammars using the log-loss. We derive sample complexity bounds in this framework that apply both to the supervised setting and the unsupervised setting. By making assumptions about the underlying distribution that are appropriate for natural language scenarios, we are able to derive distribution-dependent sample complexity bounds for probabilistic grammars. We also give simple algorithms for carrying out empirical risk minimization using this framework in both the supervised and unsupervised settings. In the unsupervised case, we show that the problem of minimizing empirical risk is NP-hard. We therefore suggest an approximate algorithm, similar to expectation-maximization, to minimize the empirical risk. Learning from data is central to contemporary computational linguistics. It is in common in such learning to estimate a model in a parametric family using the maximum likelihood principle. This principle applies in the supervised case (i.e., using annotate

An Unusual Moving Boundary Condition Arising in Anomalous Diffusion Problems

In the context of analyzing a new model for nonlinear diffusion in polymers, an unusual condition appears at the moving interface between the glassy and rubbery phases of the polymer. This condition, which arises from the inclusion of a viscoelastic memory term in our equations, has received very little attention in the mathematical literature. Due to the unusual form of the moving-boundary condition, further study is needed as to the existence and uniqueness of solutions satisfying such a condition. The moving boundary condition which results is not solvable by similarity solutions, but can be solved by integral equation techniques. A solution process is outlined to illustrate the unusual nature of the condition; the profiles which result are characteristic of a dissolving polymer

Empirical Risk Minimization with Approximations of Probabilistic Grammars

Probabilistic grammars are generative statistical models that are useful for compositional and sequential structures. We present a framework, reminiscent of structural risk minimization, for empirical risk minimization of the parameters of a fixed probabilistic grammar using the log-loss. We derive sample complexity bounds in this framework that apply both to the supervised setting and the unsupervised setting.

Joint Morphological and Syntactic Disambiguation

In morphologically rich languages, should morphological and syntactic disambiguation be treated sequentially or as a single problem? We describe several efficient, probabilistically interpretable ways to apply joint inference to morphological and syntactic disambiguation using lattice parsing. Joint inference is shown to compare favorably to pipeline parsing methods across a variety of component models. State-of-the-art performance on Hebrew Treebank parsing is demonstrated using the new method. The benefits of joint inference are modest with the current component models, but appear to increase as components themselves improve

Bifurcation of Localized Disturbances in a Model Biochemical Reaction

Asymptotic solutions are presented to the nonlinear parabolic reaction-diffusion equations describing a model biochemical reaction proposed by I. Prigogine. There is a uniform steady state which, for certain values of the adjustable parameters, may be unstable. When the uniform solution is slightly unstable, the two-timing method is used to find the bifurcation of new solutions of small amplitude. These may be either nonuniform steady states or time-periodic solutions, depending on the ratio of the diffusion coefficients. When one of the parameters is allowed to depend on space and the basic state is unstable, it is found that the nonuniform steady state which is approached may show localized spatial oscillations. The localization arises out of the presence of turning points in the linearized stability equations