Approximation with Random Bases: Pro et Contra

In this work we discuss the problem of selecting suitable approximators from families of parameterized elementary functions that are known to be dense in a Hilbert space of functions. We consider and analyze published procedures, both randomized and deterministic, for selecting elements from these families that have been shown to ensure the rate of convergence in $L_2$ norm of order $O(1/N)$, where $N$ is the number of elements. We show that both randomized and deterministic procedures are successful if additional information about the families of functions to be approximated is provided. In the absence of such additional information one may observe exponential growth of the number of terms needed to approximate the function and/or extreme sensitivity of the outcome of the approximation to parameters. Implications of our analysis for applications of neural networks in modeling and control are illustrated with examples.Comment: arXiv admin note: text overlap with arXiv:0905.067

CARD-660: Cambridge rare word dataset - A reliable benchmark for infrequent word representation models

Rare word representation has recently enjoyed a surge of interest, owing to the crucial role that effective handling of infrequent words can play in accurate semantic understanding. However, there is a paucity of reliable benchmarks for evaluation and comparison of these techniques. We show in this paper that the only existing benchmark (the Stanford Rare Word dataset) suffers from low-confidence annotations and limited vocabulary; hence, it does not constitute a solid comparison framework. In order to fill this evaluation gap, we propose CAmbridge Rare word Dataset (CARD-660), an expert-annotated word similarity dataset which provides a highly reliable, yet challenging, benchmark for rare word representation techniques. Through a set of experiments we show that even the best mainstream word embeddings, with millions of words in their vocabularies, are unable to achieve performances higher than 0.43 (Pearson correlation) on the dataset, compared to a human-level upperbound of 0.90. We release the dataset and the annotation materials at https://pilehvar.github.io/card-660/

Metastable phases and "metastable" phase diagrams

The work discusses specifics of phase transitions for metastable states of substances. The objects of condensed media physics are primarily equilibrium states of substances with metastable phases viewed as an exception, while the overwhelming majority of organic substances investigated in chemistry are metastable. It turns out that at normal pressure many of simple molecular compounds based on light elements (these include: most hydrocarbons; nitrogen oxides, hydrates, and carbides; carbon oxide (CO); alcohols, glycerin etc) are metastable substances too, i.e. they do not match the Gibbs' free energy minimum for a given chemical composition. At moderate temperatures and pressures, the phase transitions for given metastable phases throughout the entire experimentally accessible time range are reversible with the equilibrium thermodynamics laws obeyed. At sufficiently high pressures (1-10 GPa), most of molecular phases irreversibly transform to more energy efficient polymerized phases, both stable and metastable. These transformations are not consistent with the equality of the Gibbs' free energies between the phases before and after the transition, i.e. they are not phase transitions in "classical" meaning. The resulting polymeric phases at normal pressure can exist at temperatures above the melting one for the initial metastable molecular phase. Striking examples of such polymers are polyethylene and a polymerized modification of CO. Many of energy-intermediate polymeric phases can apparently be synthesized by the "classical" chemistry techniques at normal pressure.Comment: 5 pages, 4 figure

Trans-sonic propeller stage

We follow the approach used by Davies and Pringle (1981) and discuss the trans-sonic substage of the propeller regime. This substage is intermediate between the supersonic and subsonic propeller substages. In the trans-sonic regime an envelope around a magnetosphere of a neutron star passes through a kind of a reorganization process. The envelope in this regime consists of two parts. In the bottom one turbulent motions are subsonic. Then at some distance $r_\mathrm{s}$ the turbulent velocity becomes equal to the sound velocity. During this substage the boundary $r_\mathrm{s}$ propagates outwards till it reaches the outer boundary, and so the subsonic regime starts. We found that the trans-sonic substage is unstable, so the transition between supersonic and subsonic substages proceeds on the dynamical time scale. For realistic parameters this time is in the range from weeks to years.Comment: 8 pages with figures, submitted to Astron. Astroph. Transaction

Impact of the strong electromagnetic field on the QCD effective potential for homogeneous Abelian gluon field configurations

The one-loop quark contribution to the QCD effective potential for the homogeneous Abelian gluon field in the presence of external strong electromagnetic field is evaluated. The structure of extrema of the potential as a function of the angles between chromoelectric, chromomagnetic and electromagnetic fields is analyzed. In this setup, the electromagnetic field is considered as an external one while the gluon field represents domain structured nonperturbative gluon configurations related to the QCD vacuum in the confinement phase. Two particularly interesting gluon configurations, (anti-)self-dual and crossed orthogonal chromomagnetic and chromoelectric fields, are discussed specifically. Within this simplified framework it is shown that the strong electromagnetic fields can play a catalysing role for a deconfinement transition. At the qualitative level, the present consideration can be seen as a highly simplified study of an impact of the electromagnetic fields generated in relativistic heavy ion collisions on the strongly interacting hadronic matter.Comment: 9 pages, 4 figure