A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces

Recently Alday, Gaiotto and Tachikawa proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on P^2. More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties. We propose a "finite analog" of the (above corollary of the) AGT conjecture. Namely, we replace the Uhlenbeck space with the space of based quasi-maps from P^1 to any partial flag variety G/P of G and conjecture that its equivariant intersection cohomology carries an action of the finite W-algebra U(g,e) associated with the principal nilpotent element in the Lie algebra of the Levi subgroup of P; this action is expected to satisfy some list of natural properties. This conjecture generalizes the main result of arXiv:math/0401409 when P is the Borel subgroup. We prove our conjecture for G=GL(N), using the works of Brundan and Kleshchev interpreting the algebra U(g,e) in terms of certain shifted Yangians.Comment: minor change

Spatial and temporal characterization of a Bessel beam produced using a conical mirror

We experimentally analyze a Bessel beam produced with a conical mirror, paying particular attention to its superluminal and diffraction-free properties. We spatially characterized the beam in the radial and on-axis dimensions, and verified that the central peak does not spread over a propagation distance of 73 cm. In addition, we measured the superluminal phase and group velocities of the beam in free space. Both spatial and temporal measurements show good agreement with the theoretical predictions.Comment: 5 pages, 6 figure

Maximum likelihood estimation of cloud height from multi-angle satellite imagery

We develop a new estimation technique for recovering depth-of-field from multiple stereo images. Depth-of-field is estimated by determining the shift in image location resulting from different camera viewpoints. When this shift is not divisible by pixel width, the multiple stereo images can be combined to form a super-resolution image. By modeling this super-resolution image as a realization of a random field, one can view the recovery of depth as a likelihood estimation problem. We apply these modeling techniques to the recovery of cloud height from multiple viewing angles provided by the MISR instrument on the Terra Satellite. Our efforts are focused on a two layer cloud ensemble where both layers are relatively planar, the bottom layer is optically thick and textured, and the top layer is optically thin. Our results demonstrate that with relative ease, we get comparable estimates to the M2 stereo matcher which is the same algorithm used in the current MISR standard product (details can be found in [IEEE Transactions on Geoscience and Remote Sensing 40 (2002) 1547--1559]). Moreover, our techniques provide the possibility of modeling all of the MISR data in a unified way for cloud height estimation. Research is underway to extend this framework for fast, quality global estimates of cloud height.Comment: Published in at http://dx.doi.org/10.1214/09-AOAS243 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fixed Points of Hopfield Type Neural Networks

The set of the fixed points of the Hopfield type network is under investigation. The connection matrix of the network is constructed according to the Hebb rule from the set of memorized patterns which are treated as distorted copies of the standard-vector. It is found that the dependence of the set of the fixed points on the value of the distortion parameter can be described analytically. The obtained results are interpreted in the terms of neural networks and the Ising model.Comment: RevTEX, 19 pages, 2 Postscript figures, the full version of the earler brief report (cond-mat/9901251

Bohl-Perron type stability theorems for linear difference equations with infinite delay

Relation between two properties of linear difference equations with infinite delay is investigated: (i) exponential stability, (ii) \l^p-input \l^q-state stability (sometimes is called Perron's property). The latter means that solutions of the non-homogeneous equation with zero initial data belong to \l^q when non-homogeneous terms are in \l^p. It is assumed that at each moment the prehistory (the sequence of preceding states) belongs to some weighted \l^r-space with an exponentially fading weight (the phase space). Our main result states that (i) $\Leftrightarrow$ (ii) whenever $(p,q) \neq (1,\infty)$ and a certain boundedness condition on coefficients is fulfilled. This condition is sharp and ensures that, to some extent, exponential and \l^p-input \l^q-state stabilities does not depend on the choice of a phase space and parameters $p$ and $q$, respectively. \l^1-input \l^\infty-state stability corresponds to uniform stability. We provide some evidence that similar criteria should not be expected for non-fading memory spaces.Comment: To be published in Journal of Difference Equations and Application

Essential self-adjointness of magnetic Schr\"odinger operators on locally finite graphs

We give sufficient conditions for essential self-adjointness of magnetic Schr\"odinger operators on locally finite graphs. Two of the main theorems of the present paper generalize recent results of Torki-Hamza.Comment: 14 pages; The present version differs from the original version as follows: the ordering of presentation has been modified in several places, more details have been provided in several places, some notations have been changed, two examples have been added, and several new references have been inserted. The final version of this preprint will appear in Integral Equations and Operator Theor

Reconsidering the generation time hypothesis based on nuclear ribosomal ITS sequence comparisons in annual and perennial angiosperms

17 pages, 3 figures, 5 tables.-- PMID: 19113991 [PubMed].[Background] Differences in plant annual/perennial habit are hypothesized to cause a generation time effect on divergence rates. Previous studies that compared rates of divergence for internal transcribed spacer (ITS1 and ITS2) sequences of nuclear ribosomal DNA (nrDNA) in angiosperms have reached contradictory conclusions about whether differences in generation times (or other life history features) are associated with divergence rate heterogeneity. We compared annual/perennial ITS divergence rates using published sequence data, employing sampling criteria to control for possible artifacts that might obscure any actual rate variation caused by annual/perennial differences.[Results] Relative rate tests employing ITS sequences from 16 phylogenetically-independent annual/perennial species pairs rejected rate homogeneity in only a few comparisons, with annuals more frequently exhibiting faster substitution rates. Treating branch length differences categorically (annual faster or perennial faster regardless of magnitude) with a sign test often indicated an excess of annuals with faster substitution rates. Annuals showed an approximately 1.6-fold rate acceleration in nucleotide substitution models for ITS. Relative rates of three nuclear loci and two chloroplast regions for the annual Arabidopsis thaliana compared with two closely related Arabidopsis perennials indicated that divergence was faster for the annual. In contrast, A. thaliana ITS divergence rates were sometimes faster and sometimes slower than the perennial. In simulations, divergence rate differences of at least 3.5-fold were required to reject rate constancy in > 80 % of replicates using a nucleotide substitution model observed for the combination of ITS1 and ITS2. Simulations also showed that categorical treatment of branch length differences detected rate heterogeneity > 80% of the time with a 1.5-fold or greater rate difference.[Conclusion] Although rate homogeneity was not rejected in many comparisons, in cases of significant rate heterogeneity annuals frequently exhibited faster substitution rates. Our results suggest that annual taxa may exhibit a less than 2-fold rate acceleration at ITS. Since the rate difference is small and ITS lacks statistical power to reject rate homogeneity, further studies with greater power will be required to adequately test the hypothesis that annual and perennial plants have heterogeneous substitution rates. Arabidopsis sequence data suggest that relative rate tests based on multiple loci may be able to distinguish a weak acceleration in annual plants. The failure to detect rate heterogeneity with ITS in past studies may be largely a product of low statistical power.This work was supported by a doctoral fellowship to D. F. Soria-Hernanz from the Spanish Ministerio de Educación y Ciencia, graduate support from Georgetown University and the Department of Biology, the Cosmos Foundation, and a National Science Foundation grant to M.B.H. (DEB9983014). Publication charges supported by the Department of Biology, Georgetown University.Peer reviewe

Crystal constructions in Number Theory

Weyl group multiple Dirichlet series and metaplectic Whittaker functions can be described in terms of crystal graphs. We present crystals as parameterized by Littelmann patterns and we give a survey of purely combinatorial constructions of prime power coefficients of Weyl group multiple Dirichlet series and metaplectic Whittaker functions using the language of crystal graphs. We explore how the branching structure of crystals manifests in these constructions, and how it allows access to some intricate objects in number theory and related open questions using tools of algebraic combinatorics

Vortex structure of thin mesoscopic disks in the presence of an inhomogeneous magnetic field

The vortex states in a thin mesoscopic disk are investigated within the phenomenological Ginzburg-Landau theory in the presence of different ''model'' magnetic field profiles with zero average field which may result from a ferromagnetic disk or circulating currents in a loop near the superconductor. We calculated the dependences of both the ground and metastable states on the magnitude and shape of the magnetic field profile for different values of the order parameter angular moment, i.e. the vorticity. The regions of existence of the multi-vortex state and the giant vortex state are found. We analysed the phase transitions between these states and studied the contribution from ring-shaped vortices. A new transition between different multi-vortex configurations as the ground state is found. Furthermore, we found a vortex state consisting of a central giant vortex surrounded by a collection of anti-vortices which are located in a ring around this giant vortex. The limit to a disk with an infinite radius, i.e. a film, will also be discussed. We also extended our results to ''real'' magnetic field profiles and to the case in which an external homogeneous magnetic field is present.Comment: 17 pages, 23 figures. Submitted to PR

Representing complex data using localized principal components with application to astronomical data

Often the relation between the variables constituting a multivariate data space might be characterized by one or more of the terms: ``nonlinear'', ``branched'', ``disconnected'', ``bended'', ``curved'', ``heterogeneous'', or, more general, ``complex''. In these cases, simple principal component analysis (PCA) as a tool for dimension reduction can fail badly. Of the many alternative approaches proposed so far, local approximations of PCA are among the most promising. This paper will give a short review of localized versions of PCA, focusing on local principal curves and local partitioning algorithms. Furthermore we discuss projections other than the local principal components. When performing local dimension reduction for regression or classification problems it is important to focus not only on the manifold structure of the covariates, but also on the response variable(s). Local principal components only achieve the former, whereas localized regression approaches concentrate on the latter. Local projection directions derived from the partial least squares (PLS) algorithm offer an interesting trade-off between these two objectives. We apply these methods to several real data sets. In particular, we consider simulated astrophysical data from the future Galactic survey mission Gaia.Comment: 25 pages. In "Principal Manifolds for Data Visualization and Dimension Reduction", A. Gorban, B. Kegl, D. Wunsch, and A. Zinovyev (eds), Lecture Notes in Computational Science and Engineering, Springer, 2007, pp. 180--204, http://www.springer.com/dal/home/generic/search/results?SGWID=1-40109-22-173750210-