32,330 research outputs found
N=1 G_2 SYM theory and Compactification to Three Dimensions
We study four dimensional N=2 G_2 supersymmetric gauge theory on R^3\times
S^1 deformed by a tree level superpotential. We will show that the exact
superpotential can be obtained by making use of the Lax matrix of the
corresponding integrable model which is the periodic Toda lattice based on the
dual of the affine G_2 Lie algebra. At extrema of the superpotential the
Seiberg-Witten curve typically factorizes, and we study the algebraic equations
underlying this factorization. For U(N) theories the factorization was closely
related to the geometrical engineering of such gauge theories and to matrix
model descriptions, but here we will find that the geometrical interpretation
is more mysterious. Along the way we give a method to compute the gauge theory
resolvent and a suitable set of one-forms on the Seiberg-Witten curve. We will
also find evidence that the low-energy dynamics of G_2 gauge theories can
effectively be described in terms of an auxiliary hyperelliptic curve.Comment: 27 pages, late
Mixtures of Spatial Spline Regressions
We present an extension of the functional data analysis framework for
univariate functions to the analysis of surfaces: functions of two variables.
The spatial spline regression (SSR) approach developed can be used to model
surfaces that are sampled over a rectangular domain. Furthermore, combining SSR
with linear mixed effects models (LMM) allows for the analysis of populations
of surfaces, and combining the joint SSR-LMM method with finite mixture models
allows for the analysis of populations of surfaces with sub-family structures.
Through the mixtures of spatial splines regressions (MSSR) approach developed,
we present methodologies for clustering surfaces into sub-families, and for
performing surface-based discriminant analysis. The effectiveness of our
methodologies, as well as the modeling capabilities of the SSR model are
assessed through an application to handwritten character recognition
Randomized Riemannian Preconditioning for Orthogonality Constrained Problems
Optimization problems with (generalized) orthogonality constraints are
prevalent across science and engineering. For example, in computational science
they arise in the symmetric (generalized) eigenvalue problem, in nonlinear
eigenvalue problems, and in electronic structures computations, to name a few
problems. In statistics and machine learning, they arise, for example, in
canonical correlation analysis and in linear discriminant analysis. In this
article, we consider using randomized preconditioning in the context of
optimization problems with generalized orthogonality constraints. Our proposed
algorithms are based on Riemannian optimization on the generalized Stiefel
manifold equipped with a non-standard preconditioned geometry, which
necessitates development of the geometric components necessary for developing
algorithms based on this approach. Furthermore, we perform asymptotic
convergence analysis of the preconditioned algorithms which help to
characterize the quality of a given preconditioner using second-order
information. Finally, for the problems of canonical correlation analysis and
linear discriminant analysis, we develop randomized preconditioners along with
corresponding bounds on the relevant condition number
Discretely normed orders of quaternionic algebras
Tato práce shrnuje autorův výzkum v oblasti teorie kvaternionových algeber, jejich izomorfismů a maximálních řádů. Nový úhel pohledu na tuto problematiku je umožněn využitím pojmu diskrétní normy. Za hlavní výsledky práce je možná považovat důkaz jednoznačnosti diskrétní normy pro celá čísla, kvadratická rozšíření těles a řády kvaternionových algeber. Dále větu, která umožňuje mezi dvěma kvaternionovými algebrami konstruovat izomorfismy explicitně vyjádřené v maticovém tvaru. A v neposlední řadě důkaz existence nekonečně mnoha různých maximálních řádů kvaternionové algebry. Výsledky uvedené v této diplomové práci budou dále publikovány ve vědeckém článku.This thesis summarizes author's research on the field of theory of the quaternion algebras, their isomorphisms and maximal orders. The new point of view to this issue is received by using the concept of the discrete norm. The three following statements could be taken as the main results of the thesis: - Proof of the uniqueness of the discrete norm for integers, for the orders of the quadratic field extension and also for the orders of quaternion algebra - Theorem, which enables us to construct isomorphisms between quaternion algebras in explicit matrix form - Proof of the existence of infinitely many mutually distinct orders of the quaternion algebra Results given in this thesis will be also used in a scientific article.
A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion
Low-rank matrix completion (LRMC) problems arise in a wide variety of
applications. Previous theory mainly provides conditions for completion under
missing-at-random samplings. This paper studies deterministic conditions for
completion. An incomplete matrix is finitely rank- completable
if there are at most finitely many rank- matrices that agree with all its
observed entries. Finite completability is the tipping point in LRMC, as a few
additional samples of a finitely completable matrix guarantee its unique
completability. The main contribution of this paper is a deterministic sampling
condition for finite completability. We use this to also derive deterministic
sampling conditions for unique completability that can be efficiently verified.
We also show that under uniform random sampling schemes, these conditions are
satisfied with high probability if entries per column are
observed. These findings have several implications on LRMC regarding lower
bounds, sample and computational complexity, the role of coherence, adaptive
settings and the validation of any completion algorithm. We complement our
theoretical results with experiments that support our findings and motivate
future analysis of uncharted sampling regimes.Comment: This update corrects an error in version 2 of this paper, where we
erroneously assumed that columns with more than r+1 observed entries would
yield multiple independent constraint
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