arXiv subjects

Wojciech Czaja

Publications and source records attributed to Wojciech Czaja.

33 records · Page 2Linked to original sources

arXiv2016

Discrete Directional Gabor Frames

We develop a theory of discrete directional Gabor frames for functions defined on the $d$-dimensional Euclidean space. Our construction incorporates the concept of ridge functions into the theory of isotropic Gabor systems, in order to develop an anisotropic Gabor system with strong directional sensitivity. We present sufficient conditions on a window function $g$ and a sampling set $Λ$ for the corresponding directional Gabor system $\{g_{m,t,u}\}_{(m,t,u)\inΛ}$ to form a discrete frame. Explicit estimates on the frame bounds are developed. A numerical implementation of our scheme is also presented, and is shown to perform competitively in compression and denoising against state-of-the-art anisotropic methods.

math.FA↗

arXiv2014

Kaczmarz Algorithm and Frames

Sequences of unit vectors for which the Kaczmarz algorithm always converges in Hilbert space can be characterized in frame theory by tight frames with constant 1. We generalize this result to the context of frames and bases. In particular, we show that the only effective sequences which are Riesz bases are orthonormal bases. Moreover, we consider the infinite system of linear algebraic equations $A x = b$ and characterize the (bounded) matrices $A$ for which the Kaczmarz algorithm always converges to a solution.

cs.IT↗

arXiv2012

Schroedinger Eigenmaps for the Analysis of Bio-Medical Data

We introduce Schroedinger Eigenmaps, a new semi-supervised manifold learning and recovery technique. This method is based on an implementation of graph Schroedinger operators with appropriately constructed barrier potentials as carriers of labeled information. We use our approach for the analysis of standard bio-medical datasets and new multispectral retinal images.

cs.CE↗

arXiv2011

Consistent initial data for CMD perturbations

We investigate the initial condition which are simultaneously consistent with perturbation equations in both the radiation-dominated and the mater-dominated epochs. The exact formula for the spectrum transfer-function is derived.

astro-ph.CO↗

arXiv2011

Momentum of cosmological acoustic field

The acoustic spacetime corresponding to perturbed Friedman-Lemaitre-Robertson-Walker universe inherit the space isometries from the original FLRW model, but essentially differs in dynamics. The scale factor manifestly depends on the equation of state of the matter content. Despite the higher complexity of the background evolution the perturbation equation in this space is substantially simpler: the density perturbations obey d'Alembert equation. Canonical formalism reconstructed in the acoustic spacetime enables one to employ the Klein-Gordon scalar product. Consequently, the Fourier decomposition of the perturbation field provide the time-independent Fourier coefficients and the time-independent spectrum. The perturbation spectrum does not depend of the choice of the Cauchy hypersurface from which the data are collected. Noether constants associated with the six-parameter isometry group define the components of the momentum, hyperbolic momentum and angular momentum of sound.

gr-qc↗

arXiv2009

The acoustic wave equation in the expanding universe. Sachs-Wolfe theorem

In this paper the acoustic field propagating in the early hot ($p=ε/$) universe of arbitrary space curvature ($K=0, \pm 1$) is considered. The field equations are reduced to the d'Alembert equation in an auxiliary static Roberson-Walker space-time. Symbolic computation in {\em Mathematica} is applied.

cs.SC↗

arXiv2007

An efficient algorithm for positive realizations

We observe that successive applications of known results from the theory of positive systems lead to an {\it efficient general algorithm} for positive realizations of transfer functions. We give two examples to illustrate the algorithm, one of which complements an earlier result of \cite{large}. Finally, we improve a lower-bound of \cite{mn2} to indicate that the algorithm is indeed efficient in general.

math.CA↗

arXiv2005

Phantom cosmology as a simple model with dynamical complexity

We study the Friedmann-Robertson-Walker model with phantom fields modelled in terms of scalar fields. We apply the Ziglin theory of integrability and find that the flat model is nonintegrable. Then we cannot expect to determine simple analytical solutions of the Einstein equations. We demonstrate that there is only a discrete set of parameters where this model is integrable. For comparison we describe the phantoms fields in terms of the barotropic equation of state. It is shown that in the contrast to the phantoms modelled as scalar fields, the dynamics is always integrable and phase portraits are constructed. In this case we find the duality relation.

astro-ph↗

arXiv2004

Remarks on Naimark's duality

We present an extension of Naimark's duality principle which states that complete systems in a Hilbert space are projections of $ω$-linearly independent systems of elements of an ambient Hilbert space. This result is presented in the context of other known extensions of Naimark's theorem.

math.FA↗

arXiv2004

Transition redshift from the V-reconstruction method

We show the effectiveness of using the procedure of the potential reconstruction (the V-method) in the estimation of the transition redshift corresponding to switch from deceleration to acceleration phase of the Universe. We investigate the FRW models with dark energy by applying the particle-like description of their dynamics. In this picture the evolution of the universe is represented by motion of a fictitious particle in the one-dimensional potential $V(a)$ or V(z)$ where $a$ is the scale factor and $z$ is the redshift. The V-method solves the inverse problem where we reconstruct the potential function from empirical data. We use the Riess et al.'s gold and gold+silver samples of SN Ia data in this reconstruction. In the same framework we obtain both the estimates of the form of the equation of state for dark energy and the transition redshift. We obtain that transition redshift is $0.38_{-0.07}^{+0.10}$ (gold subset) and $0.37_{-0.07}^{+0.06}$ (gold+silver sample) when the linear model of dark energy ($w(z)=w_{0} + w_{1}z$) and $Ω_{m,0}=0.3$ are assumed. We compare the estimation of transition redshift with the Riess et al. results. The V-method also alows us to find the value of Hubble function at the moment of transition.

astro-ph↗

arXiv2004

Modified Friedmann Cosmologies - theory & observations

We investigate a class of Cardassian scenarios of the universe evolution in notions of the qualitative theory of dynamical systems. This theory allows us to analyze all solutions for all possible initial conditions on the phase plane. In the Cardassian models we find that big-rip singularities are present as a typical behavior in the future if $n<0$. Some exact solutions for the flat Cardassian models as well as a duality relation were found. In turn from the statistical analysis of Knop's SNIa data, without any priors on matter content in the model, we obtain that at the 99% confidence level this big-rip scenario will reach. The potential function for the Hamiltonian description of dynamics is reconstructed from the SNIa data (inverse dynamical problem). We also pointed out the statistical analysis results depend oversensitively on the choice of the model parameter $\mathcal{M}$.

astro-ph↗

arXiv2003

Particle-Like Description in Quintessential Cosmology

Assuming equation of state for quintessential matter: $p=w(z)ρ$, we analyse dynamical behaviour of the scale factor in FRW cosmologies. It is shown that its dynamics is formally equivalent to that of a classical particle under the action of 1D potential $V(a)$. It is shown that Hamiltonian method can be easily implemented to obtain a classification of all cosmological solutions in the phase space as well as in the configurational space. Examples taken from modern cosmology illustrate the effectiveness of the presented approach. Advantages of representing dynamics as a 1D Hamiltonian flow, in the analysis of acceleration and horizon problems, are presented. The inverse problem of reconstructing the Hamiltonian dynamics (i.e. potential function) from the luminosity distance function $d_{L}(z)$ for supernovae is also considered.

gr-qc↗

arXiv2003

Toward reconstruction of dynamics of the Universe from distant supernovae type Ia

We demonstrate a model-independent method of estimating qualitative dynamics of an accelerating universe from observations of distant type Ia supernovae. Our method is based on the luminosity-distance function, optimized to fit observed distances of supernovae, and the Hamiltonian representation of dynamics for the quintessential universe with a general form of equation of state $p=w(a(z))ρ$. Because of the Hamiltonian structure of FRW dynamics with the equation of state $p = w(a(z)) ρ$, the dynamics is uniquelly determined by the potential function $V(a)$ of the system. The effectiveness of this method in discrimination of model parameters of Cardassian evolution scenario is also given. Our main result is the following, restricting to the flat model with the current value of $Ω_{m,0}=0.3$, the constraints at $2σ$ confidence level to the presence of $ρ^{n}$ modification of the FRW models are $-0.50 \lesssim n \lesssim 0.36$.

astro-ph↗

arXiv2003

Stability of FRW Cosmology with Generalized Chaplygin Gas

We apply methods of dynamical systems to study the behaviour of universe dominated by the generalized Chaplygin gas. We reduce the dynamics to a 2-dimensional Hamiltonian system and study its behaviour for various ranges of parameters. The dynamics is studied on the phase plane by using methods of qualitative analysis of differential equations. The behaviour of trajectories at infinity is studied in some convenient coordinates introduced on the phase plane. Hence we shown that FRW model with the generalized Chaplygin gas is structurally stable. We clearly find the domains of cosmic acceleration as well as conditions for which the horizon problem is solved. We also define some general class of fluids which generalize the Chaplygin gas. The dynamics of such models in terms of energy conditions is also discussed.

astro-ph↗

arXiv2002

The Balian--Low theorem for the symplectic form on ${\mathbb R}^{2d}$

In this paper we extend the Balian--Low theorem, which is a version of the uncertainty principle for Gabor (Weyl--Heisenberg) systems, to functions of several variables. In particular, we first prove the Balian--Low theorem for arbitrary quadratic forms. Then we generalize further and prove the Balian--Low theorem for differential operators associated with a symplectic basis for the symplectic form on ${\mathbb R}^{2d}$.

math.FA↗