Allgebra Liniowa 1 Kolokwia I egzaminy M. Gewert, Z. Algebra liniova 1- Definicje, threwerdzenia. Algebra abstrakcyjna w zadaniach Jerzy Rutkowski. But if you willing to look for them, you find some ants and wasps and other superhero movies as well as fiercely anticipated consequences like 2, offered on IMDb upcoming movie guide.
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We investigate the problem of an adaptive parameter choice for regularization learning algorithms. In the theory of ill-posed problems there is a long history of choosing regularization parameters in optimal way without a priori knowledge of a smoothness of the element of interest. But known parameter choice rules cannot be applied directly in Learning Theory. The point is that these rules are based on the estimation of the stability of regularization algorithms measured in the norm of the space where unknown element of interest should be recovered.
But in the context of Learning Theory this norm is determined by an unknown probability measure, and is not accessible. In the talk we are going to present a new parameter choice strategy consisting in adaptive regularization performed simultaneously in a Hypothesis space and in a space equipped with an empirical norm. Both these spaces are accessible and a new parameter choice rule called the balancing principle can be used there.
Then a parameter for the regularization in the inaccessible space is chosen as the minimal among the parameters selected for above mentioned accessible spaces. We prove that under rather mild assumptions such strategy guarantees an optimal order of the risk.
Our analysis covers the capacity independent learning algorithms, but some capacity dependent results can be also obtained in a similar way. Before the presentation of the main results we will give a brief introduction to Statistical Learning Theory.
In course of his study of the Hilbert 14th problem, in Nagata posed the following question:. Given a set of points P 1 , The conjecture remains so far open. I will put the Nagata Conjecture in a more general perspective concerned with the behavior of Seshadri constants. I will also discuss its connections with apparently distant problems in mathematics e.
We study control-affine systems with n-1 inputs evolving on an n-dimensional manifold for which the distribution spanned by the control vector fields is involutive and of constant rank equivalently, they may be considered as 1-dimensional systems with n-1 inputs entering nonlinearly. We provide a complete classification of such generic systems and their one-parameter families. We also describe all generic bifurcations of 1-parameter families of systems of the above form. West Nile WN virus is an infectious disease spreading through interacting bird and mosquito populations in North America.
In the subsequent 5 years the epidemic has spread spatially across to most of the west coast of North America. It is likely that the spread of WN virus comes from the interplay of disease dynamics and bird and mosquito movement.
Here we mathematically investigate the spread of WN virus by spatially extending the non-spatial dynamical model of Wonham et al. Diffusive movement provides the simplest possible movement model for birds and mosquitoes. Our approach is to focus on the implications of diffusive motion coupled to a dynamical model for WN virus.
Infection dynamics are based on a modified version of a model for cross infection between birds and mosquitoes. Working with a simplified version of the model, the co-operative nature of cross-infection dynamics is utilized to prove the existence of traveling waves and to calculate the spatial spread rate of infection. One of the important tools we use is the basic reproduction number. A polynomial algebra is a free object in the category of commutative algebras.
Other types of geometry, like noncommutative geometry, can be based on other types of algebras and their free objects, like associative algebras. The ad hoc tool is the notion of algebraic operad. After introducing the basics of operad theory, I will show how to handle the notions of cohomology, bialgebra, smoothness in the operadic geometry setting. Though most of the motivation comes from geometry and topology, new links from computer sciences are now expanding.
I will overview algebraic geometry arising from the study of binary symmetric trivalent phylogenetic trees. The problems discussed will involve graphs, Markov processes, group actions and their invariants, and almost elementary geometry of lattice polytopes. I will present some basic results concerning the concentration of measure phenomenon, which has recently gained much attention in various, sometimes quite distant, fields of mathematics. I am going to focus mainly on connections with the isoperimetric inequalities.
In particular I will speak about the isoperimetric problem in the Euclidean space the classical case , the sphere and in the Gauss space. The talk will be in English!
The core problem of approximation continues to be the development of efficient methods for replacing general functions by simpler functions. Some methods were invited long ago Fourier sums, Taylor polynomials, best approximation by trigonometric or algebraic polynomials etc.
More recently however, driven by several numerical applications, the directions of approximation theory have moved toward nonlinear approximation. This includes the comparatively new subject of nonlinear m -term approximation. It has found applications in numerical solution of integral equations, image compression, design of neural networks, and so on. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a fixed linear space but are allowed to depend on the function being approximated.
The fundamental question of nonlinear approximation is how to construct good algorithms of nonlinear approximation. Systolic complexes are simply connected simplicial complexes satisfying certain local combinatorial condition, resembling nonpositive curvature.
However, systolicity neither implies, nor is implied by nonpositive curvature of the complex equipped with the standard metric. In my talk I will present a short sketch of the theory of systolic complexes and prove a systolic analogue of the classical theorem in the theory of nonpositively curved metric spaces - the flat torus theorem.
Asymptotic dimension of Gromov is a large scale analog of the covering dimension. In case of a finitely generated group G the asymptotic dimension does not depend on the word metric. Therefore it is a group invariant.
Asymptotic dimension gained attention since G. The Novikov conjecture concerns the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group.
According to the Novikov conjecture, the higher signatures, which are certain numerical invariants of smooth manifolds, are homotopy invariants. The Strong Novikov conjecture states that the analytic assembly map on K-theory is injective. I will also discuss variants of asymptotic dimension Assouad-Nagata dimension, Higson property and their connection to Lipschitz extensions.
Rosasco and E. De Vito Univ. The West Nile virus model - travelling waves, spread rate, reproduction number. On geometry of phylogenetic trees Streszczenie Prezentacja. Isoperimetry and the concentration of measure phenomenon Streszczenie Prezentacja. Twierdzenie Banacha-Tarskiego z punktu widzenia algebraika Streszczenie Prezentacja.
J. Rutkowski - Algebra Abstrakcyjna w Zadaniach
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