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Holomorphic functional calculus in \(A\)-pseudoconvex algebras

100%
Łuczak M.
Commentationes Mathematicae
|
2005
|
tom 45
|
nr 2
EN
We define and study a holomorphic functional calculus for a single element in complex and real complete A-pseudoconvex algebras with unit. As a consequence of the main result we obtain the spectral mapping theorem and existence of the logarithm and the nth root of an algebra element.
2
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A characterization of linear-multiplicative functionals in topological algebras

100%
Łuczak M.
Commentationes Mathematicae
|
2006
|
tom 46
|
nr 1
EN
The classical Gleason–Kahane–Żelazko theorem for complex Banach algebras was generalized for not necessary linear functionals by Kowalski and Słodkowski. We prove a version of the Kowalski–Słodkowski theorem for real Banach algebras and also for real and complex \(A\)-pseudoconvex algebras.
3
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Some methods of constructing kernels in statistical learning

63%
Górecki T., Łuczak M.
Discussiones Mathematicae Probability and Statistics
|
2010
|
tom 30
|
nr 2
179-201
EN
This paper is a collection of numerous methods and results concerning a design of kernel functions. It gives a short overview of methods of building kernels in metric spaces, especially $R^n$ and $S^n$. However we also present a new theory. Introducing kernels was motivated by searching for non-linear patterns by using linear functions in a feature space created using a non-linear feature map.
4
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Linear discriminant analysis with a generalization of the Moore-Penrose pseudoinverse

63%
Górecki T., Łuczak M.
International Journal of Applied Mathematics and Computer Science
|
2013
|
tom 23
|
nr 2
463-471
EN
The Linear Discriminant Analysis (LDA) technique is an important and well-developed area of classification, and to date many linear (and also nonlinear) discrimination methods have been put forward. A complication in applying LDA to real data occurs when the number of features exceeds that of observations. In this case, the covariance estimates do not have full rank, and thus cannot be inverted. There are a number of ways to deal with this problem. In this paper, we propose improving LDA in this area, and we present a new approach which uses a generalization of the Moore-Penrose pseudoinverse to remove this weakness. Our new approach, in addition to managing the problem of inverting the covariance matrix, significantly improves the quality of classification, also on data sets where we can invert the covariance matrix. Experimental results on various data sets demonstrate that our improvements to LDA are efficient and our approach outperforms LDA.
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