Zbigniew Pasternak-Winiarski

On Reproducing Kernels for Holomorphic Vector Bundles

We introduce (1) the reproducing kernels of Bergman type for holomorphic sections of complex hermitian vector bundles, and (2) the maps defined by these kernels on total and base spaces of considered bundles into some Hilbert and Grassmann spaces. We present (without proofs) the main results concerning basic properties of the introduced objects.

On real-valued homomorphisms in countably generated differential structures

Abstract Real valued homomorphisms on the algebra of smooth functions on a differential space are described. The concept of generators of this algebra is emphasized in this description.

Weighted generalization of the Ramadanov’S theorem and further considerations

We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space ℂN, and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then we give a weighted generalization of the theorem given by M. Skwarczyński (1980), highlighting some special property of the domains, on which the weighted Bergman kernels converge uniformly. Moreover, we show that convergence of weighted Bergman kernels implies this property, which will give a characterization of the domains, for which the inverse of the Ramadanov’s theorem holds.

SVM Kernel Configuration and Optimization for the Handwritten Digit Recognition

The paper presents optimization of kernel methods in the task of handwritten digits identification. Because such digits can be written in various ways (depending on the person’s individual characteristics), the task is difficult (subsequent categories often overlap). Therefore, the application of kernel methods, such as SVM (Support Vector Machines), is justified. Experiments consist in implementing multiple kernels and optimizing their parameters. The Monte Carlo method was used to optimize kernel parameters. It turned out to be a simple and fast method, compared to other optimization algorithms. Presented results cover the dependency between the classification accuracy and the type and parameters of selected kernel.

Differential completions and compactifications of a differential space

Abstract Differential completions and compactifications of differential spaces are introduced and investigated. The existence of the maximal differential completion and the maximal differential compactification is proved. A sufficient condition for the existence of a complete uniform differential structure on a given differential space is given.

Weighted Szegő Kernels

In this paper we define a weighted Szegő kernel by putting a measurable almost everywhere positive function μ under the inner product integral and try to answer which conditions it must satisfy in order to give a ‘good generalization’ of a classical case.

Ramadanov Theorem for Weighted Bergman Kernels on Complex Manifolds

We study the limit behavior of weighted Bergman kernels on a sequence of domains in a manifold M and show that under some conditions on domains and weights, weighted Bergman kernel converges uniformly on compact sets.

A Characterization of Domains of Holomorphy by Means of Their Weighted Skwarczyński Distance

M. Skwarczynski (†) introduced pseudodistance on domains in Cn which under some conditions (if the domain is bounded for instance) gives rise to biholomorphically invariant distance, i.e., invariant under biholomorphic transformations. One can find a proof that completeness with respect to Skwarczynski distance implies completeness with respect to Bergman distance, which implies that the considered domain is a domain of holomorphy. In this paper we give a characterization of domains of holomorphy with the help of a weighted version of Skwarczynski pseudodistance. We will work with a special kind of weights, called “admissible weights”. Midway, we obtain a new proof (even in the unweighted case) of the theorem that the so-called Kobayashi condition implies Bergman completeness, which may be helpful in answering the (open) question if Bergman completeness and Skwarczyśnki completeness are equivalent or not.

Integration on differential spaces

Abstract The generalization of the n-dimensional cube, an n-dimensional chain, the exterior derivative and the integral of a differential n-form on it are introduced and investigated. The analogue of Stokes theorem for the differential space is given