1,721,470 research outputs found
Classical operators of harmonic analysis and Sobolev embeddings on rearrangement-invariant function spaces
No full textBoundedness properties of some classical operators of harmonic analysis (namely the Hilbert and Riesz transforms, the Riesz potentials and (fractional and nonfractional) maximal operators) as well as certain Sobolev-type embeddings on the entire space are studied. The compactness of Sobolev trace embeddings is also investigated. The focus is on the optimality of the results within the class of rearrangement-invariant function spaces. The aforementioned questions are reduced to equivalent problems concerning appropriate Hardy-type operators acting on functions of a single variable. The behavior of the Hardy-type operators on rearrangement-invariant function spaces is investigated first. The results concerning the Hardy-type operators are used as the building blocks from which together with known results from the literature the other results are obtained. To illustrate possible applications, general results are accompanied by particular exam- ples. The results presented in this thesis are based on some of the papers authored or coauthored by the author.
Francesco Altomare - the remarkable mathematician and human being
Full text linkWe laconically describe the great contributions of Professor Francesco Altomare to mathematical research and PhD education, and his unique status in the mathematical community. In particular, we present and give examples of his innovative and great achievements related to the following areas of mathematics: Functional Analysis, Operator Theory, Potential Theory, Approximation Theory, Probability Theory, Function Spaces, Choquet's Theory, Dirichlet's Problem and Semigroup Theory. Moreover, we report on and give concrete examples of his unique way to work together with PhD students, both before and sometimes also after their dissertation. Finally, we shortly describe his remarkable “class travel” from “simple” conditions with no academic traditions in his family in the small hometown Giovinazzo to finally become the broad, ingenious, and powerful mathematician he is regarded to be today
Klasické operátory harmonické analýzy a Sobolevova vnoření na prostorech funkcí s normou invariantní vůči nerostoucímu přerovnání
No full textBoundedness properties of some classical operators of harmonic analysis (namely the Hilbert and Riesz transforms, the Riesz potentials and (fractional and nonfractional) maximal operators) as well as certain Sobolev-type embeddings on the entire space are studied. The compactness of Sobolev trace embeddings is also investigated. The focus is on the optimality of the results within the class of rearrangement-invariant function spaces. The aforementioned questions are reduced to equivalent problems concerning appropriate Hardy-type operators acting on functions of a single variable. The behavior of the Hardy-type operators on rearrangement-invariant function spaces is investigated first. The results concerning the Hardy-type operators are used as the building blocks from which together with known results from the literature the other results are obtained. To illustrate possible applications, general results are accompanied by particular exam- ples. The results presented in this thesis are based on some of the papers authored or coauthored by the author. 1Je zkoumána omezenost jistých klasických operátorů harmonické analýzy (jmenovitě Hilbertova a Rieszova transformace, Rieszovy potenciály a (frakční i nefrakční) maximální operátory) a platnost jistých sobolevových vnoření na celém prostoru. Kompaktnost ope- rátoru stop pro Sobolevovy prostory je také zkoumána. Důraz je kladen na optimalitu výsledků ve třídě prostorů funkcí s normou invariantní vůči nerostoucímu přerovnání. Zmíněné problémy jsou zredukovány na ekvivalentní problémy týkající se vhodných ope- rátorů Hardyho typu, které jsou definovány na funkcích jedné proměnné. Chování těchto operátorů Hardyho typu na prostorech funkcí s normou invariantní vůči nerostoucímu přerovnání je zkoumáno jako první. Výsledky týkající se operátorů Hardyho typu jsou poté použity jako stavební kameny, ze kterých spolu se známými výsledky z literatury jsou ostatní výsledky odvozeny. Pro ilustraci možného použití jsou obecné výsledky doprová- zeny konkrétními příklady. Výsledky prezentované v této disertační práci jsou založeny na výsledcích z některých článků, jichž je autor této práce autorem či spoluautorem. 1Department of Mathematical AnalysisKatedra matematické analýzyMatematicko-fyzikální fakultaFaculty of Mathematics and Physic
Shock-wave behaviour of sedimentation in wastewater treatment: a rich problem
No full textA common industrial process for separating particles from a liquid is continuous sedimentation, which is used in the chemical, mining, pulp-and-paper and food industries. It can also be found in most wastewater treatment plants, where it is a crucial subprocess of a complex biological system. The process has provided, and will continue to provide, scientific problems that lead to fundamental research in different disciplines such as mathematics, wastewater, chemical, mineral, control and automation engineering. A selective survey of previous results within the field of pure and applied mathematics is presented with focus on a nonlinear convection-diffusion partial differential equation with discontinuous coefficients. In a model of a wastewater treatment plant, such an equation is coupled to a set of ordinary differential equations. Some new results on the steady-state solutions of such a coupled system are also presented
Váhové nerovnosti, limitní reálná interpolace a prostory funkcí
No full textV této disertační práci studujeme limitní interpolační prostory opatřené tak- zvanými pomalu se měnícími váhovými funkcemi a vlastnosti operátorů defino- vaných na těchto prostorech. V článku 1 jsme odvodili podmínky, za nichž je možné popsat K-prostory získané limitní formou reálné interpolace založené na pomalu se měnících funkcích pomocí J-prostorů, a zároveň zde nalézáme odpověd' i na opačnou otázku. Dále využíváme naše hlavní výsledky k získání vět o hustotě pro odpovídající limitní interpolační prostory. V článku 2 jsme studovali kompaktnost operátorů definovaných na limitních interpolačních prostorech a odvodili kvantitativní odhady jejich míry nekompakt- nosti. V článku 3 jsme získali odhady pro duální prostory limitních interpolačních prostorů opatřených pomalu se měnícími váhovými funkcemi. 1This thesis is focused on studying limiting interpolation spaces with weight func- tions of slowly varying type and properties of operators defined on them. In Paper 1 we establish conditions under which K-spaces in the limiting real interpolation involving slowly varying functions can be described by means of J-spaces and we also solve the reverse problem. Further, we apply our results to obtain density theorems for the corresponding limiting interpolation spaces. In paper 2 we study the properties of compactness of operators defined on lim- iting interpolation spaces and derive the quantitative estimates of measure of non-compactness. In paper 3 we estimate dual spaces of limiting interpolation spaces that involve weight functions of slowly varying type. 1Department of Mathematical AnalysisKatedra matematické analýzyMatematicko-fyzikální fakultaFaculty of Mathematics and Physic
Shape Modeling by Optimising Description Length Using Gradients and Parameterisation Invariance
No full textIn Statistical Shape Modeling, a dense correspondence between the shapes in the training set must be established. Traditionally this has been done by hand, a process that commonly requires a lot of work and is difficult, especially in 3D. In recent years there has been a lot of work on automatic construction of Shape Models. In recent papers (Davies et al., Medical Image Computing and Computer-Assisted Intervention MICCAI’2001, pp. 57–65, 2001; Davies et al., IEEE Trans. Med. Imaging. 21(5):525–537 2002; Kotcheff and Taylor, Med. Image Anal. 2:303–314 1998) Minimum Description Length, (MDL), is used to locate a dense correspondence between shapes. In this paper the gradient of the description length is derived. Using the gradient, MDL is optimised using steepest descent. The optimisation is therefore faster and experiments show that the resulting models are better. To characterise shape properties that are invariant to similarity transformations, it is first necessary to normalise with respect to the similarity transformations. This is normally done using Procrustes analysis. In this paper we propose to align shapes using the MDL criterion. The MDL based algorithm is compared to Procrustes on a number of data sets. It is concluded that there is improvement in generalisation when using MDL to align the shapes. In this paper novel theory to prevent the commonly occurring problem of clustering under correspondence optimisation is also presented. The problem is solved by calculating the covariance matrix of the shapes using a scalar product that is invariant to mutual reparameterisations. An algorithm for implementing the ideas is proposed and compared to Thodberg’s state of the art algorithm for automatic shape modeling. The suggested algorithm is more stable and the resulting models are of higher quality according to the generalisation measure and according to visual inspection of the specificity
Weighted inequalities, limiting real interpolation and function spaces
No full textThis thesis is focused on studying limiting interpolation spaces with weight func- tions of slowly varying type and properties of operators defined on them. In Paper 1 we establish conditions under which K-spaces in the limiting real interpolation involving slowly varying functions can be described by means of J-spaces and we also solve the reverse problem. Further, we apply our results to obtain density theorems for the corresponding limiting interpolation spaces. In paper 2 we study the properties of compactness of operators defined on lim- iting interpolation spaces and derive the quantitative estimates of measure of non-compactness. In paper 3 we estimate dual spaces of limiting interpolation spaces that involve weight functions of slowly varying type.
Generalizations of some classical inequalities and their applications
No full textThe paper is a survey (in a number of cases proofs are provided) of various and interesting generalizations and supplements of classical inequalities as well as inequalities important in information theory obtained in the last few years independently by the author and in collaboration with other mathematicians.</p
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