1,522 research outputs found
On Bernstein-Schnabl operators on the unit interval
No full textBernstein-Schnabl operators were first introduced by
R. Schnabl in 1968 in the context
of sets of probability Radon measures on compact Hausdorff spaces.
Subsequently Grossman proposed a general method
of constructing Bernstein-Schnabl operators on an arbitrary convex
compact subset of a locally convex space and he showed that they
are an approximation process for continuous functions.
A particular class of these operators has been also studied by the
F. Altomare and,
subsequently, by several other authors. Their construction
essentially involves positive projections and they satisfy many additional
properties useful for the study of evolution problems.
In this paper we deep the study of the Bernstein-Schnabl
operators associated with a general continuous selection of
probability Borel measures on the interval [0,1], which not
necessarily arise from a positive projection. These operators seem
to have some interest because they furnish new general
approximation processes for continuous functions and they also
approximate the solutions of the initial-boundary problems
associated with a class of degenerate diffusion equations.
In the first section we recall their definition and discuss some
examples of them. After that, we investigate their approximation
properties and show several estimates of the rate of convergence
by means of suitable moduli of smoothness.
Shape preserving properties are discussed in Section 2.
In particular, we investigate some conditions under which these
operators preserve the convexity.
In the third section we show that suitable iterates of
Bernstein-Schnabl operators converge to a Markov semigroup on
C([0,1]) whose generator is a degenerate differential
operator of the form Au(x):=\alpha(x) u''(x) (x \in [0,1]
defined on a suitable subspace of smoot functions satisfying the so-called Wentcel boundary conditions.
By means of Bernstein-Schnabl operators we establish some
qualitative properties of this semigroup and, in particular, its
asymptotic behaviour.
In the same section we also study the generation properties of
general differential operators and
determine suitable continuous selections of Borel measures such
that the iterates of the corresponding Bernstein-Schnabl operators
converge to the given Markov semigroup
Korovkin-type Approximation Theory and its Applications
No full textThe power of the original result by Korovkin impressed many mathematicians and hence a considerable amount of research extended this theorem to the setting of different function spaces or more general abstract spaces such as Banach lattices, Banach algebras, Banach spaces and so on. At the same time, strong and fruitful connections of this theory have also been revealed not only with classical approximation theory, but also with other fields such as functional analysis (abstract Choquet boundaries and convexity theory, uniqueness of extensions of positive linear forms, convergence of sequences of positive linear operators in Banach lattices, structure theory of Banach lattices, convergence of sequences of linear operators in Banach algebras and in C*-algebras, structure theory of Banach algebras, approximation problems in function algebras), harmonic analysis (convergence of sequences of convolution operators on function spaces and function algebras on (locally) compact topological groups, structure theory of topological groups), measure theory and probability theory (weak convergence of sequences of positive Radon measures and positive approximation processes constructed by probabilistic methods), and partial differential equations (approximation of solutions of Dirichlet problems and of diffusion equations).
This work, in fact, delineated a new theory called Korovkin-type approximation theory.
The reader will find a quite complete picture of what has been achieved in the field, a modern and comprehensive exposition of the main aspects of Korovkin-type approximation theory in spaces of continuous real functions together with its main applications. The function spaces we have chosen to treat play a central role in the whole theory and are the most useful for the applications in the various univariate, multivariate and infinite dimensional settings.
The book is mainly intended as a reference text for research workers in the field; a large part of it can also serve as a textbook for a graduate level course. The organization of the material does not follow the historical development of the subject and allows us to present the most important part of the theory in a concise way. Chapters 2, 3 and 4 are devoted to the main aspects of Korovkin-type approximation theory in C_0(X) and C(X)-spaces. In Chapter IV we also point out the strong interplay between KAT and Choquet's integral representation theory, as well as Stone-Weierstrass-type theorems.
Chapters 5 and 6 are mainly concerned with applications to: Approximation of continuous functions by means of positive linear operators, Approximation and representation of the solutions of particular partial differential equations of diffusion type, by means of powers of positive linear operators, More precisely, in Chapter 5 we give the first and best-known applications of Korovkin-type approximation theory. We describe different kinds of positive approximation processes. Particular care is devoted to probabilistic-type operators, discrete-type operators, convolution operators for periodic functions and summation methods. In the final Chapter 6 we present a detailed analysis of some further sequences of positive linear operators that have been studied recently. These operators play an important role in some fine aspects of approximation theory. They connect the theory of C_0-semigroups of operators, partial differential equations and Markov processes. The main examples we consider are the Bernstein-Schnabl operators, the Stancu-Schnabl operators and the Lototsky-Schnabl operators.
Subsequently we show how these operators are strongly connected with initial and (Ventcel-type) boundary value problems in the theory of partial differential equations.
Although the aim of the book is to survey both classical and recent results in the field, the reader will find a certain amount of new material. In any case, the majority of the results presented here appears in a book for the first time
Reply to the letter “The first report on the effect of sacral neuromodulation on intestinal transit time and colonic motility in chronic constipation”
No full textlette
The diagnosis of early psoriatic arthritis
No full textPsoriatic arthritis (PsA) is a chronic inflammatory joint disease with heterogeneous clinical presentation and unpredictable course but often with a tendency to irreversible joint damage. Joint damage can occur early in the disease also in the absence of significant clinical signs of arthritis. These observations and the current availability of effective treatments in controlling skin and joint disease underline the importance of early diagnosis of PsA. The use of specific questionnaires for screening patients at risk of psoriatic arthritis, knowledge of new classification criteria for PsA and especially the proper use of new imaging techniques are all important steps in achieving the goal of early diagnosis of PsA. The dermatologist may play a key role in this regard supported, when necessary, by the collaboration of the rheumatologist and radiologist
Diagnosis and management of psoriatic arthritis
No full textPsoriatic arthritis (PsA) is a chronic inflammatory joint disease with heterogeneous clinical presentation, an unpredictable course and a tendency to irreversible joint damage. Joint damage can occur early in the disease also when significant clinical signs of arthritis are absent. These observations and currently availabe effective treatments in controlling skin and joint disease underline the importance of early diagnosis of psoriatic arthritis. Use of specific questionnaires for screening at-risk patients, knowledge of new classification criteria and judicious use of new imaging techniques are key to early diagnosis. Besides traditional terapy with non-steroidal anti-inflammatory drugs and the more active disease-modifying anti-rheumatic drugs, the advent of biologics (anti-tumor necrosis factor- inhibitors) has considrerably enhanced treatment options. Collaboration between dermatologists, radiologists and rheumatologists offers a new opportunity to establish early diagnosis and effective therapy for psoriatic arthritis
Defining authorship in the era of big data collection and its consequences on the academic career
No full textOn a class of elliptic-parabolic equations on unbounded intervals
No full textWe study a class of degenerate elliptic second order differential operators acting on some
polynomial weighted function spaces on [0, +∞[. We show that these operators are the generators
of C_0-semigroups of positive operators which, in turn, are the transition semigroups associated with
right-continuous normal Markov processes with state space [0, +∞]. Approximation and qualitative
properties of both the semigroups and the Markov processes are investigated as well. Most of the
results of the paper depend on a representation of the semigroups we give in terms of powers of
particular positive operators of discrete type we introduced and studied in a previous paper
Age related changes of the mitochondrial energy metabolism in rat liver and heart.
No full textThe influence of aging on mitochondrial energy metabolism of rat liver and rat heart has been studied by analysis of (i) respiratory rate of succinate-supplemented mitochondria in state III (coupled state in the presence of ADP + Pi); (ii) the rate of synthesis of ATP in succinate-supplemented mitochondria; (iii) the ATP hydrolase activity of sonicated submitochondrial particles. The results indicate a decrease of the F(0)F(1)-ATP synthase activity in mitochondria isolated from both organs of aged (24-month-old) as compared to young (3-month-old) rats which was accompanied by a decrease of immunodetected amount of the beta-F(1) (subunit of the catalytic F(1) sector of F(0)F(1)-ATP synthase). These effects were more evident in heart than in liver mitochondria. Analysis of the mitochondrial content of glutathione (GSH), an essential intracellular antioxidant agent, shows a decrease in mitochondria of both tissues of aged animals. Exposure of submitochondrial particles to free radicals, produced either by (60)Co or by respirtory chain (in presence of the inhibitor antimycin A) mimicked the alterations of F(0)F(1) ATP synthase obsreved in submitochondrial particles of aged rats. The possible relationship between aging process, free radical production and alteration of mitochondrial oxidative phosphorylation is discussed
- …
