88,403 research outputs found
Functional analysis
No full textAs taught in 2006-2007 and 2007-2008. Functional analysis begins with a marriage of linear algebra and metric topology. These work together in a highly effective way to elucidate problems arising from differential equations. Solutions are sought in an infinite dimensional space of functions. This module paves the way by establishing the principal theorems (all due in part to the great Polish mathematician Stefan Banach) and exploring their diverse consequences. Topics to be covered will include: – norm topology and topological isomorphism; – boundedness of operators; – compactness and finite dimensionality; – extension of functionals; – weak*-compactness; – sequence spaces and duality; – basic properties of Banach algebras. Suitable for: Undergraduate students Level Four Dr Joel F. Feinstein School of Mathematical Sciences Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras. Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area.
Normed algebras of differentiable functions on compact plane sets
No full textWe investigate the completeness and completions of the normed algebras (D (1)(X), ‖ · ‖) for perfect, compact plane sets X. In particular, we construct a radially self-absorbing, compact plane set X such that the normed algebra (D (1)(X), ‖ · ‖) is not complete. This solves a question of Bland and Feinstein. We also prove that there are several classes of connected, compact plane sets X for which the completeness of (D (1)(X), ‖ · ‖) is equivalent to the pointwise regularity of X. For example, this is true for all rectifiably connected, polynomially convex, compact plane sets with empty interior, for all star-shaped, compact plane sets, and for all Jordan arcs in ℂ. In an earlier paper of Bland and Feinstein, the notion of an F -derivative of a function was introduced, where F is a suitable set of rectifiable paths, and with it a new family of Banach algebras D(1)F(X) corresponding to the normed algebras D (1)(X). In the present paper, we obtain stronger results concerning the questions when D (1)(X) and D(1)F(X) are equal, and when the former is dense in the latter. In particular, we show that equality holds whenever X is ‘ F -regular’. An example of Bishop shows that the completion of (D (1)(X), ‖ · ‖) need not be semisimple. We show that the completion of (D (1)(X), ‖ · ‖) is semisimple whenever the union of all the rectifiable Jordan arcs in X is dense in X. We prove that the character space of D (1)(X) is equal to X for all perfect, compact plane sets X, whether or not (D (1)(X), ‖ · ‖) is complete. In particular, characters on the normed algebras (D (1)(X), ‖ · ‖) are automatically continuous
S.C. Feinstein, P.L. Giovacchini, A. A. Miller. — Psychiatrie de l’adolescent. Paris, P. U. F., Le fil rouge, 1982
No full textBolzinger André. S.C. Feinstein, P.L. Giovacchini, A. A. Miller. — Psychiatrie de l’adolescent. Paris, P. U. F., Le fil rouge, 1982. In: Bulletin de psychologie, tome 36 n°360, 1983. Psychologie clinique VI. pp. 696-697
Functional analysis
No full textAs taught in 2006-2007 and 2007-2008.
Functional analysis begins with a marriage of linear algebra and metric topology. These work together in a highly effective way to elucidate problems arising from differential equations. Solutions are sought in an infinite dimensional space of functions.
This module paves the way by establishing the principal theorems (all due in part to the great Polish mathematician Stefan Banach) and exploring their diverse consequences. Topics to be covered will include:
– norm topology and topological isomorphism;
– boundedness of operators;
– compactness and finite dimensionality;
– extension of functionals;
– weak*-compactness;
– sequence spaces and duality;
– basic properties of Banach algebras.
Suitable for: Undergraduate students Level Four
Dr Joel F. Feinstein
School of Mathematical Sciences
Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras.
Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area
Mathematical analysis
No full textThis is a module framework. It can be viewed online or downloaded as a zip file. As taught in 2007-2008 and 2009-2010. This module introduces mathematical analysis building upon the experience of limits of sequences and properties of real numbers and on calculus. It includes limits and continuity of functions between Euclidean spaces, differentiation and integration. A variety of very important new concepts are introduced by investigating the properties of numerous examples, and developing the associated theory, with a strong emphasis on rigorous proof. This module is suitable for study at undergraduate level 2. Dr Joel Feinstein, School of Mathematics Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras. Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area.
Functional analysis 2010
No full textThis is a module framework. It can be viewed online or downloaded as a zip file.
As taught Autumn semester 2010.
Functional analysis begins with a marriage of linear algebra and metric topology. These work together in a highly effective way to elucidate problems arising from differential equations. Solutions are sought in an infinite dimensional space of functions.
This module paves the way by establishing the principal theorems (all due in part to the great Polish mathematician Stefan Banach) and exploring their diverse consequences. Topics to be covered will include:
– norm topology and topological isomorphism;
– boundedness of operators;
– compactness and finite dimensionality;
– extension of functionals;
– weak*-compactness;
– sequence spaces and duality;
– basic properties of Banach algebras.
Suitable for: Undergraduate students Level Four
Dr Joel F. Feinstein
School of Mathematical Sciences
Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras.
Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area
The chain rule for -differentiation
No full textLet be a perfect, compact subset of the complex plane, and let
denote the (complex) algebra of continuously
complex-differentiable functions on . Then is a normed algebra
of functions but, in some cases, fails to be a Banach function algebra. Bland
and the second author investigated the completion of the algebra ,
for certain sets and collections of paths in , by
considering -differentiable functions on .
In this paper, we investigate composition, the chain rule, and the quotient
rule for this notion of differentiability. We give an example where the chain
rule fails, and give a number of sufficient conditions for the chain rule to
hold. Where the chain rule holds, we observe that the Fa\'a di Bruno formula
for higher derivatives is valid, and this allows us to give some results on
homomorphisms between certain algebras of -differentiable
functions.Comment: 12 pages, submitte
How and why we do mathematical proofs
No full textThis is a module framework. It can be viewed online or downloaded as a zip file. As taught in Autumn Semester 2009/10 The aim of this short unit is to motivate students to understand why we might want to do proofs (why proofs are important and how they can help us) and to help students with some of the relatively routine aspects of doing proofs. In particular, the student will learn the following: * proofs can help you to really see why a result is true; * problems that are easy to state can be hard to solve (e.g. Fermat's Last Theorem); * sometimes statements which appear to be intuitively obvious may turn out to be false (e.g. Simpson's paradox); * the answer to a question will often depend crucially on the definitions you are working with; * how to start proofs; * how and when to use definitions and known results. The module is organised into three sections: Why; How (Part I); How (Part II) With practice, students should become fluent in these routine aspects of writing proofs, and this will allow them to focus instead on the more creative and interesting aspects of constructing proofs. A practice sheet is included after students have completed all three sections. Each section is suitable for a different level of audience, as described below: Suitable for: Foundation, undergraduate year one and undergraduate year two students Section 1: Why: Anyone with a knowledge of elementary algebra and prime numbers, as may be obtained by studying A level mathematics. (Foundation) Section 2: How (Part I) – Suitable for anyone with a knowledge of elementary algebra (including odd numbers, multiples of eight and the binomial theorem for expanding powers of (a+b)), and functions from the set of real numbers to itself (odd functions, even functions, multiplication and composition of functions). (Undergraduate year one) Section 3: How (Part II) – Requires some background knowledge of convergence and divergence of series of real numbers. A revision sheet is available. (Undergraduate year two) Dr Joel Feinstein, School of Mathematical Sciences Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras. Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area
An inductive proof of the Feinstein-Heath Swiss cheese “Classicalisation” theorem
No full textA theory of allocation maps has been developed by J. F. Feinstein and M. J. Heath in order to prove a theorem, using Zorn’s lemma, concerning the compact plane sets known as Swiss cheese sets. These sets are important since, as domains, they provide a good source of examples in the theory of uniform algebras and rational approximation. In this paper we take a more direct approach when proving their theorem by using transfinite induction and cardinality. An explicit reference to a theory of allocation maps is no longer required. Instead we find that the repeated application of a single operation developed from the final step of the proof by Feinstein and Heath is enough.</p
Banach function algebras with dense invertible group
No full textIn 2003 Dawson and Feinstein asked whether or not a Banach function algebra with dense invertible group can have a proper Shilov boundary. We give an example of a uniform algebra showing that this can happen, and investigate the properties of such algebras. We make some remarks on the topological stable rank of commutative, unital Banach algebras. In particular, we prove that whenever is approximately regular
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