On all of the familiar Banach spaces, it is very easy to write down a formula which defines a norm one (linear) projection. This is not true for arbitrary Banach spaces. The following rough trichotomy indicates the range of behaviour.

Most finite dimensional Banach spaces admit no norm one projections at all (except those with rank one). Such examples may be smooth, strictly convex, or polyhedral.
Some separable Banach spaces admit no continuous projections at all (except those with finite rank or co-rank).
Many non-separable Banach spaces admit many nontrivial norm one projections, even under every equivalent norm.

The seminar is going to take place at 10:00 in B3-38.

2011-06-14 Seminar

Xiao-Xiong Gan
(Mathematics Department, Morgan State University, Baltimore, USA)

Formal Power Series and the Boundary Convergence of Power Series

The behaviour of power series on boundary of convergence domain has been an interesting topic since power series was introduced. For example,

$f(x) = \sum_{n=1}^{\infty}\dfrac{(-1)^n}{n}x^n$

converges on $(-1,1]$ but diverges at $x = -1$ , and $f^{\prime}(x)$ diverges at both $x = 1$ and $x = -1$ . The composition of formal power series has been an important part of the formal power series theory. We introduce some relationship between these two subjects and provide a condition for convergence of a power series at every point in its interval of convergence, including endpoints or boundary points.

2011-06-08 Seminar

Xiao-Xiong Gan
(Mathematics Department, Morgan State University, Baltimore, USA)

Composition and Generalized Composition of Formal Power Series

Given two formal power series

$f(z) = \sum_{n=1}^{\infty} a_n z^n$ and $g(z)=\sum_{n=0}^{\infty} b_n z^n$

the composition $g \circ f$ has been a very interesting topic for many years, where $f$ is called the nonunit. If $f(z) =\sum_{n=0}^{\infty} a_n z^n$ with $a_0 \neq 0$ , such a composition $g \circ f$ is called the generalized composition if it exists. We introduce the history of these two compositions and some results of them. Both of them are applied widely although the generalized composition is much younger than the regular one.

The seminar will take place at 9:00.

2011-06-01 Seminar

The joint seminar of the Optimization and Control Theory Department and the Department of Theory of Function Spaces will take place at 13:30 in room A1-33.

13:30

Mahide Kucuk
Department of Mathematics, Faculty of Science, Anadolu University, Eskişehir, Turkey

Construction and Some Properties of a Total Ordering Cone and Scalarization and Vectorization Using This Cone

14:15

Yalcin Kucuk
Department of Mathematics, Faculty of Science, Anadolu University, Eskişehir, Turkey

Weak Conjugacy of Scalar and Set Valued Maps and Weak
Conjugate Duality of Non-convex Scalar and Vector Optimization Problems

2011-05-31 Seminar

Timothy Myers
(Department of Mathematics, Howard University, Washington, D.C., USA)

Some Applications of the Sequence: 1, 5, 22, 93, 386,... to Dyck Paths and Ordered Trees.

The integer sequence starting: 1, 5, 22, 93, 386,... occurs in a variety of situations involving Dyck paths, grand Dyck paths, and ordered trees. We will discuss some new and some known connections, and in particular explore the enumeration of vertices in ordered trees by height using generating functions and the $V = TL$ equation.

The seminar has been cancelled.

2011-05-24 Seminar

prof. UZ dr hab. Krzysztof Przesławski
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra

Sets of constant width

During the seminar we will recall the definition of a set of constant width. Furthermore, we will provide some properties of the above-mentioned sets. All the presented results will be illustrated by examples.

The seminar is taking place at 11:30 in room A2-22.

2011-05-17 Seminar

Jakub Bielawski
Embedding theorem for fuzzy sets.

During the seminar we introduce and study new fuzzy algebraic operations: $\oplus$ -sum and scalar $\odot$ -multiplication deﬁned by

$(A \oplus B)(z):=\sup_{x+y=z}A(x)\cdot B(x)$ and $(\lambda \odot A)(z):=(A(z/\lambda))^\lambda$
where $A, B$ are fuzzy subsets and $\lambda \in (0, \infty)$ . This allows us to investigate a new deﬁnition of fuzzy convexity.

The seminar will take place in room A2-22 at 11:30.

2011-05-10 Seminar

Jerzy Grzybowski
On finite exhausters

The seminar will take place in room A2-22 at 11.30

We will introduce the notion of a shadowing set, which is a generalization of the notion of separating set to the family of more than two sets. Then we will give a criterion for an upper exhauster to be an exhauster of sublinear function as well as the criterion for minimality of finite upper exhauster.

2011-04-19 Seminar

Ryszard Urbański
On some applications of Simons' Theorem

During the seminar we will discuss a few theorems concerning the extension of functionals. Moreover, we will indicate possible applications of the above-mentioned results to convex analysis.

2011-04-12 Seminar

Ryszard Urbański
Some remarks on the Hahn-Banach theorem

The seminar is taking place at 11:30 in room A2-22.

(Continuation of the 2011-01-04 and 2011-03-22 Seminars).

2011-04-05 Seminar

dr Magdalena Łysakowska
Decomposition of $\mathbb R^3$ into unit balls

The seminar is taking place at 11:30.

2011-03-29 Seminar

prof. UZ dr hab. Krzysztof Przesławski
Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra

Finitely additive measures

It will be demonstrated that numerous operations over convex sets induce linear or multiliear mappings on vector spaces spanned by characteristic functions of such sets. A theorem of McMullen, Pukhlikov and Khovanskii on polynomial finite measures on polyhedra will be discussed as an application.

The seminar is taking place at 11:30.

2011-03-22 Seminar

Ryszard Urbański

Some remarks on the Hahn-Banach theorem

(Continuation of the 2011-01-04 Seminar).

2011-03-15 Seminar

Marcin Borkowski
On some properties of almost periodic functions

(Continuation of the 2011-01-11, 2011-01-18 and 2011-03-01 seminars).

2011-03-15 Convex Seminar

Jerzy Grzybowski
Geometry of convex surfaces

The convex seminar is taking place at 11:30 in room A2-22.

2011-03-08 Seminar

Hubert Przybycień
On the completeness of the Minkowski-Radström-Hörmander space

(Continuation of the 2011-02-22 Seminar).

2011-03-01 Seminar

Marcin Borkowski
On some properties of almost periodic functions

(Continuation of the 2011-01-11 and 2011-01-18 seminars).

2011-02-22 Seminar

Hubert Przybycień
On the completeness of the Minkowski-Radström-Hörmander space

The aim of the talk is to prove the completeness of the Minkowski-Radström-Hörmander space endowed with a certain metric.

2011-01-18 Seminar

Marcin Borkowski
On some properties of almost periodic functions

(Continuation of the 2011-01-11 Seminar).

2011-01-11 Seminar

Marcin Borkowski
On some properties of almost periodic functions

During the talk we will present a construction of certain uniformity on a given topological group G. It turns out that the class of (real or complex) uniformly continuous functions with respect to this uniformity coincides with the class of Bohr almost periodic functions on G. The talk is based on the paper of E. M. Alfsen and P. Holm A note on compact representations and almost periodicity in topological groups, Math. Scand. 10 (1962), 127-136.

2011-01-04 Seminar

Ryszard Urbański

Some remarks on the Hahn-Banach theorem

New seminar