Here are the titles and abstracts of the seminars which were held in 2012:

Monika Naskręcka**Homeomorphism of the space of polynomials and the space of their roots**

(Continuation of the 2012-12-11 Nonlinear Seminar)

Monika Naskręcka**Homeomorphism of the space of polynomials and the space of their roots**

During this seminar an elementary, topological proof of continuity of the roots of complex polynomials with respect to their roots will be shown. For normed polynomials with complex coefficients and properly defined metric spaces this theorem can be expressed as a homeomorphism of corresponding metric spaces of coeffecients and roots.

Piotr Maćkowiak**The existence of surplus demand function zeros**

(Continuation of the 2012-10-09 and 2012-10-16 Nonlinear Seminars.)

Piotr Zdanowicz**Basic properties of formal series**

(Continuation of the 2012-11-20 Nonlinear Seminar)

Piotr Zdanowicz**Basic properties of formal series**

Piotr Kasprzak**On a certain class of functions of -bounded variation**

(Continuation of the 2012-10-30 Nonlinear Seminar)

The seminar will take place at 8:30 in the Faculty Council meeting room (A1-33/34)

Justyna Signerska**Firing map for integrate-and-fire models with an almost periodic drive**

Consider the *integrate-and-fire* system , , in which the continuous dynamics is interrupted by the threshold-and-reset behaviour: if , meaning that once the value of a dynamical variable reaches a certain threshold it is immediately reset to a resting value and the system evolves again according to the differential equation. The question is to describe the time series of consecutive resets (spikes) as iterations of some map , called the *firing map*, and the sequence of interspike-intervals as a sequence of displacements along a trajectory of this map. The problem appears in various applications, e.g. in modelling of an action potential (spiking) by a neuron.

However, so far properties of the firing map were analytically investigated only for the function , which is smooth enough and often also periodic in . We present a complete description of the properties of the firing map arising from the most popular models: (Leaky Integrate-and-Fire) and (Perfect Integrator), when the function is only locally integrable and/or almost periodic. In particular, we prove that a Stepanov almost periodic function induces the firing map with the uniformly (Bohr) almost periodic displacement . In this way we provide a formal framework for next studying of the interspike-intervals in almost-periodically driven integrate-and-fire models.

Piotr Kasprzak**On a certain class of functions of -bounded variation**

During the seminar we are going to define a certain class of functions of -bounded variation and prove some properties of such functions. Furthermore, we are going to show that endowed with a certain functional the class in questions becomes a strictly convex Banach space. We are also going to provide several non-trivial examples illustrating our considerations.

Piotr Kasprzak**On a certain class of functions of -bounded variation**

During the seminar we are going to define a certain class of functions of -bounded variation and prove some properties of such functions. Furthermore, we are going to show that endowed with a certain functional the class in questions becomes a strictly convex Banach space. We are also going to provide several non-trivial examples illustrating our considerations.

Piotr Maćkowiak**The existence of surplus demand function zeros**

(Continuation of the 2012-10-09 Nonlinear Seminar)

Piotr Maćkowiak**The existence of surplus demand function zeros**

Piotr Kasprzak**On a certain class of functions of bounded variation**

(Continuation of the 2012-04-24 Nonlinear Seminar)

Michał Burzyński**Alternative proofs of Arrow's impossibility theorem**

(Continuation of the 2012-05-08 Nonlinear Seminar)

Michał Burzyński**Alternative proofs of Arrow's impossibility theorem**

(Continuation of the 2012-05-08 Nonlinear Seminar)

Michał Burzyński**Alternative proofs of Arrow's impossibility theorem**

Arrow's impossibility theorem states, that if there are at least three potential social states, there is no such social welfare function (that describes the society as a whole with given individual preferences), which satisfies the following four conditions: has an unbounded domain; the strong relation of social preferences satisfies the weak Pareto principle; satisfies the property of independence of irrelevant alternatives and in a society with preferences described with the function there is no dictatorship. The original proof, given by K. Arrow, consists of two steps: to show the existence of a deciding unit and to prove, it must be a dictator. The first of the alternative proofs is based on using the Condorcet preference. The second proof applies the first step of Arrow's proof and points, that all social decisions are made in the same way. The last proof is of graphic character.

Piotr Kasprzak**On a certain class of functions of bounded variation**

We are going to define a relatively large class of functions of bounded variation defined on an open subset of , which in particular contains the Sobolev space . Furthermore, we are going to discuss the connection between the classical variation in the sense of Jordan and the generalized one.

Piotr Kasprzak**On the compactness criterion in the space of bounded real-valued continuous functions defined on a non-compact domain**

During the seminar we are going to discuss a certain compactness criterion in the space of bounded real-valued continuous functions defined on a non-compact domain related to the well-known Arzela-Ascoli theorem.

Piotr Maćkowiak**Exploding points**

We will present and prove theorem of the exploding point. This theorem states that for any function defined on a compact set X, containing the cube K centered at zero in its interior, with values in X\K, which is the identity on the boundary of the set X, there exists a point c in X, that in any surroundings of c, there is a point x such that the value of f(x) and f(c) are located on opposite walls of the cube K.

Adam Nawrocki**An Application of Continued Fractions**

During this seminar we will discuss basic properties of Continued Fractions. Next we shall use Continued Fractions to calculate the limit of:

(Continuation of the 2012-03-13 Nonlinear Seminar)

Adam Nawrocki**An Application of Continued Fractions**

During this seminar we will discuss basic properties of Continued Fractions. Next we shall use Continued Fractions to calculate the limit of:

Marcin Wachowiak**Applications of Theorem Hopf-Lefschetz fixed point**

(Continuation of the 2011-11-22 Nonlinear Seminar and 2011-11-29 Nonlinear Seminar and 2011-12-06 Nonlinear Seminar).

Jędrzej Sadowski**Definitions of the N-almost periodic functions**

(Continuation of the 2011-12-20 Nonlinear Seminar, 2012-01-03 Nonlinear Seminar, 2012-02-14 Nonlinear Seminar and 2012-02-21 Nonlinear Seminar).

Jędrzej Sadowski**Definitions of the N-almost periodic functions**

(Continuation of the 2011-12-20 Nonlinear Seminar and 2012-01-03 Nonlinear Seminar and 2012-02-14 Nonlinear Seminar).

Jędrzej Sadowski**Definitions of the N-almost periodic functions**

(Continuation of the 2011-12-20 Nonlinear Seminar and 2012-01-03 Nonlinear Seminar).

Daria Bugajewska**On differential and inetgral equations in the spaces of functions of Lambda-bounded variation.**

(Continuation of the 2012-01-10 Nonlinear Seminar and 2012-01-17 Nonlinear Seminar).

Daria Bugajewska**On differential and inetgral equations in the spaces of functions of Lambda-bounded variation.**

(Continuation of the 2012-01-10 Nonlinear Seminar).

Daria Bugajewska**On differential and inetgral equations in the spaces of functions of Lambda-bounded variation.**

We are going to consider linear differential and nonlinear integral equations in the

spaces of functions of Lambda-bounded variation. We will state the existence and the existence and uniqueness results in this space as well as in its subspace consisting of contninuous functions of Lambda-bounded variation.

Jędrzej Sadowski**Definitions of the N-almost periodic functions**

(Continuation of the 2011-12-20 Nonlinear Seminar).