2011 - Autumn semester

December 29, 10:00 AM, Room #58
Happy New sc:ala Year!

December 22, 12:00 PM, Room #60
Miroslav Nikolić (Novi Sad): Modal approximations of damped linear systems

In a recent article by Veselić, a finite dimensional damped second order system is considered, and some spectral inclusion theorems of the associated quadratic eigenvalue problem are obtained. Modally damped part of the system was taken as the unperturbed system, to exploit the features of the well known proportionally damped systems. This kind of setup is applied to derive some easily computable sufficient conditions for the overdampedness of the given system. Inclusion sets, known as Cassini ovals, greatly outperform standard Gershgorin circles. However, there is reasonable amount of doubt that even these sets can be improved, or at least leveled.

December 15, 12:00 PM, Room #60
Miloš Stojaković (Novi Sad): Small sample spaces of permutations

We start by introducing the notion of a small sample space (SSS), with few illustrative examples. Then, we take a closer look at SSS of permutations with respect to the so-called min-wise independence. En route, we will use notions and techniques from combinatorics, probability theory, linear algebra and theoretical computer science.

December 8, 12:00 PM, Room #60
Vladimir Dragović (Belgrade): From Cayley’s determinant to integrable billiard dynamics

We present a fruitful relationship between geometry of pencils of quadrics and billiard dynamics. We describe periodic trajectories and explore related discrete structures such as the Poncelet-Darboux grids, the Weyr chains and the double-reflection nets. We introduce a class of discriminantly separable polynomials and relate them to the Kowalevski top dynamics and to two-valued Buchstaber-Novikov groups.

December 1, 12:00 PM, Room #60
Dragan Mašulović (Novi Sad): Countable homogeneous linearly ordered posets

A relational structure is called homogeneous if each isomorphism between its finite substructures extends to an automorphism of that structure. A linearly ordered poset is a relational structure consisting of a partial order relation on a set, along with a total (linear) order that extends the partial order in question. We characterise all countable homogeneous linearly ordered posets, thus extending earlier work by Cameron on countable homogeneous permutations.

November 24, 12:00 PM, Room #60
Igor Dolinka (Novi Sad): Finite homomorphism-homogeneous permutations via edge colourings of chains

A relational structure is homomorphism-homogeneous if any homomorphism between its finite substructures extends to an endomorphism of the structure in question. I will discuss the characterisation of all permutations on a finite set enjoying this property, obtained recently by Éva Jungábel and myself. To this end, I will review the more traditional view of a permutation as a set endowed with two linear orders (which eventually led to the theory of permutation patterns), and then switch to a different representation by a single linear order (considered as a directed graph with loops) whose non-loop edges are coloured in two colours, thereby `splitting' the linear order into two posets.

November 17, 12:00 PM, Room #60
Mária B. Szendrei (Szeged): Factorizability in certain classes of semigroups

In the same way as the notion of a group is considered the mathematical way of describing symmetry, the notion of an inverse monoid allows us catching partial symmetry. Inverse monoids appear in many areas of mathematics as collections of partial symmetries of objects. In this sense, factorizable inverse monoids can be identified with those structures of partial symmetries where each partial symmetry is a restriction of a total symmetry. Another approach to factorizable inverse monoids is via group actions on semilattices. In the structure theory of inverse monoids, they play a fundamental role since

• the factorizable inverse monoids are just the homomorphic images of semidirect products of semilattice monoids by groups,

• each inverse monoid is embeddable in a factorizable one, and

• they form a kind of dual of another important class of inverse monoids, called E-unitary inverse monoids.

Since the mid 1970s, these results have been generalized for a number of classes: first for inverse semigroups, and later for orthodox, locally inverse and restriction semigroups. In the talk we survey these results.

November 10, 12:00 PM, Room #60
Vladimir Kostić (Novi Sad): On the stability of neural networks

Biological-relevant neural networks represent large- and multi-time-scale nonlinear dynamical systems capturing both the activity and synaptic changes. These systems form the basis for every single cognitive task and their complex dynamical behavior has been very rigorously mathematically analyzed.

Inherently, these biological networks undergo many parametric perturbations. Thus, it is imperative to understand their dynamical behavior which has, as a result of activation functions and synaptic weights, instabilities. Motivated by this, in this talk we will address the stability conditions applicable to such large scale problems that can be obtained through the theory of diagonally dominant matrices.

November 3, 12:00 PM, Room #60
Zagorka Lozanov-Crvenković (Novi Sad): P-values vs. confidence intervals

Whereas, in hypothesis testing, study results lead the researcher to reject or accept a null hypothesis, in estimation, using confidence intervals, the researcher can assess whether a result is strong or weak, definitive or not.

October 27, 12:00 PM, Room #60
Robert D. Gray (Lisbon): Structures with lots of symmetry

A relational structure is homogeneous if every isomorphism between finite substructures extends to an automorphism. Countable homogeneous structures arise as Fraïssé limits of amalgamation classes of finite structures. The subject has connections to model theory, to permutation group theory, and to combinatorics.  

Following terminology of Fraïssé, a countable relational structure M is said to be set-homogeneous if, whenever two finite substructures U and V of M are isomorphic, there is an automorphism of M that sends U to V (setwise). Clearly every homogeneous structure is set-homogeneous, and Ronse (1978) showed that for finite graphs the converse is also true. He did this by classifying the finite set-homogeneous graphs and then observing that each of them also happens to be homogeneous. Following this, Enomoto (1981) reproved Ronse's result by giving a very elegant direct proof of the fact that every finite set-homogeneous graph is homogeneous.

In this talk I will present Enomoto's argument and go on to explain how it inspired some joint with Dugald Macpherson, Cheryl Praeger and Gordon Royle, which began with the question of whether a similar approach might be applied to other kinds of relational structure. Specifically I will talk about set-homogeneous directed graphs, outlining a classification in the finite case.

October 20, 12:00 PM, Room #60
Ljiljana Cvetković (Novi Sad): A report from the SC2011 conference (Sardinia, Italy)

The lecture will be followed by a small season-opening celebration.