December 16, 11:00 AM, Room #60
Maja Nedović (Novi Sad): Geršgorin-type theorems – a different norm approach and a generalisation to partitioned matrices 

The purpose of the talk is to present a different way of deriving Geršgorin's Theorem (GT). This approach will be extended to partitioned matrices and it gives rise to different proofs of well-known matrix eigenvalue inclusions. Instead of the so-called 'classical' proof of GT, we will see another one due to Householder, stemming from the theory of norms and analyse one of the earliest generalisations of strict diagonal dominance to the partitioned matrices, introduced independently by Ostrowski, Fiedler and Ptak, and Feingold and Varga.

December 9, 11:00 AM, Room #60
Miloš Stojaković (Novi Sad): Many collinear k-tuples in pointsets with no k+1 points collinear

For a finite set S of points in the plane, let t_k(S) denote the number of lines passing through exactly k points of S. Let r_k(n)=max t_k(S), where the maximum is over all sets S of n points in the plane with no k+1 collinear points.

In 1962, Erdős formulated the problem of determining r_k(n) for a constant k, conjecturing that r_k(n)=o(n²). It is obvious that r_k(n)=O(n²), for every k, and this is still the best known upper bound. From the other side, Elkies in 2006 proved that r_k(n)=Ω(n^{1+1/log k}), for all k>4.

We will show that r_k(n)=Ω(n^2-ε), for every k>3 and every ε>0, improving considerably on the lower bound.

This is joint work with J. Solymosi.

December 2, 11:00 AM, Room #60
Dragan Mašulović (Novi Sad): Probabilistic constructions of countable homogeneous structures

A countable random graph is a graph constructed as follows: as the set of vertices take the positive integers, and then for every pair of distinct vertices u and v toss a (not necessarily fair!) coin to decide whether u and v are adjacent. In 1963 Erdős and Rényi proved that with probability 1 all countable random graphs are isomorphic to the universal ultrahomogeneous graph, providing thus a probabilistic construction of the latter.

In 2002 Vershik provided a probabilistic construction of the Urysohn's space (the universal ultrahomogeneous metric space with rational distances). This construction, however, lacks the charm and elegance of the Erdős and Rényi's construction. In this talk we present a simplification of Vershik's construction.

November 18, 11:00 AM, Room #60
Joso Vukman (Maribor): On Halperin's problem

In 1965 S. Kurepa proved a result which can be formulated as follows.

Theorem. Let X be a vector space over the complex field C. Assume there exists a mapping Q : X → C such that Q(x+y) + Q(x-y) = 2Q(x) + 2Q(y) and Q(λx) = |λ|²Q(x) holds for all x,y from X and all complex numbers λ. Under these conditions the mapping B: X x X → C defined by B(x,y) = 1/4 [Q(x+y) - Q(x-y)] + i/4 [Q(x+iy) - Q(x-iy)] is additive in both arguments and the equalities B(λx,y)=λB(x,y) and B(x,λy)=λB(x,y) hold. In addition, we have Q(x)=B(x,x) for all x from X.

A simple proof of this result was given later by Vrbová. The above theorem can be considered as a generalisation of the well-known result due to P. Jordan and J. von Neumann which characterizes complex pre-Hilbert spaces among all complex normed spaces. S. Kurepa has shown that a real analogue of this theorem does not hold. Some more recent generalisations of this theorem will be presented in the talk.

The talk is dedicated to the memory of Svetozar Kurepa (1928-2010).

November 4, 11:00 AM, Room #60
Dragan Mašulović (Novi Sad): The Kirszbraun theorem

Assume that there are m balls in the Euclidean n-space and that they have a point in common. We then shuffle the balls and thus obtain a new configuration. If we know that the distances between the centres of the balls in the new configuration are not larger than the corresponding distances in the original arrangement, is it true that the balls in new configuration must have a point in common?

The affirmative answer for the Euclidean metric (ℓ₂) in Rⁿ is provided by the Kirszbraun Theorem. It is interesting that in case of the ℓ_∞-metric the proof can be carried out by purely combinatorial tools. On the other hand, an example provided by J.T. Schwartz in 1969 shows that the answer to the above question is negative for any ℓ_p-metric on Rⁿ when p belongs to  (1,2) or (2,∞). The problem is still open for the ℓ₁-metric on Rⁿ.

October 28, 11:00 AM, Room #60
Vladimir Kostić (Novi Sad): Applications of generalised diagonal dominance

Properties of matrices related to diagonal dominance were used previously in a large number of ways, and they proved to be extremely useful in various areas of linear algebra and its applications. The principal aim of this talk is to point out some of these applications, with emphasis on modelling wireless and optical communication networks, and ecosystems. The key question that arises in this context is the one of stability of dynamical systems. Interesting answers are offered by generalisations of the notion of diagonal dominance, both from the theoretical and practical point of view.

October 21, 10:00 AM, sc:ala (Room #58)
The first (internal) meeting of sc:ala members in the fall semester of 2010/11.

October 5, 12:15 PM, Computer Room #1
Manfred Droste (Leipzig): Random constructions imply symmetry
(A German-Serbian DAAD-project lecture; members of sc:ala assisted in its organisation.)

We will argue for the claim of the title in the areas of algebra, theoretical computer science, and theoretical physics. In algebra, we will consider the random graph and Ulm's theorem for countable abelian p-groups. For theoretical computer science, we will give a probabilistic construction of Scott domains and show that with probability 1 our construction produces a universal homogeneous domain. Finally, we consider causal sets which have been used as basic models for discrete space-time in quantum gravity.