Mathematical Research Seminar - Archive

Competitive equilibrium problems are usually studied by means of fixed point theory. Alternatively, it is possible to study these problems by means of a variational approach. The variational inequality theory was introduced by Fichera and Stampacchia, in the early 1960's, in connection with several equilibrium problems originating from mathematical physics. This theory turned out to be an innovative and powerful methodology for the study of several kind of equilibrium problems.

Here we consider a new formulation of a competitive equilibrium in terms of a suitable quasivariational inequality involving multivalued maps. More precisely, it is considered a pure exchange economy where the consumer's preferences are represented by quasiconcave and non-differentiable utility functions. By relaxing concavity and differentiability assumptions on the utility functions, the subdifferential operator of the utility function, required in the variational problem, is suitably replaced by a multimap involving the normal operator to the adjusted sublevel sets. Using this variational formulation, we are able to prove the existence of equilibrium points by using arguments from the set-valued analysis and non-convex analysis.

A blocking set in a plane is a set of points that meets every line. It is called non-trivial if it does not contain a line. A blocking set is minimal if it is minimal subject to set theoretical inclusion. There are classical results on the possible sizes of minimal blocking sets, due to Bruen, Bruen-Thas, Blokhuis, Ball Gacs, Sziklai and the speaker. We will try to survey these result. An almost blocking set is a set of points which "almost" meets all lines, that is there are only few lines that are disjoint from it. We will show some conditions (specify what "few" means) that guarantee that an almost blocking set can be obtained from a blocking by deleting not too many points. Again, there are some old results by Erdos and Lovasz in this directions. We will mention recent such results by Zsuzsa Weiner and the speaker.

In the group theory, ”arithmetical” properties of a group G are the properties which are defined by some arithmetical parameters of G.

Let G be a finite group. The spectrum of G is the set omega(G) of orders of all its elements. The subset pi(G) of prime elements of omega(G) is called prime spectrum of G. The spectrum omega(G) of a group G defines its prime graph (or Grünberg–Kegel graph) Gamma(G) with vertex set pi(G), in which any two different vertices p and q are adjacent if and only if the number pq belongs to the set omega(G). The spectrum, the prime spectrum and the prime graph of a finite group G give examples of arithmetical parameters of G.

Such invariants of a group G as sets of composition and chief factors of G with details of action of G on its chief factors are called the ”normal structure” of G . Cross impact of arithmetical properties of a finite group and its normal structure is well known.

In the present talk we will discuss some questions on the finite prime spectrum minimal, prime graph minimal and spectrum minimal groups. In particular, we will describe all the cases when the prime graphs of a finite simple group and of its proper subgroup coincide and spectrums of a finite simple group and of its proper subgroup coincide.

The Ring of Fire is a problem that baffled me for 25 years. Here it is: You are given a circle of five integers; some positive, some negative, and the sum of all of them is positive. As long as there is at least one negative number in the ring, select one and then 'pop' it by replacing it with its absolute value and then adding the negative version to both of its neighbours. In symbols, replace  b  -c  d  with  b-c  c  d-c. Notice that this leaves the sum unaffected.  For instance we could have
-1  6  2  -5  1
Popping the -5 gives
-1  6  -3  5  -4
Then popping the -4 gives
-5  6  -3  1  4.
You are to give an elementary proof that this process eventually terminates, i.e., that no matter what sequence of negatives you choose to pop, eventually all of the numbers on the circle will become non-negative.  We use   rocks   and   fishes   to describe the situation.  And we propose a further unsolved problem for which rocks and fishes provide no help.

Racionalna gibanja so zelo uporabno orodje v računalniški grafiki, robotiki in sorodnih področjih. Konstrukcijo navadno razdelimo na dva dela - najprej določimo rotacijski del, nato dodamo translacijski del gibanja. Za interpolacijo danih pozicij se pogosto uporabljajo standardne interpolacijske sheme, ki imajo dve ključni slabosti. Dobljeno gibanje je odvisno od parametrizacije, ki mora biti vnaprej določena, konstruirani polinomi, ki določajo rotacijski del gibanja, pa so relativno visoke stopnje. V doktorski disertaciji bo obravnavan drug pristop - konstrukcija gibanj, ki interpolirajo dano zaporedje pozicij togega telesa (tj. pozicije centra in rotacije) s pomočjo geometrijske interpolacije.  Ta pristop ima pomembne prednosti, med drugim to, da je parametrizacija izbrana avtomatično, dobljeno gibanje pa je najmanjše možne stopnje. V doktorski disertaciji bomo predstavili nekatere interpolacijske probleme, izpeljali ustrezne enačbe in razvili interpolacijske sheme za konstrukcije gibanj togih teles z racionalnimi zlepki nizkih stopenj. Vsi teoretični rezultati bodo tudi podkrepljeni z numeričnimi zgledi.
To predavanje je namenjeno predstavitvi teme za doktorsko disertacijo in bo potekalo v slovenskem jeziku.