University of Primorska Faculty of Mathematics, Natural Sciences and Information Technologies
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Mathematical Research Seminar - Archive

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18.10.2010. ob 10:00 Seminarska soba v Galebu
Predavatelj: dr.Martin Milanič.
Naslov: Graphs of separability at most two: structural characterizations and their consequences
 
Povzetek: We introduce graphs of separability at most k as graphs in which every two non-adjacent vertices can be separated by a set of at most k other vertices. Graphs of separability at most k arise in connection with the parsimony haplotyping problem from computational biology. For k = 0 or k = 1, the only connected graphs of separability at most k are complete graphs and block graphs, respectively. For values of k greater than 2, graphs of separability at most k form a rich class of graphs containing all graphs of maximum degree k.
We prove several characterizations of graphs of separability at most 2, which generalize complete graphs, cycles and trees. The main result is that every connected graph of separability at most 2 can be constructed from complete graphs and cycles by pasting along vertices or edges, and vice-versa, every graph constructed this way is of separability at most 2. The structure theorem has nice algorithmic implications – some of which cannot be extended to graphs of higher separability – however certain optimization problems remain intractable on graphs of separability 2. Finally, we characterize graphs of separability at most 2 in terms of minimal forbidden induced subgraphs and minimal forbidden induced minors.
 
You can download slides from the lecture here. DOWNLOAD

11.10.2010. ob 10:00 Seminarska soba v Galebu
Predavatelj:  Alexandru Tomescu  (University of Udine, Italy)
Naslov: Mapping Hypersets into Numbers

Povzetek: We introduce and prove the basic properties of an encoding that generalizes to non-well-founded hereditarily finite sets the bijection defined by Ackermann between hereditarily finite sets and natural numbers. The new encoding maps its domain onto dyadic rational numbers.
 
You can download slides from the lecture here: DOWNLOAD

4.10.2010. ob 10:00 Seminarska soba v Galebu
Predavatelj: dr. Barbara Boldin
Naslov: Biološke invazije v strukturiranih populacijah

Povzetek :Kdaj je invazija nove populacije uspešna? Če dinamiko populacij opišemo kot determinističen proces in privzamemo, da so obstoječe populacije v ravnovesnem stanju, potem na to vprašanje lahko odgovorimo takole: če je osnovno reprodukcijsko razmerje   nove populacije R­0 večje kot 1, je invazija uspešna, medtem ko je invazija obsojena na neuspeh kadar je R­0 0 = 1 pride do transkritične bifurkacije. 

Z matematičnega vidika obstaja samo en tip transkritične bifurkacije. Kadar pa dinamični sistem opisuje nek biološki proces, moramo razlikovati dva primera: superkritično bifurkacijo, pri kateri netrivialna veja ravnovesnih stanj obstaja za R­0 > 1 in podkritično bifurkacijo, kjer netrivialne ravnovesne točke najdemo le ko je R­0 

Na predavanju bomo predstavili način izračuna osnovnega repodukcijskega razmerja za strukturirane populacije, izpeljali formulo za določanje smeri bifurkacije in opisali posledice tipa transkritične bifurkacije za biološki proces. Rezultati so uporabni za študij različnih invazij, tako v ekologiji, epidemiologiji kot tudi evoluciji, kar bomo prikazali na primerih.

Predavanje bo povzeto po članku:
“B. Boldin: Introducing a population into a steady community: the critical case, the centre manifold and the direction of bifurcation. SIAM Journal on Applied Mathematics, Volume 66 (2006), Issue 4, pp. 1424-1453”

 Tukaj lahko prevzamete prosojnice s predavanja:DOWNLOAD