A Markov theorem for generalized plat decomposition

We prove a Markov theorem for tame links in a connected closed orientable 3-manifold $M$ with respect to a plat-like representation. More precisely, given a genus $g$ Heegaard surface $\Sigma_g$ for $M$ we represent each link in $M$ as the plat closure of a braid in the surface braid group $B_{g,2n}=\pi_1(C_{2n}(\Sigma_g))$ and analyze how to translate the equivalence of links in $M$ under ambient isotopy into an algebraic equivalence in $B_{g,2n}$. First, we study the equivalence problem in $\Sigma_g\times [0,1]$, and then, to obtain the equivalence in $M$, we investigate how isotopies corresponding to "sliding" along meridian discs change the braid representative. At the end we provide explicit constructions for Heegaard genus 1 manifolds, i.e. lens spaces and $S^2\times S^1$.Comment: Acknowledgements added. Accepted for publication on Ann. Sc. Norm. Super. Pisa Cl. Sc

Informacijski sistem za znanstvene raziskave mikroklime v kraških jamah: Primer Postojnske jame, Slovenija

The paper presents the construction and operation of a cave information system on the example of the Postojnska jama cave. It thoroughly describes the construction of the basic modules of the cave information system, such as an automatic cave measuring station with sensors for measuring air temperature, air flow and CO2 concentration, and a central unit for collecting, saving, processing and displaying data. The system provides quality automatic measurements for the scientific study of the micrometeorology of karst caves. Its construction is based on good practices stemming from many years of experience in taking measurements in the external atmosphere and in karst caves. The paper is rich and comprehensive guideline for automatic cave measuring system construction for scientific and tourist management purposes.V članku na primeru Postojnske jame predstavimo izdelavo in delovanje jamskega informacijskega sistema. Podrobno opišemo izdelavo osnovnih gradnikov sistema. Mednje spadajo jamska merilna postaja s senzorji za merjenje temperature, vetra in koncentracije CO2 ter centralna enota za zbiranje, procesiranje in prikaz podatkov. Sistem zagotavlja kakovostne avtomatske meritve za znanstvene raziskave jamske mikroklime. Zgradba sistema temelji na dolgoletnih izkušnjah iz okoljskih meritev v jamskem in zunanjem okolju. Članek je tudi priročnik za izgradnjo avtomatskih merilnih sistemov za raziskovalne namene in za podporo pri upravljanju turističnih jam

Alexandrov polinom sklenjenih kit v lečastih prostorih

We present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid.Vpeljemo reducirano Burauovo reprezentacijo za mešano grupo kit na eni niti, ki predstavlja splet v lečastem prostoru in pokažemo kako izračunati Alexandrov polinom spleta neposredno iz mešane kite

O Aleksandrovem polinomu spletov v lečastih prostorih

We show how the Alexander polynomial of links in lens spaces is related to the classical Alexander polynomial of a link in the 3-sphere, obtained by cutting out the exceptional lens space fiber. It follows from this relationship that a certain normalization of the Alexander polynomial satisfies a skein relation in lens spaces.Pokažemo, kako je Aleksandrov polinom spletov v lečastih prostorih povezan s klasičnim Aleksandrovim polinomom spleta v 3-sferi, ki ga dobimo, če izrežemo izjemno vlakno lečastega prostora. Iz te povezave sledi, da normalizacija Aleksandrovega polinoma v lečastem prostoru ustreza premenjalni relaciji

Invariants of multi-linkoids

In this paper, we extend the definition of a knotoid that was introduced by Turaev, to multi-linkoids that consist of a number of knot and knotoid components. We study invariants of multi-linkoids that lie in a closed orientable surface, namely the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the $T$-invariant in relation with generalized $\Theta$-graphs.Comment: 15 page

Funkcije več spremenljivk in osnove vektorske analize

Učbenik je namenjen študentom drugega letnika razvojno raziskovalnega programa pri predmetu Matematika 3 na Fakulteti za strojništvo, Univerze v Ljubljani (FS UL), vendar avtorja verjameva, da bo koristen študijski pripomoček tudi študentom drugih naravoslovnih študijskih programov. V tem učbeniku obravnavana snov predstavlja pomembno matematično orodje za reševanje številnih fizikalnih in inženirskih problemov, ki so obravnavani pri mnogih predmetih na FS UL. Dobro obvladovanje in razumevanje te snovi sodi v “osnovno opremo” vsakega inženirja, ki se želi s takšnimi problemi ukvarjati na primerno visokem nivoju

Neskončno mnogo rešitev s spremenljivim predznakom za probleme Kirchhoffovega tipa v R[sup]3

In this paper, we consider the following nonlinear Kirchhoff type problem: ▫begin{cases} - Big (a+b int_{mathbb{R}^3} |nabla u|^2 Big) Delta u + V(x)u = f(u), & text{in} quad mathbb{R}^3, \ u in H^1 (mathbb{R}^3), end{cases}▫ where ▫$a,b > 0$▫ are constants, the nonlinearity ▫$f$▫ is superlinear at infinity with subcritical growth and ▫$V$▫ is continuous and coercive. For the case when ▫$f$▫ is odd in ▫$u$▫ we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity ▫$|u|^{p-2}u$▫ with ▫$p in (2, 4]$▫.Obravnavamo naslednji nelinearni problem Kirchhoffovega tipa ▫begin{cases} - Big (a+b int_{mathbb{R}^3} |nabla u|^2 Big) Delta u + V(x)u = f(u), & text{in} quadmathbb{R}^3, \ u in H^1 (mathbb{R}^3), end{cases}▫ kjer sta ▫$a,b > 0$▫ konstanti, nelinearni člen ▫$f$▫ je superlinearen v neskončnosti, s subkritično rastjo, ▫$V$▫ pa je zvezna in vsiljena funkcija. V primeru, ko je ▫$f$▫ liha funkcija za ▫$u$▫, dobimo z uporabo kombinacije invariantnih množic in mini-maks metode Ljusternik-Schnirelmanovega tipa neskončno mnogo rešitev s spremenljivim predznakom za ta problem. Kolikor je nam znano, je bilo doslej najdenih le malo eksistenčnih rezultatov za ta problem. Velja omeniti, da nelinearni člen ni nujno 4-superlinearen v neskončnosti, konkretno vključuje nelinearnost potenčnega tipa ▫$|u|^{p-2}u$▫ za ▫$p$▫ iz intervala ▫$(2,4]$▫