Raziskovalni matematični seminar
Raziskovalni matematični seminar poteka v organizaciji oddelkov za matematiko dveh članic Univerze na Primorskem - UP FAMNIT in Inštituta Andrej Marušič (UP IAM), in sicer vsak ponedeljek od oktobra do junija.
Vljudno vabljeni k udeležbi na prihodnjih seminarjih!
Consider an electrical network modeled by a digraph G each directed edge e of which is a semiconductor with a monomial conductance function ye* = fe (ye) = yer / μes if ye ≥0 and ye* = 0 if ye < 0. Here e is a directed edge, ye is the potential difference (voltage), ye* ≥0 is the current in e, and μe > 0 is the resistance of e; furthermore, r > 0 and s > 0 are two real parameters common for all edges. In particular, case r = s = 1 corresponds to the Ohm law, while r = ½, s =1 may be interpreted as the square law of resistance, which is typical of hydraulics and gas dynamics.
We show that for every ordered pair of nodes a, b of the network, the effective resistance μa,b is well-defined. In other words, any two-pole network with poles a and b can be effectively replaced by two oppositely directed edges: from a to b of resistance μa,b and from b to a of resistance μb,a. Furthermore, for every three nodes a, b, c, the inequality μa,cs/r + μc,bs/r ≥ μa,bs/r holds, in which the equality is achieved if and only if every directed path from a to b in G contains c.
In addition to case (i) s = r = 1, some limit values of parameters s and r are also of interest being related to classic triangle inequalities. Namely,
(ii) the length/time of a shortest directed path,
(iii) the inverse width of a bottleneck path,
(iv) the inverse capacity (maximum flow per unit time)
from a terminal a to a terminal b correspond to:
(ii) r = s → ∞,
(iii) r = 1, s → ∞,
(iv) r → 0, s = 1$, respectively.
These results generalize ones obtained in 1987 for monomial isotropic networks, which are modeled by undirected graphs. In this special case, effective resistances form a metric space
when s ≥ r and an ultrametric one when s/r → ∞.
Of course, for anisotropic networks, which are modeled by directed graphs, the symmetry, μa,b=μb,a may fail. For the linear isotropic networks triangle inequality is known since 1967 and it was extended to the linear anisotropic case in 2016.
