Campus Event Calendar

Event Entry

New for: D1, D3, D4, D5

What and Who

Metric and ultrametric spaces of resistances

Vladimir Gurvich
RUTCOR, Rutgers University
AG1 Mittagsseminar (own work)
AG 1, AG 3, AG 4, AG 5, SWS, RG1, MMCI  
AG Audience

Date, Time and Location

Tuesday, 30 November 2010
45 Minutes
E1 4
Rotunda 3rd floor


Given an electrical circuit
each edge $e$ of which is an isotropic conductor
with a monomial conductivity function $y_e^* = y_e^r /\mu_e^s$.

In this formula, $y_e$ is
the potential difference and $y_e^*$ current in $e$, while
$\mu_e$ is the resistance of $e$;
furthermore, $r$ and $s$ are two strictly positive real parameters common for all edges.

In particular, $r = s = 1$ correspond to the standard Ohm low.
In 1987, Gvishiani and Gurvich
[Russian Math. Surveys, 42:6(258) (1987) 235--236]
proved that, for every two nodes $a, b$ of the circuit,
the effective resistance $\mu_{a, b}$ is well-defined and
for every three nodes $a,b,c$ the following
"triangle" inequality holds

$\mu^{s/r}_{a, b} \leq \mu^{s/r}_{a, c} + \mu^{s/r}_{c, b}$.

It obviously implies the standard triangle inequality

$\mu_{a, b} \leq \mu_{a, c} + \mu_{c, b}$

whenever $s \geq r$ and it turns into the ultrametric inequality

$\mu_{a, b} \leq \max(\mu_{a, c}, \mu_{c, b})$

as $r/s \rightarrow 0$.

For the case $s = r = 1$ these results were rediscovered in 90s.
Now, in 23 years, I venture to reproduce
the original proof for the following reasons:

(i) the result is more general and one can get several interesting examples of metric and ultrametric spaces playing with parameters $r$ and $s$;

(ii) the proof is much simpler and can be easily explained
to high-school students;

(iii) the paper was written in Russian and the English translation in the internet is not free and not that easy to find out;

(iv) the last but not least: priority.


Khaled Elbassioni
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Uwe Brahm, 02/14/2011 13:37
Khaled Elbassioni, 11/29/2010 17:40 -- Created document.