Campus Event Calendar

### Event Entry

New for: D1, D3, D4, D5

### 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
English

#### Date, Time and Location

Tuesday, 30 November 2010
15:00
45 Minutes
E1 4
Rotunda 3rd floor
Saarbrücken

#### Abstract

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.

#### Contact

Khaled Elbassioni
--email hidden
passcode not visible
logged in users only

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