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AG Audience

English

Note: We use this to send email in the morning.

45 Minutes

Saarbrücken

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.

Khaled Elbassioni, 11/29/2010 17:40 -- Created document.