Poincaré–Miranda theorem

In mathematics, the Poincaré–Miranda theorem is a generalization of the intermediate value theorem, from a single function in a single dimension, to $n$ functions in $n$ dimensions. It says as follows:

Consider

n

continuous functions of

n

variables,

f_{1},\ldots ,f_{n}

. Assume that for each variable

x_{i}

, the function

f_{i}

is constantly negative when

x_{i}=-1

and constantly positive when

x_{i}=1

. Then there is a point in the

n

-dimensional cube

[-1,1]^{n}

in which all functions are simultaneously equal to

0

.

The theorem is named after Henri Poincaré, who conjectured it in 1883, and Carlo Miranda, who in 1940 showed that it is equivalent to the Brouwer fixed-point theorem.[1]

Intuitive description

A graphical representation of Poincaré–Miranda theorem for

n = 2

The picture on the right shows an illustration of the Poincaré–Miranda theorem for $n = 2$ functions. Consider a couple of functions $(f, g)$ whose domain of definition is the $[-1,+1] 2$ square. The function $f$ is negative on the left boundary and positive on the right boundary (green sides of the square), while the function $g$ is negative on the lower boundary and positive on the upper boundary (red sides of the square). When we go from left to right along any path, we must go through a point in which $f$ is $0$ . Therefore, there must be a "wall" separating the left from the right, along which $f$ is $0$ (green curve inside the square). Similarly, there must be a "wall" separating the top from the bottom, along which $g$ is $0$ (red curve inside the square). These walls must intersect in a point in which both functions are $0$ (blue point inside the square).

Generalizations

The simplest generalization, as a matter of fact a corollary, of this theorem is the following one. For every variable $x i$ , let $a i$ be any value in the range $[sup x i = 0 f i, inf x i = 1 f i]$ . Then there is a point in the unit cube in which for all $i$ :

f_{i}=a_{i}

.

The this statement can be reduced to the original one by a simple translation of axes,

x_{i}^{\prime }=x_{i}\qquad y_{i}^{\prime }=y_{i}-a_{i}\qquad \forall i\in \{1,\dots ,n\}

where

$x i$ are the coordinates in the domain of the function
$y i$ are the coordinates in the codomain of the function

Notes

(Kulpa 1997, p. 545).

gollark: If something is right beside the edge of the gamefield, you can't push it, right?

gollark: And you cannotpush thingsWHICH ARE ADJACENTto the WALL?

gollark: Wait, *is* water noun?

gollark: And water is noun, you see.

gollark: Well, you could make the water float.

References

Dugundji, James; Granas, Andrzej (2003), Fixed Point Theory, Springer Monographs in Mathematics, New York: Springer-Verlag, pp. xv+690, ISBN 0-387-00173-5, MR 1987179, Zbl 1025.47002
Kulpa, Wladyslaw (June 1997), "The Poincare-Miranda Theorem", The American Mathematical Monthly, 104 (6): 545–550, doi:10.2307/2975081, JSTOR 2975081, MR 1453657, Zbl 0891.47040.
Miranda, Carlo (1940), "Un'osservazione su un teorema di Brouwer", Bollettino dell'Unione Matematica Italiana, Serie 2 (in Italian), 3: 5–7, JFM 66.0217.01, MR 0004775, Zbl 0024.02203.

External links

Ahlbach, Connor Thomas (2013). "A Discrete Approach to the Poincare–Miranda Theorem (HMC Senior Theses)". Retrieved 18 May 2015.

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[1] (Kulpa 1997, p. 545).