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Brouwer fixed-point theorem

The Brouwer Fixed-Point Theorem states that in any continuous mapping of a compact convex shape, like a disk or a ball, there is at least one point that remains in the same position after the transformation. For example, if you push all the points of a disk into a new shape without tearing or lifting them, at least one point will land back where it started. This theorem is important in various fields, including mathematics, economics, and computer science, as it assures the existence of equilibrium states in systems.

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    The Brouwer Fixed Point Theorem states that any continuous function mapping a compact convex space to itself has at least one fixed point. In simpler terms, if you imagine stretching and bending a solid shape, like a ball, and mapping points on its surface to points inside it without tearing or creating holes, there will always be at least one point that stays in the same place. This theorem is significant in various fields, including mathematics and economics, as it applies to situations where systems return to equilibrium.