Journal•ISSN: 0003-486X

# Annals of Mathematics

Princeton University

About: Annals of Mathematics is an academic journal published by Princeton University. The journal publishes majorly in the area(s): Conjecture & Group (mathematics). It has an ISSN identifier of 0003-486X. Over the lifetime, 5088 publications have been published receiving 534031 citations. The journal is also known as: Ann. Math..

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TL;DR: In this article, it was shown that the set of equilibrium points of a two-person zero-sum game can be defined as a set of all pairs of opposing "good" strategies.

Abstract: we would call cooperative. This theory is based on an analysis of the interrelationships of the various coalitions which can be formed by the players of the game. Our theory, in contradistinction, is based on the absence of coalitions in that it is assumed that each participant acts independently, without collaboration or communication with any of the others. The notion of an equilibrium point is the basic ingredient in our theory. This notion yields a generalization of the concept of the solution of a two-person zerosum game. It turns out that the set of equilibrium points of a two-person zerosum game is simply the set of all pairs of opposing "good strategies." In the immediately following sections we shall define equilibrium points and prove that a finite non-cooperative game always has at least one equilibrium point. We shall also introduce the notions of solvability and strong solvability of a non-cooperative game and prove a theorem on the geometrical structure of the set of equilibrium points of a solvable game. As an example of the application of our theory we include a solution of a

6,577 citations

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2,896 citations

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TL;DR: The superposition principle of the wave function is defined in this article, which is the fundamental principle of quantum mechanics that the system of states forms a linear manifold, in which a unitary scalar product is defined.

Abstract: It is perhaps the most fundamental principle of Quantum Mechanics that the system of states forms a linear manifold,1 in which a unitary scalar product is defined.2 The states are generally represented by wave functions3 in such a way that φ and constant multiples of φ represent the same physical state. It is possible, therefore, to normalize the wave function, i.e., to multiply it by a constant factor such that its scalar product with itself becomes 1. Then, only a constant factor of modulus 1, the so-called phase, will be left undetermined in the wave function. The linear character of the wave function is called the superposition principle. The square of the modulus of the unitary scalar product (ψ,Φ) of two normalized wave functions ψ and Φ is called the transition probability from the state ψ into Φ, or conversely. This is supposed to give the probability that an experiment performed on a system in the state Φ, to see whether or not the state is ψ, gives the result that it is ψ. If there are two or more different experiments to decide this (e.g., essentially the same experiment, performed at different times) they are all supposed to give the same result, i.e., the transition probability has an invariant physical sense.

2,694 citations

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TL;DR: In this article, it was shown that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and in particular one can never predict both the position and the momentum of S, (Heisenberg's Uncertainty Principle) and most pairs of observations are incompatible, and cannot be made on S simultaneously.

Abstract: One of the aspects of quantum theory which has attracted the most general attention, is the novelty of the logical notions which it presupposes It asserts that even a complete mathematical description of a physical system S does not in general enable one to predict with certainty the result of an experiment on S, and that in particular one can never predict with certainty both the position and the momentum of S, (Heisenberg’s Uncertainty Principle) It further asserts that most pairs of observations are incompatible, and cannot be made on S, simultaneously (Principle of Non-commutativity of Observations)

2,315 citations

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2,037 citations