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We consider the product of infinitely many copies of a spin-$1\over 2$ system. We construct projection operators on the corresponding nonseparable Hilbert space which measure whether the outcome of an infinite sequence of $\sigma^x$…
This paper examines the quantum mechanical system that arises when one quantises a classical mechanical configuration described by an underdetermined system of equations. Specifically, we consider the well-known problem in classical…
We describe classical orbits and quantum states of a three particle system which is weakly chaotic. The quantum energies of the lowest 48 states which are invariant under permutations are given both in various approximations and to high…
In non-relativistic as well as in special relativistic quantum theory, {\em mass} and {\em charge} are {\em pure numbers} appearing in various (quantum) operators and admit {\em any values}, {\it ie}, values for these quantities are to be…
We apply classical invariant theory of binary forms to explicitly characterize isomorphism classes of hyperelliptic curves of small genus and, conversely, propose algorithms for reconstructing hyperelliptic models from given invariants. We…
This third part of the series is a brief comment to certain aspects of the theory of classical $r$-matrix and bihamiltonian formalism, which motivations lie in constructions of the previous two parts.
This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the…
We examine three probabilistic formulations of the sentence a and b are totally unrelated with respect to a given set of variables U. First, two variables a and b are totally independent if they are independent given any value of any subset…
The two envelopes paradox is discussed. By calculating the conditional probability, we arrive at a conditional expectations which differs from existing results.
Classical problem of random triangle in square is solved by simple and transparent geometrical method.
We present a conclusive answer to Bertrand's paradox, a long standing open issue in the basic physical interpretation of probability. The paradox deals with the existence of mutually inconsistent results when looking for the probability…
First we survey generating function methods for obtaining useful probability estimates about random matrices in the finite classical groups. Then we describe a probabilistic picture of conjugacy classes which is coherent and beautiful.…
The ternary Goldbach conjecture (or three-prime conjecture) states that every odd number greater than 5 can be written as the sum of three primes. The purpose of this book is to give the first proof of the conjecture, in full.
In this paper we study the Three Hat Problem which appeared in Puzzle Corner of the Technology Review magazine. This puzzle gives a scenario in which three players wearing hats are sitting together and each hat can be seen by everyone…
We introduce a notion of the ``explanation" of one (generalized) probabilistic model by another as particular kind of span in the category $\Prob$ of probabilistic models and morphisms. We show that explanations compose under a standard…
A large amount of literature in social choice theory deals with quantifying the probability of certain election outcomes. One way of computing the probability of a specific voting situation under the Impartial Anonymous Culture assumption…
The ancient unsolved problem of congruent numbers has been reduced to one of the major questions of contemporary arithmetic: the finiteness of the number of curves over $\bf Q$ which become isomorphic at every place to a given curve. We…
We describe a scheme for constructing quantum mechanics in which a quantum system is considered as a collection of open classical subsystems. This allows using the formal classical logic and classical probability theory in quantum…
The Jacobian Conjecture has been reduced to the symmetric homogeneous case. In this paper we give an inversion formula for the symmetric case and relate it to a combinatoric structure called the Grossman-Larson Algebra. We use these tools…
This paper is a case study of probabilistic approach to homological aspects of topological quantum field theory via the example of topological quantum mechanics. We propose topological correlations in terms of large variance limit. An…