Related papers: Hilbert's Sixth Problem and Soft Logic
This contribution derives from a rather extensive study on the foundations of probability. We start by discussing critically the two main models of the random event in Probability Theroy and cast light over a number of incongruities. We…
We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs.…
In this paper, we introduce a fundamental framework to create a bridge between Probability Theory and Fuzzy Logic. Indeed, our theory formulates a random experiment of selecting crisp elements with the criterion of having a certain fuzzy…
This paper investigates probabilistic robustness of nonconvex-nonconcave minimax problems via the scenario approach. Specifically, under convex strategy sets for all players, inspired by recent advances in scenario optimization, we first…
The arrival of AI techniques in computations, with the potential for hallucinations and non-robustness, has made trustworthiness of algorithms a focal point. However, trustworthiness of the many classical approaches are not well understood.…
Analytic continuation problems are notoriously ill-posed without additional regularizing constraints, even though every analytic function has a rigidity property of unique continuation from every curve inside the domain of analyticity. In…
We argue that the complex numbers are an irreducible object of quantum probability. This can be seen in the measurements of geometric phases that have no classical probabilistic analogue. Having complex phases as primitive ingredient…
This note examines the implications of randomly selecting vectors from an infinite-dimensional Hilbert space on linear independence, assuming that for all $k$, the first $k$ vectors follow an absolutely continuous law with respect to a…
This paper introduces several new classes of mathematical structures that have close connections with physics and with the theory of dynamical systems. The most general of these structures, called indivisible stochastic processes,…
There is a third way of implementing probability models and practicing. This is to answer questions put in terms of observables. This eliminates frequentist hypothesis testing and Bayes factors and it also eliminates parameter estimation.…
The quantum mechanical transition probability is symmetric. A probabilistically motivated and more general quantum logical definition of the transition probability was introduced in two preceding papers without postulating its symmetry, but…
In this work, we establish conditions ensuring convergence in distribution of a sequence admitting a Wiener-It\^o chaos representation to a nondegenerate Gaussian measure on a separable Hilbert space. Our first main result shows that,…
This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or "toy" model of quantum mechanics over sets (QM/sets). There…
We examine the fundamental aspects of statistical mechanics, dividing the problem into a discussion purely about probability, which we analyse from a Bayesian standpoint. We argue that the existence of a unique maximising probability…
This article constructs the Hilbert space for the algebra $\alpha \beta - e^{i \theta} \beta \alpha = 1 $ that provides a continuous interpolation between the Clifford and Heisenberg algebras. This particular form is inspired by the…
In quantum logic, i.e., within the structure of the Hilbert lattice imposed on all closed linear subspaces of a Hilbert space, the assignment of truth values to quantum propositions (i.e., experimentally verifiable propositions relating to…
This paper discusses a class of uncertain optimization problems, in which unknown parameters are modeled by fuzzy intervals. The membership functions of the fuzzy intervals are interpreted as possibility distributions for the values of the…
Modal interpretations have the ambition to construe quantum mechanics as an objective, man-independent description of physical reality. Their second leading idea is probabilism: quantum mechanics does not completely fix physical reality but…
We explain why and how the Hilbert space comes about in quantum theory. The axiomatic structures of vector space, of scalar product, of orthogonality, and of the linear functional are derivable from the statistical description of quantum…
This thesis deals with applications of experimental mathematics to a number of problems. The first problem is related to random graph statistics. We consider a certain class of Galton-Watson random trees and look at the total height…