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Regularity theorems are presented for cosmology and gravitational collapse in non-Riemannian gravitational theories. These theorems establish conditions necessary to allow the existence of timelike and null path complete spacetimes for…
The first part of this work deals with the development of a natural differential calculus on non-commutative manifolds. The second part extends the covariance and equivalence principle as well studies its kinematical consequences such as…
The first mathematical problems of the global analysis of dynamical models can be traced back to the engineering problem of the Watt governor design. Engineering requirements and corresponding mathematical problems led to the fundamental…
Long tails in the velocity distribution are observed in plasmas and gravitational systems. Some experiments and observations in far-from-equilibrium conditions show that these tails behave as 1/v^(5/2). We show here that such heavy tails…
The question of building a local diff-invariant effective gravitational action for the trace anomaly is reconsidered. General Relativity (GR) combined with the existing action for the trace anomaly is an inconsistent low energy effective…
We derive two-sided estimates on moments and tails of Gaussian chaoses, that is, random variables of the form $\sum a_{i_1,...,i_d}g_{i_1}... g_{i_d}$, where $g_i$ are i.i.d. ${\mathcal{N}}(0,1)$ r.v.'s. Estimates are exact up to constants…
Although stochastic optimization is central to modern machine learning, the precise mechanisms underlying its success, and in particular, the precise role of the stochasticity, still remain unclear. Modelling stochastic optimization…
Ever since we have been in the possession of quantum theories without observers, such as Bohmian mechanics or the Ghirardi-Rimini-Weber (GRW) theory of spontaneous wave function collapse, a major challenge in the foundations of quantum…
The search for a consistent and empirically established quantum theory of gravity is among the biggest open problems of fundamental physics. The obstacles are of formal and of conceptual nature. Here, I address the main conceptual problems,…
The problem of characterizing complexity of quantum dynamics - in particular of locally interacting chains of quantum particles - will be reviewed and discussed from several different perspectives: (i) stability of motion against external…
In the last four decades different programs have been carried out aiming at understanding the final fate of gravitational collapse of massive bodies once some prescriptions for the behaviour of gravity in the strong field regime are…
This paper presents nine inconsistency theorems for general relativity theory (GRT), and shows that they ultimately originate from the use of Riemannian curvature and the abandonment of universal invariance (which is stronger than the…
Let a number, N, of particles interact classically through Newton's Laws of Motion and Newton's inverse square Law of Gravitation. The resulting equations of motion provide an approximate mathematical model with numerous applications in…
The Problem of Time occurs because the `time' of GR and of ordinary Quantum Theory are mutually incompatible notions. This is problematic in trying to replace these two branches of physics with a single framework in situations in which the…
The quantum superposition principle is reexamined and reformulated based on the adiabatic theorem of quantum mechanics, nonadiabatic dressed states and experimental evidences. The collapse of the wave function and the quantum measurement…
We study a class of stochastic evolution equations of jump type with random coefficients and its optimal control problem. There are three major ingredients. The first is to prove the existence and uniqueness of the solutions by continuous…
In a scalar field theory that has a symmetric octic potential with a quartic minimum and two quadratic minima, kink and mirror kink solutions have long-range tails. We calculate the force between these kinks when their long-range tails…
Maintaining the position that the wave function $\psi$ provides a complete description of state, the traditional formalism of quantum mechanics is augmented by introducing continuous trajectories for particles which are sample paths of a…
Discontinuous dynamical systems with grazing solutions are discussed. The group property, continuation of solutions, continuity and smoothness of motions are thoroughly analyzed. A variational system around a grazing solution which depends…
Every physical theory has (at least) two different forms of mathematical equations to represent its target systems: the dynamical (equations of motion) and the kinematical (kinematical constraints). Kinematical constraints are…