Related papers: Hilbert's Sixth Problem: the endless road to rigou…
If the quantum mechanical description of reality is not complete and a hidden variable theory is possible, what arises is the problem to explain where the rates of the outcomes of statistical experiments come from, as already noticed by…
We first recall a fact which is well-known among mathematical physicists although lesser-known among theoretical physicists that the standard quantum mechanics over a complex Hilbert space, is a Hamiltonian mechanics, regarding the Hilbert…
Polynome codes and code evaluation; arithmetical theory frames; $\mu$-recursive race for decision; decision correctness; decision termination; correct termination in theory $T = PR$ of Primitive Recursion; comparison with the negative…
A quantum mechanical theory is proposed which abandons an external parameter ``time'' in favor of a self-adjoint operator on a Hilbert space whose elements represent measurement events rather than system states. The standard quantum…
Analysing Quantum Measurement requires analysing the physics of amplification since amplification of phenomena from one scale to another scale is essential to measurement. There still remains the task of working this into an axiomatic…
The "problem of time" in canonical quantum gravity refers to the difficulties involved in defining a Hilbert space structure on states -- and local observables on this Hilbert space -- for a theory in which the spacetime metric is treated…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
We consider the space of probabilities {P(x)}, where the x are coordinates of a configuration space. Under the action of the translation group there is a natural metric over the space of parameters of the group given by the Fisher-Rao…
This survey article on Hilbert's first and second problems is adapted from a one-hour colloquium lecture given at the University of Auckland in May, 2000, just three months before the 100th anniversary of Hilbert's lecture. It includes an…
Quantum technology is seeing a remarkable explosion in interest due to a wave of successful commercial technology. As a wider array of engineers and scientists are needed, it is time we rethink quantum educational paradigms. Current…
In the 1920s, Ackermann and von Neumann, in pursuit of Hilbert's Programme, were working on consistency proofs for arithmetical systems. One proposed method of giving such proofs is Hilbert's epsilon-substitution method. There was, however,…
Hilbert's first problem is of importance in relation to work being done in computational systems. It is the question of equipollence of natural and real numbers. By construction equipollence is established for real numbers in open interval…
We present a Hilbert space perspective to homogenization of standard linear evolutionary boundary value problems in mathematical physics and provide a unified treatment for (non-)periodic homogenization problems in thermodynamics,…
This paper explores multiple closely related themes: bounding the complexity of Diophantine equations over the integers and developing mathematical proofs in parallel with formal theorem provers. Hilbert's Tenth Problem (H10) asks about the…
In his book `Physics and Philosophy', Heisenberg suggested that the quantum world is one of ``potentialities or possibilities'' and that the classical realm is one of ``things or facts''. After ascertaining that his categories most…
We introduce the key ideas and mathematical tools central to modern discussions of collapse. We assume knowledge of basic quantum mechanics -- the Schr\"{o}dinger evolution of states in Hilbert space, the Born probability rule, the Dirac…
New status in quantum mechanics is connected with recent achievements in the inverse problem. With its help instead of about ten exactly solvable models which serve as a basis of the contemporary education there are infinite (!) number,…
This paper concerns periodic solutions for a 1D-model with nonlocal velocity given by the periodic Hilbert transform. There is a rich literature showing that this model presents singular behavior of solutions via numerics and mathematical…
Sphere packing, Hilbert's eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains…
These lecture notes evolve around mathematical concepts arising in inverse problems. We start by introducing inverse problems through examples such as differentiation, deconvolution, computed tomography and phase retrieval. This then leads…