Effective Quantum Observables

Thought experiments about the physical nature of set theoretical counterexamples to the axiom of choice motivate the investigation of peculiar constructions, e.g. an infinite dimensional Hilbert space with a modular quantum logic. Applying a concept due to BENIOFF, we identify the intrinsically effective Hamiltonians with those observables of quantum theory which may coexist with a failure of the axiom of choice. Here a self adjoint operator is intrinsically effective, iff the Schroedinger equation of its generated semigroup is soluble by means of eigenfunction series expansions.Comment: TeX-file, 32 page

Minimal counting systems and commutative monoids

These notes present an approach to obtaining the basic operations of addition and multiplication on the natural numbers in terms of elementary results about commutative monoids.Comment: 36 page

Solving large classes of nonlinear systems of PDEs

It is shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff continuous functions. The usual Navier-Stokes equations, as well as their various modifications aiming at a realistic modelling are included as particular cases. The same holds for the critically important constitutive relations in various branches of Continuum Mechanics. The solution method does not involve functional analysis, nor various Sobolev or other spaces of distributions or generalized functions. The general and type independent existence and regularity results regarding solutions presented here are a first in the literature.Comment: Preprin

On algebraic properties of sets of Banach ideal function spaces

It is shown that every set I(m) of Banach lattices of measurable functions defined on a measure space (Q,S,m), equipped with a some natural ordering became a modular lattice, which is Dedekind complete provided m is a probability measure. Moreover, some natural operations on considered spaces are in Galois connexion.Comment: Latex2e, corrected versio

Some Aspects of Boolean Valued Analysis

This is a survey of some recent applications of Boolean valued analysis to operator theory and harmonic analysis. Under consideration are pseudoembedding operators, the noncommutative Wickstead problem, the Radon-Nikodym Theorem for JB-algebras, and the Bochner Theorem for lattice-valued positive definite mappings on locally compact groups.Comment: 17 page

Hausdorff continuous solutions of nonlinear PDEs through the order completion method

It was shown in 1994, in Oberguggenberger & Rosinger, that very large classes of nonlinear PDEs have solutions which can be assimilated with usual measurable functions on the Euclidean domains of definition of the respective equations. In this paper the regularity of these solutions is significantly improved by showing that they can in fact be assimilated with Hausdorff continuous functions. The method of solution of PDEs is based on the Dedekind order completion of spaces of smooth functions which are defined on the domains of the given equations.Comment: Preprin

Isotone Cones in Banach Spaces and Applications to Best Approximations of Operators without Continuity Conditions

In this paper, we introduce the concept of isotone cones in Banach spaces. Then we apply the order monotonic property of the metric projection operator to prove the existence of best approximations for some operators without continuity conditions in partially ordered Banach spaces.Comment: 23 page

Extending Mappings between Posets

A variety of possible extensions of mappings between posets to their Dedekind order completion is presented. One of such extensions has recently been used for solving large classes of nonlinear systems of partial differential equations with possibly associated initial and/or boundary value problems

Temporal Type Theory: A topos-theoretic approach to systems and behavior

This book introduces a temporal type theory, the first of its kind as far as we know. It is based on a standard core, and as such it can be formalized in a proof assistant such as Coq or Lean by adding a number of axioms. Well-known temporal logics---such as Linear and Metric Temporal Logic (LTL and MTL)---embed within the logic of temporal type theory. The types in this theory represent "behavior types". The language is rich enough to allow one to define arbitrary hybrid dynamical systems, which are mixtures of continuous dynamics---e.g. as described by a differential equation---and discrete jumps. In particular, the derivative of a continuous real-valued function is internally defined. We construct a semantics for the temporal type theory in the topos of sheaves on a translation-invariant quotient of the standard interval domain. In fact, domain theory plays a recurring role in both the semantics and the type theory.Comment: 224 pages, including an inde

Ramsey for R1\mathcal{R}_{1} ultrafilter mappings and their Dedekind cuts

Associated to each ultrafilter $\mathcal{U}$ on $\omega$ and each map $p:\omega\rightarrow \omega$ is a Dedekind cut in the ultrapower $\omega^{\omega}/p( \mathcal{U})$. Blass has characterized, under CH, the cuts obtainable when $\mathcal{U}$ is taken to be either a p-point ultrafilter, a weakly-Ramsey ultrafilter or a Ramsey ultrafilter. Dobrinen and Todorcevic have introduced the topological Ramsey space $\mathcal{R}_{1}$. Associated to the space $\mathcal{R}_{1}$ is a notion of Ramsey ultrafilter for $\mathcal{R}_{1}$ generalizing the familiar notion of Ramsey ultrafilter on $\omega$. We characterize, under CH, the cuts obtainable when $\mathcal{U}$ is taken to be a Ramsey for $\mathcal{R}_{1}$ ultrafilter and $p$ is taken to be any map. In particular, we show that the only cut obtainable is the standard cut, whose lower half consists of the collection of equivalence classes of constants maps. Forcing with $\mathcal{R}_{1}$ using almost-reduction adjoins an ultrafilter which is Ramsey for $\mathcal{R}_{1}$. For such ultrafilters $\mathcal{U}_{1}$, Dobrinen and Todorcevic have shown that the Rudin-Keisler types of the p-points within the Tukey type of $\mathcal{U}_{1}$ consists of a strictly increasing chain of rapid p-points of order type $\omega$. We show that for any Rudin-Keisler mapping between any two p-points within the Tukey type of $\mathcal{U}_{1}$ the only cut obtainable is the standard cut. These results imply existence theorems for special kinds of ultrafilters