Related papers: Kochen-Specker Sets and Generalized Orthoarguesian…

We describe an explicitly non-contextual statistical model of hidden variables for the qutrit, which fully reproduces the predictions of quantum mechanics and, thus, bypasses the constraints imposed by the Kochen-Specker theorem and its…

Quadratic algebras are generalizations of Lie algebras; they include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical…

We define topological orthoalgebras (TOAs) and study their properties. While every topological orthomodular lattice is a TOA, the lattice of projections of a Hilbert space is an example of a lattice-ordered TOA that is not a toplogical…

It is shown how the 300 rays associated with the antipodal pairs of vertices of a 120-cell (a four-dimensional regular polytope) can be used to give numerous "parity proofs" of the Kochen-Specker theorem ruling out the existence of…

One of spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group $SU_q(2)$ through the Askey-Wilson polynomials, associated with the $q$-hypergeometric functions…

Quantum processes cannot be reduced, in a nontrivial way, to classical processes without specifying the context in the description of a measurement procedure. This requirement is implied by the Kochen-Specker theorem in the…

We introduce a logic modelling some aspects of the behaviour of the measurement process, in such a way that no direct mention of quantum states is made, thus avoiding the problems associated to this rather evasive notion. We then study some…

We consider the problem of existence of the diagonal representation for operators in the space of a family of generalized coherent states associated with an unitary irreducible representation of a (compact) Lie group. We show that necessary…

We prove that every Kakeya set in $\mathbb{R}^3$ formed from lines of the form $(a,b,0) + \operatorname{span}(c,d,1)$ with $ad-bc=1$ must have Hausdorff dimension $3$; Kakeya sets of this type are called $SL_2$ Kakeya sets. This result was…

We consider the Kronecker algebra $A=\mathcal{O}[X, Y]/(X^2, Y^2)$, where $\mathcal{O}$ is a complete discrete valuation ring. Since $A \otimes\kappa$ is a special biserial algebra, where $\kappa$ is the residue field of $\mathcal{O}$, one…

A subspace representation of a poset $\mathcal S=\{s_1,...,s_t\}$ is given by a system $(V;V_1,...,V_t)$ consisting of a vector space $V$ and its subspaces $V_i$ such that $V_i\subseteq V_j$ if $s_i \prec s_j$. For each real-valued vector…

For a quantum system with Hilbert space ${\cal H}$ of dimension $N$ and a set $S$ of $n$ Hermitian operators ${\cal O}_i$, a basic question is to understand the set $E_S \subset \mathbb{R}^n$ of points $\vec{e}$ where $e_i = {\rm tr}(\rho…

Here we show some results related with Kakeya conjecture which says that for any integer $n\geq 2$, a set containing line segments in every dimension in $\mathbb{R}^n$ has full Hausdorff dimension as well as box dimension. We proved here…

For a field $\mathbb{F}$ and integers $d$ and $k$, a set ${\cal A} \subseteq \mathbb{F}^d$ is called $k$-nearly orthogonal if its members are non-self-orthogonal and every $k+1$ vectors of ${\cal A}$ include an orthogonal pair. We prove…

Representing Z/N as roots of unity, we restrict a natural U(1)-action on the Heegaard quantum sphere to Z/N, and call the quotient spaces Heegaard quantum lens spaces. Then we use this representation of Z/N to construct an associated…

While causal perturbation theory and lattice regularisation allow treatment of the ultraviolet divergences in qed, they do not resolve the issues of constructive field theory, or show the validity of qed except as a perturbation theory. I…

We conjecture the factorized scattering description for OSP(m/2n)/OSP(m-1/2n) supersphere sigma models and OSP(m/2n) Gross Neveu models. The non-unitarity of these field theories translates into a lack of `physical unitarity' of the S…

Kobayashi-Ochiai's theorem says us that the set of dominant rational maps to a complex variety of general type is finite. In this paper, we give a generalization of it in the category of log schemes.

We study the Coxeter matrices and the homological quadratic forms of $n$-hereditary algebras within the framework of higher dimensional Auslander--Reiten theory. Let $\Lambda$ be a finite dimensional $n$-hereditary algebra with the Coxeter…

In this second in a series of four articles we create a mathematical formalism sufficient to represent nontrivial hamiltonian quantum dynamics, including resonances. Some parts of this construction are also mathematically necessary. The…