Embeddings into Orthomodular Lattices with Given Centers, State Spaces and Automorphism Groups

Using the general hypergraph technique developed in [7], we first give a much simpler proof of Shultz's theorem [10]: Each compact convex set is affinely homeomorphic to the state space of an orthomodular lattice. We also present partial solutions to open questions formulated in [10] - we show that not every compact convex set has to be a state space of a unital orthomodular lattice and that for unital orthomodular lattices the state space characterization can be obtained in the context of unital hypergraphs.

In recent work, the author has developed a general category-theoretic framework for decision theory. This article applies this to the category of orthomodular lattices (OMLs). Every Boolean algebra is an OML, so this yields a new (syntactic) model of decision-making with classical uncertainty. The lattice of closed subspaces of a Hilbert space is also an OML, so this also yields a new model of decision-making with quantum uncertainty. This article is part of the theme issue 'Quantum theory and topology in models of decision making (Part 2)'.

We call an orthomodular lattice L m \mathcal {L}\;m -orthocomplete for an infinite cardinal m m if every orthogonal family of ≦ m \leqq m elements from L \mathcal {L} has a join in L \mathcal {L} , and we call L m \mathcal {L}\;m -complete if every family, orthogonal or not, of ≦ m \leqq m elements from L \mathcal {L} has a join in L \mathcal {L} . We prove that an m m -orthocomplete orthomodular lattice is m m -complete. Since a Boolean algebra is a distributive orthomodular lattice, we obtain as a special case the Smith-Tarski theorem: An m m -orthocomplete Boolean algebra is m m -complete.

Article A Helly-type theorem for hyperplane transversals to well-separated convex sets Share on Authors: Boris Aronov Polytechnic University, Brooklyn, NY Polytechnic University, Brooklyn, NYView Profile , Jacob E. Goodman City College, City University of New York, New York, NY City College, City University of New York, New York, NYView Profile , Richard Pollack Courant Institute of Mathematical Sciences, New York University, New York, NY Courant Institute of Mathematical Sciences, New York University, New York, NYView Profile , Rephael Wenger The Ohio State University, Columbus, OH The Ohio State University, Columbus, OHView Profile Authors Info & Claims SCG '00: Proceedings of the sixteenth annual symposium on Computational geometryMay 2000 Pages 57–63https://doi.org/10.1145/336154.336178Online:01 May 2000Publication History 0citation233DownloadsMetricsTotal Citations0Total Downloads233Last 12 Months4Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

Let S be a finite collection of compact convex sets in \R d . Let D(S) be the largest diameter of any member of S . We say that the collection S is ?-separated if, for every 0 < k < d , any k of the sets can be separated from any other d-k of the sets by a hyperplane more than ? D(S)/2 away from all d of the sets. We prove that if S is an ? -separated collection of at least N(?) compact convex sets in \R d and every 2d+2 members of S are met by a hyperplane, then there is a hyperplane meeting all the members of S . The number N(?) depends both on the dimension d and on the separation parameter ? . This is the first Helly-type theorem known for hyperplane transversals to compact convex sets of arbitrary shape in dimension greater than one.

Let K and L be compact convex sets in ℝ n . The following two statements are shown to be equivalent: (i) For every polytope Q ⊆ K having at most n + 1 vertices, L contains a translate of Q. (ii) L contains a translate of K. Let 1 ≤ d ≤ n - 1. It is also shown that the following two statements are equivalent: (i) For every polytope Q ⊆ K having at most d + 1 vertices, L contains a translate of Q. (ii) For every d-dimensional subspace ξ, the orthogonal projection L ξ of the set L contains a translate of the corresponding projection K ξ of the set K. It is then shown that, if K is a compact convex set in ℝ n having at least d + 2 exposed points, then there exists a compact convex set L such that every d-dimensional orthogonal projection L ξ contains a translate of the projection K ξ , while L does not contain a translate of K. In particular, if dim K > d, then there exists L such that every d-dimensional projection L ξ contains a translate of the projection K ξ , while L does not contain a translate of K.

The aim of this paper is to design regular feedback controls for the global asymptotic stabilization gas) of affine control systems with closed (convex) control value sets cvs) $U\subseteq\mathbb{R}^{m}$ with $0\in{int}U$, in the framework of Artstein and Sontag's control Lyapunov function clf) approach. We can consider this work as a continuation and complement of our paper [SIAM J. Control Optim., 51 (2013), pp. 2152--2175]. First, we resume some of the results achieved in that paper. Then, based on a method for approximation by compact convex sets, we propose an explicit formula for regular feedback controls for the gas of affine control systems with general compact cvs $U$ with $0\in{int}U$, provided an appropriate clf is known, but at the expense of small overflows in the control values. Moreover, if we assume smoothness on the system and the clf, a redesign yields practically smooth feedback controls. Consequently, we approximately solve the synthesis problem entailed by Artstein's theorem and also (in some degree) a Sontag's open problem with respect to general compact cvs $U$ with $0\in{int}U$. Finally, tools from convex analysis and the approximation method by compact convex sets allow us to address the problem of the global clf stabilization of affine control systems with respect to unbounded closed cvs $U$ with $0\in{int}U$, and also to the design of regular feedback controls. We illustrate the results with some examples.

On the equivalence of extrapolation and Richardson’s iteration and its applications

For $$0<\lambda <\frac{1}{2}$$ we consider the product $$F_\lambda =E_\lambda \times {\mathbb R}$$ , where $$E_\lambda $$ is the attractor of the IFS $$\{x\mapsto \lambda x,\ \ x\mapsto 1\!-\!\lambda +\lambda x\}$$ on $${\mathbb R}$$ . The Huasdorff dimension of $$F_\lambda $$ is $$s=1-\frac{\ln 2}{\ln \lambda }$$ . We show that $$\sup \left\{ \frac{{\mathcal H}^s(X\cap F_\lambda )}{|X|^s}: |X|>0\right\} =1$$ and that there is a convex compact set $$A$$ ( $$=A(\lambda )$$ ) with $$\frac{{\mathcal H}^s(A\cap F_\lambda )}{|A|^s}=1$$ . Such a convex compact set $$A$$ is called an “extremal set” of $$F_\lambda $$ with respect to $$s$$ -dimensional Hausdorff measure $${\mathcal H}^s$$ . When $$\lambda $$ is small, say $$\lambda \le \frac{1}{5}$$ , we further show that there exists an extremal set $$A$$ with $$|A|\ge \frac{2}{\sqrt{3}}$$ such that $${\mathcal H}^s(A\cap F_\lambda )={\mathcal H}^s(D_{|A|}\cap F_\lambda )$$ for $$D_{|A|}=\left\{ (x,y): \left( x-\frac{1}{2}\right) ^2+y^2 \le \frac{1}{4}|A|^2\right\} $$ . As an application, we can estimate the value of $${\mathcal H}^s(E_\lambda \!\times \![0,1])$$ to any pre-set error $$\epsilon $$ .

The Dirichlet problem on compact convex sets

It is shown that the problem of the best uniform approximation in the Hausdorff metric of a continuous set-valued map with finite-dimensional compact convex images by constant set-valued maps whose images are balls in some norm can be reduced to a visual geometric problem. The latter consists in constructing a spherical layer of minimal thickness which contains the complement of a compact convex set to a larger compact convex set.

(1978). Orthomodular Lattices and Quantum Physics. Mathematics Magazine: Vol. 51, No. 5, pp. 299-303.

This paper studies rough approximation via join and meet on a complete orthomodular lattice. Different from Boolean algebra, the distributive law of join over meet does not hold in orthomodular lattices. Some properties of rough approximation rely on the distributive law. Furthermore, we study the relationship among the distributive law, rough approximation and orthomodular lattice-valued relation.

In the orthodox quantum logic approach to foundations of quantum mechanics originated by Birkhoff and von Neumann [1] the object called quantum logic is not a logic in the very sense of this word but an algebraic structure: orthomodular lattice or poset. These orthomodular structures, abstracted from structures of Hilbert spaces used to describe quantum phenomena are weakenings of Boolean algebras. They can be thought of as representing Lindenbaum algebras of systems of propositions about quantum objects, the propositions belonging to the quantum logic proper. Since Lindenbaum algebras of propositional systems which obey the rules of classical logic are Boolean algebras it is inferred that the quantum logic proper, whatever it means, is non-classical.

Introduction. Every orthomodular lattice (abbreviated : OML) is the union of its maximal Boolean subalgebras (blocks). The question thus arises how conversely Boolean algebras can be amalgamated in order to obtain an OML of which the given Boolean algebras are the blocks. This question we deal with in the present paper.The problem was first investigated by Greechie [6, 7, 8, 9]. His technique of pasting [6] will also play an important role in this paper. A case solved completely by Greechie [9] is the case that any two blocks intersect either in the bounds only or have the bounds, an atom and its complement in common. This is, of course, a very special situation. The more surprising it is that Greechie's methods, if skillfully applied, yield considerable insight into the structure of OMLs and provide a seemingly unexhaustible source for counter-examples.