513 research outputs found
State property systems and orthogonality
The structure of a state property system was introduced to formalize in a
complete way the operational content of the Geneva-Brussels approach to the
foundations of quantum mechanics, and the category of state property systems
was proven to be equivalence to the category of closure spaces. The first
axioms of standard quantum axiomatics (state determination and atomisticity)
have been shown to be equivalent to the and axioms of closure
spaces, and classical properties to correspond to clopen sets, leading to a
decomposition theorem into classical and purely nonclassical components for a
general state property system. The concept of orthogonality, very important for
quantum axiomatics, had however not yet been introduced within the formal
scheme of the state property system. In this paper we introduce orthogonality
in a operational way, and define ortho state property systems. Birkhoff's well
known biorthogonal construction gives rise to an orthoclosure and we study the
relation between this orthoclosure and the operational orthogonality that we
introduced.Comment: 10 pages, 2 figures, proceeding of the IQSA 2002 conference in Vienn
Being and Change: Foundations of a Realistic Operational Formalism
The aim of this article is to represent the general description of an entity
by means of its states, contexts and properties. The entity that we want to
describe does not necessarily have to be a physical entity, but can also be an
entity of a more abstract nature, for example a concept, or a cultural
artifact, or the mind of a person, etc..., which means that we aim at very
general description. The effect that a context has on the state of the entity
plays a fundamental role, which means that our approach is intrinsically
contextual. The approach is inspired by the mathematical formalisms that have
been developed in axiomatic quantum mechanics, where a specific type of quantum
contextuality is modelled. However, because in general states also influence
context -- which is not the case in quantum mechanics -- we need a more general
setting than the one used there. Our focus on context as a fundamental concept
makes it possible to unify `dynamical change' and `change under influence of
measurement', which makes our approach also more general and more powerful than
the traditional quantum axiomatic approaches. For this reason an experiment (or
measurement) is introduced as a specific kind of context. Mathematically we
introduce a state context property system as the structure to describe an
entity by means of its states, contexts and properties. We also strive from the
start to a categorical setting and derive the morphisms between state context
property systems from a merological covariance principle. We introduce the
category SCOP with as elements the state context property systems and as
morphisms the ones that we derived from this merological covariance principle.
We introduce property completeness and state completeness and study the
operational foundation of the formalismComment: 44 page
Functional disorganization of small-world brain networks in mild Alzheimer's disease and amnestic Mild cognitive impairment:An EEG study using Relative Wavelet Entropy (RWE)
Previous neuroscientific findings have linked Alzheimer's disease (AD) with less efficient information processing and brain network disorganization. However, pathological alterations of the brain networks during the preclinical phase of amnestic Mild Cognitive Impairment (aMCI) remain largely unknown. The present study aimed at comparing patterns of the detection of functional disorganization in MCI relative to Mild Dementia (MD). Participants consisted of 23 cognitively healthy adults, 17 aMCI and 24 mild AD patients who underwent electroencephalographic (EEG) data acquisition during a resting-state condition. Synchronization analysis through the Orthogonal Discrete Wavelet Transform (ODWT), and directional brain network analysis were applied on the EEG data. This computational model was performed for networks that have the same number of edges (N=500, 600, 700, 800 edges) across all participants and groups (fixed density values). All groups exhibited a small-world (SW) brain architecture. However, we found a significant reduction in the SW brain architecture in both aMCI and MD patients relative to the group of Healthy controls. This functional disorganization was also correlated with the participant's generic cognitive status. The deterioration of the network's organization was caused mainly by deficient local information processing as quantified by the mean cluster coefficient value. Functional hubs were identified through the normalized betweenness centrality metric. Analysis of the local characteristics showed relative hub preservation even with statistically significant reduced strength. Compensatory phenomena were also evident through the formation of additional hubs on left frontal and parietal regions. Our results indicate a declined functional network organization even during the prodromal phase. Degeneration is evident even in the preclinical phase and coexists with transient network reorganization due to compensation
Reality and Probability: Introducing a New Type of Probability Calculus
We consider a conception of reality that is the following: An object is
'real' if we know that if we would try to test whether this object is present,
this test would give us the answer 'yes' with certainty. If we consider a
conception of reality where probability plays a fundamental role it can be
shown that standard probability theory is not well suited to substitute
'certainty' by means of 'probability equal to 1'. The analysis of this problem
leads us to propose a new type of probability theory that is a generalization
of standard probability theory. This new type of probability is a function to
the set of all subsets of the interval [0, 1] instead of to the interval [0, 1]
itself, and hence its evaluation happens by means of a subset instead of a
number. This subset corresponds to the different limits of sequences of
relative frequency that can arise when an intrinsic lack of knowledge about the
context and how it influences the state of the physical entity under study in
the process of experimentation is taken into account. The new probability
theory makes it possible to define probability on the whole set of experiments
within the Geneva-Brussels approach to quantum mechanics, which was not
possible with standard probability theory. We introduce the structure of a
'state experiment probability system' and derive the state property system as a
special case of this structure. The category SEP of state experiment
probability systems and their morphisms is linked with the category SP of state
property systems and their morphismsComment: 27 page
A theory of concepts and their combinations I: The structure of the sets of contexts and properties
We propose a theory for modeling concepts that uses the state-context-property theory (SCOP), a generalization of the quantum formalism, whose basic notions are states, contexts and properties. This theory enables us to incorporate context into the mathematical structure used to describe a concept, and thereby model how context influences the typicality of a single exemplar and the applicability of a single property of a concept. We introduce the notion `state of a concept' to account for this contextual influence, and show that the structure of the set of contexts and of the set of properties of a concept is a complete orthocomplemented lattice. The structural study in this article is a preparation for a numerical mathematical theory of concepts in the Hilbert space of quantum mechanics that allows the description of the combination of concepts
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