31 research outputs found
A quantum logical and geometrical approach to the study of improper mixtures
We study improper mixtures from a quantum logical and geometrical point of
view. Taking into account the fact that improper mixtures do not admit an
ignorance interpretation and must be considered as states in their own right,
we do not follow the standard approach which considers improper mixtures as
measures over the algebra of projections. Instead of it, we use the convex set
of states in order to construct a new lattice whose atoms are all physical
states: pure states and improper mixtures. This is done in order to overcome
one of the problems which appear in the standard quantum logical formalism,
namely, that for a subsystem of a larger system in an entangled state, the
conjunction of all actual properties of the subsystem does not yield its actual
state. In fact, its state is an improper mixture and cannot be represented in
the von Neumann lattice as a minimal property which determines all other
properties as is the case for pure states or classical systems. The new lattice
also contains all propositions of the von Neumann lattice. We argue that this
extension expresses in an algebraic form the fact that -alike the classical
case- quantum interactions produce non trivial correlations between the
systems. Finally, we study the maps which can be defined between the extended
lattice of a compound system and the lattices of its subsystems.Comment: submitted to the Journal of Mathematical Physic
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure
Models for spin-dependent transport in helical molecules
Chiral molecules act as strong spin filters for transmitted electrons (chiral-induced spin selectivity). The interplay of geometry and spin mediated by spin-orbit coupling is commonly assumed as the cause of the effect, but the theoretical description remains incomplete.
In this thesis, two models for electron transport through helical molecules were investigated: an atomistic tight binding model for the molecule helicene and a simple continuum model for an electron in a helix-shaped potential.
In an attempt to cover the middle ground between phenomenological tight binding approaches and detailed first principle simulations, the helicene model starts with a lattice of carbon atoms represented by a minimal basis of local atomic s- and p-orbitals including electronic nearest-neighbor and spin-orbit interactions. Löwdin partitioning is used to reduce the model to a p-orbital tight binding representation, providing numeric values for all the couplings dependent on geometry. Transport calculations showed helicity dependent spin polarization several orders of magnitude smaller than experimentally observed.
To understand the effect on a more fundamental level, an electron moving through a helix-shaped confinement potential in 3D space with spin-orbit coupling was considered. By taking the limit of strong confinement, an approximate model with one-dimensional configuration space (the helix) was obtained. Novel onsite spin-orbit coupling terms appear in the effective Hamiltonian, leading to sizeable spin polarization in transport calculations. These new terms are thoroughly justified by the adiabatic limiting procedure which was adapted to include spin-orbit coupling and might thus provide one of the missing pieces for the theory of chiral-induced spin selectivity
The *-composition -A Novel Generating Method of Fuzzy Implications: An Algebraic Study
Fuzzy implications are one of the two most important fuzzy logic connectives, the other being
t-norms. They are a generalisation of the classical implication from two-valued logic to the multivalued
setting.
A binary operation I on [0; 1] is called a fuzzy implication if
(i) I is decreasing in the first variable,
(ii) I is increasing in the second variable,
(iii) I(0; 0) = I(1; 1) = 1 and I(1; 0) = 0.
The set of all fuzzy implications defined on [0; 1] is denoted by I.
Fuzzy implications have many applications in fields like fuzzy control, approximate reasoning,
decision making, multivalued logic, fuzzy image processing, etc. Their applicational value necessitates
new ways of generating fuzzy implications that are fit for a specific task. The generating methods
of fuzzy implications can be broadly categorised as in the following:
(M1): From binary functions on [0; 1], typically other fuzzy logic connectives, viz., (S;N)-, R-, QL-
implications,
(M2): From unary functions on [0,1], typically monotonic functions, for instance, Yager’s f-, g-
implications, or from fuzzy negations,
(M3): From existing fuzzy implications
Object-oriented data mining
EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Rules and Meaning in Quantum Mechanics
This book concerns the metasemantics of quantum mechanics (QM). Roughly, it
pursues an investigation at an intersection of the philosophy of physics and
the philosophy of semantics, and it offers a critical analysis of rival
explanations of the semantic facts of standard QM. Two problems for such
explanations are discussed: categoricity and permanence of rules. New results
include 1) a reconstruction of Einstein's incompleteness argument, which
concludes that a local, separable, and categorical QM cannot exist, 2) a
reinterpretation of Bohr's principle of correspondence, grounded in the
principle of permanence, 3) a meaning-variance argument for quantum logic,
which follows a line of critical reflections initiated by Weyl, and 4) an
argument for semantic indeterminacy leveled against inferentialism about QM,
inspired by Carnap's work in the philosophy of classical logic.Comment: 150 page