2,344 research outputs found
Homomorphisms preserving types of density
The concept of density in a free monoid can be generalized from the infix relation to arbitrary relations. Many of the properties known for density can be established over these more general notions of densities. In this paper, we investigate homomorphisms which preserve different types of density. We demonstrate a strict hierarchy between families of homomorphisms which preserve density over different types of relations. However, as with the case of endomorphisms, a similar hierarchy for weak-coding homomorphisms collapses. We also present an algorithm to decide whether a homomorphism preserves density over any relation which satisfies some natural conditions
Limits of dense graph sequences
We show that if a sequence of dense graphs has the property that for every
fixed graph F, the density of copies of F in these graphs tends to a limit,
then there is a natural ``limit object'', namely a symmetric measurable
2-variable function on [0,1]. This limit object determines all the limits of
subgraph densities. We also show that the graph parameters obtained as limits
of subgraph densities can be characterized by ``reflection positivity'',
semidefiniteness of an associated matrix. Conversely, every such function
arises as a limit object. Along the lines we introduce a rather general model
of random graphs, which seems to be interesting on its own right.Comment: 27 pages; added extension of result (Sept 22, 2004
Category forcings, , and generic absoluteness for the theory of strong forcing axioms
We introduce a category whose objects are stationary set preserving complete
boolean algebras and whose arrows are complete homomorphisms with a stationary
set preserving quotient. We show that the cut of this category at a rank
initial segment of the universe of height a super compact which is a limit of
super compact cardinals is a stationary set preserving partial order which
forces and collapses its size to become the second uncountable
cardinal. Next we argue that any of the known methods to produce a model of
collapsing a superhuge cardinal to become the second uncountable
cardinal produces a model in which the cutoff of the category of stationary set
preserving forcings at any rank initial segment of the universe of large enough
height is forcing equivalent to a presaturated tower of normal filters. We let
denote this statement and we prove that the theory of
with parameters in is generically invariant
for stationary set preserving forcings that preserve . Finally we
argue that the work of Larson and Asper\'o shows that this is a next to optimal
generalization to the Chang model of Woodin's generic
absoluteness results for the Chang model . It remains open
whether and are equivalent axioms modulo large cardinals
and whether suffices to prove the same generic absoluteness results
for the Chang model .Comment: - to appear on the Journal of the American Mathemtical Societ
Area Preserving Transformations in Non-commutative Space and NCCS Theory
We propose an heuristic rule for the area transformation on the
non-commutative plane. The non-commutative area preserving transformations are
quantum deformation of the classical symplectic diffeomorphisms. Area
preservation condition is formulated as a field equation in the non-commutative
Chern-Simons gauge theory. The higher dimensional generalization is suggested
and the corresponding algebraic structure - the infinite dimensional -Lie
algebra is extracted. As an illustrative example the second-quantized
formulation for electrons in the lowest Landau level is considered.Comment: revtex, 9 pages, corrected typo
Resource theories of knowledge
How far can we take the resource theoretic approach to explore physics?
Resource theories like LOCC, reference frames and quantum thermodynamics have
proven a powerful tool to study how agents who are subject to certain
constraints can act on physical systems. This approach has advanced our
understanding of fundamental physical principles, such as the second law of
thermodynamics, and provided operational measures to quantify resources such as
entanglement or information content. In this work, we significantly extend the
approach and range of applicability of resource theories. Firstly we generalize
the notion of resource theories to include any description or knowledge that
agents may have of a physical state, beyond the density operator formalism. We
show how to relate theories that differ in the language used to describe
resources, like micro and macroscopic thermodynamics. Finally, we take a
top-down approach to locality, in which a subsystem structure is derived from a
global theory rather than assumed. The extended framework introduced here
enables us to formalize new tasks in the language of resource theories, ranging
from tomography, cryptography, thermodynamics and foundational questions, both
within and beyond quantum theory.Comment: 28 pages featuring figures, examples, map and neatly boxed theorems,
plus appendi
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