1,437 research outputs found
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Distributed Computation as Hierarchy
This paper presents a new distributed computational model of distributed
systems called the phase web that extends V. Pratt's orthocurrence relation
from 1986. The model uses mutual-exclusion to express sequence, and a new kind
of hierarchy to replace event sequences, posets, and pomsets. The model
explicitly connects computation to a discrete Clifford algebra that is in turn
extended into homology and co-homology, wherein the recursive nature of objects
and boundaries becomes apparent and itself subject to hierarchical recursion.
Topsy, a programming environment embodying the phase web, is available from
www.cs.auc.dk/topsy.Comment: 16 pages, 3 figure
Symmetry-protected Topological Phases at Finite Temperature
We have applied the recently developed theory of topological Uhlmann numbers
to a representative model of a topological insulator in two dimensions, the
Qi-Wu-Zhang model. We have found a stable symmetry-protected topological (SPT)
phase under external thermal fluctuations in two-dimensions. A complete phase
diagram for this model is computed as a function of temperature and coupling
constants in the original Hamiltonian. It shows the appearance of large stable
phases of matter with topological properties compatible with thermal
fluctuations or external noise and the existence of critical lines separating
abruptly trivial phases from topological phases. These novel critical
temperatures represent thermal topological phase transitions. The initial part
of the paper comprises a self-contained explanation of the Uhlmann geometric
phase needed to understand the topological properties that it may acquire when
applied to topological insulators and superconductors.Comment: Contribution to the focus issue on "Artificial Graphene". Edited by
Maciej Lewenstein, Vittorio Pellegrini, Marco Polini and Mordechai (Moti)
Sege
Unsolved Problems in Virtual Knot Theory and Combinatorial Knot Theory
This paper is a concise introduction to virtual knot theory, coupled with a
list of research problems in this field.Comment: 65 pages, 24 figures. arXiv admin note: text overlap with
arXiv:math/040542
Weakly compact operators and the strong* topology for a Banach space
Peer reviewedPublisher PD
- …