1,867 research outputs found
On the representation theory of Galois and Atomic Topoi
We elaborate on the representation theorems of topoi as topoi of discrete
actions of various kinds of localic groups and groupoids. We introduce the
concept of "proessential point" and use it to give a new characterization of
pointed Galois topoi. We establish a hierarchy of connected topoi:
[1. essentially pointed Atomic = locally simply connected],
[2. proessentially pointed Atomic = pointed Galois],
[3. pointed Atomic].
These topoi are the classifying topos of, respectively: 1. discrete groups,
2. prodiscrete localic groups, and 3. general localic groups.
We analyze also the unpoited version, and show that for a Galois topos, may
be pointless, the corresponding groupoid can also be considered, in a sense,
the groupoid of "points". In the unpointed theories, these topoi classify,
respectively: 1. connected discrete groupoids, 2. connected (may be pointless)
prodiscrete localic groupoids, and 3. connected groupoids with discrete space
of objects and general localic spaces of hom-sets, when the topos has points
(we do not know the class of localic groupoids that correspond to pointless
connected atomic topoi).
We comment and develop on Grothendieck's galois theory and its generalization
by Joyal-Tierney, and work by other authors on these theories.Comment: This is a revised version of arXiv.org/math.CT/02008222 to appear in
JPA
Nonradiating Photonics with Resonant Dielectric Nanostructures
Nonradiating sources of energy have traditionally been studied in quantum
mechanics and astrophysics, while receiving a very little attention in the
photonics community. This situation has changed recently due to a number of
pioneering theoretical studies and remarkable experimental demonstrations of
the exotic states of light in dielectric resonant photonic structures and
metasurfaces, with the possibility to localize efficiently the electromagnetic
fields of high intensities within small volumes of matter. These recent
advances underpin novel concepts in nanophotonics, and provide a promising
pathway to overcome the problem of losses usually associated with metals and
plasmonic materials for the efficient control of the light-matter interaction
at the nanoscale. This review paper provides the general background and several
snapshots of the recent results in this young yet prominent research field,
focusing on two types of nonradiating states of light that both have been
recently at the center of many studies in all-dielectric resonant meta-optics
and metasurfaces: optical {\em anapoles} and photonic {\em bound states in the
continuum}. We discuss a brief history of these states in optics, their
underlying physics and manifestations, and also emphasize their differences and
similarities. We also review some applications of such novel photonic states in
both linear and nonlinear optics for the nanoscale field enhancement, a design
of novel dielectric structures with high- resonances, nonlinear wave mixing
and enhanced harmonic generation, as well as advanced concepts for lasing and
optical neural networks.Comment: 22 pages, 9 figures, review articl
Evolutionary game theory: Temporal and spatial effects beyond replicator dynamics
Evolutionary game dynamics is one of the most fruitful frameworks for
studying evolution in different disciplines, from Biology to Economics. Within
this context, the approach of choice for many researchers is the so-called
replicator equation, that describes mathematically the idea that those
individuals performing better have more offspring and thus their frequency in
the population grows. While very many interesting results have been obtained
with this equation in the three decades elapsed since it was first proposed, it
is important to realize the limits of its applicability. One particularly
relevant issue in this respect is that of non-mean-field effects, that may
arise from temporal fluctuations or from spatial correlations, both neglected
in the replicator equation. This review discusses these temporal and spatial
effects focusing on the non-trivial modifications they induce when compared to
the outcome of replicator dynamics. Alongside this question, the hypothesis of
linearity and its relation to the choice of the rule for strategy update is
also analyzed. The discussion is presented in terms of the emergence of
cooperation, as one of the current key problems in Biology and in other
disciplines.Comment: Review, 48 pages, 26 figure
Glassy dynamics of kinetically constrained models
We review the use of kinetically constrained models (KCMs) for the study of
dynamics in glassy systems. The characteristic feature of KCMs is that they
have trivial, often non-interacting, equilibrium behaviour but interesting slow
dynamics due to restrictions on the allowed transitions between configurations.
The basic question which KCMs ask is therefore how much glassy physics can be
understood without an underlying ``equilibrium glass transition''. After a
brief review of glassy phenomenology, we describe the main model classes, which
include spin-facilitated (Ising) models, constrained lattice gases, models
inspired by cellular structures such as soap froths, models obtained via
mappings from interacting systems without constraints, and finally related
models such as urn, oscillator, tiling and needle models. We then describe the
broad range of techniques that have been applied to KCMs, including exact
solutions, adiabatic approximations, projection and mode-coupling techniques,
diagrammatic approaches and mappings to quantum systems or effective models.
Finally, we give a survey of the known results for the dynamics of KCMs both in
and out of equilibrium, including topics such as relaxation time divergences
and dynamical transitions, nonlinear relaxation, aging and effective
temperatures, cooperativity and dynamical heterogeneities, and finally
non-equilibrium stationary states generated by external driving. We conclude
with a discussion of open questions and possibilities for future work.Comment: 137 pages. Additions to section on dynamical heterogeneities (5.5,
new pages 110 and 112), otherwise minor corrections, additions and reference
updates. Version to be published in Advances in Physic
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