34,174 research outputs found
Normal Forms for Symplectic Maps with Twist Singularities
We derive a normal form for a near-integrable, four-dimensional symplectic
map with a fold or cusp singularity in its frequency mapping. The normal form
is obtained for when the frequency is near a resonance and the mapping is
approximately given by the time- mapping of a two-degree-of freedom
Hamiltonian flow. Consequently there is an energy-like invariant. The fold
Hamiltonian is similar to the well-studied, one-degree-of freedom case but is
essentially nonintegrable when the direction of the singular curve in action
does not coincide with curves of the resonance module. We show that many
familiar features, such as multiple island chains and reconnecting invariant
manifolds, are retained even in this case. The cusp Hamiltonian has an
essential coupling between its two degrees of freedom even when the singular
set is aligned with the resonance module. Using averaging, we approximately
reduced this case to one degree of freedom as well. The resulting Hamiltonian
and its perturbation with small cusp-angle is analyzed in detail.Comment: LaTex, 27 pages, 21 figure
Wrinkles, folds and plasticity in granular rafts
We investigate the mechanical response of a compressed monolayer of large and
dense particles at a liquid-fluid interface: a granular raft. Upon compression,
rafts first wrinkle; then, as the confinement increases, the deformation
localizes in a unique fold. This characteristic buckling pattern is usually
associated to floating elastic sheets and as a result, particle laden
interfaces are often modeled as such. Here, we push this analogy to its limits
by comparing the first quantitative measurements of the raft morphology to a
theoretical continuous elastic model of the interface. We show that although
powerful to describe the wrinkle wavelength, the wrinkle-to-fold transition and
the fold shape, this elastic description does not capture the finer details of
the experiment. We describe an unpredicted secondary wavelength, a compression
discrepancy with the model and a hysteretic behavior during compression cycles,
all of which are a signature of the intrinsic discrete and frictional nature of
granular rafts. It suggests also that these composite materials exhibit both
plastic transition and jamming dynamics.Comment: 10 pages, including Supplementary Information. Submitted to Physical
Review Material
Fold-Saddle Bifurcation in Non-Smooth Vector Fields on the Plane
This paper presents results concerning bifurcations of 2D piecewise-smooth
dynamical systems governed by vector fields. Generic three parameter families
of a class of Non-Smooth Vector Fields are studied and its bifurcation diagrams
are exhibited. Our main result describes the unfolding of the so called
Fold-Saddle singularity
Predicate Abstraction for Linked Data Structures
We present Alias Refinement Types (ART), a new approach to the verification
of correctness properties of linked data structures. While there are many
techniques for checking that a heap-manipulating program adheres to its
specification, they often require that the programmer annotate the behavior of
each procedure, for example, in the form of loop invariants and pre- and
post-conditions. Predicate abstraction would be an attractive abstract domain
for performing invariant inference, existing techniques are not able to reason
about the heap with enough precision to verify functional properties of data
structure manipulating programs. In this paper, we propose a technique that
lifts predicate abstraction to the heap by factoring the analysis of data
structures into two orthogonal components: (1) Alias Types, which reason about
the physical shape of heap structures, and (2) Refinement Types, which use
simple predicates from an SMT decidable theory to capture the logical or
semantic properties of the structures. We prove ART sound by translating types
into separation logic assertions, thus translating typing derivations in ART
into separation logic proofs. We evaluate ART by implementing a tool that
performs type inference for an imperative language, and empirically show, using
a suite of data-structure benchmarks, that ART requires only 21% of the
annotations needed by other state-of-the-art verification techniques
Stream Fusion, to Completeness
Stream processing is mainstream (again): Widely-used stream libraries are now
available for virtually all modern OO and functional languages, from Java to C#
to Scala to OCaml to Haskell. Yet expressivity and performance are still
lacking. For instance, the popular, well-optimized Java 8 streams do not
support the zip operator and are still an order of magnitude slower than
hand-written loops. We present the first approach that represents the full
generality of stream processing and eliminates overheads, via the use of
staging. It is based on an unusually rich semantic model of stream interaction.
We support any combination of zipping, nesting (or flat-mapping), sub-ranging,
filtering, mapping-of finite or infinite streams. Our model captures
idiosyncrasies that a programmer uses in optimizing stream pipelines, such as
rate differences and the choice of a "for" vs. "while" loops. Our approach
delivers hand-written-like code, but automatically. It explicitly avoids the
reliance on black-box optimizers and sufficiently-smart compilers, offering
highest, guaranteed and portable performance. Our approach relies on high-level
concepts that are then readily mapped into an implementation. Accordingly, we
have two distinct implementations: an OCaml stream library, staged via
MetaOCaml, and a Scala library for the JVM, staged via LMS. In both cases, we
derive libraries richer and simultaneously many tens of times faster than past
work. We greatly exceed in performance the standard stream libraries available
in Java, Scala and OCaml, including the well-optimized Java 8 streams
Qualitative Analysis of Polycycles in Filippov Systems
In this paper, we are concerned about the qualitative behaviour of planar
Filippov systems around some typical minimal sets, namely, polycycles. In the
smooth context, a polycycle is a simple closed curve composed by a collection
of singularities and regular orbits, inducing a first return map. Here, this
concept is extended to Filippov systems by allowing typical singularities lying
on the switching manifold. Our main goal consists in developing a method to
investigate the unfolding of polycycles in Filippov systems. In addition, we
applied this method to describe bifurcation diagrams of Filippov systems around
certain polycycles
Two State Behavior in a Solvable Model of -hairpin folding
Understanding the mechanism of protein secondary structure formation is an
essential part of protein-folding puzzle. Here we describe a simple model for
the formation of the -hairpin, motivated by the fact that folding of a
-hairpin captures much of the basic physics of protein folding. We argue
that the coupling of ``primary'' backbone stiffness and ``secondary'' contact
formation (similar to the coupling between the ``secondary'' and ``tertiary''
structure in globular proteins), caused for example by side-chain packing
regularities, is responsible for producing an all-or-none 2-state
-hairpin formation. We also develop a recursive relation to compute the
phase diagram and single exponential folding/unfolding rate arising via a
dominant transition state.Comment: Revised versio
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