804 research outputs found
Zone Diagrams in Euclidean Spaces and in Other Normed Spaces
Zone diagram is a variation on the classical concept of a Voronoi diagram.
Given n sites in a metric space that compete for territory, the zone diagram is
an equilibrium state in the competition. Formally it is defined as a fixed
point of a certain "dominance" map.
Asano, Matousek, and Tokuyama proved the existence and uniqueness of a zone
diagram for point sites in Euclidean plane, and Reem and Reich showed existence
for two arbitrary sites in an arbitrary metric space. We establish existence
and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary
(finite) dimension, and more generally, in a finite-dimensional normed space
with a smooth and rotund norm. The proof is considerably simpler than that of
Asano et al. We also provide an example of non-uniqueness for a norm that is
rotund but not smooth. Finally, we prove existence and uniqueness for two point
sites in the plane with a smooth (but not necessarily rotund) norm.Comment: Title page + 16 pages, 20 figure
Multiscaling for Systems with a Broad Continuum of Characteristic Lengths and Times: Structural Transitions in Nanocomposites
The multiscale approach to N-body systems is generalized to address the broad
continuum of long time and length scales associated with collective behaviors.
A technique is developed based on the concept of an uncountable set of time
variables and of order parameters (OPs) specifying major features of the
system. We adopt this perspective as a natural extension of the commonly used
discrete set of timescales and OPs which is practical when only a few,
widely-separated scales exist. The existence of a gap in the spectrum of
timescales for such a system (under quasiequilibrium conditions) is used to
introduce a continuous scaling and perform a multiscale analysis of the
Liouville equation. A functional-differential Smoluchowski equation is derived
for the stochastic dynamics of the continuum of Fourier component order
parameters. A continuum of spatially non-local Langevin equations for the OPs
is also derived. The theory is demonstrated via the analysis of structural
transitions in a composite material, as occurs for viral capsids and molecular
circuits.Comment: 28 pages, 1 figur
Qubit Decoherence and Non-Markovian Dynamics at Low Temperatures via an Effective Spin-Boson Model
Quantum Brownian oscillator model (QBM), in the Fock-space representation,
can be viewed as a multi-level spin-boson model. At sufficiently low
temperature, the oscillator degrees of freedom are dynamically reduced to the
lowest two levels and the system behaves effectively as a two-level (E2L)
spin-boson model (SBM) in this limit. We discuss the physical mechanism of
level reduction and analyze the behavior of E2L-SBM from the QBM solutions. The
availability of close solutions for the QBM enables us to study the
non-Markovian features of decoherence and leakage in a SBM in the
non-perturbative regime (e.g. without invoking the Born approximation) in
better details than before. Our result captures very well the characteristic
non-Markovian short time low temperature behavior common in many models.Comment: 19 pages, 8 figure
Crystal constructions in Number Theory
Weyl group multiple Dirichlet series and metaplectic Whittaker functions can
be described in terms of crystal graphs. We present crystals as parameterized
by Littelmann patterns and we give a survey of purely combinatorial
constructions of prime power coefficients of Weyl group multiple Dirichlet
series and metaplectic Whittaker functions using the language of crystal
graphs. We explore how the branching structure of crystals manifests in these
constructions, and how it allows access to some intricate objects in number
theory and related open questions using tools of algebraic combinatorics
Time-convolutionless reduced-density-operator theory of a noisy quantum channel: a two-bit quantum gate for quantum information processing
An exact reduced-density-operator for the output quantum states in
time-convolutionless form was derived by solving the quantum Liouville equation
which governs the dynamics of a noisy quantum channel by using a projection
operator method and both advanced and retarded propagators in time. The
formalism developed in this work is general enough to model a noisy quantum
channel provided specific forms of the Hamiltonians for the system, reservoir,
and the mutual interaction between the system and the reservoir are given.
Then, we apply the formulation to model a two-bit quantum gate composed of
coupled spin systems in which the Heisenberg coupling is controlled by the
tunneling barrier between neighboring quantum dots. Gate Characteristics
including the entropy, fidelity, and purity are calculated numerically for both
mixed and entangled initial states
Effect of temperature on rheology behaviour of banana peel pectin extracted using hot compressed water
Banana peel pectin is extracted from banana peel waste using a hot compressed water extraction (140-160°C, 5 minutes, 1.18 mm particle size). Physicochemical contents of banana peel pectin were found to be in a similar range with commercial pectin, and is comprised of moisture (7.44-8.47%), ash (3.45-4.98%), protein (1.08-1.92%), fat (0.04-3.42), carbohydrate (83-86%), total sugar (1.77-3.41%), energy (353-369 kcal/100g) and specific heat (1.42-1.62 kJ/kg°C). These contents possibly related to their flow deformation of rheological behaviour. Regression analysis displayed good agreements in all models applied, apart from the Casson Model. Flow behaviour indices, n<1 and decreasing of apparent viscosity within increasing of shear rate indicates that banana peel pectin has excellent shear thinning behaviour with a presence of yield stress
Efficiently Correcting Matrix Products
We study the problem of efficiently correcting an erroneous product of two
matrices over a ring. Among other things, we provide a randomized
algorithm for correcting a matrix product with at most erroneous entries
running in time and a deterministic -time
algorithm for this problem (where the notation suppresses
polylogarithmic terms in and ).Comment: Fixed invalid reference to figure in v
Diffusive Evolution of Stable and Metastable Phases II: Theory of Non-Equilibrium Behaviour in Colloid-Polymer Mixtures
By analytically solving some simple models of phase-ordering kinetics, we
suggest a mechanism for the onset of non-equilibrium behaviour in
colloid-polymer mixtures. These mixtures can function as models of atomic
systems; their physics therefore impinges on many areas of thermodynamics and
phase-ordering. An exact solution is found for the motion of a single, planar
interface separating a growing phase of uniform high density from a
supersaturated low density phase, whose diffusive depletion drives the
interfacial motion. In addition, an approximate solution is found for the
one-dimensional evolution of two interfaces, separated by a slab of a
metastable phase at intermediate density. The theory predicts a critical
supersaturation of the low-density phase, above which the two interfaces become
unbound and the metastable phase grows ad infinitum. The growth of the stable
phase is suppressed in this regime.Comment: 27 pages, Latex, eps
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