1,521 research outputs found
Thermodynamically Consistent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities
A new diffuse interface model for a two-phase flow of two incompressible
fluids with different densities is introduced using methods from rational
continuum mechanics. The model fulfills local and global dissipation
inequalities and is also generalized to situations with a soluble species.
Using the method of matched asymptotic expansions we derive various sharp
interface models in the limit when the interfacial thickness tends to zero.
Depending on the scaling of the mobility in the diffusion equation we either
derive classical sharp interface models or models where bulk or surface
diffusion is possible in the limit. In the two latter cases the classical
Gibbs-Thomson equation has to be modified to include kinetic terms. Finally, we
show that all sharp interface models fulfill natural energy inequalities.Comment: 34 page
Lossless Analog Compression
We establish the fundamental limits of lossless analog compression by
considering the recovery of arbitrary m-dimensional real random vectors x from
the noiseless linear measurements y=Ax with n x m measurement matrix A. Our
theory is inspired by the groundbreaking work of Wu and Verdu (2010) on almost
lossless analog compression, but applies to the nonasymptotic, i.e., fixed-m
case, and considers zero error probability. Specifically, our achievability
result states that, for almost all A, the random vector x can be recovered with
zero error probability provided that n > K(x), where K(x) is given by the
infimum of the lower modified Minkowski dimension over all support sets U of x.
We then particularize this achievability result to the class of s-rectifiable
random vectors as introduced in Koliander et al. (2016); these are random
vectors of absolutely continuous distribution---with respect to the
s-dimensional Hausdorff measure---supported on countable unions of
s-dimensional differentiable submanifolds of the m-dimensional real coordinate
space. Countable unions of differentiable submanifolds include essentially all
signal models used in the compressed sensing literature. Specifically, we prove
that, for almost all A, s-rectifiable random vectors x can be recovered with
zero error probability from n>s linear measurements. This threshold is,
however, found not to be tight as exemplified by the construction of an
s-rectifiable random vector that can be recovered with zero error probability
from n<s linear measurements. This leads us to the introduction of the new
class of s-analytic random vectors, which admit a strong converse in the sense
of n greater than or equal to s being necessary for recovery with probability
of error smaller than one. The central conceptual tools in the development of
our theory are geometric measure theory and the theory of real analytic
functions
Lossless Linear Analog Compression
We establish the fundamental limits of lossless linear analog compression by
considering the recovery of random vectors
from the noiseless linear
measurements
with
measurement matrix . Specifically,
for a random vector of arbitrary
distribution we show that can be recovered with
zero error probability from
linear measurements,
where denotes the lower
modified Minkowski dimension and the infimum is over all sets
with . This achievability statement holds for Lebesgue almost all measurement
matrices . We then show that -rectifiable random vectors---a
stochastic generalization of -sparse vectors---can be recovered with zero
error probability from linear measurements. From classical compressed
sensing theory we would expect to be necessary for successful
recovery of . Surprisingly, certain classes of
-rectifiable random vectors can be recovered from fewer than
measurements. Imposing an additional regularity condition on the distribution
of -rectifiable random vectors , we do get the
expected converse result of measurements being necessary. The resulting
class of random vectors appears to be new and will be referred to as
-analytic random vectors
Lossy Compression of General Random Variables
This paper is concerned with the lossy compression of general random
variables, specifically with rate-distortion theory and quantization of random
variables taking values in general measurable spaces such as, e.g., manifolds
and fractal sets. Manifold structures are prevalent in data science, e.g., in
compressed sensing, machine learning, image processing, and handwritten digit
recognition. Fractal sets find application in image compression and in the
modeling of Ethernet traffic. Our main contributions are bounds on the
rate-distortion function and the quantization error. These bounds are very
general and essentially only require the existence of reference measures
satisfying certain regularity conditions in terms of small ball probabilities.
To illustrate the wide applicability of our results, we particularize them to
random variables taking values in i) manifolds, namely, hyperspheres and
Grassmannians, and ii) self-similar sets characterized by iterated function
systems satisfying the weak separation property
Rotational barriers in perylene fluorescent dyes
Rotational barriers in N-substituted perylene dyes have been determined. Phenyl substituents with tert-butyl groups in the o-position give rigid systems, whereas secondary alkyl groups cause low rotational barriers. In spite of that, fluorescent quantum yields are high in both cases. Conformations in solution are discussed
Directly decomposable ideals and congruence kernels of commutative semirings
As pointed out in the monographs by J. S. Golan and by W. Kuich and A.
Salomaa on semirings, ideals play an important role despite the fact that they
need not be congruence kernels as in the case of rings. Hence, having two
commutative semirings S1 and S2, one can ask whether an ideal I of their direct
product S = S1 x S2 can be expressed in the form I1 x I2 where Ij is an ideal
of Sj for j=1,2. Of course, the converse is elementary, namely if Ij is an
ideal of Sj for j=1,2 then I1 x I2 is an ideal of S1 x S2. Having a congruence
on a commutative semiring S, its 0-class is an ideal of S, but not every ideal
is of this form. Hence, the lattice Id S of all ideals of S and the lattice Ker
S of all congruence kernels (i.e. 0-classes of congruences) of S need not be
equal. Furthermore, we show that the mapping which assigns to every congruence
its kernel need not be a homomorphism from Con S onto Ker S. Moreover, the
question arises when a congruence kernel of the direct product S1 x S2 of two
commutative semirings can be expressed as a direct product of the corresponding
kernels on the factors. In the paper we present necessary and sufficient
conditions for such direct decompositions both for ideals and for congruence
kernels of commutative semirings. We also provide sufficient conditions for
varieties of commutative semirings to have directly decomposable kernels
Das Alter der Sinterkalke vom Solbad Laer i.T.W.
Aus 4 Profilen durch den Laerer Sinterkalk wurden 33 Proben pollenanalytisch untersucht. Zwei Zähltabellen geben für jede einzelne dieser Proben den Gehalt an Pollen und Sporen. Zwei Diagramme stellen die Ergebnisse dieser qualitativen und quantitativen Analysen graphisch dar. Neben den Pollenkörnern der gebräuchlichen 11 Baumarten wurden 24 verschiedene Nichtbaumpollen-Gruppen ausgewertet. Als ältester Zeitabschnitt ließ sich die mindestens 10000 Jahre alte sog. "Jüngste Dryaszeit" feststellen, gekennzeichnet durch eine subarktische Tundra. Die darauf folgenden Zeitabschnitte Präboreal, Boreal, Atlantikum, Subboreal und Subatlantikum konnten in einem oder mehreren der Profile gefaßt werden. In günstig gelagerten Fällen wurden darüber hinaus Beziehungen angedeutet, die zwischen der Bildung dieses Sinterkalklagers und der Besiedlung durch den Menschen bestehen
Completion of Matrices with Low Description Complexity
We propose a theory for matrix completion that goes beyond the low-rank
structure commonly considered in the literature and applies to general matrices
of low description complexity. Specifically, complexity of the sets of matrices
encompassed by the theory is measured in terms of Hausdorff and upper Minkowski
dimensions. Our goal is the characterization of the number of linear
measurements, with an emphasis on rank- measurements, needed for the
existence of an algorithm that yields reconstruction, either perfect, with
probability 1, or with arbitrarily small probability of error, depending on the
setup. Concretely, we show that matrices taken from a set such
that has Hausdorff dimension can be recovered
from measurements, and random matrices supported on a set
of Hausdorff dimension can be recovered with probability 1 from
measurements. What is more, we establish the existence of recovery mappings
that are robust against additive perturbations or noise in the measurements.
Concretely, we show that there are -H\"older continuous mappings
recovering matrices taken from a set of upper Minkowski dimension from
measurements and, with arbitrarily small probability of error,
random matrices supported on a set of upper Minkowski dimension from
measurements. The numerous concrete examples we consider
include low-rank matrices, sparse matrices, QR decompositions with sparse
R-components, and matrices of fractal nature
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