1,332 research outputs found
Minkowski Tensors of Anisotropic Spatial Structure
This article describes the theoretical foundation of and explicit algorithms
for a novel approach to morphology and anisotropy analysis of complex spatial
structure using tensor-valued Minkowski functionals, the so-called Minkowski
tensors. Minkowski tensors are generalisations of the well-known scalar
Minkowski functionals and are explicitly sensitive to anisotropic aspects of
morphology, relevant for example for elastic moduli or permeability of
microstructured materials. Here we derive explicit linear-time algorithms to
compute these tensorial measures for three-dimensional shapes. These apply to
representations of any object that can be represented by a triangulation of its
bounding surface; their application is illustrated for the polyhedral Voronoi
cellular complexes of jammed sphere configurations, and for triangulations of a
biopolymer fibre network obtained by confocal microscopy. The article further
bridges the substantial notational and conceptual gap between the different but
equivalent approaches to scalar or tensorial Minkowski functionals in
mathematics and in physics, hence making the mathematical measure theoretic
method more readily accessible for future application in the physical sciences
Nodal domains of the equilateral triangle billiard
We characterise the eigenfunctions of an equilateral triangle billiard in
terms of its nodal domains. The number of nodal domains has a quadratic form in
terms of the quantum numbers, with a non-trivial number-theoretic factor. The
patterns of the eigenfunctions follow a group-theoretic connection in a way
that makes them predictable as one goes from one state to another. Extensive
numerical investigations bring out the distribution functions of the mode
number and signed areas. The statistics of the boundary intersections is also
treated analytically. Finally, the distribution functions of the nodal loop
count and the nodal counting function are shown to contain information about
the classical periodic orbits using the semiclassical trace formula. We believe
that the results belong generically to non-separable systems, thus extending
the previous works which are concentrated on separable and chaotic systems.Comment: 26 pages, 13 figure
On the Complexity of Existential Positive Queries
We systematically investigate the complexity of model checking the
existential positive fragment of first-order logic. In particular, for a set of
existential positive sentences, we consider model checking where the sentence
is restricted to fall into the set; a natural question is then to classify
which sentence sets are tractable and which are intractable. With respect to
fixed-parameter tractability, we give a general theorem that reduces this
classification question to the corresponding question for primitive positive
logic, for a variety of representations of structures. This general theorem
allows us to deduce that an existential positive sentence set having bounded
arity is fixed-parameter tractable if and only if each sentence is equivalent
to one in bounded-variable logic. We then use the lens of classical complexity
to study these fixed-parameter tractable sentence sets. We show that such a set
can be NP-complete, and consider the length needed by a translation from
sentences in such a set to bounded-variable logic; we prove superpolynomial
lower bounds on this length using the theory of compilability, obtaining an
interesting type of formula size lower bound. Overall, the tools, concepts, and
results of this article set the stage for the future consideration of the
complexity of model checking on more expressive logics
Coherent frequentism
By representing the range of fair betting odds according to a pair of
confidence set estimators, dual probability measures on parameter space called
frequentist posteriors secure the coherence of subjective inference without any
prior distribution. The closure of the set of expected losses corresponding to
the dual frequentist posteriors constrains decisions without arbitrarily
forcing optimization under all circumstances. This decision theory reduces to
those that maximize expected utility when the pair of frequentist posteriors is
induced by an exact or approximate confidence set estimator or when an
automatic reduction rule is applied to the pair. In such cases, the resulting
frequentist posterior is coherent in the sense that, as a probability
distribution of the parameter of interest, it satisfies the axioms of the
decision-theoretic and logic-theoretic systems typically cited in support of
the Bayesian posterior. Unlike the p-value, the confidence level of an interval
hypothesis derived from such a measure is suitable as an estimator of the
indicator of hypothesis truth since it converges in sample-space probability to
1 if the hypothesis is true or to 0 otherwise under general conditions.Comment: The confidence-measure theory of inference and decision is explicitly
extended to vector parameters of interest. The derivation of upper and lower
confidence levels from valid and nonconservative set estimators is formalize
Singularity Structures in Coulomb-Type Potentials in Two Body Dirac Equations of Constraint Dynamics
Two Body Dirac Equations (TBDE) of Dirac's relativistic constraint dynamics
have been successfully applied to obtain a covariant nonperturbative
description of QED and QCD bound states. Coulomb-type potentials in these
applications lead naively in other approaches to singular relativistic
corrections at short distances that require the introduction of either
perturbative treatments or smoothing parameters. We examine the corresponding
singular structures in the effective potentials of the relativistic
Schroedinger equation obtained from the Pauli reduction of the TBDE. We find
that the relativistic Schroedinger equation lead in fact to well-behaved wave
function solutions when the full potential and couplings of the system are
taken into account. The most unusual case is the coupled triplet system with
S=1 and L={(J-1),(J+1)}. Without the inclusion of the tensor coupling, the
effective S-state potential would become attractively singular. We show how
including the tensor coupling is essential in order that the wave functions be
well-behaved at short distances. For example, the S-state wave function becomes
simply proportional to the D-state wave function and dips sharply to zero at
the origin, unlike the usual S-state wave functions. Furthermore, this behavior
is similar in both QED and QCD, independent of the asymptotic freedom behavior
of the assumed QCD vector potential. Light- and heavy-quark meson states can be
described well by using a simplified linear-plus-Coulomb-type QCD potential
apportioned appropriately between world scalar and vector potentials. We use
this potential to exhibit explicitly the origin of the large pi-rho splitting
and effective chiral symmetry breaking. The TBDE formalism developed here may
be used to study quarkonia in quark-gluon plasma environments.Comment: 23 pages, 4 figure
Renormalization group in Statistical Mechanics and Mechanics: gauge symmetries and vanishing beta functions
Two very different problems that can be studied by renormalization group
methods are discussed with the aim of showing the conceptual unity that
renormalization group has introduced in some areas of theoretical Physics. The
two problems are: the ground state theory of a one dimensional quantum Fermi
liquid and the existence of quasi periodic motions in classical mechanical
systems close to integrable ones. I summarize here the main ideas and show that
the two treatments, although completely independent of each other, are
strikingly similar.Comment: Plain TeX, 20 pages 7 figures (included in ps format
Informal proof, formal proof, formalism
Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened
Sharp thresholds for high-dimensional and noisy recovery of sparsity
The problem of consistently estimating the sparsity pattern of a vector
\betastar \in \real^\mdim based on observations contaminated by noise arises
in various contexts, including subset selection in regression, structure
estimation in graphical models, sparse approximation, and signal denoising. We
analyze the behavior of -constrained quadratic programming (QP), also
referred to as the Lasso, for recovering the sparsity pattern. Our main result
is to establish a sharp relation between the problem dimension \mdim, the
number \spindex of non-zero elements in \betastar, and the number of
observations \numobs that are required for reliable recovery. For a broad
class of Gaussian ensembles satisfying mutual incoherence conditions, we
establish existence and compute explicit values of thresholds \ThreshLow and
\ThreshUp with the following properties: for any , if \numobs
> 2 (\ThreshUp + \epsilon) \log (\mdim - \spindex) + \spindex + 1, then the
Lasso succeeds in recovering the sparsity pattern with probability converging
to one for large problems, whereas for \numobs < 2 (\ThreshLow - \epsilon)
\log (\mdim - \spindex) + \spindex + 1, then the probability of successful
recovery converges to zero. For the special case of the uniform Gaussian
ensemble, we show that \ThreshLow = \ThreshUp = 1, so that the threshold is
sharp and exactly determined.Comment: Appeared as Technical Report 708, Department of Statistics, UC
Berkele
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