23,871 research outputs found
Linear and Branching System Metrics
We extend the classical system relations of trace\ud
inclusion, trace equivalence, simulation, and bisimulation to a quantitative setting in which propositions are interpreted not as boolean values, but as elements of arbitrary metric spaces.\ud
\ud
Trace inclusion and equivalence give rise to asymmetrical and symmetrical linear distances, while simulation and bisimulation give rise to asymmetrical and symmetrical branching distances. We study the relationships among these distances, and we provide a full logical characterization of the distances in terms of quantitative versions of LTL and Ό-calculus. We show that, while trace inclusion (resp. equivalence) coincides with simulation (resp. bisimulation) for deterministic boolean transition systems, linear\ud
and branching distances do not coincide for deterministic metric transition systems. Finally, we provide algorithms for computing the distances over finite systems, together with a matching lower complexity bound
Quantitative Graded Semantics and Spectra of Behavioural Metrics
Behavioural metrics provide a quantitative refinement of classical two-valued
behavioural equivalences on systems with quantitative data, such as metric or
probabilistic transition systems. In analogy to the classical
linear-time/branching-time spectrum of two-valued behavioural equivalences on
transition systems, behavioural metrics come in various degrees of granularity,
depending on the observer's ability to interact with the system. Graded monads
have been shown to provide a unifying framework for spectra of behavioural
equivalences. Here, we transfer this principle to spectra of behavioural
metrics, working at a coalgebraic level of generality, that is, parametrically
in the system type. In the ensuing development of quantitative graded
semantics, we discuss presentations of graded monads on the category of metric
spaces in terms of graded quantitative equational theories. Moreover, we obtain
a canonical generic notion of invariant real-valued modal logic, and provide
criteria for such logics to be expressive in the sense that logical distance
coincides with the respective behavioural distance. We thus recover recent
expressiveness results for coalgebraic branching-time metrics and for trace
distance in metric transition systems; moreover, we obtain a new expressiveness
result for trace semantics of fuzzy transition systems. We also provide a
number of salient negative results. In particular, we show that trace distance
on probabilistic metric transition systems does not admit a characteristic
real-valued modal logic at all
Distances for Weighted Transition Systems: Games and Properties
We develop a general framework for reasoning about distances between
transition systems with quantitative information. Taking as starting point an
arbitrary distance on system traces, we show how this leads to natural
definitions of a linear and a branching distance on states of such a transition
system. We show that our framework generalizes and unifies a large variety of
previously considered system distances, and we develop some general properties
of our distances. We also show that if the trace distance admits a recursive
characterization, then the corresponding branching distance can be obtained as
a least fixed point to a similar recursive characterization. The central tool
in our work is a theory of infinite path-building games with quantitative
objectives.Comment: In Proceedings QAPL 2011, arXiv:1107.074
Linear Distances between Markov Chains
We introduce a general class of distances (metrics) between Markov chains,
which are based on linear behaviour. This class encompasses distances given
topologically (such as the total variation distance or trace distance) as well
as by temporal logics or automata. We investigate which of the distances can be
approximated by observing the systems, i.e. by black-box testing or simulation,
and we provide both negative and positive results
Image-based Quantification of 3D Morphology for Bifurcations in the Left Coronary Artery: Application to Stent Design
Background
Improved strategies for stentâbased treatment of coronary artery disease at bifurcations require a greater understanding of artery morphology. Objective
We developed a workflow to quantify morphology in the left main coronary (LMCA), left anterior descending (LAD), and left circumflex (LCX) artery bifurcations. Methods
Computational models of each bifurcation were created for 55 patients using computed tomography images in 3D segmentation software. Metrics including crossâsectional area, length, eccentricity, taper, curvature, planarity, branching law parameters, and bifurcation angles were assessed using openâsources software and custom applications. Geometric characterization was performed by comparison of means, correlation, and linear discriminant analysis (LDA). Results
Differences between metrics suggest dedicated or multistent approaches should be tailored for each bifurcation. For example, the side branch of the LCX (i.e., obtuse marginal; OM) was longer than that of the LMCA (i.e., LCXprox) and LAD (i.e., first diagonal; D1). Bifurcation metrics for some locations (e.g., LMCA Finet ratio) provide results and confidence intervals agreeing with prior findings, while revised metric values are presented for others (e.g., LAD and LCX). LDA revealed several metrics that differentiate between artery locations (e.g., LMCA vs. D1, LMCA vs. OM, LADprox vs. D1, and LCXprox vs. D1). Conclusions
These results provide a foundation for elucidating common parameters from healthy coronary arteries and could be leveraged in the future for treating diseased arteries. Collectively the current results may ultimately be used for design iterations that improve outcomes following implantation of future dedicated bifurcation stents
Non-Euclidean geometry in nature
I describe the manifestation of the non-Euclidean geometry in the behavior of
collective observables of some complex physical systems. Specifically, I
consider the formation of equilibrium shapes of plants and statistics of sparse
random graphs. For these systems I discuss the following interlinked questions:
(i) the optimal embedding of plants leaves in the three-dimensional space, (ii)
the spectral statistics of sparse random matrix ensembles.Comment: 52 pages, 21 figures, last section is rewritten, a reference to
chaotic Hamiltonian systems is adde
Convexity and the Euclidean metric of space-time
We address the question about the reasons why the "Wick-rotated",
positive-definite, space-time metric obeys the Pythagorean theorem. An answer
is proposed based on the convexity and smoothness properties of the functional
spaces purporting to provide the kinematic framework of approaches to quantum
gravity. We employ moduli of convexity and smoothness which are eventually
extremized by Hilbert spaces. We point out the potential physical significance
that functional analytical dualities play in this framework. Following the
spirit of the variational principles employed in classical and quantum Physics,
such Hilbert spaces dominate in a generalized functional integral approach. The
metric of space-time is induced by the inner product of such Hilbert spaces.Comment: 41 pages. No figures. Standard LaTeX2e. Change of affiliation of the
author and mostly superficial changes in this version. Accepted for
publication by "Universe" in a Special Issue with title: "100 years of
Chronogeometrodynamics: the Status of Einstein's theory of Gravitation in its
Centennial Year
Geometric combinatorics and computational molecular biology: branching polytopes for RNA sequences
Questions in computational molecular biology generate various discrete
optimization problems, such as DNA sequence alignment and RNA secondary
structure prediction. However, the optimal solutions are fundamentally
dependent on the parameters used in the objective functions. The goal of a
parametric analysis is to elucidate such dependencies, especially as they
pertain to the accuracy and robustness of the optimal solutions. Techniques
from geometric combinatorics, including polytopes and their normal fans, have
been used previously to give parametric analyses of simple models for DNA
sequence alignment and RNA branching configurations. Here, we present a new
computational framework, and proof-of-principle results, which give the first
complete parametric analysis of the branching portion of the nearest neighbor
thermodynamic model for secondary structure prediction for real RNA sequences.Comment: 17 pages, 8 figure
Coalgebraic Behavioral Metrics
We study different behavioral metrics, such as those arising from both
branching and linear-time semantics, in a coalgebraic setting. Given a
coalgebra for a functor , we define a framework for deriving pseudometrics on which
measure the behavioral distance of states.
A crucial step is the lifting of the functor on to a
functor on the category of pseudometric spaces.
We present two different approaches which can be viewed as generalizations of
the Kantorovich and Wasserstein pseudometrics for probability measures. We show
that the pseudometrics provided by the two approaches coincide on several
natural examples, but in general they differ.
If has a final coalgebra, every lifting yields in a
canonical way a behavioral distance which is usually branching-time, i.e., it
generalizes bisimilarity. In order to model linear-time metrics (generalizing
trace equivalences), we show sufficient conditions for lifting distributive
laws and monads. These results enable us to employ the generalized powerset
construction
- âŠ