18,323 research outputs found
Geometry of logarithmic strain measures in solid mechanics
We consider the two logarithmic strain measureswhich are isotropic invariants of the
Hencky strain tensor , and show that they can be uniquely characterized
by purely geometric methods based on the geodesic distance on the general
linear group . Here, is the deformation gradient,
is the right Biot-stretch tensor, denotes the principal
matrix logarithm, is the Frobenius matrix norm, is the
trace operator and is the -dimensional deviator of
. This characterization identifies the Hencky (or
true) strain tensor as the natural nonlinear extension of the linear
(infinitesimal) strain tensor , which is the
symmetric part of the displacement gradient , and reveals a close
geometric relation between the classical quadratic isotropic energy potential
in
linear elasticity and the geometrically nonlinear quadratic isotropic Hencky
energywhere
is the shear modulus and denotes the bulk modulus. Our deduction
involves a new fundamental logarithmic minimization property of the orthogonal
polar factor , where is the polar decomposition of . We also
contrast our approach with prior attempts to establish the logarithmic Hencky
strain tensor directly as the preferred strain tensor in nonlinear isotropic
elasticity
K\"ahlerian information geometry for signal processing
We prove the correspondence between the information geometry of a signal
filter and a K\"ahler manifold. The information geometry of a minimum-phase
linear system with a finite complex cepstrum norm is a K\"ahler manifold. The
square of the complex cepstrum norm of the signal filter corresponds to the
K\"ahler potential. The Hermitian structure of the K\"ahler manifold is
explicitly emergent if and only if the impulse response function of the highest
degree in is constant in model parameters. The K\"ahlerian information
geometry takes advantage of more efficient calculation steps for the metric
tensor and the Ricci tensor. Moreover, -generalization on the geometric
tensors is linear in . It is also robust to find Bayesian predictive
priors, such as superharmonic priors, because Laplace-Beltrami operators on
K\"ahler manifolds are in much simpler forms than those of the non-K\"ahler
manifolds. Several time series models are studied in the K\"ahlerian
information geometry.Comment: 24 pages, published versio
Models of nonlinear kinematic hardening based on different versions of rate-independent maxwell fluid
Different models of finite strain plasticity with a nonlinear kinematic hardening are analyzed in a systematic way. All the models are based on a certain formulation of a rate-independent Maxwell fluid, which is used to render the evolution of backstresses. The properties of each material model are determined by the underlying formulation of the Maxwell fluid. The analyzed approaches include the multiplicative hyperelastoplasticity, additive hypoelasto-plasticity and the use of generalized strain measures. The models are compared with respect to different classification criteria, such as the objectivity, thermodynamic consistency, pure volumetric-isochoric split, shear stress oscillation, exact integrability, and w-invariance
Nilpotent Approximations of Sub-Riemannian Distances for Fast Perceptual Grouping of Blood Vessels in 2D and 3D
We propose an efficient approach for the grouping of local orientations
(points on vessels) via nilpotent approximations of sub-Riemannian distances in
the 2D and 3D roto-translation groups and . In our distance
approximations we consider homogeneous norms on nilpotent groups that locally
approximate , and which are obtained via the exponential and logarithmic
map on . In a qualitative validation we show that the norms provide
accurate approximations of the true sub-Riemannian distances, and we discuss
their relations to the fundamental solution of the sub-Laplacian on .
The quantitative experiments further confirm the accuracy of the
approximations. Quantitative results are obtained by evaluating perceptual
grouping performance of retinal blood vessels in 2D images and curves in
challenging 3D synthetic volumes. The results show that 1) sub-Riemannian
geometry is essential in achieving top performance and 2) that grouping via the
fast analytic approximations performs almost equally, or better, than
data-adaptive fast marching approaches on and .Comment: 18 pages, 9 figures, 3 tables, in review at JMI
Entropy Production of Doubly Stochastic Quantum Channels
We study the entropy increase of quantum systems evolving under primitive,
doubly stochastic Markovian noise and thus converging to the maximally mixed
state. This entropy increase can be quantified by a logarithmic-Sobolev
constant of the Liouvillian generating the noise. We prove a universal lower
bound on this constant that stays invariant under taking tensor-powers. Our
methods involve a new comparison method to relate logarithmic-Sobolev constants
of different Liouvillians and a technique to compute logarithmic-Sobolev
inequalities of Liouvillians with eigenvectors forming a projective
representation of a finite abelian group. Our bounds improve upon similar
results established before and as an application we prove an upper bound on
continuous-time quantum capacities. In the last part of this work we study
entropy production estimates of discrete-time doubly-stochastic quantum
channels by extending the framework of discrete-time logarithmic-Sobolev
inequalities to the quantum case.Comment: 24 page
Nonlinear Dynamics of Parity-Even Tricritical Gravity in Three and Four Dimensions
Recently proposed "multicritical" higher-derivative gravities in Anti de
Sitter space carry logarithmic representations of the Anti de Sitter isometry
group. While generically non-unitary already at the quadratic, free-theory
level, in special cases these theories admit a unitary subspace. The simplest
example of such behavior is "tricritical" gravity. In this paper, we extend the
study of parity-even tricritical gravity in d = 3, 4 to the first nonlinear
order. We show that the would-be unitary subspace suffers from a linearization
instability and is absent in the full non-linear theory.Comment: 22 pages; v2: references added, published versio
Smolyak's algorithm: A powerful black box for the acceleration of scientific computations
We provide a general discussion of Smolyak's algorithm for the acceleration
of scientific computations. The algorithm first appeared in Smolyak's work on
multidimensional integration and interpolation. Since then, it has been
generalized in multiple directions and has been associated with the keywords:
sparse grids, hyperbolic cross approximation, combination technique, and
multilevel methods. Variants of Smolyak's algorithm have been employed in the
computation of high-dimensional integrals in finance, chemistry, and physics,
in the numerical solution of partial and stochastic differential equations, and
in uncertainty quantification. Motivated by this broad and ever-increasing
range of applications, we describe a general framework that summarizes
fundamental results and assumptions in a concise application-independent
manner
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