scholarly journals PATH INTEGRALS ON A COMPACT MANIFOLD WITH NON-NEGATIVE CURVATURE

2007 ◽  
Vol 19 (09) ◽  
pp. 967-1044 ◽  
Author(s):  
ADRIAN P. C. LIM
Keyword(s):  
Compact Manifold ◽  
Compact Riemannian Manifold ◽  
Path Integrals ◽  
Energy Functional ◽  
Normalization Constant ◽  
Finite Dimensional ◽  
Riemannian Volume ◽  
Volume Measure ◽  
Constant E ◽  
Geodesic Paths

A typical path integral on a manifold, M is an informal expression of the form [Formula: see text] where H(M) is a Hilbert manifold of paths with energy E(σ) < ∞, f is a real-valued function on H(M), [Formula: see text] is a "Lebesgue measure" and Z is a normalization constant. For a compact Riemannian manifold M, we wish to interpret [Formula: see text] as a Riemannian "volume form" over H(M), equipped with its natural G1 metric. Given an equally spaced partition, [Formula: see text] of [0, τ], let [Formula: see text] be the finite dimensional Riemannian submanifold of H(M) consisting of piecewise geodesic paths adapted to [Formula: see text]. Under certain curvature restrictions on M, it is shown that [Formula: see text] where [Formula: see text] is a "normalization" constant, E : H(M) → [0,∞) is the energy functional, [Formula: see text] is the Riemannian volume measure on [Formula: see text], ν is Wiener measure on continuous paths in M, and ρ is a certain density determined by the curvature tensor of M.

A monolithic phase-field model of a fluid-driven fracture in a nonlinear poroelastic medium

2018 ◽  
Vol 24 (5) ◽  
pp. 1530-1555 ◽  
Author(s):  
CJ van Duijn ◽  
Andro Mikelić ◽  
Thomas Wick
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Galerkin Approximation ◽  
Phase Field Model ◽  
Coupled System ◽  
Energy Functional ◽  
Poroelastic Medium ◽  
Existence Of A Solution ◽  
Finite Dimensional ◽  
Biot Equations

In this paper, we present a full phase-field model for a fluid-driven fracture in a nonlinear poroelastic medium. The nonlinearity arises in the Biot equations when the permeability depends on porosity. This extends previous work (see Mikelić et al. Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput Geosci 2015; 19: 1171–1195), where a fully coupled system is considered for the pressure, displacement, and phase field. For the extended system, we follow a similar approach: we introduce, for a given pressure, an energy functional, from which we derive the equations for the displacement and phase field. We establish the existence of a solution of the incremental problem through convergence of a finite-dimensional Galerkin approximation. Furthermore, we construct the corresponding Lyapunov functional, which is related to the free energy. Computational results are provided that demonstrate the effectiveness of this approach in treating fluid-driven fracture propagation. Specifically, our numerical findings confirm differences with test cases using the linear Biot equations.

scholarly journals Linearly constrained evolutions of critical points and an application to cohesive fractures

2017 ◽  
Vol 27 (02) ◽  
pp. 231-290 ◽  
Author(s):  
Marco Artina ◽  
Filippo Cagnetti ◽  
Massimo Fornasier ◽  
Francesco Solombrino
Keyword(s):  
Critical Points ◽  
Energy Functional ◽  
Two Dimensions ◽  
Infinite Dimensional ◽  
Energy Functionals ◽  
Finite Dimensional ◽  
Nonconvex Energy ◽  
Linearly Constrained ◽  
Cohesive Fracture Model ◽  
Crack Initiation Criterion

We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite-dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite-dimensional. Nevertheless, in the infinite-dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite-dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one- and two-dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.

scholarly journals Quantum ergodicity of random orthonormal bases of spaces of high dimension

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120511 ◽  
Author(s):  
Steve Zelditch
Keyword(s):  
Hilbert Spaces ◽  
Orthonormal Basis ◽  
Compact Riemannian Manifold ◽  
Random Sequence ◽  
Limit State ◽  
Flat Torus ◽  
Quantum Ergodicity ◽  
Orthonormal Bases ◽  
Finite Dimensional ◽  
Unique Limit

We consider a sequence of finite-dimensional Hilbert spaces of dimensions . Motivating examples are eigenspaces, or spaces of quasi-modes, for a Laplace or Schrödinger operator on a compact Riemannian manifold. The set of Hermitian orthonormal bases of may be identified with U ( d N ), and a random orthonormal basis of is a choice of a random sequence U N ∈ U ( d N ) from the product of normalized Haar measures. We prove that if and if tends to a unique limit state ω ( A ), then almost surely an orthonormal basis is quantum ergodic with limit state ω ( A ). This generalizes an earlier result of the author in the case where is the space of spherical harmonics on S 2 . In particular, it holds on the flat torus if d ≥5 and shows that a highly localized orthonormal basis can be synthesized from quantum ergodic ones and vice versa in relatively small dimensions.

scholarly journals Isometries of a Bergman-Privalov-Type Space on the Unit Ball

2009 ◽  
Vol 2009 ◽  
pp. 1-16 ◽  
Author(s):  
Stevo Stević ◽  
Sei-Ichiro Ueki
Keyword(s):  
Unit Ball ◽  
Holomorphic Functions ◽  
Complete Characterization ◽  
Normalization Constant ◽  
Linear Isometry ◽  
Type Space ◽  
Basic Properties ◽  
Volume Measure ◽  
New Space

We introduce a new spaceANlog⁡,α(&#x1D539;)consisting of all holomorphic functions on the unit ball&#x1D539;⊂ℂnsuch that‖f‖ANlog⁡,α:=∫&#x1D539;φe(ln⁡(1+|f(z)|))dVα(z)<∞, whereα>−1,dVα(z)=cα,n(1−|z|2)αdV(z)(dV(z)is the normalized Lebesgue volume measure on&#x1D539;, andcα,nis a normalization constant, that is,Vα(&#x1D539;)=1), andφe(t)=tln⁡(e+t)fort∈[0,∞). Some basic properties of this space are presented. Among other results we proved thatANlog⁡,α(&#x1D539;)with the metricd(f,g)=‖f−g‖ANlog⁡,αis anF-algebra with respect to pointwise addition and multiplication. We also prove that every linear isometryTofANlog⁡,α(&#x1D539;)into itself has the formTf=c(f∘ψ)for somec∈ℂsuch that|c|=1and someψwhich is a holomorphic self-map of&#x1D539;satisfying a measure-preserving property with respect to the measuredVα. As a consequence of this result we obtain a complete characterization of all linear bijective isometries ofANlog⁡,α(&#x1D539;).

Calibrations and laminations

2016 ◽  
Vol 162 (1) ◽  
pp. 151-171
Author(s):  
VICTOR BANGERT ◽  
XIAOJUN CUI
Keyword(s):  
Riemannian Manifold ◽  
Cohomology Class ◽  
Compact Manifold ◽  
Volume Form ◽  
Riemannian Volume

AbstractA calibration of degree k ∈ ℕ on a Riemannian manifold M is a closed differential k-form θ such that the integral of θ over every k-dimensional, oriented submanifold N is smaller or equal to the Riemannian volume of N. A calibration θ is said to calibrate N if θ restricts to the oriented volume form of N. We investigate conditions on a calibration θ that ensure the existence of submanifolds calibrated by θ. The cases k = 1 and k > 1 turn out to be essentially different. Our main result says that, on a compact manifold M, a calibration θ calibrates a lamination if θ is simple, of class C1, and if θ has minimal comass norm in its cohomology class.

QUASI-SURE EXISTENCE OF BROWNIAN ROUGH PATHS AND A CONSTRUCTION OF BROWNIAN PANTS

2006 ◽  
Vol 09 (04) ◽  
pp. 513-528 ◽  
Author(s):  
YUZURU INAHAMA
Keyword(s):  
Loop Space ◽  
Compact Manifold ◽  
Continuous Process ◽  
Rough Paths ◽  
Finite Dimensional ◽  
Rough Path

In this paper we will prove the quasi-sure existence of the Brownian rough path for finite-dimensional cases. As an application we will give a construction of Brownian pants, that is a certain continuous process on the continuous loop space over a compact manifold.

scholarly journals Ordering, Symbols and Finite-Dimensional Approximations of Path Integrals

1994 ◽  
Vol 92 (3) ◽  
pp. 669-685 ◽  
Author(s):  
T. Kashiwa ◽  
S. Sakoda ◽  
S. V. Zenkin
Keyword(s):  
Path Integrals ◽  
Finite Dimensional

scholarly journals Wong’s equations in Yang-Mills theory

Open Physics ◽  
2014 ◽  
Vol 12 (4) ◽  
Author(s):  
Sergey Storchak
Keyword(s):  
Dynamical System ◽  
Compact Riemannian Manifold ◽  
Scalar Particle ◽  
Orbit Space ◽  
Yang Mills ◽  
Smooth Action ◽  
Finite Dimensional ◽  
Gauge Fixing ◽  
Dimensional Dynamical System ◽  
Dependent Coordinates

AbstractWong’s equations for the finite-dimensional dynamical system representing the motion of a scalar particle on a compact Riemannian manifold with a given free isometric smooth action of a compact semi-simple Lie group are derived. The equations obtained are written in terms of dependent coordinates which are typically used in an implicit description of the local dynamics given on the orbit space of the principal fiber bundle. Using these equations, we obtain Wong’s equations in a pure Yang-Mills gauge theory with Coulomb gauge fixing. This result is based on the existing analogy between the reduction procedures performed in a finite-dimensional dynamical system and the reduction procedure in Yang-Mills gauge fields.

scholarly journals A variational formulation of the BDF2 method for metric gradient flows

2019 ◽  
Vol 53 (1) ◽  
pp. 145-172 ◽  
Author(s):  
Daniel Matthes ◽  
Simon Plazotta
Keyword(s):  
Variational Formulation ◽  
Metric Spaces ◽  
Compact Riemannian Manifold ◽  
Energy Functional ◽  
Convergence Order ◽  
Well Posedness ◽  
Gradient Flows ◽  
Wasserstein Metric ◽  
Discrete Approximations ◽  
Variational Form

We propose a variational form of the BDF2 method as an alternative to the commonly used minimizing movement scheme for the time-discrete approximation of gradient flows in abstract metric spaces. Assuming uniform semi-convexity – but no smoothness – of the augmented energy functional, we prove well-posedness of the method and convergence of the discrete approximations to a curve of steepest descent. In a smooth Hilbertian setting, classical theory would predict a convergence order of two in time, we prove convergence order of one-half in the general metric setting and under our weak hypotheses. Further, we illustrate these results with numerical experiments for gradient flows on a compact Riemannian manifold, in a Hilbert space, and in the L2-Wasserstein metric.

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