scholarly journals GENERALIZED TWISTS AS ELASTIC ENERGY EXTREMALS ON ANNULI, QUATERNIONS AND LIFTING TWIST LOOPS TO THE SPINOR GROUPS

2013 ◽  
Vol 11 (02) ◽  
pp. 1350008 ◽  
Author(s):  
M. S. SHAHROKHI-DEHKORDI ◽  
A. TAHERI
Keyword(s):  
Elastic Energy ◽  
Lagrange Equation ◽  
Energy Functional ◽  
Double Cover ◽  
Explicit Representation ◽  
Exponential Map ◽  
Dirichlet Energy ◽  
Smooth Solutions ◽  
Low Dimensions ◽  
Admissible Maps

Let X = {x ∈ ℝn : a < |x| < b} be a generalized annulus and consider the Dirichlet energy functional [Formula: see text] over the space of admissible maps [Formula: see text] where φ is the identity map. In this paper we consider a class of maps referred to as generalized twists and examine them in connection with the Euler–Lagrange equation associated with 𝔽[⋅, X] on [Formula: see text]. The approach is novel and is based on lifting twist loops from SO(n) to its double cover Spin(n) and reformulating the equations accordingly. We restrict our attention to low dimensions and prove that for n = 4 the system admits infinitely many smooth solutions in the form of twists while for n = 3 this number sharply reduces to one. We discuss some qualitative features of these solutions in view of their remarkable explicit representation through the exponential map of Spin(n).

Curvature-dependent energies: a geometric and analytical approach

2017 ◽  
Vol 147 (3) ◽  
pp. 449-503 ◽  
Author(s):  
Emilio Acerbi ◽  
Domenico Mucci
Keyword(s):  
Bounded Variation ◽  
Analytical Approach ◽  
Total Curvature ◽  
Representation Formula ◽  
Energy Functional ◽  
Continuous Functions ◽  
Explicit Representation ◽  
Functions Of Bounded Variation ◽  
One Dimensional ◽  
Relaxed Energy

We consider the total curvature of graphs of curves in high-codimension Euclidean space. We introduce the corresponding relaxed energy functional and prove an explicit representation formula. In the case of continuous Cartesian curves, i.e. of graphs cu of continuous functions u on an interval, we show that the relaxed energy is finite if and only if the curve cu has bounded variation and finite total curvature. In this case, moreover, the total curvature does not depend on the Cantor part of the derivative of u. We treat the wider class of graphs of one-dimensional functions of bounded variation, and we prove that the relaxed energy is given by the sum of the length and total curvature of the new curve obtained by closing the holes in cu generated by jumps of u with vertical segments.

scholarly journals Regularity of Weak Solution of Variational Problems Modeling the Cosserat Micropolar Elasticity

2020 ◽  
Author(s):  
Yimei Li ◽  
Changyou Wang
Keyword(s):  
Weak Solution ◽  
Harmonic Maps ◽  
Lagrange Equation ◽  
Variational Problems ◽  
Energy Functional ◽  
Geometrically Nonlinear ◽  
Micropolar Elasticity ◽  
One Dimensional ◽  
The Poisson Equation ◽  
System Coupling

Abstract In this paper, we consider weak solutions of the Euler–Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling between the Poisson equation and the equation of $p$-harmonic maps ($2\le p\le 3$). We show that if a weak solution is stationary, then its singular set is discrete for $2&lt;p&lt;3$ and has zero one-dimensional Hausdorff measure for $p=2$. If, in addition, it is a stable-stationary weak solution, then it is regular everywhere when $p\in [2, \frac{32}{15}]$.

Multi-Frame Super-Resolution Reconstruction Algorithm Based on Diffusion Tensor Regularization Term

2014 ◽  
Vol 543-547 ◽  
pp. 2828-2832 ◽  
Author(s):  
Xiao Dong Zhao ◽  
Zuo Feng Zhou ◽  
Jian Zhong Cao ◽  
Long Ren ◽  
Guang Sen Liu ◽  
...  
Keyword(s):  
Diffusion Tensor ◽  
Lagrange Equation ◽  
Super Resolution ◽  
Reconstruction Algorithm ◽  
Energy Functional ◽  
L1 Norm ◽  
Regularization Term ◽  
Sr Reconstruction ◽  
Image Edges ◽  
Anisotropic Diffusion Equation

This paper presents a multi-frame super-resolution (SR) reconstruction algorithm based on diffusion tensor regularization term. Firstly, L1-norm structure is used as data fidelity term, anisotropic diffusion equation with directional smooth characteristics is introduced as a prior knowledge to optimize reconstruction result. Secondly, combined with shock filter, SR reconstruction energy functional is established. Finally, Euler-Lagrange equation based on nonlinear diffusion model is exported. Simulation results validate that the proposed algorithm enhances image edges and suppresses noise effectively, which proves better robustness.

scholarly journals Nonlocal Variational Model for Saliency Detection

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Meng Li ◽  
Yi Zhan ◽  
Lidan Zhang
Keyword(s):  
Visual Attention ◽  
Diffusion Equation ◽  
Nonlinear Diffusion ◽  
Temporal Evolution ◽  
Lagrange Equation ◽  
Saliency Detection ◽  
Energy Functional ◽  
Experimental Results ◽  
Variational Model ◽  
Still Images

We present a nonlocal variational model for saliency detection from still images, from which various features for visual attention can be detected by minimizing the energy functional. The associated Euler-Lagrange equation is a nonlocalp-Laplacian type diffusion equation with two reaction terms, and it is a nonlinear diffusion. The main advantage of our method is that it provides flexible and intuitive control over the detecting procedure by the temporal evolution of the Euler-Lagrange equation. Experimental results on various images show that our model can better make background details diminish eventually while luxuriant subtle details in foreground are preserved very well.

New Model with Spring and Simulation of Flexible Manipulator

2014 ◽  
Vol 513-517 ◽  
pp. 3840-3843
Author(s):  
Ying Tian ◽  
Jian Hua Qian ◽  
Qing Song Liu ◽  
Jie Yuan
Keyword(s):  
Computer Simulation ◽  
Dynamic Model ◽  
Elastic Energy ◽  
Lagrange Equation ◽  
Rigid Bodies ◽  
Flexible Manipulator ◽  
Dynamic Parameters ◽  
New Model ◽  
Real Model ◽  
Model Computer

For easily calculated and more accurate dynamic model of single flexible manipulator, a new model was built through connection of spring and two rigid bodies. It is approximate to the real model of single manipulator in trajectory of end point. With simplifying manipulator and introducing simplified and Predetermined elastic energy of manipulator, Lagrange equation was used to built dynamic model based on the new model. And based on dynamic model, computer simulation result of dynamic parameters with Matlab software proved that the new model is available simple and easily-adjusted.

General Elasticity Theory for Graphene Membranes Based on Molecular Dynamics

2007 ◽  
Vol 1057 ◽  
Author(s):  
Kaveh Samadikhah ◽  
Juan Atalaya ◽  
Caroline Huldt ◽  
Andreas Isacsson ◽  
Jari Kinaret
Keyword(s):  
Molecular Dynamics ◽  
Elasticity Theory ◽  
Elastic Energy ◽  
Poisson Ratio ◽  
Md Simulations ◽  
Energy Functional ◽  
Linear Description ◽  
Continuum Description ◽  
External Forces ◽  
Graphene Membranes

ABSTRACTWe have studied the mechanical properties of suspended graphene membranes using molecular dynamics (MD) and generalized continuum elasticity theory (GE) in order to develop and assess a continuum description for graphene. The MD simulations are based on a valence force field model which is used to determine the deformation and the elastic energy of the membrane (EMD) as a function of external forces. For the continuum description, we use the expression Econt = Estretching + Ebending for the elastic energy functional. The elastic parameters (tensile rigidity and Poisson ratio) entering Econt are determined by requiring that Econt = EMD for a set of deformations.Comparisons with the MD results show excellent agreement. We find that the elastic energy of a supported graphene sheets is typically dominated by the nonlinear stretching terms whereas a linear description is valid only for very small deflections. This implies that in some applications, i.e. NEMS, a linear description is of limited applicability.

scholarly journals Heteroclinic connections and Dirichlet problems for a nonlocal functional of oscillation type

2021 ◽  
Author(s):  
Annalisa Cesaroni ◽  
Serena Dipierro ◽  
Matteo Novaga ◽  
Enrico Valdinoci
Keyword(s):  
Lagrange Equation ◽  
Discontinuous Solution ◽  
Energy Functional ◽  
Dirichlet Problems ◽  
Piecewise Constant ◽  
Local Oscillation ◽  
External Data ◽  
Regularity Properties ◽  
Classical Setting ◽  
Nonlocal Functional

AbstractWe consider an energy functional combining the square of the local oscillation of a one-dimensional function with a double-well potential. We establish the existence of minimal heteroclinic solutions connecting the two wells of the potential. This existence result cannot be accomplished by standard methods, due to the lack of compactness properties. In addition, we investigate the main properties of these heteroclinic connections. We show that these minimizers are monotone, and therefore they satisfy a suitable Euler–Lagrange equation. We also prove that, differently from the classical cases arising in ordinary differential equations, in this context the heteroclinic connections are not necessarily smooth, and not even continuous (in fact, they can be piecewise constant). Also, we show that heteroclinics are not necessarily unique up to a translation, which is also in contrast with the classical setting. Furthermore, we investigate the associated Dirichlet problem, studying existence, uniqueness and partial regularity properties, providing explicit solutions in terms of the external data and of the forcing source, and exhibiting an example of discontinuous solution.

The Correlated Basis Function Method and its Application to Liquid 3He, Nuclear Matter and Neutron Matter

1975 ◽  
Vol 30 (8) ◽  
pp. 923-936
Author(s):  
J. Nitsch
Keyword(s):  
Correlation Function ◽  
Nuclear Matter ◽  
Lagrange Equation ◽  
Expansion Method ◽  
Liquid Structure ◽  
Energy Functional ◽  
Basis Functions ◽  
Neutron Matter ◽  
Expectation Values ◽  
Energy Expectation

Abstract The method of correlated basis functions is studied and applied to the Fermi systems: liquid 3 He, nuclear matter and neutron matter. The reduced cluster integrals xijkl... and so the sub-normalization integrals Iijkl... are generalized to coinciding quantum numbers out of the set {i, j, k, I,...}. This generalization has an important consequence for the radial distribution function g (r) (and then for the liquid structure function) ; g(r) has no contributions of the order O (A-1). For 3 He the state-independent two-body correlation function g(r) is calculated from the Euler-Lagrange equation (in the limit of r → 0) for the unrenormalized two-body energy functional. For nuclear matter and neutron matter we adopt the three-parameter correlation function of Bäckman et al. Then the energy expectation values are calculated by including up to the three-body terms in the unrenormalized and renormalized version of the correlated basis functions method. The experimental ground-state energy and density of liquid s He can be well reproduced by the present method with the Lennard-Jones-(6 -12) potential. The same method is applied to the nuclear matter and neutron matter calculations with the OMY-potential. The results of the energy expectation values indicate a practical superiority of the unrenormalized cluster expansion method over the renormalized one.

scholarly journals Sphere-valued harmonic maps with surface energy and the K13 problem

2019 ◽  
Vol 12 (4) ◽  
pp. 363-392
Author(s):  
Stuart Day ◽  
Arghir Dani Zarnescu
Keyword(s):  
Liquid Crystals ◽  
Surface Energy ◽  
Boundary Conditions ◽  
Critical Point ◽  
Partial Regularity ◽  
Harmonic Maps ◽  
Nematic Liquid Crystals ◽  
Energy Functional ◽  
Dirichlet Energy ◽  
Surface Term

AbstractWe consider an energy functional motivated by the celebrated {K_{13}} problem in the Oseen–Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional surface term. It is known that this energy is unbounded from below and our aim has been to study the local minimisers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity of the global minimisers.

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