The negative gradient flow for the L 2 -integral of Ricci curvature

2003 ◽  
Vol 111 (2) ◽  
pp. 163-186 ◽  
Author(s):  
Zheng Yu
Keyword(s):  
Ricci Curvature ◽  
Gradient Flow ◽  
Negative Gradient

scholarly journals Convergence of the Yang–Mills–Higgs flow on Gauged Holomorphic maps and applications

2018 ◽  
Vol 29 (04) ◽  
pp. 1850024
Author(s):  
Samuel Trautwein
Keyword(s):  
Uniqueness Theorem ◽  
Gradient Flow ◽  
Holomorphic Section ◽  
Holomorphic Maps ◽  
Yang Mills ◽  
Stable Case ◽  
Finite Dimensional ◽  
Closed Riemann Surface ◽  
The Moment ◽  
Negative Gradient

The symplectic vortex equations admit a variational description as global minimum of the Yang–Mills–Higgs functional. We study its negative gradient flow on holomorphic pairs [Formula: see text] where [Formula: see text] is a connection on a principal [Formula: see text]-bundle [Formula: see text] over a closed Riemann surface [Formula: see text] and [Formula: see text] is an equivariant map into a Kähler Hamiltonian [Formula: see text]-manifold. The connection [Formula: see text] induces a holomorphic structure on the Kähler fibration [Formula: see text] and we require that [Formula: see text] descends to a holomorphic section of this fibration. We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the [Formula: see text]-topology when [Formula: see text] is equivariantly convex at infinity with proper moment map, [Formula: see text] is holomorphically aspherical and its Kähler metric is analytic. As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang–Mills–Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet’s Kobayashi–Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment–weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.

scholarly journals Numerical Simulation of Physical Transportation Network

2021 ◽  
Vol 2 (1) ◽  
Author(s):  
Alexander Liang
Keyword(s):  
Energy Consumption ◽  
Biological Networks ◽  
Transportation Network ◽  
Circulatory System ◽  
Gradient Flow ◽  
Electric Circuit ◽  
Transportation Systems ◽  
Pipeline Network ◽  
Flow Method ◽  
Negative Gradient

Prior experiments with physarum polycephalum, a plasmodial slime mold, have shown its inherent biological ability to adapt to the surroundings. When placed in an environment with food surrounding it, it builds a network of biological pipelines to the food that reinforces heavily used pipelines and withdraws those rarely used. Interestingly, the human body displays a similar ability. The development of our circulatory system is in the absence of our brain or nervous system, which suggests humans mimic low level organisms in the regard that our biological transportation systems are self-regulating and self-constructing. Like slime molds, this adaptive quality suggests that our biological networks follow a method of lowest energy consumption: energy is allocated to maintaining and constructing only the most efficient paths. Thus, the purpose of our model is to explain the biological mechanism of how organisms construct a pipeline network through a mathematical model. The model is developed using 2 main assumptions: (1) The network exists in a hexagonal grid system and (2) there only exists nodes that are sinks, sources, or neither. Borrowing Kirchhoff’s Law and other electric circuit principles, we determine an energy function for the construction and maintenance of a biological pipeline network in our model dependent on the current, conductivity, and pressure within the pipelines. The method we develop in this paper is called the negative gradient flow method. Given any initial values for the pipeline, the negative gradient flow method is an algorithm for unconstrained nonlinear optimization that finds at each instant the next set of values to most minimize the function until eventually reaching the minimizer values. Accomplishing the main goal of this paper, the negative gradient flow improves upon existing models at replicating natural phenomena since it is based on the evolutionary advantage of lowest energy consumption.

On a class of parametric Neumann problems with indefinite and unbounded potential

2015 ◽  
Vol 27 (3) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
George Smyrlis
Keyword(s):  
Critical Point ◽  
Morse Theory ◽  
Critical Point Theory ◽  
Gradient Flow ◽  
Point Theory ◽  
Three Solutions ◽  
Neumann Problems ◽  
Unbounded Potential ◽  
All Solutions ◽  
Negative Gradient

AbstractWe consider a parametric Neumann problem with an indefinite and unbounded potential. Using a combination of critical point theory with truncation and comparison techniques, with Morse theory and with invariance arguments for a suitable negative gradient flow, we prove two multiplicity theorems for certain values of the parameter. In the first theorem we produce three solutions and in the second five. For all solutions we provide sign information. Our work improves significantly results existing in the literature.

scholarly journals The convergence of a numerical method for total variation flow

2021 ◽  
Vol 15 ◽  
pp. 174830262110113
Author(s):  
Qianying Hong ◽  
Ming-jun Lai ◽  
Jingyue Wang
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Method ◽  
Convergence Analysis ◽  
Total Variation ◽  
Iterative Algorithm ◽  
Finite Difference Scheme ◽  
Gradient Flow ◽  
Time Dependent ◽  
Total Variation Flow

We present a convergence analysis for a finite difference scheme for the time dependent partial different equation called gradient flow associated with the Rudin-Osher-Fetami model. We devise an iterative algorithm to compute the solution of the finite difference scheme and prove the convergence of the iterative algorithm. Finally computational experiments are shown to demonstrate the convergence of the finite difference scheme.

scholarly journals Role of social integration in the protection against depression in the Chilean elderly population

2020 ◽  
Vol 30 (Supplement_5) ◽  
Author(s):  
L de Albuquerque Araújo ◽  
N Bello Escamilla ◽  
V Sabando Franulic
Keyword(s):  
Mental Health ◽  
Older Adults ◽  
Social Integration ◽  
Old Age ◽  
Protective Factor ◽  
Depression Scale ◽  
Mental Illnesses ◽  
National Study ◽  
Close Relatives ◽  
Negative Gradient

Abstract Chile has improved survival however this process occurs under a framework of socioeconomic and gender inequalities, which results in an impact of mental health, especially in vulnerable groups. The objective was to determine the association between depression and social integration in Chilean older adults. Cross-sectional study from the National Study of Dependence in Old Person 2010. The risk of depression was amount with Geriatric Depression Scale (>5). The social integration were quantity as frequency of meeting with close relatives (child, partner, daughter/ son-in-law, grandchildren); with other relatives (brother, brothers-in-law, nephews or other relatives) and with friends and neighbors in the last 12 months in 5 categories (never visit; less frequently; 1-2 times a month; 1-2 times a week; every day or almost). Logistic regression models considered the sampling design of the survey to identify association with odd ratio (OR) (never as reference category), adjustment for sex, age, ethnicity, household income, education, housing arrangement and chronic diseases (p ≤ 0,5). Total of 4179 older adults 25,3% reported risk of depression, the significative association with close relatives was in daily or almost frequency OR:0.42 (95%CI 0.27-0.67), 1-2 times a week OR: 0,57 (95%CI 0,33-0,99), 1-2 times/month OR: 0,56 (95%CI 0,99); other relatives were lower frequency OR: 0.54 (95%CI 0.38-0.78); 1-2 times/month OR; 0.50 (95%CI 0.31-0.81); 1-2 times/week OR:0.35 (95%CI 0.22-0.55); daily or almost OR:0.27 (95%CI 0.18-0.42). And meeting with friends and neighbors in the same frequency order were OR: 0.66 (95%CI 0.44-0.99); OR:0.43 (95%CI 0.26-0.73); OR:0.4 (95%CI 0.25-0.62); OR: 0.32 (95%CI 0.21-0.47). There is a negative gradient between depression and the frequency of meeting with friends, neighbors and family, independent of sociodemographic and health characteristics. Social integration must be promoted as a protective factor of mental health in elderly. Key messages Depression is one of the most common mental illnesses in old age and we found a negative gradient between the frequency of meeting friends, neighbors and family and the possibility of depression. It seems essential for public health to have strategies that address social life in old age to strengthen quality of live and mental health.

scholarly journals Relating a Rate-Independent System and a Gradient System for the Case of One-Homogeneous Potentials

2021 ◽  
Author(s):  
Alexander Mielke
Keyword(s):  
Gradient Flow ◽  
Careful Analysis ◽  
Flow Equation ◽  
Uniqueness Result ◽  
Gradient System ◽  
Independent System ◽  
Exact Relation ◽  
Flow Equations ◽  
Total Variation Flow ◽  
Energetic Solutions

AbstractWe consider a non-negative and one-homogeneous energy functional $${{\mathcal {J}}}$$ J on a Hilbert space. The paper provides an exact relation between the solutions of the associated gradient-flow equations and the energetic solutions generated via the rate-independent system given in terms of the time-dependent functional $${{\mathcal {E}}}(t,u)= t {{\mathcal {J}}}(u)$$ E ( t , u ) = t J ( u ) and the norm as a dissipation distance. The relation between the two flows is given via a solution-dependent reparametrization of time that can be guessed from the homogeneities of energy and dissipations in the two equations. We provide several examples including the total-variation flow and show that equivalence of the two systems through a solution dependent reparametrization of the time. Making the relation mathematically rigorous includes a careful analysis of the jumps in energetic solutions which correspond to constant-speed intervals for the solutions of the gradient-flow equation. As a major result we obtain a non-trivial existence and uniqueness result for the energetic rate-independent system.

scholarly journals Genetic recombination as a generalised gradient flow

2021 ◽  
Author(s):  
Frederic Alberti
Keyword(s):  
Arbitrary Number ◽  
Genetic Recombination ◽  
Reaction Network ◽  
Gradient Flow ◽  
Mass Action ◽  
Gradient System ◽  
Gradient Structure ◽  
Law Of Mass Action ◽  
Reversible Chemical Reaction ◽  
Time Partitioning

AbstractIt is well known that the classical recombination equation for two parent individuals is equivalent to the law of mass action of a strongly reversible chemical reaction network, and can thus be reformulated as a generalised gradient system. Here, this is generalised to the case of an arbitrary number of parents. Furthermore, the gradient structure of the backward-time partitioning process is investigated.

Sign in / Sign up

Export Citation Format

Share Document