Mathematics

Questions tagged [linearization]

On the many different ways to turn non-linear systems of equations into linear ones.

384 questions
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What is the 'linearization' of a PDE?

Specifically I am looking at the proof of Lemma 4.1 on page 9 here , where the graphical form of curve shortening flow is given, and then its 'linearization'. I am struggling to find any online resources that explain what this means, and what the…
jl2
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Set a variable the max of two other variables

Suppose I have two real variables, $A$ and $B$ , and another one $C$ . I want to store the max between $A$ and $B$ in $C$ for a problem I am modeling. I can neither use a max function nor multiply variables. What can I do?
Magic code
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Lyapunov stability question from Arnold's trivium

V.I. Arnold put the following question in his Mathematical Trivium : Can an asymptotically stable equilibrium position become unstable in the Lyapunov sense under linearization? It puzzled me for a while, since my experience doesn't include such a…
H. M. Šiljak
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Noether's theorem in the critical heat equation

I am watching a serie of lectures on "Blow up solution for the energy critical heat equation" from Monica Musso on YouTube and at some point she states a result I do not understand. Let met recall the setting. We are studying the following system:…
Falcon
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Linearization of scalar curvature: $DR|_g(h)=-\Delta_g(\mathrm{tr}_g h)+\mathrm{div}_g(\mathrm{div}_g h)-\langle\mathrm{Ric}_g,h\rangle_g$

I'm working on an exercise from Geometric Relativity by Dan A. Lee, but things didn't go well: Following Lee's hint, I was trying to use Exercise 1.12 and view $\color{red}{g}$ as the background metric ( $\bar g$ in Exercise 1.12). In Exercise 1.12,…
Boar
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Formulate the Least Norm (Using ${L}_{1}$ Norm) Problem as Linear Programming Problem

Problem Statement Show how the $L_1$ -sparse reconstruction problem: $$\min_{x}{\left\lVert x\right\rVert}_1 \quad \text{subject to} \; y=Ax$$ can be reduced to a linear program of form similar to: $$\min_{u}{b^Tu} \quad \text{subject to} \; Gu=h,…
p.koch
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Converting sum of infinity norm and L1 norm to linear programming

So I'm trying to convert this minimization problem, min $\parallel Ax-y \parallel_{\infty}$ + $\parallel x \parallel_{1}$ where $A$ is $m$ by $n$ , $y$ is $m$ by $1$ and $x$ is $n$ by $1$ . into a linear program. My attempted solution is to rewrite…
Eagle1992
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What is the derivative of $\ddot x = x + A \dot x^2$ with respect to $A$?

Find the derivative of the solution of the equation $\ddot x = x + A \dot x^2$ , with initial conditions $x(0) = 1, \dot x(0) = 0$ , with respect to the parameter $A$ for $A = 0$ . -- Vladimir Arnold This is a great ODE question posed by Vladimir…
SRobertJames
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Proper linearization of ODEs of the form $\dot{x}(t) + f(x(t)) + \sigma(t) = 0$?

For a scalar ODE of the form $$\dot{x}(t) + f\left(x(t)\right) = 0 \label{1}\tag{1}$$ where $f \colon \mathbb R \to \mathbb R$ is some smooth function admitting a unique root $x^*$ such that $f(x^*) = 0$ . The linearization of \eqref{1} is…
Fei Cao
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How to write boolean expressions as linear equations

I want to convert a set of boolean expressions to linear equations. In some cases, this is easy. For example, suppose $a, b, c$ $\in$ {0,1}. Then if the boolean expression is: $a$ $\ne$ b, I could use the linear equation $a + b = 1$. To give a more…
user66360
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What is the significance of the linearization of a non-linear PDE?

This may be too general a question so please let me know if I need to make it more specific. I am a first year graduate student in PDEs, and as such have not had much exposure to non-linear PDEs. I am starting to look at research papers, in many of…
jl2
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Understanding a proof about Riemannian metrics in three dimensions always being diagonalizable

I've recently been working through Deturck's and Yang's Existence of elastic deformations with prescribed principal strains . First and formost, I'm interested in it's proof that Riemannian metrics can in three dimensions always be diagonalized,…
moran
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Linearization of Gradient Flow

As someone who has only "theoretical" knowledge in Riemannian geometry, I have a hard time trying to wrap my head around how to actually compute the so called "linearization" of a gradient flow on a manifold. The setting is: We have a symplectic…
Emilia
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Example for Carleman Linearization resulting in a linear system

The Carleman linearization came to my attention due to this article . I tried to understand this method but so far i wasn't succesful i tried to understand page 39 of this presentation however the example didn't make sense for me. Could someone…
worldsmithhelper
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Newtons method for finding reciprocal

Define a function 1 which is $f_1(x)=a-1/x$ and function 2 which is $f_2(x)=1-ax $ If I set both to zero I am looking for when $x=1/a$ as the root using Newtons method. When I do this I get two different answers however and they should surely both…
user129299
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