# Questions tagged [perturbation-theory]

The perturbation-theory tag has no usage guidance.

99
questions

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### Hopf normal form of 2D system – How to calculate parameter $\beta$?

In the Scholarpedia article on the Hopf bifurcation, it is discussed that any generic (nondegeneracy and transversality) 2D system exhibiting a Hopf bifurcation has a locally topologically equivalent ...

7
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2
answers

160
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### Finding $\theta$ such that at least one eigenvalue of $A(\theta)$ is real

Is there a known method to find a set of $\theta$ such that at least one eigenvalue of $A(\theta)$ is purely real?
Assume $A(\theta)$ is a real square matrix whose elements are linear functions of a ...

0
votes

1
answer

98
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### Local differentiability of eigenvalues and eigenvectors of a real symmetric matrix

Let $A(x)\in\mathbb{R}^{n\times n}$ be a real symmetric matrix depending on the point $x\in\mathbb{R}^n$, where the eigenvalues are not necessarily simple. Can we say that for all $x$ there exists an ...

0
votes

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117
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### Could variable be still function in x and y after performing Reynolds averaging over area

All,
Let $S(x,y,t)$ be a variable function in $x$, $y$, and $t$. After
performing Reynold averaging over area $\frac{1}{A}\int S(x,y,t) dA$, could $S$ still be a function in $x$, and $y$?
Equations (1-...

4
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1
answer

202
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### Asymptotics of integral representation of distribution

I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community.
Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...

2
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0
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118
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### Choosing the derivative of a flow

I am looking for something like the Franks' Lemma for flows. The celebrated Franks' Lemma states that: Let $f:M \rightarrow M$ be a $C^1$ diffeomorphism and $S=\{p_1,...,p_k\}$ be a finite set of ...

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73
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### Is there an infinite combinatorics of common transseries expansions?

By now there is a rich understanding of generating functions in combinatorics, and the way that operations in power series are 'shadows' of richer constructions on combinatorial objects. This lifting ...

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21
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### Diagonal of the difference of two orthogonal matrices who are close to the identity matrix

Suppose that $\mathbf{A} = \mathbf{I}_{n} + \mathbf{E}_{1}$ and $\mathbf{B} = \mathbf{I}_{n} + \mathbf{E}_{2}$ are two orthogonal matrices where $\mathbf{E}_{1}$ and $\mathbf{E}_{2}$ are two small ...

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36
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### Small perturbation to a commuting family of hermitian matrices will hurt the nice properties?

Let $A_1, \dotsc A_N$ be a collection of finite Hermitian matrices that commute with one another and all have the matrix $2$-norm as $1$. Here $N$ is large but fixed.
Then, they are simultaneously ...

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69
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### Perturbation theory for $UV^*$ in singular value decomposition

There is a fair amount of research into perturbation theory for singular value decompositions (e.g. Liu et al's 'First-Order Perturbation Analysis of Singular Vectors
in Singular Value Decomposition' ...

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### Is there a treatment to relate the Multiple scale analysis or scale separation to the usage of the CFT especially in the perturbation?

In the recent years one started to think weather the abstract group treatment to the Multiple-scale analysis and the scale separation could be uniformly obtained. Especially, one attempted to search ...

2
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58
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### Bounding eigenvalue/eigenspace perturbations for hermitian matrices

Let $H$ be a Hermitian $n \times n$ matrix. Let $V$ be another such matrix.
For real $t$, let us consider the one-parameter family
$$ H(t) = H + t V$$
of Hermitian matrices.
Kato's perturbation theory ...

2
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0
answers

54
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### Is there a generalisation of this perturbation result about rank-one modifications of diagonal matrices?

In Theorem 1 of [1] we have the following result: Let $D$ be a real $n \times n$ diagonal matrix and consider the rank-one modification $C = D + \rho z z^T$, where $\rho > 0$ is a real scalar and $...

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2
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192
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### Solution of parabolic partial differential equation using singular perturbation method

Consider the following parabolic partial differential equation (PDE)
\begin{align}
\label{eq:42}
\left(\cos\psi \frac{\partial}{\partial r} + \frac{\gamma}{r} \sin\psi \frac{\partial}{\partial \psi} + ...

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133
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### Essential spectrum under perturbation

Given a Banach space $X$ and a bounded linear operator $T$ on $X$.
It's well known that the essential spectrum of $T$ is invariant under additive compact perturbation.
My question is about minimal ...

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208
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### Eigenvalue perturbation under sparse perturbations

Let $A \in \{0,1\}^{n \times n}$ be an irreducible matrix whose entries are in $\{0,1\}$, and let $\lambda_1(A)$ be the eigenvalue with the largest magnitude. By Perron–Frobenius theorem, we know that ...

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107
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### Is Sun's spectral variation bound for normal matrices optimal?

In On the variation of the spectrum of a normal matrix, Sun proves the following result (Corollary 1.2):
Let $A$ be an $n$-square normal matrix and $B$ an arbitrary $n$-square matrix. Then $$ \min_{\...

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118
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### Seeking a precedent – two-stage Gaussian integration?

Sometimes, by iteration, linear algebra can be used to solve non-linear equations. For example, consider the system
$$Ax=a \qquad B(x)y=b(x), $$
where $a$ is a vector with scalar entries, $A$ is a ...

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2
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446
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### Lipschitz continuity of eigenvalues and eigenvectors of Hermitian matrices

It is well-known that the eigenvalues (in decreasing order) of a Hermitian matrix $A$ are Lipschitz continuous functions of $A$.
Do there exist orthonormal eigenvectors that vary in a Lipschitz ...

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166
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### Proof (or reference) about $\lambda_i(A+\epsilon e_je_j^*) = \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2).$

I'm looking for a proof (or a reference in a textbook) about the fact that
$$
\lambda_i(A+\epsilon e_je_j^*) =_{\epsilon \to 0} \lambda_i(A) + \epsilon |v_{i,j}|^2 + O(\epsilon^2),
$$
where $A$ is a ...

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133
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### Reference for action-angle coordinates [closed]

Does anyone know a good reference to start studying Action-Angle coordinates?
Thank you in advance !

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68
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### Perturbative approach starting from a probability distribution approximated form

I approximate a probability distribution $P_x(x)$ with a $P_x^{app}(x)$,
such that $P_x(x)-P_x^{app}(x) = O(\epsilon)$ uniformly in x, where epsilon is a small positive quantity.
Consider the generic ...

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42
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### Stability in coefficients for the rescaled radiative tranport equation

One form of the radiative transport equation is as follows:
$$ v\cdot \nabla_x u + \left(\epsilon \sigma_a(x) + \frac{1}{\epsilon}\sigma_s(x)\right) u - \frac{1}{\epsilon}\sigma_a(x)\int_{S^{n-1}} p(v,...

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72
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### Sum of positive self-adjoint operator and an imaginary "potential": literature request

To keep things simple, let us consider the following: $L$ is a positive, unbounded S.A. operator on $L_2(\mathbb{R},f(x))$, where $f(x)$ is a Gaussian. Assume that we know the spectrum and ...

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81
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### Is a basis that almost diagonalizes a matrix 'close' to its eigenbasis?

Let $A\in\mathbb{R}^{n\times n}$ be diagonalisable (over $\mathbb{C}$) with pairwise distinct eigenvalues $\lambda_1,\ldots,\lambda_n\in\mathbb{C}$, and suppose that
$$\tag{1}S^{-1}\cdot A\cdot S = \...

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### Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?

Wikipedia calls resolvent formalism a useful tool for relating complex analysis to studying the spectra of a linear operator on a Banach space. Sure, I believe you because I've seen results that use ...

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### Perturbation in the equation $u_t=\epsilon Pu$, where $P$ is an elliptic partial differential operator

Let $Pu=\sum_{ij} \partial_j(a_{ij}(x) \partial_{i} u)$ be an elliptic operator. Consider the equation
$$
(u=u_\epsilon)\\
\partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\
u(0,x)=u_0(...

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2
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166
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### Convergence of the eigenvector matrix for an analytic perturbation of a singular matrix

Let $A$ be an $n\times n$ matrix of all ones. Consider the analytic perturbation of $A$ as $$\tilde{A} = A + \epsilon H_1 + \epsilon^2 H_2 + \epsilon^3 H_3 + ... $$ All matrices are symmetric. Assume $...

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168
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### How to prove the following linearized operator is positive?

In $L^2(\mathbb{R}^d)$, let $Q$ be the solution to
\begin{equation}
-\Delta Q+\alpha^2 Q = |Q|^{2\sigma} Q,
\end{equation}
and $Q$ satisfies that it is positive, radial, and exponentially decaying (...

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1
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111
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### Perturbation of the value of a general-sum game at a equilibirium

Consider a general-sum game with $N$ players. Let $u_i(a_1, \ldots, a_N)\colon \prod_{i=1}^N A_i \rightarrow \mathbb{R} $ be the payoff of the player $i\in \{ 1, \ldots, N \}$ when each player takes ...

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49
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### Two-variable singular perturbation analysis

I am having difficulty with a two-variable singular perturbation analysis on a set of ODE's. The key difficulty is also present in the following, embarrassingly simple problem:
If $x\sim \mathcal{O}(...

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208
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### Poking into a Lie group with your finger

I consider this as a differential geometry problem. I have asked some
of my classmates who are more interested in that, and also looked into
some literature, but none of what I've found seems to help.
...

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864
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### Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have high rank?

Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth.
Suppose that $df$ is invertible outside a set of Hausdorff dimension $\le n-1$, and ...

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880
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### A toy model in 0-d QFT

Questions
For any positive integer $r$, compute $$(\frac{d}{dY})^r e^{(Y^2)}| _{Y=0}.$$ The answer should directly relates to a counting problem about Feynman diagrams.
Is there a tutorial for how ...

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### Can we perturb the Dirichlet boundary conditions to make harmonic maps locally invertible?

While analyzing a variational problem, I came to the following question:
Let $\mathbb D^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball, and let $f: \mathbb D^n \to \mathbb{R}^n$ ...

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### Coexistence of different solutions in a nonlinear matrix differential equation

I've faced a system of first-order nonlinear matrix differential equation, and I have tried to use perturbation method to approach the solutions.
The differential equation has the form:
\begin{...

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votes

1
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442
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### Understanding Gillman's proof of the Chernoff bound for expander graphs

My question is about the proof of Claim 1 in this paper: Gillman (1993).
At the end of the proof, the author says:
The matrix product $U^\top\sqrt{D^{-1}}(P+(\mathrm{e}^x-1)B(0)-\mu I)\sqrt{D}U$, ...

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### Perturbations of the eigen/singular directions

Let $A = U_A \Sigma_A V_A^\top$ and $B = U_B \Sigma_B V_B^\top$, and $A+B = U \Sigma V^\top$ be the respective singular vector decompositions.
Is there some known relationship of the form
$$\| U_A ...

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1
answer

453
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### $C^1$ perturbation of diffeomorphism is diffeomorphism?

if $f \in $ diff($M$), where $M$ is manifold, if $C^1$ perturbation $f_{\epsilon} $ of $f$ s.t. $||f_{\epsilon}-f||_{C^1} < \epsilon $.
Can we prove $f_{\epsilon} \in $ diff($M$) if $\epsilon$ is ...

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### stochastical stable

Given dynamic $f: S^1 \to S^1$ with Lebegue measure $dm$ on $S^1$. Assume it has unique SRB probability measure $\frac{d\mu_f}{dm} dm $. Given left shift space $([-\epsilon, \epsilon]^{\otimes \...

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### Determinant of a rank r perturbation

In the following paper:
Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations
on page 79, Golub et al. have the following set of equations:
$f(\lambda) = \...

3
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1
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375
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### Is the kernel of a Fredholm operator stable under perturbation?

This is a follow-up of this question.
In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator?
Let $(X,\| \cdot \|)$ be an infinite-dimensional normed vector space.
...

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### Bounds on $\|(P+\Delta)^n - P^n\|_F$ for stochastic matrices

Let us suppose that $P$ is a stochastic matrix (non-negative matrix with $P \mathbf{1} = \mathbf{1}$).
Let $\Delta$ such that $P + \Delta$ is a stochastic matrix (which means $P + \Delta$ is non-...

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### Analytic families of compact self-adjoint operators: eigenvalue extension

Suppose that $A(t), t \in \mathbb{R}$, is an analytic family of compact self-adjoint operators on a Hilbert space. The Kato-Rellich theorem says that every non-zero eigenvalue of $A(t)$ splits into ...

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### Eigenvalue Argument Perturbation

Given two square matrices $A$ and $B$. There are quite some results on the distance between the eigenvalues, e.g.,
$$
| \lambda_A - \lambda_B | \leq \| A - B \|_F,
$$
where $A$ and $B$ are Hermitian ...

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3
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609
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### Perturbation of a stochastic differential equation

Suppose we have the following two stochastic differential equations for $x_0$ and $x$ respectively
\begin{align}
dx_0 &= -k_0(t)(x_0-1)dt+\eta_0(t) x_0\,dB \tag1\\
dx &= -(k_0(t)+\epsilon ...

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votes

3
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### Differentiability of Eigenvalues - Perturbation Theory

first, I have a general question. In perturbation theory, I saw perturbations in eigenvalues and eigenvectors of square, non-symmetric matrices and the calculations were all right but no one ever ...

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73
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### Showing a modified system of quadratic equations is stable

I have and $n$ dimensional dynamical system, given by
$\dot{x} = M D(x) P x - \frac{c}{2}x$
$P$ is a full rank $n \times n$ matrix, with $p_{ij} \in [0,c]$, such that $p_{ij}=c-p_{ji}$ for some ...

2
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### Region of attraction of simple ODE with perturbation

Consider the following simplest example:
$$\dot{x} = x(x-1)(x+1)$$ $[-1,1]$ is the ROA.
Now consider the two dimensional case:
\begin{equation}
\begin{aligned}
&\dot{x} = x(x-1)(x+1)\\
&...

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### Questions on "The condition number of a randomly perturbed matrix"

This question is about the two vectors $w'$ and $y$ that are necessary for the argument in section $7$ (page 6) of this paper by Terence Tao and Van Vu,
https://arxiv.org/abs/math/0703307 (that ...