# Questions tagged [perturbation-theory]

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### How to get perturbation bounds of singular vectors

Let an adjacency matrix $A={A^\top}\in {\mathbb{R}^{n \times n}}$ (a binary matrix) of a simple undirected graph and its degree matrix $D$ be given. When adding $Q$ edges into the graph, which is ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $$\| U_A ... 2 votes 1 answer 626 views ### C^1 perturbation of diffeomorphism is diffeomorphism? if f \in  diff(M), where M is manifold, if C^1 perturbation f_{\epsilon}  of f s.t. ||f_{\epsilon}-f||_{C^1} < \epsilon . Can we prove f_{\epsilon} \in  diff(M) if \epsilon is ... 1 vote 0 answers 100 views ### stochastical stable Given dynamic f: S^1 \to S^1 with Lebegue measure dm on S^1. Assume it has unique SRB probability measure \frac{d\mu_f}{dm} dm . Given left shift space ([-\epsilon, \epsilon]^{\otimes \... 2 votes 0 answers 263 views ### Determinant of a rank r perturbation In the following paper: Restricted Rank Modification of the Symmetric Eigenvalue Problem: Theoretical Considerations on page 79, Golub et al. have the following set of equations: f(\lambda) = \... 3 votes 1 answer 413 views ### Is the kernel of a Fredholm operator stable under perturbation? This is a follow-up of this question. In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator? Let (X,\| \cdot \|) be an infinite-dimensional normed vector space. ... 3 votes 1 answer 92 views ### Bounds on \|(P+\Delta)^n - P^n\|_F for stochastic matrices Let us suppose that P is a stochastic matrix (non-negative matrix with P \mathbf{1} = \mathbf{1}). Let \Delta such that P + \Delta is a stochastic matrix (which means P + \Delta is non-... 5 votes 1 answer 280 views ### Analytic families of compact self-adjoint operators: eigenvalue extension Suppose that A(t), t \in \mathbb{R}, is an analytic family of compact self-adjoint operators on a Hilbert space. The Kato-Rellich theorem says that every non-zero eigenvalue of A(t) splits into ... 2 votes 1 answer 268 views ### Eigenvalue Argument Perturbation Given two square matrices A and B. There are quite some results on the distance between the eigenvalues, e.g.,$$ | \lambda_A - \lambda_B | \leq \| A - B \|_F,  where $A$ and $B$ are Hermitian ...
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 ...