All Questions
Tagged with perturbation-theory real-analysis
33
questions
0
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40
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Convergence rate of eigenvectors for perturbed matrices
Let $f(s)= (f_{1}(s),\ldots,f_{d}(s)), s \in \mathbb{R}^d, \textbf{o} \le s \le \textbf{1}$, be vector of probability generating functions, were each entry has finite third moments. Consider matrix of ...
- 1
3
votes
1
answer
57
views
Finding a counterexample of approximating the solution to $x'=f(x,t)$ by $x_{n+1}'=f(x_n,t)$
Suppose we have a Cauchy problem
$$x'=f(x(t),t),\quad x(t_0)=C,$$
where $f$ is Lipschitz continuous in its first argument and continuous in its second argument.
By Picard theorem, there must exist a ...
- 61
4
votes
2
answers
91
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A fluke or deep reason for this nice result: rigorously identify derivative from algebraic perturbation
I came across a cool result and was wondering whether what I saw is a fluke or has a deeper reason. Long story short: can we find derivatives with algebraic perturbations?
Consider the algebraic ...
- 457
1
vote
1
answer
39
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How to perturb nonzero elements to get nonzero elements
Let $x \in \mathbb{R}^n$ be such that $x_i\neq 0$ for all $i\in \{1,\dots,n\}$. How can we perturb $x$ so that the perturbed vector $y$ has the same property, i.e., $y_i\neq 0$ for all $i\in \{1,\dots,...
- 175
1
vote
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82
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Perturbed Gram Matrix
Let $x_t \in \mathbb{S}^{d-1}$, $\forall t\in \mathbb{N}$ and let $e_1$ be the first cannonical basis vector of $\mathbb{R}^d$, ie, $e_1 = (1,0,\cdots,0)$. Let us form a Gram Matrix
$$\sum_{t=1}^T(...
- 477
0
votes
0
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286
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Taylor expansion for a Bessel function with complex argument
If we have a Bessel function of the first kind and the $m$-th order as $J_{m}(x+i\epsilon x)$, where $m$ is integer, $x, \epsilon$ are real and $\epsilon$ is a small parameter ($0<\epsilon\ll 1$), ...
- 1,309
7
votes
1
answer
250
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How to Taylor expand a scalar quantity about a unit sphere?
Consider a vector field $\mathbf{v} (r,\theta)$ expressed in the system of (axisymmetric) spherical coordinates with $r$ denoting the radial distance and $\theta$ the polar angle.
We consider a ...
1
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0
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38
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Fractional form that approximately defines exponential function
How can we prove that, in the limit of $|1/y| \leq |c|$, where $c,y$ are generally complex, the following approximation holds?
$$ \frac{1-yc}{1+yc} \rightarrow e^{-2/cy} $$
My attempt: if we call $...
- 1,309
1
vote
0
answers
28
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Conjecture on openness of an analytic mapping
Consider a real analytic function $g: \mathbb{R}^m \rightarrow \mathfrak{M}_{\mathbb{R}}(n, k)$ to the set of $n \times k$ matrices where $k \ge n + 1$ such that $\forall x \in \mathbb{R}^m \quad \...
- 155
2
votes
1
answer
59
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Endpoint Perturbation Theory
So, suppose we want to evaluate the integral
$$\int_{a}^{b+\epsilon c}f(x)\, dx$$
where $f:\mathbb{R}\to\mathbb{R}$ is assumed to be smooth and regular in the integration region $[a,b+\epsilon c]\...
- 334
2
votes
0
answers
83
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Inverse of Sum of Matrices
Show that $(A+B)^{-1} -A^{-1} = A^{-1}\sum_{k=1}^{\infty}({BA^{-1}})^{k}$ goes to zero as $B$ goes to zero.
This leads rise to another question of mine, can I say that, if $||BA^{-1}|| \leq r < 1$,...
- 964
0
votes
1
answer
26
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Are approximate minimisers the minimisers of a perturbed function?
Suppose we have a convex function $f(x)$ defined on some compact convex set $X \subset \mathbb R$ with minimiser $x^* \in X$. Without loss of generality $X$ contains the origin.
We run some algorithm ...
- 10.4k
1
vote
1
answer
49
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ODE satisfied by the error vector in differential geometry
Consider a control system of the form $\dot x(t) = f(x(t),u(t))$ where $x$ takes his values in $\mathbb{R}^n$ and $u$ in $\mathbb{R}^m$.
Here is what I read:
Let $x^*$ be a reference curve ...
- 2,062
1
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0
answers
96
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Analytic non-zero eigenvalues of two-parameter (periodic) family of positive semi-definite hermitian matrices
I am aware of results about eigenvalues of one-parameter families of Hermitian matrices sharing analytic properties with the entries of the matrix, and how this result does not extend to the two-...
- 374
2
votes
2
answers
2k
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Find the two term asymptotic expansion of the solution
Find the two term asymptotic expansion of the solution of
$$
1 +\sqrt{x^2 + \epsilon} = e^x
$$
My approach: i tried solutions of the form $x = x_0+\epsilon ^ \alpha x_1$ and plug it in. Then using ...
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