# Questions tagged [contraction-operator]

Use this tag for questions about operators whose norm is at most 1.

102 questions
1answer
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### Is log-sum-exp a contraction

For $x \in \mathbb{R}^n$, the log-sum-exp (lse) function $\mathbb{R}^n \rightarrow \mathbb{R}$ is defined as $lse(x) = \tau \log \sum_{i=1}^n \exp(\frac{x_i}{\tau})$, where $\tau > 0$ is a ...
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### I want to understand the triangle inequality about contraction operator

I read the Chow's paper, Multigrid algorithms and complexity results, https://dspace.mit.edu/handle/1721.1/14254 I have a question on page 42. Let me write the Lemma 2.4.3 on the page. Lemma 2.4.3 ...
1answer
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### Variant of the Contraction Mapping Theorem

Let (C, ||.||) be a closed subset of a Banach space, and let f: C -> C be a mapping such that ||f(x) - f(y)|| < ||x - y|| for all x, y in C. Must there exist a fixed point in C that f maps to ...
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### Using the contraction mapping principle for the solution of an ODE

Suppose we want to solve a simple ODE $$\dot{x}=2t(1-x)=f(t,x)\qquad (1)$$ with initial condition $x_0=2$ using the contraction mapping principle. I showed that $f$ is uniformly Lipschitz continuous ...
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### Using Banach fixed-point theorem to show existence and uniqueness of the following system of linear equations

Consider the following system of linear equations : $$\begin{cases}2x_1 + x_2 = 5, \\ x_1 - 5x_2 = -3.\end{cases}$$ Using basic linear algebra, it is easy to show that $x_1 = 2$ and $x_2 = 1$ is the ...
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### Contraction and $\max$ function

$f: \Bbb R \mapsto \Bbb R$ $g: \Bbb R \mapsto \Bbb R$ $h: \Bbb R \mapsto \Bbb R$ $h:=\max\{f(x), g(x)\}$ Is $h$ a contraction on $\Bbb R$ if $f$ and $g$ are both so? First attempts of ...
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### Tensor contraction computation

I am currently trying to implement tensors as multidimensional arrays in C++, which is why i am asking this with regards to algorithmic computability. I would like to implement the contraction of two ...
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### Example of a nowhere differentiable contraction mapping

The Weierstrass function https://en.wikipedia.org/wiki/Weierstrass_function is a pathological example of a continuous nowhere differentiable function.. Since a conttaction mapping is necessarily ...
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### Picard's method does not solve first order differential equation?

I have the following first order differential equation $$x^\prime(t)=-(x(t))^2+2x(t),\quad t\geq 0,\quad x(0)=1$$ Now I want to obtain an approximation of $x(t)$ by using Picard's method. Then the ...
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### can someone explain me in simpler way of what this paper has conveyed

I do not have a thorough knowledge of measure-theoretic probability and Markov chain but I would start to learn by myself soon, but for few research-related works, I have to understand the theme of ...
1answer
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### Solve differential equation via successive approximations (contraction mapping principle)

I'm working on a problem (self study) to solve the differential equation $$\frac{dx}{dt} = tx^2 + x^3$$ using successive approximations within a neighborhood interval of $\delta < 1/5$. I'm given ...
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### Finding a contraction of a transformation by Contraction Mapping Theorem

I want to demonstrate that this transformation is a contraction: $$Tx(t)=\frac{1}{2}\left (x^2(t) + 1 -t^2 \right )$$ In the closed set $S$: $$S=\{x\in C([-1,1; \mathbb{R} ]):\Vert x\Vert_0 \leq 1\}$$ ...
1answer
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