Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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2
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What is the norm of the pre-multiplication by a fixed matrix operator?

Let $A \colon= \left(\alpha_{ij} \right)_{m\times n}$ be a given $m \times n$ matrix of complex numbers, and let the operator $T \colon \mathbb{C}^n \to \mathbb{C}^m$ be defined by $$T(x) \colon= Ax ...
6
votes
1answer
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Nonlinear heat equation $u_{t} = \Delta(u^{4})$

Consider the nonlinear heat equation $u_{t} = \Delta(u^{4})$ in $\{x \in \mathbb{R}^{3}: |x| < 1\}$ with $u = 0$ on $\{x \in \mathbb{R}^{3}: |x| = 1\}$. The problem I am working on is to show that ...
0
votes
0answers
69 views
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estimate an equality on sine function

I want to prove the following: For any given $\epsilon>0$, there exists a $\delta>0$ such that for any fixed $0<\theta<\frac{\pi}{2}$ with $\frac{\pi}{2}-\theta<\delta$, there exists ...
1
vote
0answers
42 views
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Consider the equation: $x' = f(t,x)$. Prove that there is a two-way correspondence between the initial and the limits of the solutions.

Consider the equation: $$x' = f(t,x)$$ wherein, $$|f(t,x)| \leq \phi(t)x, \forall(t,x) \in \mathbb{R}\times \mathbb{R} $$ $$ \int^{\infty}_a\phi/(t)< \infty$$ where $a \in \mathbb{R}$. If in ...