Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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Solution to a certain moment problem

I'm looking for a function $f$ that satisfies $f(x)\geq0$ $\int f(x) \mathrm{d}x=1$ $\int xf(x) \mathrm{d}x=0$ $\int x^2f(x)\mathrm{d}x=1$ $\int x^4f(x)\mathrm{d}x=\delta$ $\int ...
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Two ODEs, why is one solution the solution of the other?

This question is based on Zeidler II/B, Problem 30.2. Consider the ODE: find $u:[0,T] \to \mathbb{R}^n$ s.t. $$u'(t) = F(t,u(t))$$ $$u(0) = u_0$$ given $F:[0,T]\times \mathbb{R}^n \to \mathbb{R}^n$ ...
10
votes
0answers
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Linear differential equations of the $n$th order

$$ L(x)=x^{(n)}+a_1(t)x^{(n-1)}+\cdots +a_{n-1}(t)x'+a_n(t)x;\qquad a_1(t),a_2(t),\ldots\in C$$ $$U_j(\varphi)= \sum_{k=0}^{n-1}(M_{jk} \varphi^{k}(\alpha)-N_{jk} \varphi^{k}(\beta))= \gamma_j\quad ...
4
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Proving the continuity of these maps

Backstory: I am having an exam soon, and these are the assignments that keep coming up, I cannot finish any of them to the end, but have ideas about solving them, and would like to hear your thoughts ...
5
votes
2answers
110 views
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Real Induction Over Multiple Variables?

I've seen in several different places* that one can use normal mathematical induction to prove the truth of a statement that relies not on just one variable (say, $x$,) but multiple variables (for ...