# Tagged Questions

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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### Rademacher complexity of regularized linear function class: does it depend on dimension or not?

I am going through some lecture notes on Learning Theory here: http://ttic.uchicago.edu/~tewari/LT_SP2008.html trying to learn about Rademacher complexities. I'm getting confused about the Rademacher ...
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### Hausdorff dimension of closed intervals is not changed under $f(x)$

Let $f(x)=x^2$. Prove that for any $E\subseteq\mathbb{R}$ the dimension of image is not changed i.e $$\dim_HE=\dim_H(f(E))$$ Any set in $\mathbb{R}$ can be represnted as a countable union of ...
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### What does it mean for an integral to “vanish”?

I had a question; What does it mean for an integral to vanish in complex analysis? There is supposedly something, which says if the integral "vanishes," the sum of the residues is 0. But what does ...
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### How to prove that $\lim_{u \downarrow 1} (u-1) \zeta(u) =1$?

I would like to prove that $$\lim_{u \downarrow 1} (u-1) \zeta(u) =1 \quad .$$ However, I am not sure which form of the Riemann-zeta function I ought to pick in order to compute this limit. I ...
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### Baby Rudin Problem 2.29

Here's is Prob. 29 in the Exercises following Chap. 2 in PRINCIPLES OF MATHEMATICAL ANALYSIS by Walter Rudin, 3rd edition: Prove that every open set in $\mathbb{R}^1$ is the union of an at most ...
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### How prove this usefull identities with equations

let $f(x):R\to R$ be $C^{k+1}$, show that ...
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### an injective map can not take several intersecting arcs onto line segment

I read a result in the theory of harmonic mappings, and i think it might be true in general setting as well. But i am unable to get a proof of this. Can anyone help me with proving it. The statement ...
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### Checking for a function to be homeomorphism

Let $h=(h_1,h_2):U\to \mathbb{R}^2$ be one-one and continuous in a neighborhood $U$ of the origin in $\mathbb{R}^2$ with $h(0)=0$ and $h_1$ harmonic. Then $h:U\to h(U)$ is bijective and continuous. If ...
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### Intergral involving arcus sinus and square root

While computing certain triple integral, I met the following one which I was unable to compute. I would be greatful for any suggestions: $$\int x\arcsin{\frac{\sqrt{2-x^2}}{x}} \,dx$$
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### Primitive of an analytic function - Proof verification

Check out the proof for the following corollary. This is only the first part of the proof and I have a issue with this. $F(z)$ is defined by integrating $f$ along a line segment. But isn't it ...
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### Another type of derivative, another type of differential equation

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Is it possible to find a continuous function $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so ...
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### How to define an explicit bijection from P(N) to 2^N [closed]

How do I define an explicit bijection between the power set of N and $2^N$ with $2^N =\{f|f:N\to\{0,1\} \text{ is a function} \}$?
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### Prove that these two definitions are equivalent

While answering this question I have used that $$\sin x=\displaystyle\sum_{n=0}^\infty\dfrac{(-1)^nx^{2n+1}}{(2n+1)!}$$ Nwe my question is that how can it be shown that the ...
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### elementry properties of closure

Definition: Let X⊂R and let x'∈R, we say that x' is an adherent point of X iff ∀ε>0 ∃x∈X s.t.d(x′,x)≤ε. the closure of X is denoted as \overline(X) and is defined to be the set of all the adherent ...
I'm proving the statement of some limit which has a form of $$\lim_{\|\mathbf{m}\| \to \infty} f(m_{1},m_{2},\cdots,m_{k}) = S$$ where $\mathbf{m} = (m_{1}, \cdots, m_{k}) \in \mathbb{R}^{k}$. I've ...