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        Questions tagged [real-analysis]

        Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.

        0
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        0answers
        26 views

        Sub-Gaussian decay of the measure of Euclidean balls

        Let $X$ be a random vector in $\mathbb{R}^d$ satisfying the following property: there exists $C_1,C_2>0$ such that $$\int_0^{+\infty}\mathbb{P}(\|X-\mu_0\|\leq\sqrt{t})\exp(-t)dt\leq C_1\exp(-C_2\|\...
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        vote
        1answer
        60 views

        Relation between the measures of two sets defined via Lebesgue integration

        I posted this question on StackExchange, people have upvoted it but I have not received any response. I read up here that it is okay to post unanswered StackExchange questions on Mathoverflow. So, ...
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        votes
        0answers
        31 views

        Energy-minimizing set of discrete points in a bounded domain

        Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain. Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize $$ \sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|} $$ over all ...
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        vote
        1answer
        46 views

        Subadditive function with special growth

        Related to one of my previous question (for which I have received an answer, thanks) I have the following new one. Maybe I am describing the empty set but not being a specialist at all of the domain I ...
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        votes
        1answer
        176 views

        Is this sequence convergent? [on hold]

        suppose $\exists S \subset \mathbb{R}$ and a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x_0 \in S $ the sequence $x_{n+1} = f(x_n)$ converge to $x \in S$ now, let $\alpha \...
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        votes
        0answers
        48 views

        Can we approximate this matrix field with an invertible matrix field?

        Let $\mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\}$ be the closed unit disk, and set $$\begin{equation*} A(x,y)=\left( \begin{array}{cc} x & -y \\ y & x \end{array} \right) \end{...
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        votes
        1answer
        136 views

        Is $π:\mathcal{C}^∞(M,N)→\mathcal{C}^∞(S,N)$, $π(f)=f|_S$ a quotient map in the $\mathcal{C}^1$ topology?

        This question was previously posted on MSE. Let $M, N$ be smooth connected manifolds (without boundary), where $M$ is a compact manifold, so we can put a topology in the space $\mathcal C^\infty(M, N)...
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        185 views

        An implication of the Zagier et al result on the hyperbolicity of Jensen polynomials for the Riemann zeta function?

        In their paper recently published in the PNAS, Zagier et al demonstrated that The Jensen polynomials $J_{\alpha}^{d,n}(X)$ of the Riemann zeta function of degree $d$ and shift $n$ are hyperbolic for ...
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        votes
        0answers
        57 views

        Find limit of sequence defined by sum of previous terms and harmonics

        I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function? I tried to ...
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        votes
        1answer
        178 views

        Bounding higher derivatives of $f(x) = 1/(1+x^2)^r$

        Let $r\in \lbrack 0,\infty)$. Define $f(x) = 1/(1+x^2)^r$. It would seem to be the case that $$|f^{(k)}(x)|\leq \frac{2r \cdot (2r+1) \dotsb (2r + k-1)}{(1+x^2)^{r + k/2}}$$ for all even $k\geq 0$. ...
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        votes
        2answers
        297 views

        Is the composition of two nowhere differentiable functions still nowhere differentiable?

        Let $f,g:\mathbb R\to\mathbb R$ be two continuous but nowhere differentiable functions. By the Denjoy–Young–Saks theorem for almost every point $x_0\in\mathbb R$ one has $$ \limsup\limits_{x\to x_0}\...
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        vote
        0answers
        29 views

        Convergence acceleration of a series by using optimal parameters

        One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ...
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        1answer
        333 views

        A maximization problem

        Consider the minimization problem described this paper. Let $f_{\lambda}$ be the minimizer. As a part of extending my work, I am able to show the following facts $$\lim_\limits{\lambda \to 0}\|f_{\...
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        votes
        0answers
        44 views

        Differential Eq [closed]

        Ok my first try of explaining my problem was not that understandable, I'll give it another try Suppose you have n persons sitting in a circle, next to each other. At time t0 everyone starts clapping ...
        -2
        votes
        1answer
        91 views

        Continuity of the Restriction Map Between Function Spaces [on hold]

        Let $X,Y,Z$ be Hausdorff spaces and suppose that $Z\subset X$. Endow $C(X,Y)$ and $C(Z,Y)$ with the compact-open topologies and define the map $\rho$ as \begin{align} \rho:&C(X,Y)\rightarrow C(Z,...

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