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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.

2,650 questions
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Let $X$ be a random vector in $\mathbb{R}^d$ satisfying the following property: there exists $C_1,C_2>0$ such that $$\int_0^{+\infty}\mathbb{P}(\|X-\mu_0\|\leq\sqrt{t})\exp(-t)dt\leq C_1\exp(-C_2\|\... 1answer 60 views Relation between the measures of two sets defined via Lebesgue integration I posted this question on StackExchange, people have upvoted it but I have not received any response. I read up here that it is okay to post unanswered StackExchange questions on Mathoverflow. So, ... 0answers 31 views Energy-minimizing set of discrete points in a bounded domain Let \Omega \subset \mathbb{R}^3 be a smooth, bounded domain. Let x_1,\ldots,x_n \in \overline{\Omega} be chosen so as to minimize$$ \sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|} $$over all ... 1answer 46 views Subadditive function with special growth Related to one of my previous question (for which I have received an answer, thanks) I have the following new one. Maybe I am describing the empty set but not being a specialist at all of the domain I ... 1answer 176 views Is this sequence convergent? [on hold] suppose \exists S \subset \mathbb{R} and a function f : \mathbb{R} \rightarrow \mathbb{R} such that \forall x_0 \in S  the sequence x_{n+1} = f(x_n) converge to x \in S now, let \alpha \... 0answers 48 views Can we approximate this matrix field with an invertible matrix field? Let \mathbb{D}^2=\{ x \in \mathbb{R}^2 \, | \, |x| \le 1\} be the closed unit disk, and set$$\begin{equation*} A(x,y)=\left( \begin{array}{cc} x & -y \\ y & x \end{array} \right) \end{...
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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 $\mathcal C^\infty(M, N)... 0answers 185 views An implication of the Zagier et al result on the hyperbolicity of Jensen polynomials for the Riemann zeta function? In their paper recently published in the PNAS, Zagier et al demonstrated that The Jensen polynomials$J_{\alpha}^{d,n}(X)$of the Riemann zeta function of degree$d$and shift$n$are hyperbolic for ... 0answers 57 views Find limit of sequence defined by sum of previous terms and harmonics I came across this sequence as part of my work. Could someone indicate me the methodology I should follow to solve it? I guess it involves harmonic numbers and/or the digamma function? I tried to ... 1answer 178 views Bounding higher derivatives of$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$k\geq 0$. ... 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 $$\limsup\limits_{x\to x_0}\... 0answers 29 views Convergence acceleration of a series by using optimal parameters One of the ways of accelerating the convergence of a series is by transforming into a faster series using optimal parameters. Examples of this approach can be found in this paper. I obtained a ... 1answer 333 views A maximization problem Consider the minimization problem described this paper. Let f_{\lambda} be the minimizer. As a part of extending my work, I am able to show the following facts$$\lim_\limits{\lambda \to 0}\|f_{\... 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 ... 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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