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

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5 views

Definition of $C^{m,k}$-capacity of a point

I have come across the following notation and a new term $C^{m,k}$-capacity of a point. I'd appreciate some reference, where I can find the definition and relevant theory.
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0answers
27 views

Nth root of a being greater than 1 when a is greater than 1 [on hold]

In a part of a bigger proof, we need to prove that when $a > 1$ then $\sqrt[n]{a} > 1$. We were told to do it using the binomial theorem, so I tried to do it like this. If $a > 1$ then $a = 1 ...
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0answers
35 views

Do closed bounded intervals with a common end point non overlap each other? [on hold]

Source: Introduction to Real Analysis, Fourth Edition, Robert G. Bartle, Donald R. Sherbert, Page 199 In the above extracted text, intervals $I_1, I_2, …, I_n$ are said to be non-overlapping, but ...
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0answers
56 views

the limits of series [on hold]

let $S_n=\sum^n_{i=1} \frac{3 i^2}{4^i-1}$ $$\frac{3 i^2}{4^i-1} \leq \frac{3 i^2}{3^i}=\frac{i^2}{3^{i-1}}$$ put $A_i=\ln \frac{i^4}{3^{i-1}}$ $$A_i=4 \ln i- (i-1)\ln 3 $$ for $i>1$ $A_i=\frac{1}{...
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0answers
34 views

Finding bounding in a formula [on hold]

Here's my solution, where did I make a mistake? The formula: My solution:
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0answers
46 views

Can we apply L'Hopital's rule with respect parameter $u$? [on hold]

If we know that $\int_1^\infty g(x,t_o)dx=0$ and $\int_1^\infty h(x,t_o)dx=0$ $ t $ is real variable here $$f(x,t_o)=\lim_{t \rightarrow t_o} \frac{\int_1^\infty g(x,t)dx}{\int_1^\infty h(x,t)dx}= ...
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1answer
100 views

A specific problem on : Can bounding the Sobolev norm, bound a higher derivative?

Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{...
8
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0answers
69 views

Borel-Ecalle re-summation and resurgence: Criteria and Results

This is about the theory of Borel-$\acute{\textrm{E}}$calle re-summation and resurgence, see Refs below. This states that the perturbative series (say of the vacuum expectation value of an operator $\...
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0answers
51 views

$L^{\infty}-L^{\infty}$ estimates on the Schrödinger evolution

A classical estimate for solutions to Schrödinger equations are $L^1-L^{\infty}$ estimates, also known as dispersive estimates. I wonder whether there are also $L^{\infty}-L^{\infty}$ estimates? ...
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1answer
137 views

Counter example about blow-up solution of DEs

Let $f(\cdot)$ be a continuously differentiable function over $\mathbb{R}$, and $u\in L^2_{loc}(0,\infty)$, $a\in \mathbb{R}$, and $x(t)$ solves the integral of $$\dot{x}(t)=ax(t)+f(x(t))+u(t), \quad ...
3
votes
1answer
223 views

Constant “periodization” of a function

Let $w$ be a rapidly decaying function on $\mathbb{R}$ such that $$ \sum_{n \in \mathbb{Z}} w(x+n) = 0$$ for all $x \in \mathbb{R}$. Does that imply that $w$ is identically zero? What if we assume ...
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0answers
34 views

Probability estimate with a Lipschitz, weak* semicontinuous function on the $\ell^\infty$ unit ball

Suppose that $X_i$ for $i=0,1,\dots$ is an i.i.d. sequence of uniformly distributed random variables taking on values in $[-1,1]$. Fix a real number $L>0$ and suppose that $f_n:[-1,1]^n\rightarrow [...
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2answers
197 views

Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ and distinct pairwise distances?

Short version of question. Is there a set $S\subseteq [0,1]$ with $|S|=2^{\aleph_0}$ such that all points of $S$ have distinct pairwise distances? Formal version of question. If $X$ is a set, let $[X]...
9
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1answer
174 views

The discrete Hardy-Littlewood-Sobolev inequality

Let $p>1$, $q>1$, $0<\lambda<1$ be such that $\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that $(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$. It is known ([1,2,3]...
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0answers
133 views

Function of two sets intersection

Let $U$ be the set of all subsets of $[0,1]$ that are a union of finitely many closed intervals (not allowing intervals that are single points). Does there exist a function $f:U\times U\rightarrow U$ ...
23
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4answers
888 views

show this nice and hard inequality with $ \prod_{i=1}^{n}|x_{i}-y_{i}|<e^{\frac{n}{2}}$

I saw the following results in a book. The author said it was not difficult to prove how I felt it was difficult to prove, so I asked here. The result comes from a book that has no electronic version....
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1answer
97 views

Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence?

Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
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2answers
213 views

Bounded deformation vs bounded variation

Let $BV(\mathbb R^n; \mathbb R^n)$ be the space of (vector-valued) functions of bounded variation and let $BD(\mathbb R^n;\mathbb R^n)$ the space of functions with bounded deformation. They are made ...
3
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1answer
217 views

Is there any Menelaus-type theorem for polynomials?

Consider $n+3$ polynomials of degree $n$, say $P_1(x) , ... P_{n+3}(x)$. In addition, consider that there are distinct (it is not necessary that all of them be distinct) numbers $x_{ij}$ for $1 \...
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3answers
1k views

Adventure with infinite series, a curiosity

It is easily verifiable that $$\sum_{k\geq0}\binom{2k}k\frac1{2^{3k}}=\sqrt{2}.$$ It is not that difficult to get $$\sum_{k\geq0}\binom{4k}{2k}\frac1{2^{5k}}=\frac{\sqrt{2-\sqrt2}+\sqrt{2+\sqrt2}}2.$$ ...
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0answers
48 views

Extension of a derivation on w [duplicate]

Let I be a closed left ideal of a Banach algebra A such that A has a bounded approximate identity and I just has a right approximate identity. let $D:I\to I^*$ be a derivation. Does $D$ extend to a ...
3
votes
1answer
103 views

Log concavity of the maximum of dependent Gaussians

Let $Z_1,\dots,Z_n$ be dependent Gaussian random variables. Is it true that $X=\max\{Z_1,\dots,Z_n\}$ has a log-concave distribution function? This is true for the independent case, but is it true in ...
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0answers
61 views

The Poisson equation

I see the following theorem in Lihe Wang's A geometric approach to the Calderon--Zygmund estimates $$\triangle u=f,in \> B_2 \>(1)$$ Lemma 7: There is a constant $N_1$ so that for any $ε > ...
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0answers
39 views

Convergence to the probability generating function of a Poisson process

I'm working currently with a Poisson process trying to proove Renyi's Theorem, so far I want to show that $\prod_{i=1}^{k_n}[z + (1-z)e^{-\mu(A_{n_i})}] \to e^{-(1-z)\mu(A)}$ as $\mu(A_{n_i}) \to 0$, ...
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2answers
146 views

A min-max approximation

Let $n\ge 1$ be an integer, $\mathcal P_n$ be the vector space of all polynomial functions over $[a,b]$, of degree at most $n$. My question is : Is it true that $$\inf_{x_0,x_1,...,x_n\in[a,b], x_0&...
2
votes
1answer
87 views

Removable set for Sobolev space

It is well known that if $\Omega\subset\mathbb{R}^{N}$ open, $F\subset\Omega$ closed, such that $\mathcal{H}^{N−1}(F)=0$,where $\mathcal{H}^{N−1}$ denotes (N-1) dimensional Hausdorff measure, then $W^{...
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4answers
644 views

How does the parity of $n$ affect the properties of $\mathbb{R}^n$? [closed]

Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?
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0answers
34 views

Defining Boundary Conditions for Spline Interpolation via the Euler–Maclaurin Formula

The Euler–Maclaurin formula states an interdependency between \begin{align} I\quad:=&\quad\int_m^nf(x) \, dx;\ m,n\in\mathbb{Z}\\[6pt] S\quad:=&\quad\sum_{k=m}^n f(k) \\[6pt] D\quad:=&\...
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0answers
75 views

Lebesgue density theorem for “doubling uniformly covering collections of subsets”

I am looking for a version of Lebesgue density theorem that works when restricting to "good" collections of balls with respect to (not necessarily doubling) metric measure spaces. Specifically Let $(...
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1answer
50 views

Does differentiating an integro-differential equation results in equivalent stability of the solution?

I have a dynamical system in the form of an integro-differential equation which I want to analyze in terms of stability. To demonstrate my problem consider the following integro-differential equation: ...
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1answer
41 views

Strict positive type function on hypersurface also of positive type in neighborhood?

Let $u\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ be symmetric and of strictly positive type on some hypersurface $S \subset \mathbb{R}^n$ diffeomorphic to $\{0\}\times\mathbb{R}^{n-1}$. This means ...
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1answer
173 views

Bounded growth of functions vs bounded growth of functions on countable sets

I am wondering if the boundedness of growth can be characterized by sequences. I am not sure if I use the term "growth" correctly, or use the correct tags for this question. Here is what I mean. Let $...
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0answers
78 views

Does this sequence have a convergence subsequence?

Let $w_n\in C([0,\tau];L^2(\Omega))$, and $\Omega$ be an open bounded set of $\mathbb{R}^2$. For every $t\in [0,\tau]$, $w_n(t)$ has a convergent subsequence in $L^2(\Omega)$. Does the sequence $$\...
3
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2answers
213 views

Infinite sum of reciprocals of squares of lengths of tangents from origin to the curve $y=\sin x$

This question is actually from MSE. I had to post it here due to the lack of response there even after placing a bounty. Here goes the question Let tangents be drawn to the curve $y=\sin x$ from ...
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0answers
96 views

Are $C^1$ immersions dense in $C^1$?

Let $M$ be a closed compact manifold. Is the space of all $C^1$ immersions from $M$ to $\mathbb{R}^m$ ($m> \dim M$) dense in $C^1(M; \mathbb{R}^m)$ (in the $C^1$ topology)?
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27 views

Existence of solutions to NLS: Local existence and boundedness

I was wondering when the following argument is valid: Consider a nonlinear Schrödinger equation $$i \partial_t \varphi = -\Delta \varphi+ N(\varphi)$$ where $N$ is a nonlinearity. Often it is ...
1
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1answer
80 views

Optimal estimate in trace norm

Let $x,y$ be vectors of some Hilbert space of unit length. Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$ Assume then that we know that $\left\lVert x-...
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2answers
591 views

Searching for a proof for a series identity

The below identity I have found experimentally. Question. Is this true? If so, may you provide a "slick" (or any) proof. $$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
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0answers
34 views

On different norms of the interpolating operator

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
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1answer
64 views

Find $p$ s.t. there is a sequence of nodes in $[0,1]$ s.t. sequence of interpolating polynomials of every continuous function converges in $p$-norm

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
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1answer
161 views

Strong estimates for the zeta function on natural numbers

Let $$\zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s}$$ be the Riemann zeta function (here we just consider real $s$). We do have a description given by $$\zeta(s) = \frac{s}{s-1}-s\int_{1}^\infty \frac{...
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1answer
196 views

Background on the functional equation $F(x+1)+F(x)=f(x)‎$ [closed]

In the theory of indefinite sums, anti-differences and finite calculus, ‎the following ‎difference ‎functional ‎equation ‎and ‎its ‎solutions ‎are ‎very ‎important: ‎$$‎\bigtriangleup ‎F(x):=F(x+1)-...
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1answer
86 views

Quotient with positive second derivative in the limit?

I am studying the quotient of $$f(\varepsilon) = \sum_{i=1}^{\infty} \frac{i^2}{2^{\varepsilon i^2}}$$ and $$g(\varepsilon) = \sum_{i=1}^{\infty} \frac{1}{2^{\varepsilon i^2}}$$ for some $\...
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0answers
87 views

Injectivity of product functions on natural number sequences

Let $M = \{ a = (a_i)_{i} : a_i \in \mathbb{N}, a_1 \geq 2, a_i > a_j \forall i>j\}$ the set of all ascending natural number sequences, with $a_1$ at least 2. We now define for each $k \geq 2$ ...
3
votes
1answer
158 views

Given any sequence of interpolating nodes, can we find a continuous function $f$ whose interpolating polynomials doesn't converge to $f$ point-wise

Let $[a,b]$ be an interval in real line . Given any function $f:[a,b]\to \mathbb R$ and set $A \subseteq [a,b]$ of size $n+1$, there exists a unique polynomial $p_{f,A,n}(x)$ of degree $n$ such that $...
3
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0answers
186 views

Eudoxus real numbers

I recently remembered the eudoxus construction of the real numbers. Does anyone know what how the rational numbers $\mathbb Q$ can be characterised inside this construction? Clearification: The ...
2
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1answer
129 views

Chain rules for Dini Derivative

Could someone provides some references for the chain rule concerning Dini derivatives. For example, let $f(\cdot) \in \mathcal{C}^1\left( \mathbb{R} ; \mathbb{R}\right)$, and $g(\cdot) \in \mathcal{C}\...
3
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0answers
66 views

Anderson Localization and Homogenization theory

I originally asked this on Mathematics Stack Exchange but realized it might be better to ask it here. The question is mostly related to homogenization theory in mathematical physics. $\textbf{...
2
votes
1answer
285 views

Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$

I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples? This seems to be a well-known result, but I can ...
6
votes
2answers
443 views

Summing Bernoulli numbers

Consider the Bernoulli numbers denoted by $B_n$, which are rational numbers. It is known that the harmonic numbers $H_n=\sum_{k=1}^n\frac1k$ are not integers once $n>1$. I am curious about the ...