All Questions
5,659 questions
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Elementary calculus estimate or not?
Does there exist a constant $C>0$ such that for all $f \in H^3(\mathbb R)$
$$\int_{\mathbb R} \vert x f''(x) \vert^2 \ dx \le C \int_{\mathbb R} \vert f'''(x) \vert^2 + \vert x^3f(x) \vert^2 + \...
user69109
5
votes
2
answers
925
views
How to show this symmetric function inequality
Question: let $x_{i}>0$ $(i=1,2,\cdots,n)$, such that $x_{i}\neq x_{j},\forall i\neq j$, find all real numbers $p$ that satisfy the following inequality
$$\sum_{i=1}^{n}\dfrac{x^p_{i}}{\prod_{j\neq ...
- 4,280
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votes
5
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623
views
Elementary inhomogeneous inequality for three non-negative reals
I need the following estimate for something I am working on, but I don't immediately see how to establish it.
For $x, y, z \in \mathbb{R}_{\ge 0}$, show that
$$2xyz + x^2 + y^2 + z^2 + 1 \ge 2(xy + yz ...
- 543
5
votes
3
answers
5k
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Zeros of "exponential" function
Define ${f}_{i}(x) = \sum_{j=1}^{i} (-1)^{i-j}{i \choose j}j^x$, where $i=1,2,3,...$ and $x \in \mathbb{R}$.
For integer $x \geq i$, ${f}_{i}(x)$ reduces to ${f}_{i}(x)=i!S(x,i)$, where $S(x,i)$ is ...
- 2,619
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votes
4
answers
589
views
Looking for a reference on conformal mapping on $\Bbb R^n$
A mapping $T: \Bbb R^n\to \Bbb R^n$ is said to be conformal if it is bijective and preserves angles, i.e.,
if $x, y: [0,1]\to \Bbb R^n$ are curves with $x(t_0)=y(t_0)$ then
$$\cos (Tx(t_0),Ty(t_0))= \...
- 1,101
5
votes
2
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875
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}...
- 43.2k
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5
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Cardinality of Equivalence Classes of Cauchy Sequences
What's the cardinality of a single equivalence class of Cauchy sequences in ℚ?
To clarify, I'm not asking for the cardinality of the real numbers, but for the cardinality of the set of Cauchy ...
- 153
5
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2
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3k
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Are there functions which are neither convex nor concave everywhere but are continuous? [closed]
By convex/cave I mean by the definition for an interval $(x,y)$ of $f$ is convex iff $f(\frac{a+b}{2})\geq\frac{f(a)+f(b)}{2}$ and is concave if $f(\frac{a+b}{2})\leq\frac{f(a)+f(b)}{2}$ where $a,b\in(...
- 675
5
votes
2
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1k
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Derivatives of $C^{\infty}$ non analytic function
Question: Given $f\in C^{\infty}$ which is not analytic on a bounded domain $\Omega \subseteq \mathbb{R}$. What can we say about the sequence $\lbrace f^{(m)} \rbrace _{m=1}^{\infty} $? Specifically - ...
- 3,574
5
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2
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957
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Dirichlet's approximation only using prime power as denominator
I am not sure whether this is a suitable question for MO. We know the classical version of Dirichlet's approximation theorem that if $x$ is a real number and $Q>0$ there exist $p,q\in \mathbb{Z}$ ...
- 1,692
5
votes
3
answers
1k
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Non-continuous differentiability for differential forms
Generally when working with differential forms, one assumes that they are continuously differentiable, i.e. $C^r$ for some $1\le r \le \infty$. Under this hypothesis, one can define the exterior ...
- 66.8k
5
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2
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363
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Can one show that $(-1)^{n-1} {(1/\zeta)}^{(n)}(x) >0$ for all real $x>1$?
Is it true that $(-1)^{n-1} {(1/\zeta)}^{(n)}(x) >0$ for all real $x>1$ ?
Or in other words can you show that the higher order derivatives of the reciprocal of the Riemann zeta function ...
- 178
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1
answer
1k
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Analytic functions where all derivatives vanish at infinity and which are bounded
Let $C_0(\mathbb{R})$ denote the analytic functions $f : \mathbb{R} \rightarrow \mathbb{R}$.
I wonder whether there a functions $f \in C_0(\mathbb{R})$ with $f \neq 0$, such that there is a constant $...
- 749
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2
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560
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Compute $ \int_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx $ [closed]
How can I compute this integral?
$$
\int_{0}^{+\infty} \left( \frac{\ln(x)}{e^x}\right)^2 dx
$$
- 71
5
votes
5
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1k
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What are the local maxima and minima of $\frac{\sin(nx)}{\sin(x)}$
FYI: I asked this question here couple of days ago but got no answer yet.
$n$ is an integer
We know the global maximum of the function $\sin(nx)/\sin(x)$ is $n$ (thanks to this question), but what are ...
- 113
5
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4
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362
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Dual norm of a subspace of $\ell_\infty^3$
We define a norm on $\mathbb C^2$ as $\|(\alpha,\beta)\|:=\max\left\{|\alpha|,|\beta|,\big|\frac{\alpha+\beta}{\sqrt{2}}\big|\right\}.$ Can the dual norm be calculated explicitly?
- 1,879
5
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1
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410
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Is there always a real $x$ such that $\cos n_1 x + \cos n_2 x + \cos n_3 x < -2$?
Problem: Given three positive integers $0 < n_1 < n_2 < n_3$. Is there always a real number $x$ such that
$$\cos n_1 x + \cos n_2 x + \cos n_3 x < -2?$$
- 1,053
5
votes
2
answers
503
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Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$
Look at the expression
$$
f(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1.
$$
The numbers $x_1,x_2,x_3$ are non-negative, and I assume that $x_1+x_2+x_3=3$. This is a sum of squares and "...
- 2,207
5
votes
3
answers
772
views
Arzelà–Ascoli for equi-Lebesgue continuous functions
Given a measurable subset $A$ of $[0, 1]$, a sequence of functions $f_n: [0, 1] \to \mathbb R$ is said to be equi-Lebesgue continuous on $A$ if for every $x \in A$, and $\varepsilon > 0$, there ...
- 6,273
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1
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2k
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A rather curious equality: is this true?
I came across (coincidentally) two integral evaluations, which seem to agree according to numerical tests. It did not seem easy to convert one into the other.
QUESTION. Is this true?
$$\int_0^1\...
- 43.2k
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2
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1k
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real analyticity, Fourier coefficients [duplicate]
Question. Suppose $f$ is periodic in $[0,2\pi]$. What conditions on the Fourier coefficients of $f$ would guarantee real analyticity of $f$? Please provide me with a reference.
- 43.2k
5
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3
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1k
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Non-continuous higher differentiability
The standard definition is that a function $f:\mathbb{R}^n\to \mathbb{R}$ is differentiable at a point $x$ if there exists a linear map $\mathrm{d}f_x: \mathbb{R}^n \to \mathbb{R}$ such that
$$f(x+h) ...
- 66.8k
5
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3
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630
views
If the Fourier coefficient $\hat{f}(k)$ of $f\in C^1(\mathbb T)$ is zero for all $|k|<N$, then $\|f\|_{L^\infty}\leq \frac CN \|f'\|_{L^1}$?
Let $f\in C^1(\mathbb T)=C^1(\mathbb R/\mathbb Z)$ be a function such that
$$\hat f(k):=\int_{\mathbb T}f(x)e^{-2\pi ikx}\,dx=0,\qquad \forall k\in\{-N+1,\cdots,-1,0,1,\cdots, N-1\}.$$
Do we have $\|f\...
- 517
5
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2
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319
views
Nowhere negative polynomials form a semialgebraic set
Let $P_{d, n}$ be the space of polynomial maps $\mathbb{R}^n\to \mathbb{R}$ of degree at most $d$.
Is the subset $S\subset P_{d, n}$ of nowhere negative polynomials semialgebraic?
- 71
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2
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708
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Approximation of Hölder continuous functions "from below"
We assume that we have a $\alpha$-Hölder continuous function $f$ on an interval $[0,1]$ with $f(0)=0$.
I am wondering if there exists an explicit construction of a sequence $f_{n} \in C_c^{\infty}(\...
5
votes
2
answers
352
views
Locating the maximum point $x_n$ of $f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)$ in $(0,1)$
I am trying to observe the behavior of $x_n \in (0,1)$ defined such that the function
\begin{equation}
f_n(x):=e^{-1/x}\Bigl(1+\frac{1}{n^2 x^n} \Bigr)
\end{equation}
attains its maximum inside the ...
- 3,477
5
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3
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164
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Reference for a Grünwald–Letnikov-type definition of the $n$-th derivative of a function
Let $U\subset\mathbb R$ be an open set. Let $n\in\mathbb N$ and suppose that $f\in\mathcal C^n(U)$, i.e. that $f$ is $n$-times continuously differentiable on $U$. The $n$-th derivative of $f$, denoted ...
- 1,326
5
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2
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421
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Inequality of two variables
Let $a$ and $b$ be positive numbers. Prove that:
$$\ln\frac{(a+1)^2}{4a}\ln\frac{(b+1)^2}{4b}\geq\ln^2\frac{(a+1)(b+1)}{2(a+b)}.$$
Since the inequality is not changed after replacing $a$ on $\frac{1}{...
- 1,054
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2
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1k
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An example of an open discontinuous function
Consider the following simple example of a function $f: \mathbb{R}\to\mathbb{R}$ which is open and discontinuous at all points. If $x\in\mathbb{R}$ is represented as something.$x_1x_2x_3\dots$ in the ...
- 1,897
5
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1
answer
903
views
Uncountable Pre-Image
I've been reading about space filling curves, and been asking myself this question.
If $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a continuous open map, is it true that $\forall x \in$ range$(f)$ ...
- 53
5
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1
answer
279
views
Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?
Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $:
$$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...
- 541
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1
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234
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Can a continuous bounded variation function be $C^0$-reparametrized to be continuously differentiable?
Let $f: [0, 1] \to \mathbb R$ be a function of bounded variation. We say that $g$ is a $C^0$ reparametrization if $g = f \circ s$ for $s$ a continuous increasing bijection from a finite interval $I$ ...
- 6,273
5
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1
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362
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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 $k\geq 0$. ...
- 20.2k
5
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4
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496
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Integral of the distance function to the boundary of a planar set
I have been stuck for a few days in a seemingly harmless question.
Given a simply connected open set $\Sigma\subset\mathbb{R}^2$, with smooth boundary $\partial\Sigma$, I am interested in estimating
$...
- 165
5
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2
answers
1k
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Stone-Weierstrass for monotone functions
Let $\; f : [0,1] \to \mathbb{R} \;$ be continuous and non-decreasing. $\;\;$ Let $\epsilon$ be a real number such that $\; 0 < \epsilon \;$.
Does it follow that that there exists a real ...
user5810
5
votes
2
answers
564
views
Stone-Weierstrass without the "subalgebra" condition
Suppose I consider $C_0(\mathbb{N})$ consisting of function on the natural numbers vanishing at $\infty$. For an irrational $1<\alpha<2$, let $p_{m\alpha}(\cdot)$ be the function $p_{m\alpha}(n)=...
- 161
5
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2
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788
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Do non-zero derivatives imply tangent lines (and vice versa)?
Let $\gamma : \mathbb{R} \rightarrow \mathbb{R}^2$ be any continuous function, with image given by $C_\gamma$.
We can say that $\gamma$ has an image tangent at $t \in \mathbb{R}$ if there exists $\...
- 700
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3
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2k
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Fourier transform of periodic distributions
Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (...
- 595
5
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2
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565
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Geometry of Level sets of elliptic polynomials in two real variables
Updated:
A polynomial $P(x,y)\in \mathbb{R}[x,y]$ is called an elliptic polynomial if its last homogeneous part does not vanish on $\mathbb{R}^2\setminus\{0\}$.The two answers to this post provide a ...
- 356
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3
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637
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Uniqueness of minimizers in a problem in the Calculus of Variations - Part II
Take any convex set $A\subset\mathbb{R}^n$ which contains a neighborhood of the origin and let $f_A$ be the associated Minkowski functional
$$
f_A(x) = \inf\{\lambda>0\mid x\in\lambda A\},
$$
which ...
- 391
5
votes
1
answer
599
views
Extending continuous functioms defined on the irrationals
Lavrentieff proved a Theorem which implies that every real valued continuous function defined on a dense subset $D\subseteq \mathbb R$ admits a continuous extension to some $G_\delta $ subset of $\...
- 2,263
5
votes
1
answer
958
views
Does a nonlinear additive function on R imply a Hamel basis of R?
A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of ...
- 4,597
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1
answer
3k
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Are all topological (finite-dim) real vector spaces homeomorphic to a coordinate space?
I know that all real, finite-dimensional topological vector spaces are isomorphic to $\mathbb{R}^n$ for some $n$, but are they also homeomorphic?
The reason I'm asking this is because I was wondering ...
- 3,079
5
votes
2
answers
3k
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Closedness of finite-dimensional subspaces
Is the (algebraic) span a finite set of vectors in a Hausdorff topological vector space over a complete field always closed?
I suspect yes, but I can't come up with a proof, and it seems like locally ...
user5810
5
votes
1
answer
351
views
Does the Poincaré inequality hold on annular domains?
Does the following Poincaré inequality hold
$$\int_{B_{r_2}\setminus B_{r_1}} |f-\bar{f}|^2 dx \leq C(r_2-r_1)^2 \int_{B_{r_2}\setminus B_{r_1}} |\nabla f|^2 dx,$$
where $B_r$ denotes a ball of radius ...
- 537
5
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1
answer
674
views
Is this infinite product entire?
Let $(z_i)$ be a square-summable sequence which is even summable but not absolute summable, i.e. $\sum_{i=1}^{\infty} \vert z_i \vert = \infty$,$\sum_{i=1}^{\infty} \vert z_i \vert^2 < \infty$ and $...
- 73
5
votes
1
answer
727
views
Which continuous, differentiable a.e. functions have $f’(x) = f(x)$ a.e.?
Question:
Consider the set of continuous, differentiable a.e. functions from $\mathbb R \to \mathbb R$. Can we characterise the subset of these that satisfy $f’(x) = f(x)$ for almost every $x \in \...
- 6,273
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1
answer
2k
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Question on an exercise from Terry Tao's blog
I've been reading Tao's An introduction to measure theory, a draft can be found here. An exercise from it is
Exercise 30 (Rising sun inequality) Let ${f: {\bf R} \rightarrow {\bf R}}$ be an absolutely ...
- 95
5
votes
2
answers
1k
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Baire class 1 and discontinuities
Is it true that a bounded real function $f:[0,1]\to[0,1]$ with only countably many discontinuities has to be of Baire class 1, that is pointwise limit of a sequence of continuous functions? Is there a ...
- 4,459
5
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1
answer
680
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When does this interesting sum diverge?
For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$
I don't know of any references or methods for this -- not even for $x=1$, for which the ...
- 3,443