All Questions
5,934 questions
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1
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638
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Analysis I, simpler proof of Tao's construction of the integers [closed]
In chapter 4 of Analysis I by Terence Tao, we have the following note about the set theoretic construction of the integers:
In the language of set theory, what we are doing here is starting with the ...
- 23
-3
votes
1
answer
315
views
Are the injective functions dense in $C([0,1]^n,\mathbb R^n) $?
Let $n\geq 2$. Are injective functions dense in $C([0,1]^n,\mathbb R^n) $ with the uniform norm?
- 4,074
-3
votes
2
answers
7k
views
Continuous map from $\mathbb R^2$ to $\mathbb R$? [closed]
There must be a map from $\mathbb R^2$ to $\mathbb R$, since they are the same cardinality. But is there a construction for a continuous map from $\mathbb R^2$ to $\mathbb R$?
I guess what I mean is ...
-3
votes
2
answers
317
views
When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ with $\Re(\zeta(s))\neq 0$ and $\Im(\zeta(s))\neq 0$? [closed]
When is $\Re(\zeta(s)) - \Im(\zeta(s)) = 0 $ for $0<\Re(s)<1$. Here $\zeta$ denotes the Reimann zeta function. Does the solution live on a vertical line? Or is this another coincidence when both ...
- 315
-3
votes
1
answer
200
views
Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$
The Euler equations are given as $$ \pmb{u}_t +\pmb{u}\cdot D\pmb{u} = Dp$$ $$div\mbox{ }\pmb{u} = 0$$
Where $$u = [u_1,u_2,\ldots u_n]^T$$
Now I want to rewrite these same equations but with a new ...
- 59
-3
votes
2
answers
260
views
On \ell_3 norm in R^2
Let $v,w\in\mathbb{R}^{2}$ and $v\perp w$. Is it true that $\left\Vert v\right\Vert _{3}\leq\left\Vert v+w\right\Vert _{3}$,
in which $\left\Vert \left(x,y\right)\right\Vert _{3}:=\sqrt[3]{\left|x\...
- 65
-3
votes
1
answer
361
views
Basis for space of continuous, surjective monotone functions on $\mathbb{R}$ [closed]
$\DeclareMathOperator\CM{CM}$
I recently came across Okhezin - Study of families of monotone continuous functions on Tychonoff spaces describing monotone functions on general topological spaces and I ...
- 5,405
-3
votes
1
answer
125
views
Does the Hadwiger-Nelson graph have a perfect matching?
The Hadwiger-Nelson graph on $\mathbb{R}^n$ is defined to be $(\mathbb{R}^n,E_n)$ where $$E_n = \big\{\{x,y\}: x,y\in \mathbb{R}^n \text{ and } |x-y|=1\big\},$$ where $|\cdot|$ denotes the Euclidean ...
- 48.9k
-3
votes
1
answer
77
views
Sobolev embedding [closed]
I was trying to understand Sobolev embedding, some results about this topic are not clear to me.
My question is the following:
what are the condition on $p_1 , \alpha_1, p_2 $and $\alpha_2$ for
$W^{...
- 35
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votes
1
answer
350
views
Can we find a closed form formula for this function?
I'm interested in this function
$$
h(m,n) = \operatorname{sgn}\Bigl(\lim_{w \to \infty}{\sum_{x_1=1}^{w}\dotsi\sum_{x_u=1}^{w}\dfrac{1}{1 + wp^2(m,n,x_1,\dotsc,x_u)}}\Bigr)
$$
where $p$ is a ...
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-3
votes
1
answer
124
views
Periodical functions with $f^{(n)} = f$, but $f^{(k)} \neq f$ for $k\in \{1,\ldots,n-1\}$ [closed]
For any infinitely differentiable function $f: \mathbb{R}\to \mathbb{R}$ and positive integer $k\in\mathbb{N}$, let $f^{(k)}$ denote the $k$-th derivative of $f$.
For which $n\in\mathbb{N}$, $n>1$, ...
- 48.9k
-3
votes
1
answer
227
views
Equation $\ \binom xn'\ =\ \log(n)$ [closed]
Problem: Solve equation
$$ \binom xn'\ =\ \log(n) $$
Here prime stands for the derivative with respect to $x$.
Observe that:
$\quad$ for integer $n$ large,
the approximate solution is $\ ...
- 4,786
-3
votes
1
answer
332
views
Convergence Question [closed]
If $\alpha _{n}\rightarrow \alpha$, then how does one show that for any j=1,2,... and $\epsilon> 0$, if $sup\int \left | x \right |^{j+\epsilon }d\alpha _{n}<\infty$, then $\int x^{j}d\alpha _{n}...
- 1
-3
votes
1
answer
194
views
Bounding a number-theoretic integral
Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH.
My attempt here is ...
- 107
-3
votes
1
answer
392
views
A generalization of Chebyshev's sum inequality
From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference.
Inequality: Let $y=f(x,y)$ is ...
- 1,776
-3
votes
1
answer
590
views
A problem regarding definition of p-norm [closed]
Let ${\bf x}=(x_1,...,x_n)$, the p-norm of x is $(|x_1|^p+...+|x_n|^p)^{1/p}$. If one of the components of x is 0, there will be exponential of the form $0^p$. If p is an irrational, $x^p$ is only ...
- 764
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votes
1
answer
167
views
Is there a simple function similar to exp? [closed]
As far as I know exp have such properties:
$f'(x) >0$
$f''(x) >0$
$\lim_{x \to -\infty}f(x)=0$
$\lim_{x \to +\infty}f(x)=\infty$
$f(x)f(-x)=1$
Let's say f(x) comply such rules.
The closest I ...
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-3
votes
1
answer
148
views
A proposition about power series
Is this proposition established?
Suppose that $0<\nu<1$, $x\in[0,1]$ and absolutely converge power series
$$p(x)=\sum_{n=0}^\infty a_nx^n,$$
$$P(x)=\sum_{n=0}^\infty \frac{\Gamma(n+1)}{\Gamma(n+...
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-3
votes
1
answer
227
views
Is this sequence convergent? [closed]
suppose $\exists S \subset \mathbb{R}$ and a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x_0 \in S $ the sequence $x_{n+1} = f(x_n)$ converge to $x \in S$
now, let $\alpha \...
- 159
-3
votes
1
answer
230
views
Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]
Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)?
If so, please show me how to construct it.
- 3
-3
votes
1
answer
451
views
Exponential decay of kernel
Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
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-4
votes
2
answers
228
views
An elementary-looking integral inequality
This might seem a bit easy but I still like to ask it for pedagogical reasons.
QUESTION. Is this inequality true for non-negative integers $n$?
$$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
- 43.2k
-4
votes
2
answers
6k
views
Factorizing polynomials of several variables (in a different perespective)
I am looking for factorization of polynomials of several variables in the way outlined below.
Consider a second degree polynomial of two variables over the complex numbers.
"P(x,y) = Ax^2 + Bxy + Cy^...
-4
votes
1
answer
200
views
How I can choose $(t_1,t_2,...,t_{r}) \in (0,1)^{r}$ such that $f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)=0$?
Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily ...
- 1,197
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votes
1
answer
302
views
A Question in Fourier Analysis proposing a conjecture
Let $f$ be a $2\pi$ periodic BV function whose derivative is also BV.Let the amount of jump at a point $x$ is denoted as $\lfloor f \rfloor (x) = f(x+0)-f(x-0)$ Define function $J:\mathbb{R} \to\...
- 698
-5
votes
1
answer
753
views
Why calculus textbooks do not include the natural integration constants in the tables of integrals? [closed]
The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...
- 10.1k
-5
votes
1
answer
270
views
Calculus based on pdf [closed]
Is there a calculus, i.e. an analytical framework, that deals with probability distributions as its variables? Measure theory goes in that direction, and Hewitt/Stromberg (Real and Abstract Analysis, ...
- 201
-5
votes
1
answer
184
views
Two inequalities in $\mathbb{R}$ [closed]
How to prove that for real numbers $a$ and $b$, the following inequalities hold?
$(a|a|^{p-2}-b|b|^{p-2})(a-b)\geq 2^{2-p}|a-b|^p$,if $p\geq 2$
$(a|a|^{p-2}-b|b|^{p-2})(a-b)\geq (p-1)\frac{|a-b|^2}{(|...
-5
votes
1
answer
184
views
a question of definite integral [closed]
1.$$\int_{0}^{1} \frac{1}{1+e^{-(x+\ln(u/(1-u)))/\tau}}\, du$$
2.$$\frac{1}{\sqrt{2}\pi}\int_{-\infty}^{+\infty}\frac{e^{-u^{2}/2}}{1+e^{-(x-u)/\tau}}\,du$$
please help me. I tried to use MATLAB but ...
-6
votes
2
answers
2k
views
Is there a transformation or a proof for these integrals?
Here are certain weighted Gaussian integrals I have encountered for which numerical computation reassures equality.
Question. Is this true? If so, is there an underlying transformation or just a ...
- 43.2k
-6
votes
1
answer
614
views
Proof of formula for $\pi$ [closed]
The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...
- 16.6k
-6
votes
1
answer
175
views
Continuous function $f:\mathbb{R}\to\mathbb{R}$ with fixed size finite fibers [closed]
During a business meeting, I was trying to find a continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $|f^{-1}(\{y\})| = 2$ for all $y\in \mathbb{R}$, and after some experimentation I found $$f:\...
- 48.9k
-6
votes
1
answer
141
views
Behavior of $f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right)$ [closed]
Consider the following function defined on $x \in \mathbb{R}^+ \cup\{0\}$
$$
f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right),
$$
...
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-8
votes
2
answers
1k
views
why do we need algorithms, and why is non-convex optimization difficult? [closed]
A simple question, but (I'm quite sure) not a superficial one: is the basic distinction between algorithms and much of the rest of math that algorithms try to tackle problems for which we lack global ...
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