All Questions
5,657 questions
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votes
1
answer
214
views
About infinite products and Euler Gamma functions [closed]
I am interested in knowing how to calculate infinite products like (or reading any reference about it):
$$\prod_{j=1}^{\infty}\left( 1-\left( \frac{x}{a+j\pi} \right) ^2 \right)$$
Inserting it into ...
-2
votes
1
answer
248
views
Upper and lower limits [closed]
Find the following limits:
(1) $\limsup_{n\to\infty } \sin (n!) $
(2) $\liminf_{n\to\infty } \sin (n!) $
(3) $\limsup_{n\to\infty } \cos (n!) $
(4) $\liminf_{n\to\infty } \cos (n!) .$
- 469
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votes
1
answer
93
views
Express the connection between roots [closed]
$\DeclareMathOperator\elim{lim}\DeclareMathOperator\Lim{Lim}\DeclareMathOperator\lmb{lmb}\DeclareMathOperator\Lmb{Lmb}\DeclareMathOperator\mts{mts}$There are two similar functions; they determine the ...
- 55
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votes
1
answer
217
views
If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]
This seems like it should be true but I was wondering if anyone could prove it. Thanks!
- 109
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votes
1
answer
100
views
Understanding the performed change of variable in this integration [closed]
I'm stuck on a passage I do not understand, which reads:
$$\int_{r<|y|<1} \bigg| \frac{1}{(|y|^2 - r^2)^s |y|^n} - \frac{1}{|y|^{n+2s}}\bigg|\ \text{d}y$$
$$\int_1^r \bigg| \frac{1}{(t^2 - r^2)^...
- 99
-2
votes
1
answer
475
views
Is a function piecewise continuous if it has a left-limit and a right-limit at every point in its interval domain and equals at least one of these? [closed]
Suppose a real-valued function f, whose domain is an interval, has the property that
at every point in its domain it has a left-limit and a right-limit, and it equals at least one of these. Is it ...
-2
votes
1
answer
968
views
Bounding the product of lipschitz function [closed]
Assume we know that $f(x,y): \mathbb{R}^{a+b} \to \mathbb{R}^d$ is lispchitz w.r.t. $y$, i.e.
$$\|f(x,y) - f(x,y')\| \leq L \|y-y'\|.$$
Is there a way to bound the product $f(x,y)\cdot f(x,y')^T \in\...
- 245
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votes
1
answer
134
views
Proof of: If $f(x)=p(x)+o(x^n)$ for $x \to 0$, then $b_{k}=\frac{f^{(k)}(0)}{k !} $ for $ k=0,1, \ldots, n$ [closed]
Before the current problem I work on, I proved the following:
Let $q$ be a polynomial with $\deg(q) \le n$. If $q(x)=o(x^n)$ for $x \to 0$, then $q$ is the zero polynomial.
I have to use the ...
- 99
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votes
1
answer
147
views
Asymptotics for certain integrals
I stumbled on the following problem, if you can see a way through it.
Let $x$ be a real variable and fix a real value $\frac14\leq\nu\leq\frac34$.
QUESTION. For $x\rightarrow0$, does there exist a ...
- 43.2k
-2
votes
1
answer
165
views
Relationship between "Radial" Fourier transform and Fourier transform, especially at infinity
Let $\phi:\mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function with compact support.
What is the relationship between
$$
\widehat{\phi}(k) = \int e^{-2\pi i x \cdot k} \phi(x) dx, \quad k \in \...
- 469
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votes
2
answers
325
views
$f\in (W^{1,p}(\Omega)\cap C(\Omega) \cap L^{\infty}(\Omega))\setminus C(\bar{\Omega})$, $f=0$ on $\partial \Omega$ imply $f\in W^{1,p}_{0}(\Omega)$?
Q1:
Let $p\geq 1$, and let $f\in W^{1,p}(\Omega)\cap C(\Omega)$. Assume also
$f\in L^{\infty}(\Omega)$ and $f=0$ on $\partial \Omega$. Is it true that
$f\in W^{1,p}_{0}(\Omega)$ even if $f\notin C(\...
- 852
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votes
1
answer
100
views
Is every implicit function reparametrized? [closed]
Consider a continously differentiable non-constant function $f:\mathbb{R}^2\to\mathbb{R}$. Define
$$
K=\{x\in\mathbb{R}^2|f(x)=0\}.
$$
I wish to know whether there is a continuously differentiable ...
- 133
-2
votes
2
answers
119
views
Systems of ODEs that fulfill a matrix relationship at steady state [closed]
It is well known that for a system of linear ODE $$x'(t) = A(t) \cdot x(t) + b(t)$$
with initial condition $x(t_0) = x_0$, that for a solution at any other time point $t_1$, $x(t_1) = (z_1, \ldots, ...
- 749
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votes
1
answer
314
views
Series representation for $\log(|\zeta(\frac{1}{2}+it)|)$
(Question is short and straight-forward. )
What is/are "nice and non-trivial" series representation/s of $\log(|\zeta(\frac{1}{2}+it)|)$ ??
By "nice and non-trivial" I mean contains no ...
- 375
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votes
1
answer
209
views
Modulus of continuity an exponential type function [closed]
Fixed $0<a<1$, define $f(x):=(1-x)^{a}$ for every $x\in [0,1]$. Recalling that the modulus of continuity of $f$ of order $\varepsilon$ is given by
$\omega(f,\varepsilon):=\sup\{|f(x)-f(y)|:|x-y|\...
- 101
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votes
1
answer
158
views
About local maxima of multivariable polynomials
Lets say I have a real valued function which is writable as a polynomial in terms of Frobenius norms of a pair of matrices as in it is of the form, $f_B(A) = f(||A||_F^2, ||AB||_F^2, ||A^TAB||_F^2)$ ...
- 2,246
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votes
1
answer
116
views
Is this intergral inequality valid? [closed]
Does the inequality $\int_2^{\infty} \dfrac{\sqrt x(\log x)^3 + (1+ \log x^2) x}{x(\log x)^2(x^2 - 1)} \,\mathrm {d}x > \ln \dfrac{17}{10}$ hold ?
- 53
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votes
1
answer
80
views
Suppose a real differentiable function with its derivative not infinity, it is sure that its second symmetric derivative should exist? [closed]
Suppose a real differentiable function $h(x)$ with its derivative not infinity, it is sure that its second symmetric derivative $\lim_{\epsilon->0}\frac{h(x+\epsilon)-2h(x)+h(x-\epsilon)}{\epsilon^...
- 39
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votes
1
answer
193
views
Analysis of Sobolev spaces [closed]
I just wanted to know wthether the following is OK or not.
Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
- 157
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votes
1
answer
212
views
A calculus question [closed]
Fix $q>1$. Define the function
$$
f_q(c):=\int_e^\infty \frac{e^{-c r^2}r}{\log(r)^q}d r.
$$
The problem is whether the following is true,
$$
\lim_{c\rightarrow 0} c \log(1/c)^q f_q(c) = C \in ...
- 1,649
-2
votes
1
answer
395
views
non-trivial convergent sequence [duplicate]
I have reached a deadlock to find a example to show that a compact Hausdorff space does not need to have a no non-trivial convergent sequence.(except $\beta\omega$)
can you give me a example of ...
- 147
-2
votes
0
answers
64
views
A Problem using Limits of Sequences of Functions
Suppose $\{f_n\}$ is a sequence of nonnegative extended real-valued functions on $X$ and $\lim_{n\to\infty}f_n=f$. Take a simple function $0\leq\varphi\leq f$. If $X_{\infty}=\{x\in X: \varphi(x)=a>...
- 1
-2
votes
1
answer
209
views
Strong estimates for the zeta function on natural numbers
Let $$\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{...
- 749
-2
votes
1
answer
92
views
An inequality between two real-valued concave functions
Can anyone help me prove the following inequality? Thanks!
- 9
-3
votes
1
answer
638
views
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 ...
- 317
-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^{...
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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 ...
- 1
-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
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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 ...
- 1
-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
$$p(x)=\sum_{n=0}^\infty a_nx^n,$$
$$P(x)=\sum_{n=0}^\infty \frac{\Gamma(n+1)}{\Gamma(n+...
- 13
-3
votes
1
answer
227
views
Is this sequence convergent? [closed]
suppose $\exists S \subset \mathbb{R}$ and a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x_0 \in S $ the sequence $x_{n+1} = f(x_n)$ converge to $x \in S$
now, let $\alpha \...
- 159
-3
votes
1
answer
230
views
Homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1) [closed]
Can one construct homeomorphism between (-1,1)×[-1,1) and [-1,1]×[-1,1)?
If so, please show me how to construct it.
- 3
-3
votes
1
answer
451
views
Exponential decay of kernel
Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
- 11
-4
votes
2
answers
228
views
An elementary-looking integral inequality
This might seem a bit easy but I still like to ask it for pedagogical reasons.
QUESTION. Is this inequality true for non-negative integers $n$?
$$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
- 43.2k
-4
votes
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
-4
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