All Questions
15 questions
1
vote
2
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102
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About the recursive inequality $w_p \geq (1-\frac {\pi}n)w_{p-2n} + 2\pi + o(1)$
Suppose we have a non-decreasing sequence of positive real numbers that tend to infinity: $0<w_1\leq w_2\leq w_3\leq...$ It is known that:
For every $n$ and $p\geq 2n$, we have $w_p \geq (1-\frac {...
- 525
1
vote
1
answer
143
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$L^1$ error between indicator function and smoothed out version
For a large parameter $r>0$, consider the indicator function $1_{[-r,r]}$ and its convolution with the (normalized) Gaussian $\frac{1}{\sqrt{\pi}}e^{-x^2}$, that is,
$$f_r(x) = \frac{1}{\sqrt{\pi}}\...
- 101
0
votes
1
answer
165
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Positive, monotone decreasing function, with derivative limit in 0 equal to ∞ submultiplicative up to an factor?
Related to this question.
For $x_+ \in (0,\infty)$, $a \in \mathbb{R}$ let $F\colon[0,x_+] \to [a,\infty)$ be a twice continuous differentiable (in $(0,x_+)$) function with $f := F'$, $f(x) > 0$, ...
- 15
1
vote
1
answer
110
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Positive, monotone decreasing function, with limit in 0 equal to ∞ submultiplicative up to an factor?
For $x_+ \in (0,\infty)$ let $f\colon(0,x_+] \to (0,\infty)$ be a continous differentiable function with $f(x) > 0$ and $f'(x) < 0$ for all $x \in (0,x_+]$.
Moreover, we assume that
$$\lim_{x \...
- 15
2
votes
0
answers
150
views
Closeness of a rational approximation
What is
$$p_*:=\inf\big\{p\in\mathbb R\colon\,\inf_{n\in\mathbb N}n^p\,\inf_{k\in\mathbb N}
|2\sqrt{3n}-9\pi/4-k\pi|>0\big\},$$
where $\mathbb N:=\{1,2,\dots\}$?
In other words, I would like to ...
- 128k
0
votes
0
answers
32
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Minimization of a palindromic-like sequence and asymptotics
Suppose that I have a sequence of $n$ numbers $x_1, \ldots, x_n$ all taken from
the real interval $[0,1]$.
I am interested in minimizing the infinity norm of the vector
$$ v = \left( \frac{x_{1}}{x_2},...
- 313
1
vote
1
answer
225
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Existence of an asymptote for $g(x)=\frac{f(x)f'(x)+f(1)f'(1)}{f'(x)+f'(1)}-f\left(\frac{xf'(x)+f'(1)}{f'(x)+f'(1)}\right)$
Working with Slater's inequality (a companion of Jensen's inequality) I found this statement:
Let $f(x)$ be a continuous, twice differentiable function, convex or concave and non constant on $(0,\...
2
votes
2
answers
214
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A question about asymptotic affinity and strict convexity with unbounded means
Let $F:[0,\infty) \to [0,\infty)$ be a $C^1$ strictly convex function.
Let $\lambda_n \in [0,1],a_n\le c<b_n \in [0,\infty)$ satisfy
$$ \lambda_n a_n +(1-\lambda_n)b_n=c_n \tag{1}$$ and assume that
...
- 6,741
1
vote
2
answers
2k
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Simple bound on $\log(x)/x$
I would like to pick $x$ as small as possible while guaranteeing that $\log(x)/x \leq \epsilon$ where $\epsilon \ll 1$. Clearly $x$ should (roughly) be of the order $1/\epsilon$; I would like simple ...
- 1,703
3
votes
2
answers
127
views
Comparing the tails of two related convergent series
Let $b_1,b_2,\dots$ be positive real numbers such that
$$s_1<\infty\quad\text{and}\quad z_1<\infty,
$$
where
$$s_k:=\sum_{j=k}^\infty b_j\quad\text{and}\quad z_k:=\sum_{j=k}^\infty\frac{b_j}{\...
- 128k
1
vote
1
answer
299
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Examples of Steffensen's inequality at undergraduated level studies
I've known few days ago the known as Steffensen's inequality, see the article Steffensen's inequality from Wolfram MathWorld and the cited bibliography. It seems that there are applications (I don't ...
1
vote
0
answers
87
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An oscillatory integral estimate
Let $n \geq 3$ and consider two sequences of strictly monotone functions $\{\mu_l(t)\}_{l=1}^{n}$ and $\{\lambda_l(t)\}_{l=1}^n$ on the interval $[-1,1]$ with $\mu_l(0)=0$ and $\lambda_l(0)=1$ for all ...
- 4,135
0
votes
0
answers
112
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On certain integrals of exponential functions with respect to Gaussian measures
I have questions about the integral
$$F(a,b,c)=\sqrt{\frac{a}{\pi}}\int_{0}^{\infty}e^{-bx^4+cx^3-ax^2}dx$$
for $a,b,c>0$.
What is the asymptotic behavior of $F(a,b,c)$ for small $a,b,c$? In ...
- 505
4
votes
1
answer
388
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Proofs of inequalities used by Erdos-Renyi in their Random Graphs Paper 1
Please refer to this, it is Erdos-Renyi 1959 paper 1 on Random Graphs. I am currently working on this, but I am stuck on the fifth page, where they use two estimates. More specifically, here's the ...
- 319
0
votes
1
answer
247
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Proving a complicated inequality with powers of logarithms
I am currently formalising some results from complexity theory with a theorem prover. For that, I have to prove the following statement:
Let $p, b, \varepsilon \in \mathbb R$ with $\varepsilon>0$ ...
- 1,241