7:19 AM
I went through questions tagged only chebyshev-functions. I have added some tag, but if you have idea for better tags, please, go ahead and retag them. — Martin Sleziak 18 secs ago
They were:
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Suppose the mean noon-time temperature for September days in San Diego is 24∘ and the standard deviation is 4.6. (Temperature in this problem is measured in degrees celsius) Using Chebyshev’s theorem, what is the minimal probability (in percents) that the noon-time temperature of a september day...

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Prove that the Gaussian $Q$ function is bounded on the top by $1/2x^2$, i.e. $Q(x)\le 1/2x^2$. for $x\ge 0$ using the Chebyshev inequality and the Nakagami $m$ distribution with $m=0.5$ that reduces it to half normal distribution.

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Does that sound about right? Assume that $X$ is uniform on the interval $[0,1]$. Plot $P\big(\big|X\big|\geq \varepsilon\big)$ and $\dfrac{E\big(X^2\big)}{\varepsilon^2}$ as functions of $\varepsilon$. How does this relate to Chebushev's inequality? Attempt: $f(x) = \left\{ \begin{array}{l... 3 hours later… 10:44 AM I wonder what would good tags for this question be: 2 Give an example to a function$ f: (a,b) \to \mathbb R $where$ f_x $is not bounded, but every point$x \in (a, b) $has a neighborhood where the function is bounded. (If I translated neighborhood wrong, please correct me) I gave:$ f_x = \frac 1{x - b} $Because the only point having no ne... Originally it was tagged and , the latter clearly being incorrect. After some hesitation I have added . Does anybody has a better idea for choice of tags? 4 hours later… 2:39 PM 3 Since nobody objected to the idea that this tag - if we keep it - should only be for questions about Chebyshev functions in number theory, I went ahead and created tag-wiki and tag-exceprt. I have also started retagging questions which do not belong here, see this list. They are mostly about Che... 3:16 PM Some questions which come to mind when seeing some of the questions that were tagged . Do we have a tag for Q-function? In statistics, the Q-function is the tail probability of the standard normal distribution . In other words, Q(x) is the probability that a normal (Gaussian) random variable will obtain a value larger than x standard deviations above the mean. If the underlying random variable is y, then the proper argument to the tail probability is derived as: which expresses the number of standard deviations away from the mean. Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally. Because of its relation to the... Should I use for such questions? Perhaps the tag would deserve a tag-wiki. The Gaussian integral, also known as the Euler–Poisson integral is the integral of the Gaussian function e−x2 over the entire real line. It is named after the German mathematician and physicist Carl Friedrich Gauss. The integral is: This integral has a wide range of applications. For example, with a slight change of variables it is used to compute the normalizing constant of the normal distribution. The same integral with finite limits is closely related both to the error function and the cumulative distribution function of the normal distribution. In physics this type of integral appears frequently... 3:37 PM 0 I added this post because of some hot discussions on tagging Ali Sadegh Daghighi's recent question here. The revision history shows that Ali's original post is tagged as a 'set-theory' question but Asaf Karagila removed the tag. Again Ali added the tag and Asaf removed it and this process happene... 1 hour later… 4:45 PM After removing the tag from the questions from this list, there are 13 questions tagged . 2 I am trying to prove the following If$\pi(x) = \operatorname{Li}(x) + O(x^{\frac{1}{2}}\log(x))$then$\psi(x) = x + O(x^{\frac{1}{2}} \log^2(x))$I have tried using$\psi(x) = \theta(x) + O(x^{\frac{1}{2}} \log^2(x))$and then bounding$\theta(x)$but could not reach anywhere. I have tried ... 1 Let$\pi, \vartheta$be respectively the prime counting function and the first chebyshev function. As you know,$ \pi(x) \sim x/\log x$, and$\vartheta(x) \sim x$, so that, at first order, seems$\pi(x) \log x \sim \vartheta(x)$. Is it easy to state that$\pi(x) \log x > \vartheta(x)$(just use ... 2 How is Chebyshev's first function $$\vartheta(N)=\sum_{p\leq N}\log p$$ useful in counting primes? Can it alone be used to analytically derive the prime number theorem? 1 How do I show that if$\psi(x)=x+O(x^{1/2}\log^2(x))$then$\pi(x)=\int_2^x \frac{dt}{logt} + O(x^{1/2}\log x)$Where$\psi(x)$is Chebyshev's second function and$\pi(x)$is the prime counting function 0 Chebyshev's Theta Function is defined as$\vartheta(x)=\sum_{p\le x} \log p$I am trying to find a table of this function for large-ish values of$x$, preferably in a form I could download to a spreadsheet. By "large-ish" I mean for$x$over 16000. I have found a table in a 1962 paper by J.Ross... 1 What are the best known (unconditional) bounds on the following: $$\mid\psi(x) - x\mid$$ (With a known constant factor) 4 In February 2013, Sadegh Nazardonyavi and Semyon Yakubovich posted on arxiv: Sharper estimates for Chebyshev's functions$\vartheta$and$\psi$. I have a question about Theorem 2.27 on page 22. My question regards the argument for this: $$\vartheta(x) < 1.000027651, \;\;(x > 0)$$ I can follow... 4 I've been told that the approach below will not work. I would be interested if someone could help me to understand what will go wrong. Let: $$\psi(x) = \sum\limits_{p^k \le x} \ln p$$ So that (see here): $$\ln(x!) = \sum_{k=1}\psi(\frac{x}{k})$$ So that: $$\ln(x!) - \ln\left(\left\lfloor\f... 3 Is this a valid application of the Möbius Inversion Formula: Define:$$\psi\left(x\right) = \sum\limits_{p^k \le x} \log p$$So that:$$\log x! = \sum\limits_{k=1}^{\infty}\psi\left(\frac{x}{k}\right)$$Then, applying the Möbius Inversion Formula gives:$$\psi\left(x\right) = \sum_{k=1}^{\in... 4 I've been reading through Jitsuro Nagura's classic proof that there is a prime between$x$and$\frac{6x}{5}$and it seems to me that it should be possible to improve on his upper bound for the second Chebyshev function. In Nagura's paper, he establishes the following inequality: $$\psi\left(x\... 4 I'm working on understanding Nagura's analysis of the upper bound for \psi(x) which is done in Lemma 2. I am unclear on one step of his reasoning. With Lemma 1, he establishes for x \ge 2000:$$\log\Gamma(\lfloor{x}\rfloor+1) - \log\Gamma(\lfloor{\frac{x}{2}}\rfloor+1) - \log\Gamma(\lfloor... 8 I was recently told that the Möbius Inversion Formula can be applied to the Chebyshev Function. Let$\vartheta(x)$,$\psi(x)$be the first and second Chebyshev functions so that: $$\vartheta(x) = \sum_{p\le{x}}\log p$$ $$\psi(x) = \sum_{n=1}^{\infty}\vartheta(\sqrt[n]{x})$$ Then applying the M... 5 I am going through the proof by Jitsuro Nagura which shows that there is always a prime between$x$and$\frac{6x}{5}$where$x \ge 25$. Nagura uses the following definitions: $$\vartheta(x) = \sum_{p \le x} \log p$$ $$\psi(x) = \sum_{m=1}^{\infty}\vartheta(\sqrt[m]{x})$$ Then as part of the ... 3 hours later… 7:33 PM Among new tags I have noticed 0 Let$X$be a Banach space and$T\in\mathscr{L}(X,X)$with$\|T\|_*<1$. Use the Contraction Mapping Principle to show (where$I$is the identity map on$X$) that$I-T\in\mathscr{L}(X,X)$is injective and surjective. Attempt: Since$\mathscr{L}(X,X)$is a normed linear space and$I,T\in\mathscr{L}...

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I'm doing an introductory linear algebra course and I'm stuck on this question. Show that with respect to any inner product, u+v is orthogonal to u-v if and only if ||u|| = ||v||. I'm trying to prove the forward implication and I don't know where to go from < u+v,u-v >=0 I tried working with t...

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Functions $g$ and $f$ are linear and injective. How do I go about finding the kernel of $g \circ f$? I'm asking because I want to prove that $\ker(f) = \ker(g \circ f)$.

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I am attempting to use LU Decomposition to calculate the determinant of a matrix. Given: $$A = \begin{pmatrix} 1 & 2 \\ 5 & 6 \end{pmatrix}$$ When using this calculator: Here the values give me: $$L = \begin{pmatrix} 1 & 0 \\ 0.2 & 1.0 \end{pmatrix}$$  U = \begin{pmatrix} 5.00 & 6.00 \...

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If you got a Matrix $A$. Is there a estimation how big the largest element in the inverse of the matrix is? If it helps the matrix is unimodular.

I don't think it is a useful tag. I'm going to remove it/replace with (linear-algebra)/whatever else seems suitable.
We also have one untaggged question - after migration.
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I am trying to determine a formula to calculate how efficient honeybee comb is at space usage. I want to determine the number of full and partial cells that exist within a defined volume of honey comb. The rectangular box area is 46.5cm long, 15cm wide, 20cm high Each cells is 4.9mm across fro...

What about ? Or maybe ?
I've added both, feel free to retag that question.

4 hours later…
11:47 PM
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This is just a copy-paste of a question posted in this meta post of Haim Shamir in connection with a tagging disagreement between Asaf Karagila and me in this question of mine at the main forum. Unfortunately Haim's post wasn't well-designed enough to remain open and as it seems despite of all ...