« first day (976 days earlier)      last day (3391 days later) » 

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:
Q: Chebyshev theorem bounds

Matthew CSuppose 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...

Q: Chebyshev inequality and $Q$-Function

user99455Prove 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.

Q: Chebyshev's inequality epsilon

OlgaDoes 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:
Q: Verify: An example to a function f:(a,b) -> R where f is not bounded, but every point has a neighborhood where the function is bounded + proof

DeanGive 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
A: What is (chebyshev-function) tag for?

Martin SleziakSince 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
Q: In support of Asaf Karagila's tagging on this question

Haim ShamirI 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 .
Q: Riemann Hypothesis: Proving a relation between $\psi(x)$ and $\pi(x)$

user12345I 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 ...

Q: Second-order asymptotics for $\pi(n), \theta(n)$

frame95Let $\pi, \vartheta$ 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 ...

Q: Chebyshev's first function prime count

TurboHow 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?

Q: Prime Counting: Relationship between Chebyshev's function and the Prime counting function

user144057How 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

Q: Table of Chebyshev's Theta Function for Large-ish Values of x?

gjhChebyshev'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...

Q: Tightest constant factor for error term of the prime number theorem

Mayank PandeyWhat are the best known (unconditional) bounds on the following: $$\mid\psi(x) - x\mid$$ (With a known constant factor)

Q: Trying to understand Theorem 2.27 in a recent paper on the Chebyshev function

Larry FreemanIn 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...

Q: Trying to show that $\ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x)$

Larry FreemanI'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...

Q: Can the Möbius inversion formula be applied to the second Chebyshev function?

Larry FreemanIs 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...

Q: Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?

Larry FreemanI'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\...

Q: Looking for help in understanding Jitsuro Nagura's analysis of the upper bound for $\psi(x)$

Larry FreemanI'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...

Q: Looking for help understanding the Möbius Inversion Formula

Larry FreemanI 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...

Q: Understanding a famous proof by Jitsuro Nagura: Need help understanding one step in the main theorem

Larry FreemanI 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
Q: Contraction Mapping Principle

Laars HeleniusLet $X$ 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}...

Q: inner product and orthogonal vectors

KevinI'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...

Q: How do I find the kernel of a composition of functions?

EmilyFunctions $g$ 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 $\ker(f) = \ker(g \circ f)$.

Q: LU Decomposition - Are there multiple ways to calculate?

PhorceI 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 \...

Q: Estimations for the size of the biggest entry in an inverse Matrix

testyIf 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.
Q: Determine the efficiency of honeycomb

Jeff SteenbergenI 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
Q: How to deal with tagging disagreement between OP and editor?

Ali Sadegh DaghighiThis 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 ...


« first day (976 days earlier)      last day (3391 days later) »