Questions tagged [constants]
For questions about mathematical constants, that are "significantly interesting in some way".
569
questions
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Matrix Norm Inequality: $c \sum_{k=1}^{n} \left| \sum_{l=1}^{n} \epsilon_{l} A_{k,l} \right| \geq \sum_{l=1}^{n} \sqrt{\sum_{k=1}^{n} A_{k,l}^{2}}$
Question
Show that there is a real universal constant $c$ with the property that for all
positive integers $n$, and all nonzero $n \times n$ real matrices $A$, there are signs
$\epsilon_1, ..., \...
- 59
1
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0
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41
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(Im)possibility of closed-form expression of Clausen functions
When I started learning Riemann zeta function, I was fascinated that $\zeta(2n)$ can be expressed with finite integers and $\pi$ while $\pi$ has no obvious relation with the sum-$\zeta(2n)$ but no &...
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Algebra problems involving constant and coefficient of exponential equations
The function $f$ is defined by $f(x)=a(3.7^x+3.7^b),$ where $a$ and $b$ are integer constants and $0<a<b$.
The functions $g$ and $h$ are equivalent to function $f$ where $k$ and $m$ are ...
5
votes
2
answers
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The "seashell constant": closed form for $\frac12\exp\int_0^1-\log(\sin(\frac{\pi}{6}+\frac{2\pi}{3}x))\mathrm dx$?
I am looking for a closed form for
$$
R = \frac{1}{2}\,\exp\left(-\int_{0}^{1}
\log\left(\sin\left(\frac{\pi}{6} +
\frac{2\pi}{3}\,x\right)\right){\rm d}x\right)\approx 0.6159
$$
Wolfram does not give ...
- 25.7k
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0
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41
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Is $c = \frac{\sqrt 3}{4} \frac{\pi}{4} \prod_{p = 2 \mod 3} \sqrt{\frac{p^2}{p^2-1}} \prod_{q = u^2 + 3 v^2} \sqrt{\frac{q^2}{q^2-1}}$?
Consider the sum of $2$ squares and Gauss circle problem
https://en.wikipedia.org/wiki/Gauss_circle_problem
and also
The Landau-Ramanujan Constant that relates to the sum of 2 squares. See : http://en....
- 16.4k
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0
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a cool optimization problem involving cubes surfaces and volumes
Consider a codimension one surface of revolution $S$ and an embedding $e:S \hookrightarrow X^n$ for $X^n=[0,1]^n$ with points $p,q$ elements of $\partial X^n$ where $\partial X^n=X^n-(0,1)^n$ for $\...
- 756
2
votes
1
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Is it "ok" to see constants as a "section" in $n$ dimentional space to get to lower dimentionality space?
Given that in $y=5x$, $5$ is a constant (let's call it $a=5$) in a 2d graph.
I thought that it could be seen as a simplification of a more general case of some 3d space where $y$ is a function of both ...
- 91
2
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2
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Prove $\int_0^\infty (1-\exp(-\operatorname{Ei}(t)))dt=\int_0^\infty\exp(-t-\operatorname{Ei}(t))dt$
How to prove $$\int_0^\infty (1-\exp(-\operatorname{Ei}(t)))dt=\int_0^\infty\exp(-t-\operatorname{Ei}(t))dt$$ where $\operatorname{Ei}(t)=\int_1^\infty \frac{\exp(-xt)}{x} dx$? This constant value is ...
- 1,213
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Function for the Champernowne constant that returns the digit position of any number.
In this paper, an equation is provided for locating the first occurance of any 10^n number:
My question is if there is a generalized version of this equation that locates any number (eg, 314159), not ...
5
votes
2
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452
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Integration by parts does not work for this complex integral. Why?
There is a longer integral for which integration by parts $\displaystyle\int udv=uv-\int vdu$ was attempted as it came across in research:
$$
\frac i{2\pi}\int_0^{2\pi}\underbrace{\ln\left(1+\frac{e^{-...
- 12.5k
25
votes
7
answers
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Show by hand : $e^{e^2}>1000\phi$
Problem:
Show by hand without any computer assistance:
$$e^{e^2}>1000\phi,$$
where $\phi$ denotes the golden ratio $\frac{1+\sqrt{5}}{2} \approx 1.618034$.
I come across this limit showing:
$$\...
1
vote
2
answers
103
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Are all constants of integration $C$ equal?
This is a question from very (very) basic calculus, but it concerns indefinite integrals and the constant $ C $ we always add when we find the antiderivative. This concerns some problems with ...
user1280826
1
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2
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Is a function $f$ defined on a closed interval $[a,b]$ constant, if $f'(x)=0$ for all $x∈(a,b)$ but $f'(a)$ or $f'(b)$ are nonzero real numbers?
Suppose we have a function $f$ that is defined on a closed interval $[a,b]$.
The following can be proven from the Mean Value Theorem:
If
$f$ is continuous on the interval $[a,b]$
$f'(x)=0$ for all $...
- 113
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0
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75
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Differentiation over the hole PDE gives contradiction?
I have two PDE:
$$\bigg(\frac{1}{R_1}\partial_rR_2+K(r)\bigg)+\bigg(-\frac{1}{S_1}\partial_\theta S_2-T(\theta)\bigg)+A(r)\sin\theta\frac{S_2}{S_1}=0\,, \\ \bigg(-\frac{1}{R_2}\partial_rR_1-K(r)\bigg)+...
- 65
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Bernstein uniform approximation by polynomials : What is $E_n(f)$?
I am having a difficult time understanding what Bernstein's constant is. Wikipedia states "Let $E_n(f)$ be the error of the best uniform approximation to a real function $f(x)$ in the interval $[-...
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