Newest Questions
1,693,659 questions
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A polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ has four distinct roots $x_1,x_2,x_3,x_4>0,$ and $a,b,c,d\in \mathbb Z$ then find the smallest value of $b.$
The question is: assuming a polynomial $P(x)=x^4+ax^3+bx^2+cx+d$ has four distinct roots $x_1,x_2,x_3,x_4>0,$ and $a,b,c,d\in \mathbb Z$ then find the smallest value of $b.$
By using Vieta's ...
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Intensity Distribution Function for Light Diffracting from Digital Micromirror Device
I am currently struggling to plot the intensity distribution function presented below using GNU Octave. There is a world where this question would be more appropriate for Stack Overflow, but I think ...
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Is the function $f(n)= \left \lfloor 2^{n+3/2} \right \rfloor - 2 \left \lfloor 2^{n+1/2} \right \rfloor$ periodic? [closed]
Let the function $f(n)$ be defined on the set of natural numbers by:
$$
\Large f(n)= \left \lfloor 2^{n+\frac{3}{2}} \right \rfloor - 2 \left \lfloor 2^{n+\frac{1}{2}} \right \rfloor$$
where $\left\...
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Confusion Regarding Fractional Calculus [closed]
I was playing around with half-derivatives and found the puzzling idea that $\mathrm{}_{0}^{}\mathrm{D}_{t}^{0.5}(\mathrm{}_{0}^{}\mathrm{D}_{t}^{1}(e^{t}))\neq \mathrm{}_{0}^{}\mathrm{D}_{t}^{1}(\...
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The subset of functions that vanish at one point is a maximal ideal [duplicate]
Let $C$ be the ring of functions $\mathbb{R}^{\mathbb{R}}$ with the pointwise operations addition and product. For each $r\in\mathbb{R}$, let $M(r)$ be the subset of $C$ given by $M(r)=\{f\in C: f(r)=...
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Solve $f(x) - f(\frac{1}{x}) = \ln(x)$
While trying to find ways to compute the natural logarithm* I stumbled upon this simple looking equation :
$$f(x) - f(\frac{1}{x}) = \ln(x)$$
Find some valid $f$.
[*] here : Approximating the natural ...
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Caratheodory's Theorem Interiors (Steinitz's Theorem)
In Barany's book Combinatorial Convexity he proves the following theorem at the end of chapter 2:
Theorem Let $A \subset \mathbb{R}^d$. If $a \in \mathrm{int}(\mathrm{conv}(A))$ then there exists a $B\...
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Definite Integration with Lambert W-Function $\int_0^\infty e^{-W(x)^2} \, dx$
I am interested in (elegantly) solving the „Lambert-Gaussian-Integral“ ;) $$I:=\int_0^\infty e^{-W(x)^2} \, dx$$ I tried the substitution $u = W(x)$, so $x = u e^u$, $dx = e^u (u + 1) \, du$.
The ...
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A question regarding complex conjugate of a holomorphic vector bundle [closed]
Let $V$ be a holomorphic vector bundle on a Riemann surface $X$. $\overline{V}$ be the corresponding complex conjugate vector bundle on $X$.
Is it true that:$\quad$ i)$(\overline{V})_x\simeq{V_x}$? (...
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Proving inequality by PHP [closed]
I encountered the following question and I'm pretty sure that pigeonhole principle is required:
Let $n\ge 2$ be a positive integer and let $S$ be a subset of $\{1,\cdots,3n\}$. Supposing $|S|=n+2$, ...
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Solving $(1 + \tfrac{1}{23})^{23} = \left(1 + \tfrac{1}{x}\right)^{x+1}$ for integer $x$ [closed]
There was this question I saw as a challenge question in a textbook, it looks impossible, but there is apparently an integer solution. $$\left(1 + \frac{1}{23}\right)^{23} = \left(1 + \frac{1}{x}\...
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Prove that if $d^k = (3^n - 1)/2$, then $d^{k+1}$ cannot be of the form $(2^m + 1)/3$
Suppose that $d^k = \frac{3^n - 1}{2}$ for positive integers $d,n,k$, with $d,k \geq 2$. Prove that it is impossible for $d^{k+1}$ to be of the form $(2^m + 1)/3$.
I originally asked a similar ...
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Finding the homogeneous polynomial associated with a symmetric tensor
Consider the order $3$, rank $2$ tensor
$$
T = 2e_1^{\otimes 3} + 5(e_1 \otimes e_2^{\otimes 2}) \in \operatorname{Sym}^3(\mathbb{R}^2)
$$
This is rank $2$ in the sense that it is written as a sum of $...
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On the rate of convergence of $\log(\|f\|^p_p)$ as $p\rightarrow\infty$
Throughout this posting $(X,\mathcal{F},\mu)$ is a probability space and $f\in L_\infty(\mu)\setminus\{0\}$.
It is well known that $f\in L_p(\mu)$ for all $0<p<\infty$, $\phi(p)=\|f\|^p_p$ is ...
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A product of a quotient map with an identity
Let $Y = \mathbb{Q}/\mathbb{Z}$ be the quotient space of the rationals obtained by collapsing the subset $\mathbb{Z}$ to a point, and let $q \colon \mathbb{Q} \to \mathbb{Q}/\mathbb{Z}$ be the ...
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