# All Questions

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### Limit of $\displaystyle\int_1^\infty \frac{dt}{t^4+t^2+1}$

I have too find the limit of $\displaystyle\int_1^\infty \frac{dt}{t^4+t^2+1}$ and i have as a hint: Divide by $t^2+t+1$ Then i get that $(t^2+t+1)(t^2-t+1)= t^4+t^2+1$ I i could do this by partial ...
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### Does this second order non-linear PDE have a closed-form solution?

I'm looking for real-valued solutions $f(x,y)$ that satisfy: $$f_{xy}=\alpha\frac{f_xf_y}{f}$$ for $0\le x\le Y$ with boundary conditions $f(0,y)=0$ and $f(Y,Y)=K(Y)$. It might be of interest to not ...
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### Creating solenoidal vector field from another 2D vector field

I know that I can create a solenidal vector field by taking the curl of another (lets say KNOWN) "vector field" because the divergence of curl is zero. However 2D vector fields result in a scalar ...
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### $x^2+(a-b)x+1=a+b$, seemingly simple but interesting

The roots of the Eqation: $$x^2+(a-b)x+1=a+b$$ are real and distinct for all real value of $x$, I have to prove that $a>0$ Now obviously, the discriminant has to be more than $0$ so i tried it ...
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### How does All metric functions on R look like?

Is it possible to know how all metric functions on the reals look like? All I found was that any d(x,y) involving multiplication of x and y cannot be a metric. If it is impossible, some examples of ...
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### A sequence $\int_{0}^{1} | f_i - f_j | dx = const.$ for $\forall i \neq j$ exists (?)

Is there a such a sequence of function $f_i: [0,1]^2 \rightarrow \mathbb{R}$ with $\int_{0}^{1} | f_i | \ dx \le 1$ and $\int_{0}^{1} | f_i - f_j | dx = const.$ for $\forall i \neq j$ ? I ...
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### nodes equation cant find formula.

Given the level $N$ at which a node $X$ is located in a binary tree, to search for node $X$ according to level-order traversal, we can use the knowledge of level N where $X$ is located to narrow our ...
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### Motivation of acute angle principle

Acute angle principle: Let $\Omega$ is open subset of $\mathbb{R}^n$,$\theta\in \Omega$ . $f:\overline\Omega\rightarrow\mathbb{R}^n$ is continuous.And $\forall x\in\partial \Omega,(f(x),x)\ge0$,then ...
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### Can any one explain in simple terms what is Ricci flow? Any books to refer please.

Please explain what is ricci flow in simple terms. I have to give a presentation to general audience.
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### How to write this sentence formally

I have this sentence Prediction is a numerical value, $P_{a,j}$, expressing the predicted likeliness of item $i_{j} \notin I_{u_{a}}$ for the active user $u_a$. This predicted value is within the ...
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### Is a polynomial always reducible?

I'm studying abstract algebra. The following seems to be too difficult for me: Let $p(x)$ be a second degree polynomial with integer coefficients. Is the polynomial $p(p(x)+x)$ always reducible in ...
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### Evaluate the improper integral $\int_0^{\pi/2}(\sec^2x-\sec x \tan x)dx$

Evaluate the improper integral $\int_0^{\pi/2}(\sec^2x-\sec x \tan x)dx$ I got $\int_0^{\pi/2}(\sec^2x-\sec x \tan x)dx=\int_0^{\pi/2} \frac{1-\sin x}{\cos^2x}dx$ The singularity occurs at the ...
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### Vectors (finding magnitude)

$C$ and $D$ are points with position vectors $c$ and $d$ respectively. If the magnitude of $c=5$ and magnitude of $d=7$, and the dot product; $c\cdot d=4$. Find $|CD|$ (vector connecting $C$ and $D$) ...
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### Proving the convergence of recurrent sequence $a_{n+1}=\sqrt{a_n} + \sqrt{a_{n-1}} + \sqrt {a_{n-2}}$

Given $a_0,a_1,a_2$ positive numbers and the sequence defined by $a_{n+1}=\sqrt{a_n} + \sqrt{a_{n-1}} + \sqrt {a_{n-2}}$ Proof that if $a_0=a_1=a_2$, the sequence $(a_n)_{n\ge0}$ is convergent.
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### Unitary and non-unitary

I have a problem where the optimum is achieved when non-unitary is equal to the unitary? Given the unitary matrix $\mathbf{U} \in \mathcal{C}^{N \times d}, N>d$, \$\mathbf{G} \in \mathcal{C}^{N ...