All Questions

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Why zeta(2) in these inifinite sums?

The infinite sum of the reciprocals of these two sequences have zeta(2) in the result. The value is not in OEIS. A000326 A002411 Edit---rolled back the changes. Both $\frac{1}{2}$ and $2$ are ...
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Multiway tree to Binary Tree

A multiway tree T can be represented as a binary tree T~ by using the firstChild and nextSibling pointers. If we think of the firstChild link as being the left link and the nextSibling link as being ...
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Unbounded sequence that does not diverge to $+ \infty$ or to $- \infty$

I'm trying to find an example of sequences such that $$a_n \to + \infty \quad \text{and} \quad b_n \to 0$$ $$a_n b_n \text{ is unbounded but does not diverge to }+ \infty \text{ or} - \infty$$ Is ...
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Time-optimal control to the origin for two first order ODES - But wait, the node is unstable? Hard-mode active!

I want to find the time optimal control to the origin of the system: $$\dot{x}_1 = 3x_1+ x_2$$ $$\dot{x}_2 = 4x_1 + 3x_2 + u$$ where $|u|\leq 1$ I ran straight into the problem full strength, hit it ...
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Schemes to the rescue?

I am reading chapter 1 of Gille and Szamuely's book Central Simple Algebras and Galois Cohomology, on quaternion algebras. In it they prove (remark 1.3.1 on p. 18) that the conic associated to a ...
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Discriminant of $x^2+x-1$

I'm working on a homework problem, and am worried I'm going crazy. I believe $x^2+x-1$ is irreducible, but it's discriminant is $1^2-4(1)(-1)=5$ is positive, which would make it reducible (since ...
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This equation define a regular surface?

Consider the function: $f(x,y,z)=xyz^2$ Its gradient is $\nabla f=(yz^2, xz^2, 2xyz)$ then the critical points are all in the sets $\{(x,y,0): x,y\in \mathbb{R}\}, \{(0,0,z): z\in \mathbb{R}\}$. My ...
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How to find the function $f$ that satisfies $f(x, y) = f(x^{-1}, y^{-1})^{-1}$ and $f(x, y)$ is $\approx$ $average(x, y)$?

Fist of all, I'm a programmer, not a mathematician, and I'm sorry for my non native English. And I'm sorry if the question is not appropriate, it is my first time here. Or if the question has no ...
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Solution of this nonlinear equation

How do I solve this equation for $y$? I can see there is a trivial solution $y=0$ but how do I get $y$ as a function of other variables? Does anyone know how to use MATLAB's fzero function to find ...
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A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding $A^{-1}$

The question is: A is a square matrix where $3A^9- 7A^4 + 4A = I$. Prove that A is invertible by finding A^-1. I have looked at other similar questions on this site: 1. Here 2. and Here But they use ...
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In what base does the equation $x^2 - 11x + 22 = 0$ have solutions $6$ and $3$?

If we have below equation and know that $6$ and $3$ are answers of this equation, how to obtain the base used in the equation? $$x^2 - 11x + 22 = 0$$ Partial result The base is not $10$. (Because ...
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Law of large numbers

Consider the following question: A coin has the probability of landing of head equal to 1/4 and is flipped 2000 times. Use the law of large numbers, find a lower bound to the probability ...
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Finding the domain of this trigonometric function

how can I find the domain of this function? f(x) = (xsin(x) + cos(x) / 1 - cos(x)) + (|X| - 2 / x^2 -4) I assume we don't want the dominator to be zero so f(x)1 ...
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ergodic system has a.e. dense orbits

one more question: Let X be a metric space with probability measure $\mu$ and T. X $\to$ X ergodic. => f.a.e. x the orbit $O_x=\{T^n(x) : n \in Z\}$ is dense in X so I have to show that the set B ...
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Analysis of IQ scores given mean, median, sd, quartiles

The statistics below provides a summary of IQ scores of 100 children Mean: 100 Median: 102 Standard Deviation: 10 First Quartile: 84 Third Quartile: 110 About 50 of the children in this sample have ...
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Fourier series of $\sqrt{1 - k^2 \sin^2{t}}$

I'm struggling with a Fourier series. I need to find the Fourier series of the following function. That's the function under study: $f(t)=\left[\sqrt{1-k^2\sin^2t}\,\right]$. The function ...
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If $|G|=p^2$ and $p$ is prime then $G\simeq \mathbb Z_{p^2}$ or $G\simeq \mathbb Z_p\times \mathbb Z_p$?

Let $G$ be a finite group of order $|G|=p^2$ where $p$ is a prime. How can I show $G\simeq \mathbb Z_{p^2}$ or $G\simeq \mathbb Z_p\times \mathbb Z_p$? Notice $|G|=p^2$ implies $G$ is abelian ...
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Minimize distance to a given point subject to a number of linear inequality

I'm trying to find a point that has minimal distance to a known point and satisfies a number of linear inequality. Example in two dimensions and one inequity: $min\{$distance to $(50,70)$ | ...
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Sequences of polynomial functions converging uniformly on $[a,b]$ to a continuous function not a polynomial

What is (are) the necessary and sufficient condition(s), if any, for a sequence of polynomial functions to converge uniformly on a given (finite) closed interval $[a,b]$ to a continuous function not a ...
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Checking if transformation T(p(x)) is diagonalizable?

Say you have a transformation of $P_3$ to $P_3$ defined by, say, $T(p(x)) = p'(x) + p''(x) + p'''(x)$. How would you determine if this is diagonalizable? Do I sub in a standard basis of ...
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Why is every open in $\mathbb{A}^1$ necessarily principal?

Let $U\subseteq\mathbb{A}^1$ be an open set in affine $1$-space. Why is $U$ necessarily a principal open set? Since $U$ is the complement of a closed set, I write $U=\mathbb{A}^1\setminus V(S)$ for ...
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Unconstrained Optimal Control - $J = \frac{1}{2}x^2(2) + \frac{1}{2} \int_{0}^{2}(u^2 - 2xu)dt$

I've been given the following unconstrained optimal control problem, but I feel like I've made a mistake at some point. The system $\dot x = -x + u$, where u = u(t) is not subject to any ...
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Formulating recurrence relation

Alice and Bob worked in a restaurant and received n currency notes in total as tips. Every note has a value (either 1 dollar, 5 dollar or 10 dollar) written on it. The currency notes are arranged from ...