# All Questions

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### Slope of the tangent to a line

$f(x)=(x+2)^2$ for $x\geq-2$. Graph the function $f$. Find the slope of the tangent to the graph of $f^{-1}$ at the point with coordinates $(25,3)$ on the graph $f^{-1}$. The answer according to a ...
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### Convergence of some integrals

I need to determine whether the following integrals converge: $\displaystyle \int_1^{\infty} (x+\sqrt{x})^{-1}$ $\displaystyle \int_0^1 \dfrac{x}{\sqrt{1-x^2}}$ For the first I think i need to use ...
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### Contraction mapping and $\epsilon$-shadowing of pseudo-orbits

I'm not able to prove this theorem: Let a map $f:M \rightarrow M$ be a contraction. Then $\forall \varepsilon > 0$ there is a $\delta > 0$ such that every $\delta$-pseudo-orbit is ...
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### Cartesian Product for not a finite number of elements has how many elements

Suppose P is a set that has m elements and Q is a set with n elements. How many elements will their Cartesian product, PxQ have?
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### Questions on normalizer

I'm trying to do these and I'm a bit stuck on (a). I know that $a \in G$ and $H$ is a subgroup of $G$ so $aH = Ha \in G$. Notice that from this we can deduce that $H$ is a normal subgroup of $G$. ...
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### Why is this a quadratic programming problem?

I am sorry if this is a stupid question, I'm very new. How would I minimize the following objective? $\sum_{k=1}^p\| I_{k} - M_{k}A \|^2$ Each I and M are known. I am told I can use a quadratic ...
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### Second part of Eloi's Conjecture

We know that "There exist some real k such that ∀ integer n>1 the integer part of k∗nln(n) is always prime?" is false (prove here Is there a $k$ for which $k\cdot n\ln n$ takes only prime values? ) ...
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### Find the Exact sum

Give the fourier series representation of $f(x) = x$ on $[-\pi, \pi]$. Use the result to give the exact sum of... $$\sum_{n=1}^\infty \frac{(-1)^{n+1}}{2n-1}$$ $$\text{ where } x \in [-\pi,\pi]$$
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### Is there a “geometric” interpretation of inert primes?

I am currently learning about étale morphisms and how they behave a lot like covering maps. Now if I'm in the example of the ring of integers of a number field, it is possible to think of it as a ...
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We have $P(x) = x^2 + 3x + 9 ≡ 0 \mod(3^i)$, for $i=1,2,3,4$ for $\mod(3): x^2 + 3x + 9 ≡ x^2$, so $x ≡ 0 \mod(3)$ is a solution. From Hensel lifting, we have $s = x + a\times 7,\ a = [-P(0)\times ... 0answers 99 views ###$D_6$, regular hexagon. Find a subgroup of$D_6$where$D_6$is the regular hexagon, with 12 symmetries. $$D_6 = \{ e, r, r^2, r^3, r^4, r^5, a, ar, ar^2, ar^3, ar^4, ar^5\}$$ where$r^6 = e$. And the$r^n$represent ... 2answers 55 views ### Distribution of$U=\frac{X}{\| X \|}$and$R^2 = \| X \|^2$where$X=(X_1, \dots , X_n)$,$X_1, \dots, X_n \sim$N(0,1) i.i.d. Independence? I have the following problem: Let$X=(X_1, \dots , X_n)$,$X_1, \dots, X_n \sim N(0,1)$i.i.d. What is the distribution of$U=\frac{X}{\| X \|}$and$R^2 = \| X \|^2$. Are$U$and$R^2$independent? ... 2answers 67 views ### Computing the lengths of the obtained trapezium$ABCD$is a quadrilateral. A line through$D$parallel to$AC$meets$BC$produced at$P$. My book asked me to show that the area of$APB$and$ABCD$, are the same, which I did. But it aroused my ... 1answer 45 views ### Combinatorics and bijective function exercise One group of$8$friends are going to the theatre, and have tickets to sit in$8$consecutive places. Between them,$J$is mad at$M$and$P$. In how many different forms can they sit, in a way that ... 1answer 24 views ### Time taken to reach height below initial point I am trying to solve the following. It is in my notes, I believe I need to solve:$-H = v_0cos\alpha*t\bf{j}$$+(v_0sin\alpha*t-gt^2/2)\bf{k} as a quadratic equation? If so, how do I solve a ... 1answer 75 views ### Bridge Number , Knot Theory I had been reading some knot theory lately and got to know about a whole classification of 2-bridge knots , does their exist any such extensive study over 3-bridge knots? 2answers 511 views ### Matrix Exponential of Identity Matrix I was just wondering what would the sum be of e^{I_n} where I_n is the identity matrix. I know the maclaurin series for e^x is 1+\frac x{1!}+\frac {x^2}{2!}+.... I know that e^0 is 1 right? ... 1answer 34 views ### Finding the Power Series of a Complex fuction. Find a power series expression \sum_{n=0}^\infty A_n z^n  for  \frac{1}{z^2-\sqrt2 z +2}  I'm completely stuck on this question. I know how to manipulate power series but I've never had to find ... 1answer 758 views ### Combinatorics - homework I've got the following problem which I am slightly confused on two parts and wondering if anyone could help me better understand how to solve this: ... 4answers 888 views ### What exactly does \frac{dx}{dy} mean? I asked 3 professors at my university and none gave me a clear cut answer, but instead merely told me qualities of this notation. Here is what I understand so far from what they told me: 1)Treat the ... 0answers 79 views ### If f(z) = \sum a_n z^n, what is \sum n^3 a_n z^n? Problem: If f(z) = \sum a_n z^n, what is \sum n^3 a_n z^n? Attempt: First we note that$$ f'(z) = \sum_{n=1}^\infty n a_n z^{(n-1)} $$so that$$ z f'(z) = \sum_{n=1}^\infty n a_n z^n  ...
Sorry for the non-descriptive title: it was a rough choice between 3 lines or less description. Let $O$ be the origin and the centre of a circle with radius $a$. Let $T$ be a point on the circle so ...