# All Questions

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### Are there functions for which the cyclic integration-by-parts technique does not work?

There are a lot of functions where you can use what my teacher has described as the 'cyclic' method of integration. An example is $$\int e^x\sin x\,dx$$ where you designate $u=\sin x$ and ...
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### Solve ${z_1/\overline{z_2}} = z^3$

Let $z_1,z_2$ be complex numbers such that: $$z_1= 4\sqrt{2}-48\sqrt{2}$$ $$z_2= \cos{135^\circ} +i\sin{135^\circ}$$ Find all the complex numbers $z$ that fulfill the following equation: ...
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### can any identity involving integers be proved by mathematical induction

Hello mathematics community, Today I was studying mathematical induction which is an axiom. I was wondering Can "ANY" identity or inequality involving integers which is already proven can also be ...
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### A tangent circle element between 2 intersecting vectors

2 Vectors, which are originating from one point I. I want to the replace the sharp corner (I) with an arc (circle element) with a radius of r. The arc touches the vectors at T1 & T2. What is the ...
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### Showing the product of two normal subgroups is normal

Prove that if $H$ or $K$ are normal subgroups then $HK=\{hk\mid h\in H,k\in K\}$ is a subgroup. Then if both are normal subgroups, prove that HK is normal.
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### Spans and Dot Product: Findin the linear combination

Suppose $(v_1, v_2, v_3)$ is a set of vectors mutually perpendicular. Assume that $\|v_1\|= \sqrt{27}\quad \|v_2\| = \sqrt{14}\quad \|v_3\|= \sqrt{ 4}\$ Let $w$ be a vector in ...
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### Increasing annuity problem

I learned increasing/decreasing annuity and am tackling the following problem for hours now without success. Amy deposits $Z$ into a bank account that has effective annual interest rate of $5\%$ ...
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### if each $(i,j)\space:\space g_ig_j=g_jg_i$, then $G$ is abelian

Let $G$ be finite group. say that $a,b\in G$ hold that $(a,b)\in R\subseteq G\times G$ iff $\exists g\in G \space:\space gag^{-1}=b$ note that $R$ is an equivalence relation. let $g_1,...,g_n$ be the ...
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### Splitting of Primes in a Given Field

Find how $p=2,3,5,7$ splits in $\mathbb{Q}(\sqrt{-5})$ (i.e. find those $e_i,f_i$ for $1 \leq i \leq r$). Can somebody please explain how this is done? My attempt is the following: Let K = ...
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### Good books and lecture notes about category theory.

What are the best books and lecture notes on category theory?
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### How to prove that an M-matrix is inverse-positive?

Wikipedia says that The inverse of any non-singular M-matrix is a non-negative matrix." To be more precise, if $A$ is an M-matrix, then the entries of the inverse of $A$ are all non-negative, ...
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### Simplify |2x^2+3x-2| so we can obtain it and control it in terms of |x-1|

First example I worked, I had $|2x^2 + x - 3|$ after some manipulations and simplifications I obtain: $|x-1|(2|x-1|+5)$. The final answer is in terms of $|x-1|$ with multiplication between two ...
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### Probability Question

A casino game has two dice, each with faces numbered 1 to 6. One of the dice is fair. The other die is biased such that a 6 is twice as likely to appear on top as any one of the other faces. ...
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### Conditional expectation, indication function

I am given that $X,Y$ are independent Bernoulli RVs with parameter $p\in (0,1)$. I am also told $Z=1_{(X+Y=0)}$. I am asked to find $E[X\mid Z]$ and $E[Y\mid Z]$. I can see that the expected values ...
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### Characterizing $\text{PGL}_2(\mathbb F_p)$

Where can I find a description and proof of the basic structure of $\text{PGL}_2(\mathbb{F}_p)$ (Number of elements with each order, conjugacy classes, etc.) which is understandable by an ...
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### Factorials with fractions

I don't understand how $$\frac {n(n-1)!+(n+1)n(n-1)!}{n(n-1)!-(n-1)!}$$ becomes $$\frac {(n-1)![n+(n+1)n]}{(n-1)!(n-1)}$$ and then how it becomes $$\frac {n+n^2+n}{n-1}$$ I've tried applying ...
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### How to prove $D^2\setminus\{0\}$ is not homeomorphic to $\mathbb{R}^2\setminus\{0\}$?

Here $D^2$ denotes the closed unit disk in $\mathbb{R}^2$. I know that $D^2$ is not homeomorphic to $\mathbb{R}^2$ as $D^2$ is compact. Intuitively I believe that $D^2\setminus\{0\}$ is not ...
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### Submagmas of natural numbers

What is known about submagmas of natural numbers under addition/multiplication? For example, all submagmas of integers under addition are of the form $~n \mathbb{Z}~$. Are there similar results for ...
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### proving that for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ (verify)

for any vectors $u,v,w \in \mathbb{R}^n$ prove $\|u+v+w\| \leq \|u\| +\|v\|+\|w\|$ I wasn't sure how to go about this correctly so what I did was set $v+w$ to $v$, yielding $w = v-v = 0$, since it ...
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### Probability of one device among 6 failing

A certain component of an electronic device has a probability of 0.1 of failing. If there are 6 such components in a circuit. What is the probability that at least one fails? This is not a duplicate ...
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### Generators of the intersection of prime monomial ideals

Let $[n] = \lbrace 1,2,\dots,n \rbrace$, and $F \subset [n]$. We denote by $P_F \subset K[X_1,\dots,X_n]$ the monomial ideal generated by the variables $X_i$ with $i \in F$. Given an ...
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### Statement about $(I-A)^{-1}$ matrices

Let $A \in \mathbb{R}^{n \times n}$ and let denote $I$ the $n \times n$ identitiy matrix. Theorem. If $(I-A)$ is invertible and $(I-A)^{-1}$ is a nonnegative matrix and there is a diagonal element in ...
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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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### Analysis of Data 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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### Working of selections

There are eight finalists in the 400 m athletics at the world championships. Three of the finalists are from the USA, and the others are from five different countries. The rules for allocating a lane ...

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