# All Questions

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### Understanding Dirac delta integrals?

I'm confused as to how exactly to integrate using the Dirac delta function. I have the following example: $$\int \delta (x-4)(x^3-4x^2-3x+4)dx$$ and am told this evaluates to 8. Can anyone please ...
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### Approximation of basis vectors of the matrix

Is there any particular matrix $(B)$ such that after multiplying to orginal matrix ($A_{m*n}$) gathering information of the matrix on diagonal of resulting matrix $(C)$?? in other words, "dependent ...
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### Showing that $\int_x^{\infty}\frac{1}{u^2}e^{(-u^2/2)}du=\frac{1}{x}e^{(-x^2/2)}-\int_x^{\infty}e^{{(-u^2/2)}}du$

My texbook claims that integration by parts of the integral $\int_x^{\infty}\frac{1}{u^2}e^{\frac{-u^2}{2}}du$, with the hint that $d(-\frac{1}{u})=\frac{du}{u^2}$, gives ...
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### Need help with proving that group is not finitely-generated [duplicate]

I need to prove that $(\mathbb{Q}^*, \times)$ (i.e rationals, zero excluded, under multiplication) is not finitely generated. So, suppose that G is finitely-generated. That means there exist a ...
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### Compactly supported smooth function with Laplace transform bounded on a cone

My question is if it is possible to find a compactly supported smooth function $\varphi:\mathbf{R}\to \mathbf{R}$ s.t. the following integration $\int_{\mathbf{R}}\varphi(t)e^{itx}e^{tx}dt$ stays ...
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### A Fantabulous integer is an integer which has another fantabulous integer smaller than it

BdMO 2013 problem-7: A positive integer is called “Fantabulous” if there is another fantabulous positive integer smaller than it. Find the number of fantabulous integers. I am bamboozled at ...
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### What is the sample space, outcomes, event space, random variables in Machine Learning problem?

Reading through materials of Machine Learning problems, I saw people treated things like they are doing with probability. Particularly consider the linear regression, I cannot figure out what is the ...
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### $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ is a cyclic group

I would like to prove that the group $(\mathbb{Z}/p^r\mathbb{Z})^{\ast}$ of the invertible elements of $\mathbb{Z}/p^r\mathbb{Z}$ with $p>2$ prime and $r>0$ is cyclic. My text suggests to ...
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### What do we learn from Mathematics? [closed]

I have been doing mathematics for some time now. I am currently a UG in Mathematics and Computer Science. I like doing mathematics but as I progress, I notice that I don't remember certain theorems at ...
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### Direct limit of topoloical spaces

Let $X$ be a topological space. Suppose $X_n$ are subspaces of X with $X_1 \subset X_2 \subset ... \subset X$. I'm going to prove $\varinjlim X_n =\cup_n X_n$. I have some trouble in proving that ...
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### Does $f^3$ integrable imply $f$ integrable?

Let $f$ be a function defined on the close interval $[a,b]$. Does the riemann stieltjes integrability of $f^3$ imply the riemann stieltjes integrability of $f$ ? The answer is trivially no in the ...
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### $A\in\mathrm{M}_{n\times n}(\mathbb{C})\implies A^TA$ is diagonalisable?

I have been set some work to do over the holidays, and one of the questions gives a hint that is as follows: $A\in\mathrm{M}_{n\times n}(\mathbb{C})\implies A^TA\text{ is diagonalisable}$. I know ...
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### Prove $(\forall n \in \Bbb N)[\gcd\left(n,(16n+1)^3\right)=1]$

Prove $(\forall n \in \Bbb N)[\gcd(n,(16n+1)^3)=1]$ Knowing that $\gcd(a,b)=\gcd(a,b+a\times k)$ with $k \in \Bbb Z$ $$\gcd\left(n,(16n+1)^3\right)=\gcd\left((16n+1)^3,n\right)=d$$ ...
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### area of rectangle whose vertices are roots of equation $z\overline{z}^3+ \overline{z}z^3=350$

We have a complex number $z=x+iy$ where $i=\sqrt{-1}$ and $\overline{z}$ represents conjugate $$z\overline{z^3}+ \overline{z}z^3=350$$ so i proceeded by taking $z\overline{z}$ common thus ...
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### Prove this equality: $det(\mathrm I_n-\mathbf A\mathbf B)=2^n$

If $\mathbf A$ and $\mathbf B$ are square matrices ($n$ dimensional) which verifies $\mathbf A\mathbf B=-\mathrm I_n$, then prove that: $$det(\mathrm I_n-\mathbf A\mathbf B)=2^n$$ I'm struggling on ...
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### How to integrate $\left(\frac{x^{n+1}}{n+1}\right)\left(\frac{1}{\sqrt{1-x^2}}\right)$ [closed]

How to integrate $$\int \left(\frac{x^{n+1}}{n+1}\right)\left(\frac{1}{\sqrt{1-x^2}}\right)\,dx$$ Please help as possible... Thank you
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### Dividing by zero in the cubic formula

See my previous question for a refresher on how Cardano's method for the cubic works. The gist is that where $x$ is any root of the cubic, we know that $x = u+v$, where we have explicit formulae for ...
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### References in Sylow's theorems, solvable groups, nilpotent group and Frattini subgroups

I'm very interested in group theory, i.e Sylow's theorems, solvable groups, nilpotent group and Frattini subgroups. Can anyone tell me some articles, textbooks, notes ... about them? Thanks a lot.
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### Find maximal ideals of a ring

I'm trying to find all the maximal ideal of the ring $\mathbb{Z}[\sqrt{3}] = \{a+b\sqrt{3} : a,b\in \mathbb{Z}\}$. I have found one, that is $A = \{3a + b\sqrt{3} : a,b \in \mathbb{Z} \}$, and I ...
Let $L$ and $M$ be two rigidified line bundles (see below for the definition) over a scheme $X\to S$, and assume we know that $L \cong M\otimes F$ for some line bundle $F$. \text{Is it true ...
### Prove $\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0$ has $n$ negative roots
Let's $n \in \mathbb{Z^+}$, how to $\text{prove}|\text{disprove}$ that: the equation $\boxed{\sum_{i=0}^n \binom{n}{i}^2x^{n-i} = 0}$ has exactly $n$ distinct negative roots. My friend get ...