# All Questions

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### Sums of partially dependent Bernoulli random variables

I am looking for any kind of Chernoff type large deviation bound for the following random variable: $$X = \sum_{i=1}^NX_i$$ where each $X_i$ is an identically distributed Bernoulli random variable ...
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### What is $0\div0\cdot0$?

We all know that multiplication is the inverse of division, and therefore $x\div{x}\cdot{x}=x$ But what if $x=0$? $0\div0$ is undefined so $0\div0\cdot0$ should be too, but whatever happens when we ...
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### Induction to prove regular expression

Prove that is if S and T are any regular expressions over the one-letter alphabet, (for example: Σ = {a}), and if n is any natural, then the languages (ST)^n and (S^n)(T^n) are equal. I have to use ...
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### Show that the smallest positive value of $ax+by$ is equal to $gcd(a, b)$.

I already read the proof of the following theorem (Abstract Algebra by T. W. Judson): Let $a$ and $b$ be nonzero integers. Then there exist integers $r$ and $s$ such that $gcd(a,b)=ar+bs$. ...
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### Number of left cosets equals number of right cosets?

So I've been working on abstract algebra out of John B. Fraleigh's 3rd edition text. In the exercises of chapter 11, I came upon a question which I cannot even begin to solve. "Show that there are ...
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### Fourier transform to find an harmonic function (Strauss)

I am trying to solve one of the problems of section 12.4 of the book "Partial Differential Equations" by Strauss. The problem says: Use the Fourier transfor in the $x$ variable to find the harmonic ...
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### Open sets in $\mathbb{R}$ are countable unions of disjoint intervals--not a duplicate

I have seen many proofs of this and they all seemed somewhat technical to me. I'd like to argue as follows, and would appreciate your comments: We use only these two easy facts: $(1)$: The union of ...
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### What is the difference between orthogonal and orthonormal in terms of vectors and vector space?

I am beginner to linear algebra. I want to know detailed explanation of what is the difference between these two and geometrically how these two are interpreted?
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### How can I show that and $n\times{n}$ matrix of the form in the description has a determinant of zero for $n>2$?

In General, $n>2$, $a_{i,j}=a_{i,j-1}+1$ and the matrix will be of the following form: ...
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### Differentiate vector function wrt vector

I have a function $\frac{df(\mathbf{y})}{d\mathbf{y}}=\mathbf{y}g(\kappa)$ where $\kappa=||\mathbf{y}||_2$ and $g(\cdot)$ is a scalar function. Thing is when I differentiate this function I get a ...
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### Additional assumptions on function to ensure uniform convergence

Given a sequence $u=(u_n)_{n\geq1}$ converging to $1$, I would like to prove uniform convergence of the sequence of functions $f_n$ defined by $f_n(x)=f(u_n x)$ for $x\in\mathbb{R}_+$ to the function ...
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### Positive hyperplane? Is there a name for these type of hyperplanes?

Consider an affine hyperplane $\{ x : \langle x,v\rangle=a \}$ where $a\ge 0$ and $v\in\mathbb{R}^n_{+}$. That is, both the level $a$ and the vector $v$ are non-negative. Is there any special name ...
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### The simplification of divided difference of cosine function

What is the following limit? $$\lim_{h \to 0}\frac{\cos(\pi/2+h)-\cos(\pi/2)}{h}$$ Why when simplified do you get $(-\sin(h))/h)$?
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### Geometry of images of maps $f: \mathbb R \to \mathbb C$?

I am having trouble seeing what a continuous map $f: \mathbb R \to \mathbb C$ might look like. If it was linear it would look like a line but it's not clear to me what happens if it's any map. I ...
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### Is it possible to factor a quadratic equation when $a$, $b$, and $c$ are all equal?

I have the equation $4x^2+4x+4$ to factor. I know that need to start with $$(2x \quad )(2x \quad )$$ to make $4^2$, but I can't seem to factor the rest of the way. What should I do?
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### How do you evaluate elements of the Drinfeld double $D(H)$ against elements of $H^\ast$ or $H$?

Proposition 8.2 of Majid's primer on quantum groups says that if $H$ is a finite dimensional Hopf algebra with quantum double $D(H)$, then this is a factorizable Hopf algebra with quasi-triangular ...
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### If I have a function that's continuous and it's limits at $\pm \infty$ are $\pm \infty$ is it surjective?

I was trying out some problems where I needed to prove that a function was surjective, and I thought I could do this, is this true? Intuitively, it seems so. If I have a function that's continuous ...
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### For a root system, why does $\beta\in\Delta_+\setminus\{\alpha_i\}$ imply $(\beta+\mathbb{Z}\alpha_i)\cap\Delta\subset\Delta_+$?

Let $\mathfrak{g}(A)$ be a Kac-Moody algebra for a matrix $A$, with root basis $\{\alpha_1,\dots,\alpha_n\}$. There is a remark on the bottom of page 6 of Kac's Infinite Dimensional Lie Algebras ...
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### Visualizing the factorial

Often in basic mathematics, we can visualize things very easily, which I believe helps understanding (instead of just working out a number theoretical proof). For example: $$(n+1)^2 - n^2 = (n+1) +n$$ ...
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### For any $N$ and $B$, is there always a $B$-smooth relation $x + y \equiv 0 \pmod{N}$?

Let $N$ be any integer and $B \geq 2$ be a smoothness bound. Does there always exist $B$-smooth integers $x,y$ such that: $$x + y \equiv 0 \pmod{N}\text{ ?}$$ My only progress is that I know the ...
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### There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x^b_{1}+x^b_{2}+\cdots+x^b_{k}|\ge 1$

This problem is a 2014 Sydney mathematics competition problem (11 grade). It seems difficult to solve. (I previously posted the n=2 case for which André Nicolas and Dan Robertson proposed solutions) ...
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### Of strings and substrings: A problem of probability

Problem Let $\Sigma=\{a, b\}$. Let $\Sigma^*$ denote the Kleene star of $\Sigma$: \begin{equation*} \Sigma^* = \{\varepsilon, a, b, aa, ab, ba, bb, aaa, aab, \ldots\} \end{equation*} where ...
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### What is the motivation behind this (tested, working) 2d coordinate transform?

I am working with programming and 2d geometry and need to transform between two different coordinate systems. I have two different representations of the same world, where I can sample any point and ...
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### Partial D.E. Change of Variables

Can anyone tell me how to derive the value for alpha and beta, I'm guessing 6 and 1 in one order or another - "via quadratic". In transforming the equation, could someone show also how the operator ...
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### How to show that action of an algebra $H$ on a vector space is the same as the coaction of $H^*$ on $V$?

Let $H$ be a Hopf algebra and $V$ a finite dimensional $H$-module. How to show that action of an algebra $H$ on a vector space is the same as the coaction of $H^*$ on $V$? Thank you very much.
Show that if $A,B,C,$ and $D$ are sets with $\lvert A\rvert=\lvert B\rvert$ and $\lvert C\rvert=\lvert D\rvert$, then $\lvert A\times B\rvert=\lvert B\times D\rvert$. Prove that a natural ...
Edit: I think I misunderstood the problem. Upon reading my textbook again, I think what they mean by $F(x,y,z)=<yz,2xz,e^{xy}>$ ; C is the circle $x^2+y^2=16, z=5$ is just literally a ...