# All Questions

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### Is the density of an absolutely continuous distribution necessarily unique?

Is the density of an absolutely continuous distribution necessarily unique?
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### Is $W^{1,2}_0$ a Hilbert space?

I came across the term $W^{1,2}_0$. Just a quick question on what the 0 means and is it a Hilbert space? I know $W^{1,2}$ is a Hilbert space. Thanks!
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### Is the sum of $z_1 = a + ib$ and $z_2 = a + i(-b)$ real or complex?

$z_1 + z_2 = 2a + i(0)$ Is $z_1 + z_2$ real or complex?
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### How to find the inverse of the Haar (4) matrix?

H4 = (1,1,1,1)(1,1,-1,-1)(1,-1,0,0)(0,0,1,-1)
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### prove limit of exponential function without concept of logarithm

The question is, prove that if $x>1$, then $\lim_{n\to\infty} = \infty$, where $n \in \mathbb N$, without using the logarithmic concept. I came up with a proof, but I'm not so sure about it. So I ...
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### Enough Projectives in Category of Groups

Working on homology and completion a question has arisen in my head. I know that $R$-mod as a category has enough projectives in it, and as such the category of abelian groups has it as they are in ...
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### A good book for beginning Group theory

I am new to the field of Algebra and so far it's looking to me quite tough. So far I have encountered the following books in group theory - Contemporary abstract algebra by Joseph Gallian and Algebra ...
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### EM algorithm for objective HMM M-step

How did step 2 in the derivation below arrive at step 3?
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### Are these two definition of boundedness equivalent?

Definition 1: A set $S \subset M$ is bounded if $\forall x \in M, \exists r > 0,$ such that $S \subset B_r(x) = \{y \in M | d(x,y) < r\}$ Definition 2: A set $S \subset M$ is bounded if ...
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### On the definition of commuting self adjoint operators.

I'm reading Mathematical Methods in Quantum Mechanics by Gerald Teschl and I came across the following exercise whose statement is causing me some troubles. It goes like this: Let $A$ and $B$ two ...
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### Looking for details on historical math anecdote

My memory is very sketchy here so bear with me. A fairly prominent 19th or 20th century mathematician was captured by a military force, probably invaders. He claimed that he was just a civilian, a ...
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### Efficiency of quasiconvex optimization

Summary: Could we minimize quasiconvex objectives in polynomial time? Whenever an objective function of an optimization problem can be formed as a convex function, this is considered as victory. ...
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### Are planes in $3$-dimensions two-dimensional?

Are planes in $3$-dimensions two-dimensional? The reason I ask is because mathematically the $xy$-plane exists in $3$D space but appears to be $2$D, but how can something $2$D be in $3$D space? I ...
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### Is it possible to write $\cos \left( \frac{1}{3}\arccos \frac{37}{64}-\frac{\pi }{3} \right)$ as a radical expression of real number

$\cos \left( \frac{1}{3}\arccos \frac{37}{64}-\frac{\pi }{3} \right)=\frac{{{\left( -37-3\text{i}\sqrt{303} \right)}^{1/3}}+{{\left( -37+3\text{i}\sqrt{303} \right)}^{1/3}}}{2}$, the number inside the ...
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### Limit of a derivative is 1/2

How do I show that $$\lim_{x \rightarrow b} \frac{d}{dx} \frac{xn^x-bn^b}{n^x-n^b} = \frac{1}{2}$$ where n and b are constants and $n>1$. I saw that it is 1/2 graphing it but I think i still ...
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### If $v$ is a valuation and $v(x)<v(y)$ then $v(x+y)=v(y)$.

I was studying from some book and I came across something I haven't been able to justify. Let's suppose se have a field $K$ and an ordered group $G$ (with multiplicative notation) with an extra ...
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### f: R → R and $|f'(x)| ≤ |f(x)|$

Let $f: R → R$ be a function such that $f'(x)$ is continuous and $|f'(x)| ≤ |f(x)|$ for all $x ∈ R$ , if $f(0)=0$ the maximum value of $f(5)$ is My Attempt: I proved that $f'(x)=0$ for $x ∈ [0,1]$ ...
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### prove that $MN \parallel BC$ in an equilateral triangle

$\Delta ABC$ is equilateral with $M$ and $N$ being interior points. if $\angle MAB=\angle MBA=40^{\circ}$ $\angle NAB=20^{\circ}$ and $\angle NBA=30^{\circ}$. Prove that $MN \parallel BC$ from ...
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### Relation between error of estimate and rate of convergence

How is bounds on estimated error of an iterative algorithm related to rate of convergence?
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### Simplify the expression and leave answer in terms of $\sin x$ and/or $\cos x$

$1-\sin^2 x = \cos^2 x$. However, $1-\sin^2 x$ can also be factored using the difference of two squares. I am stuck on whether $1- \sin^2 x$ should turn into $\cos^2 x$ or be factored by using the ...
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### Integral, partial fractions, need explanation for how to get from one step to another.

Can someone explain how they go from the red step to the blue one?
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### Difference quation to state space form

I have a difference equation that I am trying to convert to a linear state space. The equation is $$y(k)=\frac{1}7{(u(k)+u(k-3))}$$ It would be nice if someone can go through the steps. I'm using ...
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### Is this proof correct? Show $\mathbb{Q}$ is dense in $\mathbb{R}$

I like proof by contradictions in showing that $\mathbb{Q}$ is dense in $\mathbb{R}$. But I can't understand this one> https://math.dartmouth.edu/archive/m54x12/public_html/m54densitynote.pdf ...
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### I want to self study systematically pure mathematics? Where do I start? [on hold]

I am an undergraduate student in Mechanical Engineering and I am highly interested in studying pure mathematics systematically.I have a fair amount of knowledge on real and complex analysis, ...
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### Convergence or divergence of infinte power towers of complex numbers

Let $s$ be any complex number, $t = e^s$ and $z = t^\frac{1}{t}$. Define the sequence $a_n$ by $a_0 = z$ and $a_{n+1} = z^{a_n}$. I want to show that the sequence $a_n$ converges to $t$ if and ...
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### The Triple Number Game (n + n + n)

I work at the Science Museum on weekends, and I sometimes present the following puzzle: $$0\;0 \; 0 = 6$$ $$1 \; 1 \; 1 = 6$$ $$2 \; 2 \; 2 = 6$$ $$3 \;3 \; 3 = 6$$ $$\vdots$$ $$n\;n\;n=6$$ The idea ...
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### Laplace Transform Injectivity

Intuitively how can the Laplace transform be injective? You are taking an integral with limits $0$ and $\infty$. So you don't care about the function before $0$. Define $g(x)=x^2$ for $x>0$ and ...
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### Evaluating a Difficult limit!

I have to evaluate a very complicated limit, I've done this task already but I wanna make sure I did it right. The function I have in my hands is  F(\omega)= \tanh \Big[a\cdot ...
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### Are all finite sets measurable?

In my textbook, it says: "Let E be any set with m*(E) < $\infty$. Then E is measurable if and only if there exists a measurable set B with m(B) = m*(E)." There always exists a measurable set of ...