# All Questions

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### Octave file extension problem

Ok for my math project, my professor requires us to use .m file for octave. And everytime I open my .m file, it opens up with a mathematica file instead of an octave file. Should I uninstall ...
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### What is the sum of these numbers?

I have this series: $$A=1×\frac{n}{2^0}+2×\frac{n}{2^1}+...+k×\frac{n}{2^{k-1}}$$ How can I calculate $A$? I know that the answer must be $2n$. But I do not remember how I did it then! Thanks.
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### Are Euclidean distances a monotone function of inner products?

Does the sum of all pairs of inner-products of k vectors (real) have to decrease if the sums of Euclidean distances between all pairs of $k$ vectors happens to decrease? Similarly-if decrease is ...
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### Quadratic Reciprocity as a consequence of Eisenstein Reciprocity

I was recently looking at the wikipedia page on Eisenstein Reciprocity, which says it "extends Quadratic Reciprocity." However, though the two do seem to be related, I don't completely understand how ...
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### Do we need AC to prove Principle of Dependent Choices

For any nonempty set $X$ and any entire binary relation $R$ on $X$, there is a sequence $(x_n)$ in $X$ such that $x_nRx_{n+1}$ for each $n \in \mathbb{N}$. (Here an entire binary relation on $X$ is ...
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### Variance of an Iterative Normally Distrubuted Function

What is the variance of the following function?? $f(x_n)=f(x_{n-1})(1+\alpha (1/m)+\beta\epsilon_n\sqrt (1/m))$ Hence, $f(x_n)=f(x_0)\prod_1^n(1+\alpha (1/m)+\beta\epsilon_n\sqrt (1/m))$ Where ...
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### freely generating elements in an algebra

Let $(\mathfrak{M}, \tau)$ be a W${}^{\ast}$-Algebra with (finite, normal, etc., whichever nice conditions one may find need for) tracial state. Elements $(a_{i})_{i\in I}\subseteq\mathfrak{M}$ shall ...
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### How to proof that $\lim_{h \to 0}\frac{e^h-1}{h} = 1$ using the definition $e = \lim_{n \to \infty}(1+\frac{1}{n})^n$?

In other words, how I can proof that the two definitions of $e$ is equal? I saw these two definitions while trying to find proofs for $\frac{d}{dx}e^x$ and ${d\over dx}\ln x$, some uses the former ...
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### Principal direction in Dynamical System

Given a dynamical system of the form $$\dot x = F(x)$$ how are the principal directions at the fixed points defined? What is the geometrical meaning? Thanks a lot.
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### Integration Using Tables

I'm trying to solve the following integral via table substitution. $$\int\frac{\cos{x}}{\sin^2{x}-9}\space dx$$ These tables look a little different from my book, but they're more or less identical. ...
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### About the Fundamental Theorem of Vector Space

In a book I found the following: "In a vector space $V$ of all real valued continuous functions of $x$ defined in the interval $[0,1]$, then $(f+g)(x)=f(x)+g(x),\forall f,g \in V$ ...
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### How to evaluate the integral: $I=\int_1^e\frac{\ln^2x-3\ln x+3}{x\ln x-2x}dx$

Evaluate this integral: $$I=\int_1^e\frac{\ln^2{x}-3\ln{x}+3}{x\ln{x}-2x}dx$$ Help me, thanks :/
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### Relationship between output and poles/zeros in the complex plane

Context There are lots of videos online which explain the time domain equivalent of poles depending on their place in the complex plane, but it's only useful for the simplest examples for which we ...
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### domain using level curves?

Let $f(x,y)=4x^2-y^2$. The problem is to determine the range of this function using the idea of level curves. So one sets $f(x,y)=D$, where $D\in\mathbb{R}$. What values can $D$ take?
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### Using Archemedian property prove that $1/n\to 0$ as $n\to \infty.$

Using Archemedian property prove that $$\frac{1}{n}\to 0$$ as $$n\to \infty$$
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### How do I find the Jacobi matrix?

I've never done questions like these, so I would very much like some help. We are given a function $f: \mathbb R^n \to \mathbb R$ given by $f(x)=\langle x,\xi\rangle^2$ where $\langle\,,\rangle$ is ...
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### shorter proof of generalized mediant inequality?

Show $\frac{a_{1}+...+a_{n}}{b_{1}+...+b_{n}}$ is between the smallest and largest fraction $\frac{a_{i}}{b_{i}}$, where $b_{i}>0$. Attempt Assume the largest is $\frac{a_{n}}{b_{n}}\Rightarrow$ ...
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### Will anyone check my primality test?

The proof is very straightforward and simple. We all know that all prime numbers have a last digit of 1, 3, 7 or 9, and I found that any composite number with a last digit of 1,3,7 or 9 is a product ...
Suppose that we have $n$ vectors $v_1,v_2...v_n$ with same magnitude in plane s.t. the angle between $v_i$ and $v_{i+1}$ is $2\pi/n$ then $v_1+v_2+...v_n=0$ for all $n \geq 2$. I can show this by ...