# All Questions

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### How to get/approximate distance between 2 close points (given in latitude/longitude)?

I have 2 points with their latitude/longitude coordinates and I know that they are in a X miles radius circle (let's say 10 miles radius) somewhere on earth where it's populated (I mean not near the ...
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### What is the “limit point of a function?”

I am asked to prove that Show that a sequence $x: \mathbb{N} \to \mathbb{R}$ has a limit point iff there $\lim_{n\to\infty} x(n)$ exists as a limit point of a function from a subset of metric ...
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### expanding requirements for equivalent conditions

We have all seen statements about equivalent conditions, such as If any one of the following three conditions hold, then all three conditions hold. Are there any examples of three conditions which ...
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### Prove a sequence converges using the definition of a limit?

Here is the sequence: $$a_n = \frac{n^2}{cn^2 + 1} \mbox{ where } c < 0.$$ If I prove this function has a limit using the limit definition, as $n$ goes to infinity, does that prove the sequence ...
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### Good lecture optimization problem involving $\ln x$ or $e^x$

I am teaching a Calc 1 of sorts, like a slightly easier version of Calc 1 with no trig. I want a good optimization/practical problem to do in lecture that involves $\ln x$ or $e^x$, to combine review ...
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### Heisenberg Uncertainty Principle

The uncertainty principle (UP) comes up in engineering and physics, but it is a mathematical idea. An old text describes it as "reciprocal spreading." If $f$ is a well-behaved function, the UP might ...
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### Separability of $l^\infty$

I am trying to prove that $l^\infty$ is not separable. However, I have proved it is separable. Can you help find the flaw in my logic? (In case my teacher uses different definitions from everyone ...
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### Prove that $\lim \limits_{n\to\infty}\frac{n}{n^2+1} = 0$ from the definition

This is a homework question: Prove, using the definition of a limit, that $$\lim_{n\to\infty}\frac{n}{n^2+1} = 0.$$ Now this is what I have so far but I'm not sure if it is correct: Let ...
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### Burnside's Formula to determine rotationally indistinguishable necklaces

Given M beads on a string and N colours, determine using Burnside's formula, the number of rotationally indistinguishable necklaces, where the group acting is a cyclic group. Any tips/ hints would be ...
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### Subspace generated by permutations of a vector in a vector space

Let $K$ be a field. Consider the vector space $K^n$ over the field $K$. Suppose $(a_1,a_2, ... ,a_n) \in K^n$. What is the dimension of the subspace generated by all the permutations of ...
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### Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere?

Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere? I found this question here, the question seems much interesting but for obvious reason it is closed there, I was ...
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### Limit at infinity of a complex function

If $f(z)$ is an entire function such that $$\lim_{x \rightarrow -\infty}\frac{f(x)}{|f(x)|}=1$$ where $|f(x)|$ is the modulus of $f$, and $f(x)$ is just evaluating $f$ at real $x$. What can we say ...
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### Set Theory- Uncountable sets

Can someone help me finish my solution? Question: Show that there are sets $A_{ij}$ for $i,j$ ∈ $\mathbb N$ such that for no countable $\space$H$\subseteq\mathbb N^{\mathbb N}$ ...
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### bitwise operations

so my question is what is the order of operations for bitwise operators << & | and also to see if my logic is right with the problem below (x03 << x08)+ x00 = 300 ...
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### The convergence in $L^p$ space

If a sequence ${f_i},f \in {L^p}([0,1]){\kern 1pt} {\kern 1pt} (1 < p < \infty )$ such that ${f_i}$ converges weakly to $f$ and ${\left\| {{f_i}} \right\|_p} \to {\left\| f \right\|_p}$, then is ...
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### Uniqueness of the Quotient Topology

Let $q:X\to Y$ be a surjective map, where $(X,\tau_X)$ is a topological space. The quotient topology $\tau_q$ on $Y$ is given as $U\in \tau_q$ iff $q^{-1}(U)\in \tau_X$. Suppose that there is ...
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### Power function for this test

Suppose you have a sample from a normal population with mean mu, and known variance $\sigma^2$. What is the power function for $H_0: \mu = 0$ versus $H_a: \mu \ne 0$ at $\alpha = 0.05$? Attempt: If ...
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### Why is a simple group of order $60$ embedded in $A_6$?

Let $H$ be a simple group of order $60$. I am trying to see why it is embedded in $A_6$. $H$ must have $6$ Sylow-$5$ subgroups and $H$ acting by conjugation on these subgroups gives an embedding of ...
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### Terminology concerning a certain solution to a certain system of equations

Say you have a solution $\textbf{x}=(x_1,x_2,\ldots,x_n)$ to a system of equations. It turns out that $-\textbf{x}$ is also a solution. Is there accepted terminology for such a pair of solutions? (I ...
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### Simple Bayes Network with Conditional Independence Problem

I am currently taking a free on-line AI class offered by Stanford (ai-class.com). It is the first time I am exposed to Bayes Network/Probability. I am having a little problem with the following Quiz ...
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### The Pigeon Hole Principle and the Finite Subgroup Test

I am currently reading this document and am stuck on Theorem 3.3 on page 11: Let $H$ be a nonempty finite subset of a group $G$. Then $H$ is a subgroup of $G$ if $H$ is closed under the ...
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### The Galois orbit of an algebraic number

Let $\alpha$ be an algebraic number and let $S$ be the orbit of $\alpha$ under the action of $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$. Do we have that $\# S$ is bounded from above by the ...
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### Proof $f(f^{-1}(x))=x$

Ok, given $f: A\rightarrow B$ is bijective. How can I prove now that $f(f^{-1}(x))=x$? It must be injective and surjective, but how is it possible to pick an element from $A$ and show after applying ...
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One knows by standard Algebraic Geometry that for any morphism $f:X \rightarrow Y$ of schemes one has canonical bijections $$\operatorname{Hom}_X(f*G,F)\simeq \operatorname{Hom}_Y(G,f_{*}F).$$ ...
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### Expected energy of a 2 state system

Suppose I have a 2 state system s.t. $H\psi_1=E_1\psi_1$ and $H\psi_2=E_2\psi_2$. And I have $\phi_+={1\over \sqrt2}(\psi_1+\psi_2)$ and $\phi_-={1\over \sqrt2}(\psi_1-\psi_2)$ Would I be right to ...
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### Mean ergodic theorem

Let $U$ be a unitary operator and $H$ a Hilbert space. $I := \{ v \in H | Uv = v\}$ and $A := \{ Uw - w | w \in H\}$. I would like to show that $A$ is dense in the ...
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### Searching name for string concatenation problem

I am basically only looking for a name of a problem so I can find information about it. A friend of mine explained it to me like this: Given is a set of string variables $(x_1, \ldots, x_n)$, and a ...
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### Irreducible representations of a cyclic group of order p over a field of q elements when p and q are distinct primes

What is a sufficient condition for the existence of an irreducible representation of degree $n$ of the cyclic group of order $p$ over the field of $q$ elements when $p$ and $q$ are distinct primes? ...
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### Calculating new score weight

See I have a test combined from 3 segments. The first two are 15% each, and the third is 70%. I can't do the second part, so they told me the scores would be 20% and 80% instead. Now, the scores I ...
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### Problem with RSA encryption

Recently I was looking at the RSA encryption scheme and decided to do some examples but this seemingly simple one is bugging me a lot. I chose $p=13$, $q=17$. Let $e=131$, be the encryption key. So, ...
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### Prove that there exists a nearest point in a closed set $A \subset \mathbf{R}^n$ to a point outside of $A$

I know that the assertion can be proved by a direct application of the Bolzano-Weierstrass Theorem. I am interested in proving this using the extreme value theorem for continuous functions. Claim: ...
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### Balancing weights with weights

We have a collection of items of weight $d_i$, $$d_1, d_2, ..., d_k, \quad k \le 100$$ where some of the weights may be equal. Let $$n = \sum_{i=1}^k d_i$$ I need to figure out quickly if this ...
Would like some guidance. What I've done so far is included. Given, $$f(x,y)=\begin{cases} 0, \text{ if } (x,y)=(0,0)\\ \\ \frac{xy}{\sqrt{x^2+y^2}}, \text{ if } (x,y)\ne (0,0) \end{cases}$$ ...