0
votes
2answers
2k views

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 ...
1
vote
3answers
334 views

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 ...
2
votes
3answers
73 views

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 ...
0
votes
1answer
78 views

What is the following Calculation about?

i'm going through some homework, but there is one thing i don't understand. Our task is to explain the following calculation: Given: $h(x) = \ln(x^4)$ I. $ \begin{align} h(x) = t(x) ...
1
vote
2answers
82 views

How to show that xyz is even( 2|xyz), when $x+y=z$?

How to show that xyz is even( 2|xyz), when $x+y=z$? for example if x=5, y=12 then z=17. And $2\mid 5 \cdot 12 \cdot 17 $ <=> $2\mid 5 \cdot 2 \cdot 6 \cdot 17 $ ok. One way to show that zyz is ...
2
votes
0answers
101 views

question about definition Silverman's AEC

I am sure many of you have Silverman's book the arithmetic of elliptic curves (2nd edition). On page 417, proposition 1.2, he defines $\delta$. He also defines $\xi: G \rightarrow M:\ ...
2
votes
1answer
865 views

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 ...
5
votes
3answers
357 views

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 ...
5
votes
1answer
318 views

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 ...
3
votes
1answer
341 views

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 ...
9
votes
4answers
263 views

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 ...
1
vote
3answers
535 views

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 ...
3
votes
1answer
155 views

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 ...
0
votes
1answer
595 views

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 ...
3
votes
0answers
279 views

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 ...
1
vote
2answers
322 views

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$\subseteq\mathbb N^{\mathbb N}$ ...
0
votes
2answers
554 views

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 ...
4
votes
1answer
142 views

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 ...
2
votes
2answers
298 views

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 ...
1
vote
1answer
965 views

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 ...
2
votes
2answers
224 views

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 ...
0
votes
1answer
32 views

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 ...
2
votes
3answers
1k views

On functions with Fourier transform having compact support

I have another question from Stein & Shakarchi, Complex Analysis. The problem is the following: Suppose $\hat{f}$ has compact support contained in $\left[-M,M\right]$ and let $f(z) = ...
8
votes
2answers
108 views

Number of solutions - please check my solution?

Determine, in accordance to $k$, how many solutions does the given system of equations have: $$ \begin{cases}kx+(k+1)y=k-1\\4x+(k+4)y=k\end{cases} $$ And check, for which values of $k$ this ...
2
votes
3answers
622 views

Finding area bounded by $y=2 \sin{x} - 1$ and $y = \frac{x}{\pi}$

[UPDATE in bold] Find the area bounded by $y=2 \sin{x} - 1$ and $y = \frac{x}{\pi}$ for the range $0 \le x \le \pi$ I don't really know how to start... How can I find the intercepts of the ...
2
votes
3answers
366 views

How can I determine this limit's value?

Today at school we have to determine this limits value, but when the teacher tried, he said maybe you can't determine this limits value without using the Hospital theorem, but try to calculate it ...
4
votes
1answer
619 views

Radon Nikodym derivative proof

Theorem: Let $\mu$ and $\nu$ be two $\sigma$-finite measures on a measurable space $(X, B)$. Then $\nu$ can be decomposed as $$ \nu = \nu_\mathrm{abs} + \nu_\mathrm{sing}$$ into the sum of two ...
0
votes
2answers
3k views

Finding area bounded by 2 lines/curves by integration

When finding area bounded by 2 lines/curves by integration, I believe I 1st need to find the intersection point eg. $y=1-x$, $y=\sqrt{1+x}$ $1-x = \sqrt{1+x}$ ... $x(x+3)=0$ So $x=0 \text{ or } ...
0
votes
1answer
367 views

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 ...
3
votes
5answers
394 views

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 ...
0
votes
1answer
213 views

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 ...
0
votes
1answer
1k views

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 ...
1
vote
1answer
110 views

Elementary Question about Adjunction Isomorphism

One knows by standard Algebraic Geometry that for any morphism $f:X \rightarrow Y$ of schemes one has canonical bijections $$\operatorname{Hom}_X(f*G,F)\simeq \operatorname{Hom}_Y(G,f_{*}F).$$ ...
0
votes
1answer
38 views

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 ...
0
votes
1answer
153 views

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 ...
0
votes
1answer
88 views

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 ...
3
votes
2answers
429 views

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? ...
0
votes
1answer
61 views

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 ...
0
votes
1answer
644 views

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, ...
1
vote
1answer
487 views

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: ...
2
votes
1answer
457 views

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 ...
0
votes
1answer
272 views

Continuity and Differentiability of a function

Would like some guidance. What I've done so far is included. Given, $$f(x,y)=\begin{cases} 0, \text{ if } (x,y)=(0,0)\\ \\ \frac{xy}{\sqrt{x^2+y^2}}, \text{ if } (x,y)\ne (0,0) \end{cases}$$ ...
2
votes
1answer
97 views

Two questions regarding formal proofs

Assume that in a formal proof I have $T \cup \{ \varphi \} \vdash \varphi$ $T \cup \{ \varphi \} \vdash \lnot \varphi$ Question 1: can I then deduce $T \cup \{ \varphi ...
3
votes
2answers
569 views

Similarity of numbers

Let's take two numbers A and B, and take their prime factorizations $A=p_1^{a_1}\cdot p_2^{a_2}\cdot\dots \cdot p_n^{a_n}$ $B=p_1^{a'_1}\cdot p_2^{a'_2}\cdot\dots\cdot p_n^{a'_n}$ Now ...
1
vote
2answers
134 views

How to find $\int \frac{\sin x}{2\cos 2x} \operatorname{d}x$?

How to compute $$ \int \frac{\sin x}{2\cos 2x} \operatorname{d}x$$
3
votes
3answers
2k views

Find out the number of ways the people can be seated in a row

I have a question, There are $4$ boys and $4$ girls and there are $8$ seats. Find the number of ways that they can sit alternatively and certain two of them (a boy and a girl) should never sit ...
3
votes
2answers
107 views

Correspondences $f: X \to 2^Y$

I am reading some notes on correspondences and have a question. (The notes are here.) I have a question about something on page 1. Basically, the notes provide some motivation for why we might want ...
3
votes
1answer
113 views

Clarification on Dirac notation

I am new to the Dirac notation, so would appreciate some clarification. Suppose $\Psi=\psi_1+\psi_2$ where $\Psi$ is normalized and $H$ is a linear operator such that $H\psi_1=E_1\psi_1$ and ...
2
votes
2answers
200 views

Why is there only one way to make $M$ a $\mathbb{Q}$-module, if possible?

Suppose $M$ is a $\mathbb{Q}$-module. Why is the given action of $\mathbb{Q}$ on $M$ (whatever it may be) the only way to make $M$ a $\mathbb{Q}$-module? I know that an abelian group can only be made ...
1
vote
2answers
312 views

Is it possible to define a frame on the Möbius strip?

The Möbius strip is non-orientable, but is it still possible to define a frame of reference? For example, take the edge of the strip (it only has one edge). $$ ...

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