0
votes
1answer
59 views

How can I determine least common multiple for a given number and all numbers before it?

The wikipedia article on least common multiples only talks about determining the least common multiple between 2 numbers. I'm looking for an algorithm that will determine it for a set of numbers 1 .. ...
1
vote
0answers
64 views

A Problem about Galois Extension

This is a basic question about Galois Extension, but I want some details about it. Let $F$ be a splitting field over $\mathbb Q$ the polynomial $x^8-5\in\mathbb Q[x]$. Recall that $F$ is the subfield ...
0
votes
1answer
44 views

Find the antiderivative of the following problem

Find the antiderivative of $\pi(\frac{4}{y^2})$. This has to do with a volume problem. And I'm using the disk method to solve. So the pi needs to incorporated.
1
vote
2answers
63 views

If I remove the premise $a\neq b$ in this question, will the statement still be true?

I encountered this proving problem, I can do the proof but my question is why in the condition/premise we need $a$ to be unequal to $b$? My guess is that even $a=b$, the statement is still true, is it ...
2
votes
1answer
126 views

Behavior of the resolvent near the boundary of the spectrum

My question is, in some sense, a continuation of the question below. Isolated singularities of the resolvent Suppose $T\in B(H)$ has no eigenvalues, pick $x\in H$, $x\neq 0$, and consider the ...
3
votes
3answers
177 views

Sylow $p$ subgroups of $S_{2p}$ and $S_{p^2}$ - Dummit foote - $4.5.45; 4.5.46$

Question is to find Sylow $p$ subgroups of $S_{2p}$ for odd prime $p$ and show that this is an abelian group of order $p^2$ Sylow $p$ subgroups of $S_{p^2}$ for odd prime $p$ and show that this is ...
1
vote
1answer
55 views

Finding number of primes of the form $1010\ldots 101$ (in base 10) [closed]

How many primes among the positive integers, written as usual in the base $10$, are such that their digits are alternating $1$’s and $0$’s, beginning and ending with $1$?
2
votes
1answer
88 views

Name for numbers expressible as radicals

What is the name (there must be one!) for real (or perhaps complex) numbers expressible as radicals? Radical numbers? Solvable numbers? (following the same logic as ‘solvable group’). In other ...
0
votes
1answer
133 views

Parseval's Theorem and Resistance/Current

When current I flows through resistance $R$, energy dissipated per second is the ave. vale of $RI^2$. Let a periodic ( but not sinusoidal) current be expanded in a Fourier series $I(t) = ...
0
votes
1answer
53 views

Find the antiderivative.

What is the antiderivative of x=(2/y)? I have to find the antiderivative to do a volume problem, revolving around the y-axis. I tried doing it, and I think it's (4y^3)/(3)pi
1
vote
0answers
63 views

Prove $t_{n+1} = \frac{(A-1)t_n^2 + (A-2)\sqrt{A}t_n}{A(t_n + \sqrt{A})}$ converges for $A > 1$

I want to show that the following sequence converges, given that $A$ is any real number greater than $1$: $$t_1 = A-\sqrt{A}$$ $$t_{n+1} = \frac{(A-1)t_n^2 + (A-2)\sqrt{A}t_n}{A(t_n + \sqrt{A})}$$ ...
1
vote
2answers
266 views

Does $\{nz^n\}_{n\in\mathbb N}$ converge uniformly in the open unit disc of $\mathbb{C}$?

Let $E$ be the open unit disc about the origin in $\mathbb{C}$. Consider the sequence of functions $\{nz^n\}_{n\in\mathbb N}$ on $E$. I'm trying to show that $\{nz^n\}_{n\in\mathbb N}$ converges ...
2
votes
2answers
35 views

question about norms of linear maps

Suppose $A = (a_{ij})$, $1 \leq i \leq m$, $1 \leq j \leq n$ is an $m \times n$ matrix. then $A: \mathbb{R}^n \to \mathbb{R}^m$ given by $$ A(x) = A(x_1,...,x_n) = ( \sum_{j=1}^n a_{1j}x_j, ..., ...
1
vote
0answers
43 views

Prove following function is continuous

Suppose $f:G \rightarrow \mathbb{C}$ is analytic and define $:\phi: G \times G \rightarrow \mathbb{C}$ by $\phi(z,w)=\frac{f(z)-f(w)}{z-w}$ if $z \neq w$ and $\phi(z,z) = f'(z)$. Prove that $\phi$ is ...
1
vote
1answer
379 views

Using a probability tree to find the probability

In a certain college, 55% of the students are women. Suppose we take a sample of two students. Use a probability tree to find the probability a) that both chosen students are women. b) that at least ...
2
votes
1answer
105 views

What is cardinal of set of all Cauchy sequences?

Here are basically two questions. The first, what is the cardinal of equivalent Cauchy sequences of rationals? I know it's $\beth_1$ because of the set is essentially real numbers. But I want to know ...
5
votes
1answer
324 views

Field extensions of finite degree and primitive elements

Over a field $F$ of characteristic $0$, if every every element of an extension field $K$ has degree less than $n$ over $F$, does this tell us that $K$ is a finite degree extension of $F$? So it would ...
0
votes
3answers
108 views

Top Cohomology of $\mathbb{P}^2$ via Sphere

I am trying to use the cohomology of the sphere to calculate $H^2(\mathbb{P}^2)$. My professor just mentioned there's an argument using the projection $\pi: \mathbb{S}^2 \to \mathbb{P}^2$ and the ...
2
votes
0answers
54 views

arithmetic progression by Dirichlet

The arithmetic progression $a_N=(p-1)N+1$ contains infinitely many primes $q$ by Dirichlet. I have searched this part in wiki, but I din't get any relevant proof. Can any one prove it how $a_N$ ...
3
votes
2answers
110 views

Limit Comparison Test vs Comparison Test

In order to test the convergence of $\sum_{k=1}^{\infty}\frac{\sin^2k}{k^2}$, it is rather easy to comapre $ \frac{\sin^2k}{k^2}$ with $ \frac{1}{k^2}$ and use the Comparison Test ($ 0\le ...
2
votes
1answer
90 views

Exact Sequence Splits

Let $$1\rightarrow N \rightarrow G\rightarrow Q\rightarrow 1$$ be an exact sequence of groups. Question If it splits, then the extension group is a semi-direct product of the quotient by the ...
1
vote
1answer
1k views

Find the volume generated by revolving the shaded region bounded by the given lines and curves about the y-axis.

The region enclosed by x=(y^2)/(4), x=0, y=-4, and y=4. I know my limits are 0, 4. And I have the integral set up. But I'm having issues finding the antiderivative of the functions.
4
votes
0answers
62 views

Group structure on pointed homotopy classes [X,S^1]

Let $[X,S^1]$ denote the set of pointed homotopy classes of maps $f:X\to S^1$. I need to show that, when $S^1$ is viewed as a subset of $\mathbb{C}$, complex multiplication induces a group structure ...
0
votes
1answer
39 views

Function composition, range and domain

I'm not good with word problems, so any input would be appreciated. A and B wrote letters in a code $f$ which consists of spelling every word backwards and interchanging every letter $s$ with $t$. ...
1
vote
1answer
284 views

Finding acceleration at a certain velocity

A race car starts from rest and travels east along a straight and level track. For the first $5.0s$ of the car's motion, the eastward component of the car's velocity is given by ...
1
vote
2answers
74 views

$A^m = r_m(A)?$ Power of a matrix!

In my Linear Algebra textbook we are reading, the following is stated for computing the power of a matrix in one of the advanced chapters as an exercise, $A^m = r_m(A)$. $r_m (A)$ is the remainder ...
1
vote
5answers
124 views

Showing something is not onto?

Quick question..: If I have a linear transformation $T:\mathbb R^2 \to \mathbb R^4$, can I say that since $\mathbb R^4$ is greater than $\mathbb R^2$, it can never be onto - since the elements in ...
0
votes
2answers
51 views

What determines the completeness of a metric space

I am considering whether how we choose a metric function that determines the completeness of a metric space,i.e. we can define a metric function that makes $R^n$ not complete, is that true?
2
votes
2answers
31 views

Simple conditional probability

I think that P(X|Z)=P(X|Y)P(Y|Z) is true, and doing some calculations in two different ways, it appears to be correct. However, I'm not sure and can't seem to prove it. Is it true, and why?
1
vote
1answer
28 views

existance of group with certain unique orders , 6 diferent ones

Does there exist a group of 12 elements such that the orders of its elements are: 12; 12; 12; 12; 6; 6; 4; 4; 3; 3; 2; 1; thought process: I know that order of an element g in a group is the least ...
1
vote
3answers
241 views

Algebra 2 Bonus Question

This was a bonus question on a test I just took: Write a function with a domain of all real numbers and a range of only $2$ numbers. The closest I got to an answer was $f(x)=\frac{x}{|x|}$, which ...
0
votes
2answers
32 views

Statistics: If $X_1$ and $X_2$ are both normally distributed then explain why $X_1 - X_2$ can be standardized with mean 0 and standard deviation of 1

I am currently studying hypothesis testing for two populations and I would like a math major or someone experienced to explain to me why this particular statistic has a mean of 0 and a standard ...
0
votes
0answers
92 views

Condition for existence of solution to a power-tower equation.

"For two positive number a, b which satisfies the condition $\ln a \ln b <0$. equation $a^{b^{a^{b^x}}}=x$ has only one root if and only if ${\frac{d}{dx}a^{b^x}}_{x=t}\geq-1$, where $a^{b^t}=t$" ...
2
votes
1answer
40 views

Every truth function of the inderterminates X and Y is an iterated composition of negations and disjunctions.

I'm reading K.T.Leung and Doris L.C.Chen's Elementary set theory.I can't solve exercise 10: Prove that every truth function of the inderterminates X and Y is an iterated composition of negations and ...
0
votes
3answers
776 views

$4^x-4^{x-1}=24$ what is the value of $(2x)^x$?"

So, every week our teacher gives us a very difficult question worth 1 point and I can never get them right, so I would highly appreciate the person who tells and explains, in good detail, why this ...
-1
votes
1answer
39 views

Showing $\mathbb{Z}_2 \times \mathbb{Z}_2$ and $\mathbb{Z}_4$ have a different number of subgroups of order $2$.

Looking at this example, can someone explain to me what is $_1$, $H_2$, $H_3$, and how they came about it?
0
votes
2answers
54 views

Showing that $\{nz^n\}$ Diverges for $|z| \ge 1$

Goal: I'm looking to verify that everything I've said here is correct. Let $z \in \mathbb{C}$ satisfy that $|z| \ge 1$. Consider the sequence $\{nz^n\}$. I'm trying to show rigorously that this ...
1
vote
2answers
73 views

How to solve this simple trignometric problem?

So this is the question that was given in a textbook and i attempted to win from the book which was saying i was wrong? If $$\frac{\sin\theta + \cos\theta}{\sin\theta - \cos\theta} = ...
2
votes
1answer
108 views

Integrating an exponential times an error function; expansion needed

In an answer to Integrating a product of exponentials and error functions the integral $I(\gamma)$ below is evaluated using a differentiation technique: $I(\gamma)= \int_0^\infty ...
0
votes
2answers
261 views

Number of intermediate fields in non-separable extensions that are also not purely inseparable

If a field extension is separable then we know there are only finitely many intermediate fields. If a field extension is purely inseparable then it is possible for there to be infinitely many ...
2
votes
6answers
106 views

How to integrate $\int^{\pi/2}_0\sin^4xdx$

Sorry if the question is lame but here it The following was given in my textbook $$\int^{\pi/2}_0\sin^4xdx$$ so i integrated it this way $$\implies\int^{\pi/2}_0\sin^4xdx = ...
7
votes
2answers
3k views

What exactly do the sin, cos, tan buttons do on a calculator?

I understand they mean sine, cosine, tangent, but what exactly is the calculator doing when I enter an angle and press those buttons? Edit: To help others better understand my question, my question ...
8
votes
3answers
324 views

Is Ramanujan's approximation for the factorial optimal, or can it be tweaked? (answer below)

Ramanujan's famous factorial approximation, $$n!\approx\sqrt{\pi}\left(\frac{n}{e}\right)^n\root\LARGE{6}\of{8n^3+4n^2+n+\frac{1}{30}}$$ is far more accurate that the Stirling approximation when ...
0
votes
2answers
100 views

How to find $\det(-6A)$, if $\det A=-4$?

How do I solve this? Assume that $A$ and $B$ are $6 \times 6$ matrices, such that $\det A = -4$ and $\det B = -2$. Find $\det(-6A)$.
2
votes
0answers
51 views

When does one need $k$ to be algebraically closed to compute the (co)tangent space with the Jacobian?

Here is a quote from @AlexYoucis who my other question about Zariski (co)tangent spaces: "Let $X=k[x_1,\ldots,x_n]/(f_1,\ldots,f_r)$ be our affine finite type $k$-scheme and ...
1
vote
1answer
51 views

$y'=(1+y^2)^k\quad y(0)=0$ functions

I always noticed that the graphs for $\sinh$ and $\tan$ look very similar. Then I realized it's because they solve the differential equations $$\begin{cases}y'=\sqrt{1+y^2}\\y(0) = 0\end{cases}\quad ...
1
vote
1answer
41 views

Is a simple curve which is nulhomotopic the boundary of a surface?

Let $C$ be a simple curve in an open subset $U$ of $\mathbb R^3$. Suppose that $C$ is nulhomotopic in $U$. Must there exist a homeomorphism $f$ from the closed unit disk $D$ in $\mathbb R^2$ to $U$ ...
3
votes
3answers
65 views

Writing $x^2+y^2+z^2$ as a polynomial combination of $xyz$, $x+y+z$, and $\dfrac1x+\dfrac1y+\dfrac1z$

Can we write $x^2+y^2+z^2$ as a polynomial combination of $xyz$, $x+y+z$, and $\dfrac1x+\dfrac1y+\dfrac1z$? What about $x^3+y^3+z^3$?
1
vote
2answers
63 views

Continuous bijective function

If $f:\mathbb R\to \mathbb R$ is a continuous function satisfying $\vert f(x)-f(y)\vert\geq \dfrac{1}{2}\vert x-y\vert$ for all $x,y\in \mathbb R$, then is $f$ bijective? I believe that $f$ is ...
1
vote
0answers
44 views

A question about choice of signs in expectation

Let $ f_1,...,f_N $ a collection of fucntions, $ \epsilon_1,...,\epsilon_N$ randomized signs ( $\pm1$) with same probability and $ N\in\mathbb{N}$. If $ ...

15 30 50 per page