0
votes
1answer
38 views

Boolean rings, subring of the ring its elements come from?

If $B=\{x \in C:x=x^2\}$ then is the boolean ring $B$ a subring of the ring $C$?
1
vote
1answer
98 views

finding the number of sub fields such that $(K : Q) = 2$

Consider the polynomial $f(x) = x^5 - 4x + 2$. Let $L$ be the complex splitting field of $f(x)$ over $\mathbb{Q}$. I want to find the number of subfields $K$ of $L$ such that $(K : \mathbb{Q}) = 2$. ...
0
votes
2answers
48 views

How was this distributed? (Trig equations)

How did this: $2(1-\sin^2x)=1+\sin x$ Become this: $2\sin^2x+\sin x-1=0$ Wouldn't it be: $2 -2\sin^2x-1+\sin x=0 ?$
1
vote
0answers
77 views

Classification problem: admissible rule is a Bayes rule for some prior $\pi$

I have a classification problem where I want to place an observation $X$ into a population described by a pdf equal to either $f_1$ or $f_2$. Given $P_{f_i}(\frac{f_1(X)}{f_2(X)}=j)=0$ for all $j\in ...
5
votes
1answer
177 views

Jech's proof of Silver's Theorem on SCH

Jech's textbook proves Silver's Theorem–that if the Singular Cardinals Hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals–by breaking up the ...
8
votes
2answers
216 views

What is the simplest mathematical concept that does not map to a physical phenomenon?

One of my colleagues argues that everything in math proves something in the physical world. For instance, he claims that the existence of math to describe fractals proves the infinite divisibility of ...
4
votes
2answers
113 views

Regularity of a basis over a local ring

The following Theorem (Thm. 19.9 in Matsumura, CRT) will be necessary for the understanding of my question: Theorem: Let $A$ be a Noetherian local ring and $I$ a proper ideal of $A$ with ...
1
vote
1answer
168 views

.cauchy integral formula.

$\int \frac{e^{zt}}{\sinh z} \mathrm{d}z$ and $|z|=8$ this is the problem about Cauchy integral formula. I keep making a mistake and find the wrong solution. Can you help me? So, we know the Cauchy ...
1
vote
1answer
61 views

Cardinality of a set of functions

Let $A=\{f: \mathbb N \to \{0,1\} : |f^{-1}\{0\}|<\infty\}$. Calculate the cardinal of $A$. The attempt at a solution: It's clear that the set $A$ can be seen as the sequences of natural numbers ...
5
votes
1answer
349 views

Folland PDE chapter 6.C problem 1

Problem 6.C.1: Suppose $0 \neq \phi \in C^\infty_c(\mathbb{R}^n)$ and $\{ a_j \}$ is a sequence in $\mathbb{R}^n$ with $|a_j| \to \infty$, and let $\phi_j(x) = \phi(x - a_j)$. Show that $\{\phi_j\}$ ...
3
votes
1answer
94 views

Polycyclic groups are finitely generated

The definition of a group $G$ being polycyclic that I'm currently learning is: G has a normal series : $e = G_n \triangleleft G_{n-1} \triangleleft ... \triangleleft G_1 \triangleleft G_0 = G$ such ...
0
votes
2answers
66 views

Rational calculator

I am calculating a recursive equation: $X_n = \frac{1}{2}(X_{n-1} + \frac{3}{X_{n-1}})$ I need a free online calculator that can calculate for $n=7$ $(1/2)\times(88063572/50843527+(3\times ...
2
votes
1answer
305 views

Taylor series for $\log(1+x)$ and its convergence

I know the taylor series of $\log(1+x)$. However I don't understand how to find the convergence for $x>1$ and divergence if $x<1$.
2
votes
1answer
45 views

Alternating Series

A professor of mine gave me this problem and asked me to figure it out. I can not seem to figure it out. Express ln($2/3$) as an alternating series and use alternating series estimates to find ...
1
vote
1answer
37 views

Limiting values for logistic function

Given the logistic function (map) $x_{n+1} = r\cdot{xn}\cdot (1 - {xn})$and an initial value $x_{0} = 0.4$ When r = 0.5, i worked out $x_{1}=0.12$ and $x_{2}=0.0528$ How do i work out the ...
2
votes
1answer
41 views

Computing the Fourier transform of a function not in $L^1$?

The Fourier transform is defined on $L^2$ by a density argument. It doesn't seem like it's constructive. So how would one go about computing the Fourier transform of a function in $L^2$ but not $L^1$? ...
3
votes
1answer
224 views

poisson's equation with robin's boundary, boundary value problem

Consider the Poisson’s equation with Robin’s boundary conditions as follows \begin{array}{ll} −\Delta u = f, &\text{in $U$,}\\ \frac{\partial u}{\partial \nu}+u=g, &\text{on $\partial U$,} ...
3
votes
1answer
355 views

Unknown number of colours Bernoulli Urn

Okay, so, in the traditional Bernoulli Urn problem, we have an urn with a number N, possibly infinite, of coloured balls, and there are k possible colours. That one I grok. However, what if I don't ...
2
votes
1answer
43 views

Easy probability ..

I want to make sure these answers are correct A class has 10 freshmen, 8 sophomores, and 12 seniors. On a recent test, 3 freshmen, 5 sophomores and 3 seniors got an A. a) What is the probability ...
0
votes
1answer
2k views

How many flips to get at least one heads and at least one tails?

A coin having probability $p$ of coming up heads is continually flipped until both heads and tails have appeared. Find the expected number of flips. Here's my guess: $$E[\text{at least one head and ...
0
votes
1answer
238 views

How can I derive euclidean distance matrix from gram-schmidt matrix?

This is my first post, sorry for my naiveness.. I know a basic equation that relates Gram-schmidt matrix and Euclidean distance matrix: $XX'=-0.5*(I-J/n)*D*(I-J/n)'$ Where $X$ is centered data (is ...
1
vote
3answers
143 views

Urn Probability (need multiple ways)

I need multiple ways to solve this question. Thank you! There are $A$ black balls, and $B$ white balls in an urn. You select balls one by one from this urn randomly without replacement. What is the ...
6
votes
1answer
87 views

Embedding of continuous functions into differentiable functions

This question refers to a solution printed in the current (December 2013, 120(10)) issue of The American Mathematical Monthly, p. 944. There, the authors intend to show that any ring homomorphism ...
0
votes
0answers
39 views

Question about zero-divisors , rings and polynomials.

Let $i,n,m$ be positive integers. For every nonnegative integer $k<i+1$ , let $a_k$ be elements of a ring $A$ that satisfies : $1)$ The ring $A$ is isomorphic to ${\Bbb R}^{n}\times{\Bbb ...
0
votes
1answer
236 views

Conditional probability involving a geometric random variable

Let $X_1 , . . .$ be independent random variables with the common distribution function $F$, and suppose they are independent of $N$, a geometric random variable with parameter $p$. Let $M = ...
-1
votes
1answer
28 views

Permutation combination questions?

An access pad has 7 buttons. An access code is a sequence of 2, 3, 4 buttons. How many access codes are possible if: buttons may be repeated buttons may not be repeated I need help getting this ...
1
vote
3answers
288 views

Series convergence of $\sin(\frac \pi{n^2})$

I would like to know how, using the limit comparison test, to show that the series $$\sum_{n=1}^{\infty} \sin \left(\frac{\pi}{n^2}\right)$$ converges. (The ratio test in inconclusive. And the ...
-1
votes
2answers
28 views

Discrete Math Functions

$f:\mathbb N\to\mathbb N$ such that $f(x) = 2x$. $f:\mathbb Z\to\mathbb Z$ such that $f(x) = 2x$ How are these two different? And also $h:\mathbb R\to\mathbb R$ where $h(x) = \sqrt x$ $f:\mathbb ...
1
vote
0answers
79 views

Simplification after chain rule differentiation of: $\frac{1+\frac{x}{\sqrt{x^2+1}}}{x+\sqrt{x^2+1}}$

I'm hitting my head against the wall here trying to figure this out... had to determine the derivative of $\,\,\ln\left(x+\sqrt{x^2+1}\right)$. That part was easy enough, chain rule a bit and ended up ...
3
votes
2answers
117 views

2 is a primitive root mod $3^h$ for any positive integer $h$

It's easy to verify that 2 is a primitive root mod $3^2$. But then why does it follow that 2 is a primitive root mod $3^h$ for any positive integer $h$? This was used in the solution of 2009 Putnam ...
3
votes
1answer
508 views

Good book for logic self-study

I know a similar question has already been asked, but can anyone suggest a good book on mathematical logic that includes answers to exercises? I am looking for something that is conducive to ...
1
vote
1answer
98 views

completely continuous implies compact

I'm searching for a proof of the fact that if: $T$ is a bounded operator in a reflexive Banach space that maps weakly convergent sequences onto convergent sequences then $T$ is compact. If we let ...
0
votes
1answer
318 views

Growth and Decay

I am having trouble with this problem. In 1994, rock climbers in southern France stumbled on a cave containing prehistoric cave paintings. A C^14 analysis carried out by French archaeologist Helene ...
1
vote
1answer
76 views

Laguerre polynomials question

Laguerre polynomials $L_n(x)$ can be calculated using the Rodriguez formula $$L_n(x)=\frac{e^x}{n!}\frac{\mathrm{d}^n}{\mathrm{d}x^n}(x^n e^{-x})$$ Show that $L_n(x)$ in the Rodriguez formula ...
0
votes
1answer
34 views

Find $\vec{QB}$ in terms of $\mathbf c$

I've managed to work out “$\vec{AM}$ in terms of $\mathbf a$ and $\mathbf b$” to be $3\mathbf a+\mathbf b$. But how can I work out “$\vec{QB}$ in terms of $\mathbf c$”?
9
votes
2answers
224 views

Transfinite derivatives

I don't know if this is exactly research level, as I am only starting college. But I feel like this is the best place to ask the question. We all know of 1st, 2nd, 3rd, nth derivatives. Is there a way ...
4
votes
2answers
77 views

$p$-adic completion of $\mathbb{Z}[X]$ and $\mathbb{Z}[[X]]$.

Let $p$ be prime. The $p$-adic completion of $\mathbb{Z}$ is the ring $\mathbb{Z}_p$ of $p$-adic integers, and its elements can be thought of as power series in $p$. Is there a nice description of the ...
1
vote
2answers
117 views

Number of subgroups and elements

I have a question that I feel I am going about in a roundabout way, and would like some help on. I am preparing for an exam. Problem: Let $G$ be a group with $|G|=150.$ Let $H$ be a non-normal ...
-1
votes
2answers
80 views

Differential Equation: Determine $x''$ when $x'=xt^2+x^3+e^xt$

Determine $x''$ when $x'=xt^2+x^3+e^xt$ I haven't worked with DE's in a while...would the best approach be to use separation of variables?
1
vote
2answers
34 views

problem with quadratic equation two variable

I have following equation $a^2+4.8ab-b^2=0$ and I have problem with solving it, I don't know why $a=-5 $ or $ a=0.2 $
1
vote
2answers
209 views

(Obvious?) Half-Space Poisson Kernel Estimate

In Stein's Singular Integrals and Differentiability Properties of Functions, Theorem 1 (a) of Chapter 8 (pg. 197) he makes the claim without proof that the Poisson Kernel for the half-space ...
3
votes
1answer
82 views

The diophantine equation $a^n - 1 = (a-1)^m$.

Let $a,n,m$ be odd integers larger than one. The diophantine equation $a^n - 1 = (a-1)^m$ fascinates me. I know that Catalan's conjecture has been proven and that Pillai's conjecture has not been ...
0
votes
2answers
71 views

Curve is a submanifold

In our class today we said today that an 1-times continuously differentiable immersion $\phi:[0,1] \rightarrow \mathbb{R}^n$ is a submanifold of dimension 1 if it is closed such that $\phi(0)=\phi(1)$ ...
2
votes
1answer
71 views

Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n f(x_i)$ exists

Problem: Let $(x_i)$ be a sequence in (0,1) such that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n x_i^k$ exists for every $k=0,1,2,...$ and $f\in C[0,1]$. Show that $\lim_{n \to \infty} (1/n)\sum_{i=1}^n ...
3
votes
1answer
55 views

irreducibility of $m$th cyclotomic polynomial in $\mathbb Z[x]$ implies irreducibility in $\mathbb Q[x]$? (Ireland and Rosen)

In Chapter 13 of Ireland and Rosen's An Introduction to Classical Modern Algebra, they prove that the $m$th cyclotomic polynomial is irreducible in $\mathbb Z[x]$. Immediately afterwards they state a ...
0
votes
0answers
67 views

Geometrically, what is the difference between a “flat face” and a “non-flat” face?

I was curious when I was checking sites like MathisFun, and I came across a pretty unclear system that defines a "flat face" and as a "non-curving" face of a shape; a polyhedron. However, I have to ...
1
vote
3answers
68 views

Prove, by induction, that $5^n + 5 < 5^{n+1}$ for all $n\in\Bbb N$.

Prove, by induction, that $5^n + 5 < 5^{n+1}$ for all $n\in\Bbb N$. Attempt: If $n = 1$, then $5^1 + 5 < 5^{2}$ => $10 < 25$ which is a true statement so the base case holds. Assume $5^k + ...
3
votes
2answers
90 views

Is this operator compact and how do I prove it? [duplicate]

I have a very big problem with the following question: Is the operator $T$ defined by $(Tx)t=tx(t)$, $(0<t<1)$ compact in $L_2(0,1)$? My guess is no and I've tried 3 different approaches to ...
1
vote
1answer
122 views

Proving a subset $A$ of a metric space $(X,d)$ is open

Let $(X,d)$ be a metric space and let $A \subset (X,d)$. Prove that $A$ is open iff for every sequence $\{a_n\}_{n \in \mathbb N}$ such that $lim_{n \to \infty} a_n \in A$, there exists $n_0 : a_n \in ...
0
votes
1answer
53 views

Show $\int_a^b \frac{sin(xt)}{t} \,dt$ is strictly increasing over $[0,\frac{\pi}{a+b}]$

Let $0<a<b$ and $f: \mathbb{R} \to \mathbb{R}$, $f(x) = \int_a^b \frac{sin(xt)}{t} \,dt$, $\forall x \in \mathbb{R}$. Prove that $f$ is strictly increasing on $[0,\frac{\pi}{a+b}]$. Attempt: ...

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