1
vote
1answer
105 views

Question on Uniform Integrability

Let $\mu(\cdot)$ be a probability measure on $W \subseteq \mathbb{R}^p$, so that $\int_W \mu(dw) = 1$. Consider a function $f: X \times Y \times W \rightarrow \mathbb{R}_{\geq 0}$, with $X \subset ...
8
votes
5answers
1k views

Proving that surjective endomorphisms of Noetherian modules are isomorphisms and a semi-simple and noetherian module is artinian.

I am revising for my Rings and Modules exam and am stuck on the following two questions: $1.$ Let $M$ be a noetherian module and $ \ f : M \rightarrow M \ $ a surjective homomorphism. Show that $f ...
1
vote
2answers
681 views

Finding Transition Matrix

Problem: Find the transition matrix P such that $P^{-1}AP=B$ where: $$A=\begin{bmatrix} 3 & -1 & 0 \\ -1 & 0 & -1 \\ 0 & 1 & 1 \end{bmatrix} \quad\text{and}\quad ...
2
votes
1answer
151 views

Probability Questions

I need help with part (1) of this problem and confirmation I've done the right thing for (2). 1) A survey carried out by a firm found 60% of clients buy products every month and 20% buy high-tech ...
7
votes
3answers
497 views

Equivalence of three properties of a metric space.

Another question about the convergence notes by Dr. Pete Clark: http://math.uga.edu/~pete/convergence.pdf (I'm almost at the filters chapter! Getting very excited now!) On page 15, Proposition 4.6 ...
1
vote
1answer
107 views

$k$-basis of a quotient of ideals in polynomial ring

Let $k$ be a field and consider the ideal $I=(x,y) \subset k[x,y]$. Am I correct in saying that $(x,y)/(x,y)^{2}$ is generated as a $k$-vector space by the class of $x$ and $y$?
1
vote
1answer
326 views

sample variance, co-variance and correlation coefficient

I need help with this little stats question a friend gave me. Q) A random sample of cars produced the following fuel consumption figures, in miles per gallon. For each car, we also know the maximum ...
1
vote
2answers
181 views

An inequality for two complex numbers

I recently saw the following inequality for complex numbers: If $a,b\in\mathbb C$ and $|a + b|$ and $|a-b|$ are each less than or equal to 1, then $$|a| + |b^2|/2 \leq 1.$$ How can one prove this?
4
votes
1answer
176 views

(Non-)Formality of A-infinity algebra implies derived (non-)equivalence?

Take an unital differential graded (dg) $k$-algebra $A$, we can regard it as $A_\infty$-algebra with $m_1$ as differential and $m_2$ as algebra multiplication, and $m_n=0$ or all $n\geq 0$. Take a dg ...
2
votes
1answer
224 views

Upper bound for $n^{th}$ power of a sum [duplicate]

Possible Duplicate: Showing the inequality $|\alpha + \beta|^p \leq 2^{p-1}(|\alpha|^p + |\beta|^p)$ We can use Young's inequality to show that $(a+b)^2 \leq 2a^2 + 2b^2$. Does a similar ...
1
vote
1answer
204 views

The divided polynomial algebra over a field

Let $\Gamma_R[\alpha]$ denote the divided polynomial algebra over $R$; that is, the quotient of the free $R$-algebra $R\langle \alpha_1,\alpha_2,\cdots \rangle$ by the relations $$\alpha_n \cdot ...
1
vote
1answer
500 views

Find point given a line and two angles

Let's say I have two points $p_1=(x_1, y_1)$ and $p_2=(x_2, y_2)$, which are given as two points of a triangle $T$. And let's say I know the angles of $T$ at $p_1$ and $p_2$. How do I find the third ...
4
votes
2answers
498 views

Use of the Reciprocal Fibonacci constant?

The Reciprocal Fibonacci constant ($\psi$) is defined as $$\psi=\sum_{k=1}^{\infty} \frac{1}{F_k}$$ where $F_{k}$ is the $k^{th}$ Fibonacci number. The irrationality of $\psi$ has been proven. ...
0
votes
3answers
1k views

Identifying Asymptotes of a Hyperbola

basic hyperbolic functionHow would I find the vertical and horizontal asymptotes of a $y = \frac{1}{x}$ function algebraically? For example, $y = -\frac{2}{x+3}-1$ (as you would type into a ...
23
votes
4answers
971 views

Is Banach-Alaoglu equivalent to AC?

The Banach-Alaoglu theorem is well-known. It states that the closed unit ball in the dual space of a normed space is $\text{wk}^*$-compact. The proof relies heavily on Tychonoff's theorem. As I have ...
1
vote
1answer
615 views

Greek sigma symbol correct representation?

I wish to display to people that a number is a total. I have decided to use the greek symbol for sum: Σ Do I prefix or suffix numbers with this symbol? Given ...
2
votes
1answer
265 views

Basis to a manifold by coordinate balls

I was trying to show that "Every manifold has a basis of coordinate balls". My approach was like this: Given a manifold $M$, it's well-known that for every point $\mathbf{x}\in M$ there exists a ...
3
votes
2answers
602 views

Fourier Transform of a frequency linearly modulated signal

I'm working on an oscillating signal whose trend can be modelled as a frequency linearly varying function. An example may be as follows: $$ \Gamma(t)=\sin(2\pi\nu(t)t) $$ with $$ \nu(t)=\nu_0 + at $$ ...
2
votes
2answers
604 views

Evaluating $\int y^4(1-y)^3 dy$ using integration by parts

Here is the function which could easily be solved using expansion method but how could I solve it using integration by parts $$\int y^4(1-y)^3 dy$$ The problem is, when I apply integration by parts ...
4
votes
3answers
702 views

Are questions of convergence important in real life?

In the real world, do we ever need to worry about convergence and what not? I am not talking about whether recursive functions and such terminate, but convergence in analysis. It seems like the ...
0
votes
1answer
518 views

Area of a Spherical Triangle from Side Lengths

I am currently working on a proof involving finding bounds for the f-vector of a simplicial $3$-complex given an $n$-element point set in $\mathbb{E}^3$, and (for a reason I won't explain) am needing ...
1
vote
1answer
218 views

Finite subgroup

In the following problem, the only parts I didn't understand were c) and e). The remaining I did. Please, help me! Let $A,B$ be subgroups of $G$ that normalize each other. Assume that the set ...
1
vote
1answer
2k views

Find region of integration where triple integral has maximum value

I have to find the region $E$ where $$ \iiint_E (1-x^2 - 2y^2 - 3z^2) dV $$ has maximum value, but I'm not sure how to start. I was thinking of getting the derivative of the integral and then ...
2
votes
2answers
181 views

Two Equivalent Notions of Algebraic Simple Extension (Proof)

When reading texts about field extensions I've come across the following two definitions for the simple extension $K(\alpha)$ where $\alpha$ some algebraic number over $K$. They are (1) $K(\alpha)$ ...
0
votes
2answers
362 views

Limits problem in Integration

please look at the following question, Let $X$ denote the diameter of an armored electric cable and $Y$ denote the diameter of the ceramic mold that makes the cable. Both $X$ and $Y$ are scaled so ...
1
vote
2answers
1k views

Finding Matrix Representation

Problem: Let T: $\mathbb{R}^3\rightarrow\mathbb{R}^3$ be a linear map given by $$T\left[ \begin{matrix} x\\y\\ z\end{matrix} \right]= \left[ \begin{matrix} 3x-y\\z-x\\z-y\\\end{matrix} \right]$$ ...
0
votes
1answer
72 views

Prove that $\mathrm{Aut}(L(\zeta_n)/K(\zeta_n))$ is a subgroup of $\mathrm{Aut}(L/K)$

Let $L/K$ be a Galois extension of fields and let $\zeta_n$ be a primitive $n$th root of unity in some field extension of $L$, where $n$ is not divisible by the characteristic. Prove that ...
1
vote
0answers
62 views

A dynamic programming problem with continuous states and observations

I have a dynamic programming expressed in the following Bellman backup equation form, $$ V(\boldsymbol{\theta},T)=\max_{i \in N} \mathbb{E} \left[ x_i + V(\boldsymbol{\theta}_{x_i}, T-1) \right] $$ ...
4
votes
1answer
218 views

Weak derivatives

How can I prove that $|\nabla u|=|\nabla|u||$ when $u$ is regular enough for example Lipschitz or $W^{1,1}_{loc}$. Other question is about the pointwise derivative when $f:[0,1]\to R$ is BV is that ...
4
votes
1answer
326 views

Dealing with connectness and compactness of matrices.

Consider the set of all $n\times n$ matrices with real entries, considered as the space $\mathbb{R^{n^2}}$ What can we say about connectedness and compactness of the following sets? The set of all ...
1
vote
3answers
761 views

Formula for the number of latin squares of size $n$?

Is there a "easy to compute" formula for the number of Latin Squares or the number of reduced Latin squares of size $n$?
3
votes
1answer
166 views

Coordinate change for metrics

I am rather confused by the idea of "geodesic polar coordinates", so I hope someone would kindly explain it to me. As far as my understanding goes, given a Riemannian metric ...
5
votes
2answers
620 views

$f$ is integrable, prove $F(x) = \int_{-\infty}^x f(t) dt$ is uniformly continuous.

I am not sure how to do this. I can prove it if I know $f$ is bounded, but otherwise I am stuck. $f$ is integrable, prove $F(x) = \int_{-\infty}^x f(t) dt$ is uniformly continuous.
1
vote
0answers
49 views

Implications of given solutions

This has been solved! Thanks to everyone who read and thought about it Suppose lines of the form $(x_0,y)$ and $(x,y_0)$ for any given $x_0,y_0\in \mathbb R$ are solutions to the system of ...
0
votes
1answer
193 views

Which surface is formed by rotating a hyperbola around its asymptotes?

I don't know even what a type of surface will be. And what equation will be? The equation of hyperbola - $$ xy = l. $$ Now, let's $$ x = x'cos(\varphi ) - y'sin(\varphi ), y = x'sin(\varphi ) + ...
1
vote
3answers
328 views

Extending geodesics to vector fields

Let $c$ be a geodesic on a Manifold $M$. Some books define $c$ to be a Geodesic iff $\nabla_{c'}c'=0$. Therefore for every $c(t)$ the Geodesic must be extendable into a smooth vector field on an open ...
1
vote
0answers
39 views

Goniometric simplification [duplicate]

Possible Duplicate: Proving that $ ...
11
votes
3answers
687 views

The Chinese remainder theorem and distributive lattices

In The Many Lives of Lattice Theory Gian-Carlo Rota says the following. Necessary and sufficient conditions on a commutative ring are known that insure the validity of the Chinese remainder ...
7
votes
3answers
192 views

Jordan decomposition of $A^T$ given that of $A$

Suppose I have the Jordan normal form of a matrix $A$. The decomposition involves the Jordan matrix $J$ and a similarity matrix $P$ such that $P^{-1}.J.P = A$. My question: is it possible to find the ...
2
votes
1answer
227 views

a clique and an independent set

In my textbook, they provide a graph $H$ and then list examples of the cliques and the independent sets in $H$. ...
1
vote
1answer
111 views

find an $A$ so that for all $n \ge A (2^n+3^n)/(4^n+5^n)/(1/1000)$

So $1000(2^n+3^n)\le 4^n+5^n$. I take $1000\cdot2^n\le 4^n$ and $1000\cdot3^n\le5^n$ so that adding both gives the inequality, theorem in the ordered fields. So $1000\cdot2^n\le2^2n$. This leads me to ...
2
votes
0answers
357 views

Characterization of Cauchy-Riemann operator

Let $U \subset \mathbf C$ be an open subset of the complex plane and suppose we have a differential operator of order 1, $L: \mathcal C^{\infty}(U) \to C^{\infty}(U)$ such that $Lu = 0$ if and only ...
30
votes
17answers
6k views

Examples of mathematical induction

What are the best examples of mathematical induction available at the secondary-school level---totally elementary---that do not involve expressions of the form $\bullet+\cdots\cdots\cdots+\bullet$ ...
2
votes
0answers
220 views

$L_2$-norm representation of the function

Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$(see for reference ...
3
votes
0answers
270 views

moving a basis along a curve (parallel transport)

I'm considering a Riemannian Manifold $M^m$ and a Basis $\{X_1 ,...,X_n\}$ of the tangent space $T_pM$. When I consider now the parallel transport $E_i$ of the vectors $X_i$ along a curve c, then the ...
1
vote
1answer
72 views

Multiple random variables with multiple events

I'm getting rather confused with the random variable concept and its distribution in probability, especially when it gets abstract with no actual example to base my understanding on. Take for ...
1
vote
3answers
101 views

If $a^n \equiv b \pmod m$ and $b^n \equiv c \pmod m$, is it true that $(a^n)^n \equiv c \pmod m$?

If $a^n \equiv b \pmod m$ and $b^n \equiv c \pmod m$, is it true that $(a^n)^n \equiv c \pmod m$? From testing cases it seems to be true, but I'm unsure of how to prove this.
10
votes
3answers
3k views

Can the Bourbaki series be used profitably by undergraduates?

Can the Bourbaki series be used profitably by undergraduates and high school students?Are we the target audience? I came across the N.Bourbaki texts while surfing the internet(I have not had the ...
1
vote
0answers
190 views

Application of method of continuity in partial differential equations

Consider a differential operator $$L_t:= (1-t)(\Delta-\lambda) + t L,\qquad t\in[0,1].$$ For any $u\in C^2_0(\mathbb{R}^2)$, we have $$\lambda^2 \|u\|_2^2 + 2\lambda\sum_{i}\|u_i\|_2^2 + ...
2
votes
1answer
61 views

significance test

I've analysed newspapers by counting the language distributions of the articles. The results look like that: ...

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