1
vote
1answer
242 views

Finding all simple $R$ modules of a ring.

I was hoping someone had an idea on how to go about solving the following; Find (up to isomorphism) all simple R-modules where i) $R = \begin{pmatrix} \mathbb{Z}/15 \mathbb{Z} & \mathbb{Z}/15 ...
1
vote
1answer
171 views

Question on Group Presentation

Is it true that $\mathbb{Z}^{2}*\mathbb{Z}^{2}$ has a group presentation: $\langle a,b,c,d|ab=ba,cd=dc\rangle$. If not, what is the correct group presentation. Thanks!
2
votes
1answer
227 views

Is this an alternative definition of a Borel set?

I'm trying to get my head around Borel sets (like so many before me!) IF I've understood them correctly, the definition on http://en.wikipedia.org/wiki/Borel_set seems unnecessarily complicated. Isn't ...
0
votes
1answer
165 views

How to formally prove that $f(n)=\Theta f(n+1)$

How to formally prove that $f(n)=\Theta f(n+1)$? It's supposed to be easy, but I still can't get it. Thank you very much.
1
vote
0answers
107 views

Partition of unity. Functional analysis

Need to find partition of unity in case of oparator $A_{f}(x)=(|x-1|+x)f(x)$. Operator $A \in L_{2}[0,2]$ Partition of Unity is set of operators $E_{\lambda}=E((- \infty,\lambda]) $, where ...
3
votes
1answer
307 views

Is a basis for the Lie algebra of a Lie group also a set of infinitesimal generators for the Lie group?

Let $G$ be a (EDIT: connected) Lie group of dimension $n$, and let $\mathfrak{g}$ be the associated Lie algebra. If $x_1,\ldots,x_n$ is a basis for $\mathfrak{g},$ is it necessarily true that the ...
0
votes
1answer
166 views

permutation of “counting out”

Josephus problem*: circle=1,2,3,4,5,6,7,8,9,10. count=2. (Beginning at 1) The "last man standing" in this case=9. Order of elimination or permutation (?): 2,4,6,8,10,3,7,1,9 For any size circle and ...
2
votes
1answer
87 views

Almost invariant subspaces for WOT closure of an algebra of operators

Let $X$ be a Banach space and $\mathcal{C}\subset\mathcal{L}(X)$ be a collection of bounded linear operators. A subspace $Y$ is said to be almost invariant under $\mathcal{C}$ if for each ...
1
vote
1answer
95 views

Notation question for denoting an ideal of a polynomial ring

Let $k$ be a field. Let $q=(x,y^2)$ be an ideal of $k[x,y]$. What exactly does the notation $q=(x,y^2)$ mean, i.e. what kind of elements does $q$ contain? Is it the set of all elements $\alpha x + ...
0
votes
1answer
90 views

existence of infinite abelian subgroup in infinite locally compact groups

1) Let $G$ be an infinite locally compact group. Does there exist an infinite abelian locally compact subgroup of $G$? Rem: I know that there exists an infinite abelian subgroup in every infinite ...
3
votes
2answers
2k views

Shorter proof of $R/I$ is a field if and only if $I$ is maximal

Here is a proof I saw somewhere of the fact $R/I$ is a field if and only if $I$ is maximal: $\implies$ Suppose that $R/I$ is a field and $B$ is an ideal of $R$ that properly contains $I$. Let $b \in ...
1
vote
0answers
166 views

Values of a covariance matrix

Is it possible to have values greater than 1 in a covariance matrix? Actually, I created a precision matrix Q and then inverted it to get the covariance matrix. And I have values greater than 1. Is it ...
0
votes
1answer
117 views

How to prove inequality

How to prove this inequality: $$a^2 d^2+b^2 c^2-1-4ac-4bd-2abcd \leq 0,$$ where: $a, b, c, d \in \{0, 1, 2, \ldots\}$ and $|a-c|\leq 1, |b-d|\leq 1$?
1
vote
0answers
221 views

Generating spatially correlated samples from a multivariate normal distribution

I am trying to generate some spatially correlated samples from a multivariate normal distribution following this algorithm Compute Cholesky factorization Q=LL' Sample z~N(0,I) Solve L'v=z Computer ...
0
votes
1answer
82 views

Explanation of a weird result regarding continuous conditional probabilities?

Suppose I have two continuous random variables $X, Y$ with joint probability density $f(x,y)$. Then by marginalizing over $y$ I can derive $$ f(x| Y=y) \\ = \int{f(x, \hat{y}|Y=y) d\hat{y}} \\ = ...
1
vote
0answers
108 views

Does this use the mean value theorem?

Let $f:[a,b] \to \mathbb{R}^n$ be a smooth curve such that the estimate $\|f'(t)\| \leq f'(t)$ holds. Show that the estimate $\|f(b) - f(a)\| \leq f(b) - f(a)$ also holds. I was started to use the ...
1
vote
1answer
238 views

Riemann integral Vs Ito Isometry “Paradox”

Let $Y(T)=\int_T B(t) dt$ an integral. We wold like to evaluate $E(Y(T)^2)$ Now, $Y(T)$ may be Riemann integrated because dt have finite absolute variation and $B(t)$ is continuous. Then we can take ...
0
votes
1answer
72 views

Mental math tip needed; moving decimal around on larger and smaller numbers?

When I do calculations I like to round off to say nanometers for wavelength. That means I need it in the form $whatever \times 10^{-9}$. The problem here is that regardless of the number I manage to ...
4
votes
2answers
6k views

Are Euler angles the same with pitch roll yaw

I am wondering if pitch-roll-yaw is used to represent Eular angle? If not, what's the relationship between them? From wiki, I know that Euler angle is used to represent the rotate from three axes ...
2
votes
2answers
189 views

Solving for an exponent?

$1^x = i$ I can't solve it through logs, because $\log 1 = 0$. Does this mean $x$ is undefined?
0
votes
1answer
78 views

Angle functions - Determine the angle needed to slide one curve into another

Working on the angle functions, there's a problem that says: Determine the angle needed to slide the $\cos$ curve into the $\sin$ curve. The solution is described as $\cos\Big(x - ...
1
vote
1answer
296 views

Help Understanding Solution? (Putnam and Beyond)

From Putnam and Beyond: Proofs (Question 10:) Let $n>1$ be an arbitrary real number and let $k$ be the number of positive prime numbers less than or equal to $n$. Select $k + 1$ positive ...
1
vote
0answers
82 views

Problem on the maximum of the function involving Stirling numbers of the second kind

When reading about the Coupon collector’s problem (CCP), I just came up with the following problem which I found very curious about, though it seems to have neither relation to the CCP nor practical ...
1
vote
1answer
88 views

trouble with understanding notation - partition of unity, section

I am currently working with a book ("Fourier Integral Operators" by J.J. Duistermaat) that mentions a differential geometric construction that I struggle to understand. Here is the setting: Suppose ...
0
votes
0answers
79 views

A Brownian motion starting from 0, it becomes -1 once reach -1, what is its expectation?

$X_t$ is a Brownian Motion, it reaches becomes -1 forever once it reaches -1. Mathematically, $T = \inf\{t: W_t = -1\}$ is a stopping time. When $t<T$, $X_t = W_t$ ,while $t\geq T$, $X_t = -1$. ...
2
votes
1answer
138 views

Is kernel of a function related to symmetry groups?

Define kernel of a function $f : X \rightarrow Y$: $$ \text{ker}\space f = \{\space (x, x')\space |\space f(x) = f(x') \space\} $$ The kernel contains pairs of equivalent inputs to $f$. Number of ...
1
vote
1answer
42 views

Correlation of bird feeding and bird seeing?

i have made a 3 column table for my garden data collection. The columns are: day, food given, bird seen. Day is from 1 to 31 Food given is blank if i missed that day, or 1 if i put food out and 2 if ...
2
votes
1answer
64 views

Quotients and continuous maps

Suppose $f:\mathbb R\times \mathbb R\to \mathbb R/\mathbb Z\times \mathbb R/\mathbb Z$ where the domain is given the usual topology and the latter the quotient topology. Why then is the restriction ...
1
vote
1answer
273 views

Aldous criterion for tightness in $D[0,1]$

Does anyone know where I can find some useful information about the Aldous criterion for tightness in the space of all cadlag functions $D[0,1]$?
10
votes
1answer
595 views

Sheafification of a presheaf through the etale space

I have some problems to show that the following contruction defines a sheafification: Let $\mathcal F$ be a presheaf on $X$, and let $Et(\mathcal F)$ be the etale space associated to $\mathcal F$, ...
3
votes
2answers
336 views

How do I prove that a complex number equals infinity?

I need to prove that $z=x+iy$ equals infinity is equivalent to $x = \infty$ and $y=\infty$. I also have to give an example of a complex number $z$ so that $\sin(z)=\infty$.
0
votes
1answer
195 views

Zero eigenvalue in inertia calculation

I'm computing the principal axes and moments of a triangle at its centroid. The triangle vertices are at the points $(12,0,0)$, $(0,24,0)$, $(0,0,36)$. According to my calculations, the relevant ...
7
votes
1answer
237 views

homotopy direct limits

$X$ is said to be the homotopy direct limit of the sequence of subsets $X_1\subset X_2\subset ...$ if the projection $\cup_i X_i\times [i,i+1] \rightarrow X$ is a homotopy equivalence. The ...
0
votes
1answer
182 views

How can I know the two functions of the following lines?

I didn't use math since ages , now I am building a game (I am a programmer) and I need a mathematical function to draw the following 2 lines The red thick line goes from x=1 to x=10 (it may go ...
23
votes
2answers
902 views

Are there examples that suggest the Riemann Hypothesis might be false?

Are there examples that might suggest the Riemann hypothesis is false? I mean, is there a zeta function $ \zeta (s,X) $ for some mathematical object $X$ with the properties $ \zeta (1-s,X) $ and ...
1
vote
1answer
115 views

Initial-Value Problem

$dy/dx = e^{-x^2} - 2xy $ $y(0) = 1$ expressing in the form $y = f(x)$ I was thinking of seperation of variable and the integrating factor method but I don't think it will work. What should I ...
10
votes
2answers
1k views

composition of covering maps

The origin of my question arose from a problem: Let $q: X \to Y$ and $r: Y \to Z$ be covering maps, let $p= r \circ q$. Show that if $r^{-1}(z)$ is finite for each $z \in Z$, then $p$ is a covering ...
2
votes
2answers
101 views

How can I compute this definite integral?

How to compute $$\int_0^1\frac1{1-x^n}dx\;,$$ where $n>0$?
2
votes
2answers
452 views

Converging sequence and subsequences

How might we rigorously argue that if we have a sequence $\{x_n\}\subset X$ such that every subsequence of it has a convergence subsequence that tends to $a$ and $X$ is a compact set then $\{x_n\}$ ...
4
votes
3answers
5k views

Expected Value of the maximum of two exponentially distributed random variables

I want to find the expected value of $\text{max}\{X,Y\}$ where $X$ ist $\text{exp}(\lambda)$-distributed and $Y$ ist $\text{exp}(\eta)$-distributed. X and Y are independent. I figured out how to do ...
1
vote
1answer
102 views

Topology on Euclidean Group

Wiki says that the euclidean group: http://en.wikipedia.org/wiki/Euclidean_group is a topological group. Can you explain me what is the topology we take on it? Thanks !
2
votes
1answer
77 views

Minimizing a norm to get a solution of a pde

Let $\Omega$ be a regular bounded open subset of $\mathbb{R}^3$. The problem is to solve the following pde: $$\left\{\begin{array}{c c}-\Delta u = u^3 & (\Omega)\\u = 0 ...
1
vote
1answer
85 views

Meaning of Strong Primitivity

As J.D.Dixon noted in his great book; Permutations Group, we can speak about Strong Primitivity of a group acting on a set $\Omega$ by means of orbital graphs. The way he paved employes digraphs prove ...
2
votes
1answer
356 views

Change of coordinate system on a sphere

This might take a while to explain, so bear with me: I've got a perfect sphere. I've set up an arbitrary longitude/latitude ("angle") coordinate system on it (imagine an equator around the middle, ...
2
votes
1answer
125 views

Finite Rooted Binary Trees

I am new to learning about finite rooted binary trees. This lemma below is from John Meiers book: Groups, Graphs and Trees. There is no aval proof in the book. I was just wondering is I could catch a ...
8
votes
5answers
1k views

If the localization of a ring $R$ at every prime ideal is an integral domain, must $R$ be an integral domain?

Let $R$ be a commutative ring. Suppose that for every prime ideal $p$ of $R$, the localized ring $R_p$ is an integral domain. Must $R$ be a integral domain? I was trying to think of counter-examples, ...
0
votes
2answers
86 views

Second Order differential eqn.

Given differential eqn of, $y'' + 16y = \cos(4x) - 8e^{3x}$ Initial conditions $y=0$ and $dy/dx = 0$ when $x = 0$ So therefor, $r^2 + 16 = 0$ where $r$ will be $4i$ so, $Y = C\cos(4x) + ...
-1
votes
1answer
127 views

On order automorphisms group of some abelian group

Please determine order The automorphisms group of the group $Z_{8}\times Z_{4}$, where $Z_{8}$ and $Z_{4}$ are cyclic groups of order $8$ and $4$, respectivly.
8
votes
2answers
594 views

Set and its Complement are Measure Dense

I'm going over old comprehensive exams and part of one question is giving me a bit of trouble. It asked for an example of a subset of the real numbers such that the set and its complement were measure ...
4
votes
2answers
170 views

Exposition on Modular Curves

I was recently reading this paper by Weston, whereby he talks about the modular curves $X_0(11)$ and $X_1(11)$. I was wondering if anyone can recommend a more general exposition of modular curves ...

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