0
votes
0answers
24 views

Finitely generated modules over principal ideal domain

Let A be principal ideal domain with field of fractions K. L is finite separable extension of K and B is integral closure of A in L. It is obvious that there exists constant d in A, such that dB is ...
0
votes
1answer
17 views

Demonstration with fitch notation and quantifiers

I'm tryng to demonstrate with fitch notation this: {∀x (A(x) ↔ B(x)), ∀x (A(x))} |= ∀x (B(x)) Here what I tried: http://i.stack.imgur.com/7S5Zy.png Someone can explain me how i can obtain ∀x (B(x)? ...
0
votes
1answer
28 views

Rational Funtion Integration

This looks to be a simple problem, but it has me stumped. I already have the answer, but a step-by-step solution would be appreciated. $$\int\frac{x+4}{x^2+2x+5}$$
-1
votes
0answers
15 views

What's the answer to this number and digit problem? (I'm confused at “exceeds three times the tens digit by 3”)

The units digit of a two digit number exceed three times the tens digit by 3. If the tens digit is subtracted from the units digit, the difference is 7. Find the number.
2
votes
3answers
32 views

Find tangent to trigonometric function

I want to find the tangent to the curve: $x\sin{y} + y\sin{x} = \frac{\pi}{4}(1+\sqrt{2})$ through the point $(\frac{\pi}{2}, \frac{\pi}{4})$ Now I know I can fill certain information into this ...
0
votes
0answers
14 views

Total derivative vs. partial derivative Legendre transformation

My question is about how to compute the total derivative for the function $f(x,y)$. In theory we have: $$df=\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy$$ So, if somebody asks ...
0
votes
0answers
30 views

Stoke's formula for a sphere

I have a question, raised form kinetic theory (pure mathematical). Imagine, that $\Psi (\overrightarrow{r},\overrightarrow{p},t)$ - sufficiently smooth function, where $\overrightarrow{r}$ - radius ...
0
votes
1answer
29 views

Confused About Definition of a Limit Proof

I'm working on $\epsilon-\delta$ limit proofs, and there's something about the proof I don't get. Currently doing a proof for $\lim_{x\to 3} (2x-1) = 5$. The first part of the definition says "if ...
5
votes
4answers
95 views

Graph Theory for Dummies Book

Does anyone have a good book on Graph Theory that will introduce me to some of the basic concepts without being so filled with terminology that it's hard to read? I have taken an introductory course ...
0
votes
1answer
27 views

Find isomorphism between $D_6$ symmetries of regular triangle, $S_3$, and $GL_2(F_2)$

problem: Find isomorphism between $D_6$ symmetries of regular triangle, $S_3$, and $GL_2(F_2)$. I already proved $D_6$ is isomorphic to $S_3$. And $S_3 $ is isomorphic to $GL_2(F_2)$ Am I suppose ...
1
vote
1answer
44 views

Probability of no 6 or no 5 when dice is rolled n times

Can anyone guide me in the general direction of the answer to the following: A die is rolled $n$ times $$A = \text{no $6$s}$$ $$B = \text{no $5$s}$$ $$P(A\cup B) = \;?$$ I am first finding $P(A)$ ...
1
vote
1answer
22 views

Determine measurability of E(X|N) or even $\sigma(E(X|N))$?

Suppose $(\Omega, F, P)$ is a sample space, $X$ a random variable, and $N$ a sub sigma algebra of $F$. How can we determine $\sigma(E(X|N))$? How is $\sigma(E(X|N))$ related to $\sigma(X)$ and ...
0
votes
0answers
29 views

Question: outer measure

To define the outer measure of an arbitrary subset $E\subset \mathbb{R}^n$, cover $E$ by a countable collection $S$ of intervals $I_k$, and let $$\sigma(S) = \sum_{I_k \in S} v(I_k)$$ where $v(I)$ is ...
2
votes
2answers
38 views

Non computable numbers are normals?

We know that almost all real number are normal and almost all real number are non computable. This does not suffice to deduce that all non computable numbers are normals but , intuitively (??) this ...
3
votes
1answer
49 views

$\pi_7(S^4)$ contains an element of infinite order.

Show that $\pi_7(S^4)$ contains an element of infinite order. Now, I know that I should probably use the Hopf bundle here somewhere. I also know that $\pi_3(S^7) = 0$. But I am stuck. Can anyone ...
0
votes
4answers
26 views

How can I find the length to this geometry problem?

A person 6 feet tall is standing at the base of a lamp post that is 25 feet tall and then begins to walk away from the lamp post. When the person is 10 feet from the lamp post, what is the length of ...
0
votes
2answers
31 views

Is the maximum of the eigenvalues of any symmetric positive?

Let $A$ be a symmetric matrix having dimension $n \times n \;, \; \; n\geq 2$. If one wants to pick the maximum of its eigenvalues, will the value be positive? Suppose A was an adjacency matrix, ...
0
votes
0answers
32 views

Grothendiek and singular points

My question relates to the following thread I opened some weeks ago: A question regarding Grothendieck , topos and (adelic??) points Specifically, consider this paragraph: At 1:14:30 and after, ...
5
votes
1answer
82 views

Proving $(1-x)\cdot (1-x^2)\cdots(1-x^{n-1})=n$ if $x^n=1$ and $x\neq 1$

If we have a equation $x^n=1$, then how can we prove $$(1-x)\cdot (1-x^2)\cdots (1-x^{n-1})=n $$ when $x$ is not $1$? I know that $x= e^{(2\pi + 2k\pi)/n}$ and we can get different value of $x$ when ...
-2
votes
0answers
19 views

$Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ .

Let $Y$ be the group such that it is define as $Y = < u, v | u^4 = v^3 = u= v=1, uv = v^2u^2, v^2 = v^{-1}>$ . a) Show that $v$ commutes with $u^3$. [Show that $v^2u^3v = u^3$ by writing the ...
-2
votes
2answers
30 views

Rank of an $m$ by $n$ matrix?

Can anyone state, in plain English, how to find the rank of an $m$ by $n$ matrix? Is it necessary to perform Gaussian elimination first, or translate it into upper triangle form (or however it is ...
0
votes
1answer
22 views

Evaluating a limit involving the power of specially structured matrix

Let $k\times k$ right-stochastic matrix $A$ be defined as follows: $$A=\left[\begin{array}{cccccc} p & 0 & 0 & \cdots & 0 & 1-p\\ 1 & 0 & 0 & \cdots & 0 & 0 \\ ...
1
vote
3answers
41 views

Induction proof.

Homework question, so just a pointer would be nice, for starters. I'm trying to prove $2 \mid 5^{2n} - 3^{2n}$ by induction. I use $n=0$ as the base step, and assume $5^{2n} - 3^{2n} = 2k$ as my ...
1
vote
0answers
18 views

Fibration implies inclusion is based homotopy equivalence?

If $p: E \to B$ is a fibration, does it follow that the inclusion$$\phi: p^{-1}(*) \to Fp$$specified by $\phi(e) = (e, c_*)$ is a based homotopy equivalence?
4
votes
1answer
21 views

A question about angles in the Euclidean plane

It has long been known that an arbitrary angle (in the Euclidean plane) cannot be trisected using only ruler and compass, but that this can be done using a mechanical linkage. Given any positive ...
2
votes
2answers
32 views

Binomial expansion derivative limit definition

Can someone help me with this? I am supposed to use a binomial expansion to calculate $\sqrt x$ directly from the limit definition of a derivative.
5
votes
2answers
130 views

The meaning of the Imaginary value of the Residue while Evaluating a Real Improper Integral

When evaluating the improper integral $$\int_{0}^{\infty}\frac{x^{3}\sin\left(2x\right)}{\left(x^{2}+1\right)^{2}}\,dx$$ (which is an even function, so half of the $(-\infty,\infty)$ integral), I used ...
2
votes
2answers
74 views

Is this a tautology: $\forall xP(x) \implies Q(x)$ if there's no $x$ such that $P(x)$?

Is this a tautology: $\forall xP(x) \implies Q(x)$ if there's no $x$ such that $P(x)$? I know that if there was an exists there instead of a for all, the antecedent would be false and thus the ...
-1
votes
1answer
19 views

Quick question about algebras

One of the requirements for an algebra $\mathcal{A}$ is that $\Omega$$\in$$\mathcal{A}$. But does $\Omega$ have to be it's own element in $\mathcal{A}$ or is it enough to let $\mathcal{A}$ consist of ...
2
votes
0answers
21 views

Assume that if in the power series expansion(around $0$) of $p(z)/q(z)$ all the coefficients of power series are integer then $q(z) \in \mathbb Z[z]$

I'm reading a paper in which following result is left by saying that its an easy exercise but I'm finding it a bit hard.Can someone give some ideas to complete this problem? Let $p(z),q(z) \in ...
0
votes
1answer
24 views

Quick way of writing equation of a tangent

The quick method and the more elaborate one do not reconcile. I must be making a mistake somewhere, mind pointing out? Equation: $x^2-3y^2=4y$ Tangent to a general point $(x_1,y_1)$ can be written ...
0
votes
0answers
15 views

Finding a solution vector to linear system of equations with lowest hamming weight efficiently

I'm trying to solve a linear system of equations modulo 2. After performing gaussian elimination, I can get a solution of the form $v + c_1 \cdot n_1 + c_2 \cdot n_2 + \cdots + c_k \cdot n_k, c_i ...
0
votes
1answer
20 views

Is a symmetric diagonal matrix in which every entry is non-negative positive semidefinite?

Let $A$ be a symmetric diagonal matrix in which $(A)_{ii} \geq 0$. Should one conclude that this matrix is positive semidefinite?
0
votes
0answers
73 views

Is there more than one looping sequence in the Collatz conjecture? [on hold]

Is it known whether there is more than one loop in the Collatz conjecture? Following advice and warnings on meta, I try below to claim that there is only one looping sequence in all the sequences ...
1
vote
1answer
40 views

Linear Systems: Exponentials of a Matrix

I have a rather odd question to some, but one that has stumped me for a good few minutes on a homework assignment that states: For each matrix, find the eigenvalues of $\text{exp}{(A)}$, ...
0
votes
0answers
11 views

Legendre transformation

My question is about Legendre Transformation. I need some example and some practical theory. Can you help me please with some examples or papers? Thanks!
0
votes
1answer
20 views

Prove the following using induction on d (matrices)

I manage to reach the step where I need to prove n = k + 1 but I am battling to complete the proof as I am not certain what to do with the exponents in my answer. I will run through the proof as I ...
1
vote
3answers
49 views

Is there an error in this GRE question?

I was doing a Manhattan GRE practice exam and I was sure I had cracked the twist in this one question... only to find in that there was no twist (apparently). Here is what I know: $$ \sqrt a = \pm b ...
1
vote
0answers
18 views

Examples of Lp spaces in Applied Math

I was wondering if there are examples of exotic Lp spaces being used in applied mathematics. I know that the "special" p's (1,2 , infinity ) are of use, for example in statistics, L1 is mean, L2 is ...
1
vote
1answer
13 views

Proving an identity of the resolvent set

$\{T(t)\}_{t\ge 0}$ is a $C_{0}$-semigroup with infinitesimal operator $A$. I'm trying to prove that the set $\{ z|\text{ Re }z>\omega_{0}\}$ belongs to $\rho(A)$, and for $z$ in this set, the ...
0
votes
1answer
12 views

Primal and dual feasible = optimal?

Is any primal feasible and dual feasible point of a convex function, a global min of that function? If yes, why? If no, do we need any more conditions?
0
votes
0answers
17 views

Problem on vertical light elastic string

A mass of $4$ lbs suspended from a light elastic string of natural length $3$ feet extends it to a distance $2$ ft. One end of the string is fixed and a mass of $2$ lbs is attached to other. The ...
1
vote
0answers
14 views

Supersolubility of $ G/H $ and $ H $ not deduced supersolubility of $ G $.

I want show $ S_{4} $ isn't supersoluble group. For this suppose $ 1 \leq B_{4} \leq A_{4} \leq S_{4} $ be a normal serie of $ S_{4} $, that $ B_{4} $ is Klein’s four-group. Since $ B_{4} $ isn't ...
0
votes
0answers
23 views

Every integer $n > 1$ can be written in one and only one way with a certain property

Every integer $n > 1$ can be written in one and only one way in the form $n = p_1p_2p_3 \ldots p_r$ where $p_i$ are positive primes s.t. $p_1 \le p_2 \le p_3 \le \ldots \le p_r$. $n$ is unique ...
1
vote
0answers
23 views

Maximal compactifications without the Tychonoff theorem

I once saw a neat proof in American Mathematical Monthly of the Tychonoff theorem (The Tychonoff product topology of a family of compact spaces is compact) for the special case of the product of ...
4
votes
0answers
38 views

Does every compact simply-connected subset of $\mathbb{R}^n$ have an efficient $r$-covering path for all $r>0$?

Let $A$ denote a subset of $\mathbb{R}^n$. Definition 0. Given a positive real number $r$, an $r$-covering path of $A$ is a non-negative real number $T$ together with a differentiable function ...
1
vote
0answers
7 views

About the solenoidal sets

In the book Dynamic Reported - The definition of the solenoidal sets is: Let $I_{0} \supset I_{1}\supset I_{2}\dots$ be periodic intervals with periods $m_{0}$, $m_{1}$,$\dots$. If $m_{i} \to \infty$ ...
0
votes
1answer
21 views

Question involving gradient of a function.

We are given any arbitrary ellipse with focii $F1$ and $F2$ , $T$ is the unit tangent to the ellipse through a point $P$. Let $f$ be the sum of the distances of the of $F1$ and $F2$ from $P$ , we ...
0
votes
3answers
54 views

Prove: set of the polynomials of degree $n$ with coefficients in $\Bbb Q $ is countable

Let $P_n =\{p(x)=a_n x^n+ a_{n-1} x^{n-1}+...+ a_1 x+a_0 |a_i \in \Bbb Q \}$ the set of the polynomials of degree $n$ with coefficients in $\Bbb Q $ Prove that $P_n$ is countable and tell why $P= ...
0
votes
2answers
43 views

Independence between conditional expectations

Suppose $(\Omega, F, P)$ is a sample space, $X$ and $Y$ random variables, and $N$ and $M$ sub sigma algebras of $F$. I know that $E(X\mid N)$ and $E(X\mid\{\emptyset, \Omega\})$ are independent. ...

15 30 50 per page