0
votes
0answers
18 views

About the $d\mathbf{s}$ notation

Let $\mathbf{F}$ be a vector field that changes with time, that is, written in components:$$\mathbf{F}(\mathbf{x},t)=(F_1(\mathbf{x},t),F_2(\mathbf{x},t),F_3(\mathbf{x},t))$$ where ...
1
vote
1answer
11 views

Struggling with connection between Clifford Algebra (/GA) and their matrix generators

As I thought I understood things, the Gamma matricies behave as the 4 orthogonal unit vectors of the Clifford algebra $\mathcal{Cl}_{1,3}(\mathbb C)$, (also the Pauli matricies are for the 3 of ...
0
votes
1answer
32 views

Showing $\sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty$ implies $f_{n}\rightarrow f$ almost everywhere.

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of integrable functions and $f$ an integrable function. I have to show that, if $$ \sum_{n\in\mathbb{N}}\int{|f_{n}-f|d\mu}<\infty, $$ then ...
2
votes
1answer
43 views

Why do we require that a complex manifold has the structure of a real manifold?

I am taking a course in complex manifolds, heavily influenced by Huybrechts' book "Complex Geometry", and in it we define a complex manifold $X$ to be a smooth, real manifold $M$ together with an ...
1
vote
0answers
13 views

Find Formal Proof on Loci Theorems

Please help me to prove this theorems of loci Theorem 12 The locus of a point at a given distance from a given line is two lines parallel to the given line and at the given distance from it. Theorem ...
-2
votes
0answers
6 views

Dynamical systems,forward invariant [on hold]

Show that the complement of a forward invariant set is backward invariant, and vice versa. Show that if f is bijective, then an invariant set A satisfies f t (A)= A for all t. Show that this is false, ...
0
votes
1answer
26 views

First-order nonlinear differential equation

How would I solve this differential equation for $y(x)$? $\frac{dy}{dx} = \frac{y-xy}{x-xy}$ $y -\ln(y) = x - \ln(x) + C$ I'm not sure what to do at this point. I looked it up on WolframAlpha and ...
2
votes
1answer
20 views

computing the full constant field of an algebraic function field

Let $K$ be a field such that char$(K) \neq 2$. Let $F=K(x,y)$ be an algebraic function field field of one variable $x$ where $$y^2 = f(x) \in K[x]$$. We want to compute the full constant field of $F$ ...
1
vote
3answers
24 views

Prove the intersection of a compact set and a set with no accumulation points is finite

Let $S\subset\mathbb{C}$. We say that $z_0$ is an accumulation point of $S$ if for every $r>0$, the intersection $D(z_0,r)\cap S$ is an infinite set. Let $U\subset\mathbb{C}$ be an open set such ...
0
votes
1answer
13 views

Skewness of a difference of random variables?

In this article( http://www.diva-portal.org/smash/get/diva2:302313/FULLTEXT01.pdf )page 28 explains how to derive the skewness of a sum of random variables; I haven't been able to derive this ...
3
votes
3answers
144 views

Combination Problem: Arranging letters of word DAUGHTER

The number of ways in which we can form a 8 letter word from the letters of the word DAUGHTER such that all vowels are never occur together is My approach: As ...
0
votes
0answers
16 views

Multiplication rule and regular conditional probability

I've been studying the conditions of existence of the regular conditional probability and have a question about it. Let's $(\Omega, \mathcal{B}, P)$ be a product probability space, and let's say the ...
3
votes
1answer
33 views

When a normal subgroup $N$ admits a complament?

Let $G$ be a finite group and let $N$ be a normal subgroup. I am looking for conditions on $N$ (and maybe also on $G$) such that there exist a subgroup $H$ of $G$ such that $$G=N\rtimes H.$$ Clearly, ...
0
votes
0answers
18 views

$E(X_T; T < \infty) \leq E(X_0)$ with $T$ stopping time

I'm doing this exercise: $(X_n)$ is a non-negative supermartingale and $T$ a stopping time, then $$E(X_T; T < \infty) \leq E(X_0)$$ My attempt: $(X_n)$ is a negative supermartingale, and so ...
3
votes
2answers
133 views

Limit of solution of differential equation without solving the equation.

Given $$x'(t)=A-B\left(x(t)\right)^2, \quad x(0)=0.$$ Is it possible to find $\lim\limits_{t\to\infty}x(t)$ without solving the differential equation? Assuming $\lim\limits_{t\to\infty}x'(t)=0$ gives ...
0
votes
0answers
15 views

Derivative of $Ad(c(t))X$

Let $G=SO(3)$ and $V=\{c'(0)|c:(-\epsilon,\epsilon)\to G, c\in C^{\infty} , c(0)=1\}$. For $g\in G$, define $Ad(g): V\to V$ by $Ad(g)(X)=gXg^{-1}$. The book says ...
1
vote
2answers
42 views

convergence of $\sum_{n=1}^\infty\frac{1}{n} [1+\frac{1}{\sqrt{2}}+…+\frac{1}{\sqrt{n}}]$ [duplicate]

Let $t_n= \frac{1}{n} [1+\frac{1}{\sqrt{2}}+.....+\frac{1}{\sqrt{n}}]$, n=1,2... Then I am asked whether series$ \sum t_n $ converge or diverge. Also whether sequence $ t_n $ converge to zero or not. ...
-2
votes
0answers
16 views

Find the slope of the secant line given a point [on hold]

The point P(1,0) lies on the curve y= sin(10π/x) . (a) If Q is the point (x, sin(10π/x), find the slope of the secant line PQ (correct to four decimal places) for x=2, 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, ...
0
votes
1answer
24 views

How to solve Inequality with factorials

Im reading a book in Numerial analysis and I have the following which I dont understand involving inequalities and factorials, What i have is the following: $$\frac{1}{(2n+1)!(2n+1)} \leq 5*10^{-9}$$ ...
0
votes
0answers
15 views

Maple produces only errors

I have Maple 18 installed on linux. I've been following the guide and I got to a point where I'm suppose to right click 2*x-9=0 and solve for x. This fairly simple ...
1
vote
0answers
8 views

difference between a combinatorial map and a rotation system?

Wikipedia has separate articles for combinatorial map and for rotation system, but as far as I can tell, their formal definitions are identical. Am I missing something? Or do these terms have ...
3
votes
7answers
105 views

What does $\frac{z_1-z_3}{z_3-z_2}=\frac{z_2-z_1}{z_1-z_3} $ imply?

I'm having trouble understanding what the following equality implies. $$\frac{z_1-z_3}{z_3-z_2}=\frac{z_2-z_1}{z_1-z_3}.$$ I suspect that this means that the points form the vertices of an ...
0
votes
0answers
10 views

estimating a convolution type maximal function

Let $\phi : \mathbb{R}^n \rightarrow \mathbb{R}_{+}$ be a $C^1$ function with $supp(\phi) \subset B(0,1)$ and $\int \phi = 1$. Define $$\phi_t(x) := t^{-n} \phi({x/t})$$ and set $$ M_{\phi} f(x) := ...
0
votes
0answers
17 views

Exercise on representations of the Dihedral group (Etingof 3.17)

I'm confronted once more with a problem on representation theory which I cannot fully solve (Problem 3.17 http://math.mit.edu/~etingof/replect.pdf): Let $G$ be the group of symmetries of a regular ...
0
votes
2answers
22 views

general solution of second order linear de

Let 1, x and $x^2$ be solutions of second order linear non homogeneous differential equation $-1\lt x\lt 1$. Then find the general solution. I only know that general solution is sum of complementry ...
0
votes
1answer
18 views

Prove that $\sum_{n≥0} a_k(n)x^n = \frac{1-x}{1- 2x + x^{k+1}}$

Let k be a fixed positive integer and for all n≥0 let $a_k(n)$ be the number of compositions of n where each part is at most k. Set $a_k(0) = 1$. For instance, if k = 2 then $a_k(1) = 1$, $a_k(2) = ...
0
votes
0answers
17 views

Cauchy's integral theorem and domain boundaries

On a homework assignment, I was asked if the following statement is true. If $f(z)$ is analytic in a simply connected domain $D$ and continuous in $\partial D$ then $\oint_{\partial D} f(z) = 0$. Is ...
2
votes
1answer
15 views

Exercise on formal deformations of representations (Etingof 2.24)

I'm trying to work out a few exercises in Etingof's book on representation theory of associative algebras (http://math.mit.edu/~etingof/replect.pdf) At the moment I'm looking at Problem 2.24. about ...
1
vote
0answers
23 views

Meaning of superscript following brackets in set definition

What does the $n$ mean in this set? $U = \{0,1\}^n$
2
votes
1answer
25 views

Do we have $\mathbb{C}[T] = \mathbb{C}[\Lambda] = \oplus_{\lambda \in \Lambda} \mathbb{C}[\lambda]$?

Let $\mathbb{C}[T]$ be the coordinate ring of a torus $T$. Suppose that $T$ acts on some variety $X$. Then $T$ acts on $\mathbb{C}[X]$: $t(f) = \lambda(t)f$ ($f$ is a homogenous function on ...
0
votes
2answers
59 views

Proving a subset of $l_2$ is closed

Let $l_2$ be the set of all real sequences $x=(x_n)$ such that $\sum|{x_n}|^2 <\infty$ and define the norm $||x_n||_2=(\sum\limits_{n=1}^{\infty}|x_n|)^{\frac{1}{2}}$. I want to show that $A=\{ ...
0
votes
1answer
11 views

Calculating decreased cost with increasing quantity

I have a hand made table I've been using to give customers price per unit on my items, which gives a better price for the more items that they buy. My sample table right now I need to keep the ...
0
votes
0answers
18 views

what does this notation exactly mean? $(x)_{+}$

So the question is already given in the title: I see in some mathematics proof, the following notation is used: $x=(z)_+$ and we know that $x$ should be greater than or equal to zero, i.e. $x\geq ...
1
vote
3answers
27 views

$\delta-\epsilon$ Question on Ordered Field $\mathbb{R}$

I got came across this question with the $\delta-\epsilon$ definition of a limit, but I do not know how to use it to solve the context of this problem: Problem: Let $f:\mathbb{R}\to\mathbb{R}$ be ...
1
vote
3answers
51 views

Explanation for $\lim_{x\to2} e^{\frac{1}{x-2}}$

I can't find out why is the limit from the left side = 0 and from the right = Infinity?
0
votes
2answers
26 views

Discrete maths proving a random observation

Suppose you had 6 points. Each point can choose to either visit another point, or choose not to visit another point. However, it can't visit itself. In addition, visiting another point works in both ...
0
votes
0answers
4 views

Distribution of absolute sqaured variable [on hold]

Given $P = \vert r \vert ^2$, where $P\sim N(\mu,\sigma)$, $r \in \mathbb{C}$ and $\vert * \vert $ denotes the absolute value of $*$, what is the distribution of $r$? If there is a solution to the ...
0
votes
1answer
20 views

Number of subgroups and normal subgroups

I am struggling to understand how to calculate the nunmber of subgroups with permutations, for example: How many normal subgroups does S3 have? How many subgroups of order 4 has group S4? And does ...
0
votes
1answer
14 views

Contour Integration where Contour contains singularity

There are many theorems in complex analysis which tell us about integration $\int_{\gamma} f$ where $f$ is continuous (or even differentiable) in the interior of $\gamma$ except finitely many points. ...
3
votes
1answer
47 views

Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing?

Is $a_n={\{\dfrac{1}{n^2}+\dfrac{(-1)^n}{3^n}}\}$is monotonically decreasing? In process of solving this problem, I faced to the problem of proving that $A::$: ...
1
vote
0answers
21 views

Probability of collecting all the sticker types

This question is in the context of tuning a training procedure, whereby the learner may receive random stickers for good performance. I am trying to figure out the probability of any given learner ...
5
votes
3answers
49 views

If $X + X^T$ is positive definite, is $X^{-1} + X^{-T}$ also positive definite?

Is it true or is there a counterexample?
0
votes
0answers
9 views

Conditions equivalent to Noetherianness

Let $R$ be a left Noetherian ring. We know that any direct sum of injective left $R$-modules is again injective. Since any injective module is quasinjective, we infer that (1):"any direct sum ...
1
vote
2answers
26 views

Can a differential equation with real coefficients have solution with complex coefficients?

Can a differential equation (with constant coefficients, linear or nonlinear) with real coefficients have solution(s) with complex coefficients? If so, are there any examples related to actual ...
1
vote
2answers
29 views

How to get to $5^3 \geq n^3$ in the proof by contradiction?

This is the same problem asked here. - Next step to take to reach the contradiction? Here is it again. I understand the solution - how you want to get to the fact 100 divides n^2 and then go ...
1
vote
0answers
27 views

Is there such a notion of “expansion” in groups?

Given a subset of elements of a finite group $G$, I would like it to be such that the set of all distinct words (as elements of $G$) that can be formed from this set is exponentially large in the size ...
0
votes
0answers
14 views

problem about equivalent norms. [duplicate]

Let $\|\cdot\|_1$ and $\|\cdot\|_2$ be equivalent norms on a normed field. Then (i) $\|x\|_1<1$ iff $\|x\|_2<1$; $\|x\|_1>1$ iff $\|x\|_2>1;$ (ii) $\|x\|=1$ iff $\|x\|_2=1$. I want to ...
1
vote
1answer
17 views

Integration with respect to a measure

I am trying to get an explanation in words, or math, of what the $d\mu$ means in an integration statement. Such as: $$\int f \ d\mu$$ How does the measure change our old "calculus" notion of ...
0
votes
1answer
7 views

Primitive roots and 'equivalent exponents'.

If M is a primitive root mod p and M = $\ N^T$ mod p , then the order of N mod p is also (p-1) is this true?
1
vote
4answers
42 views

Difference between $E[X^2]$ and $E[X^3]$

Hope to ask a dumb question. $Y = aX$,with $a \in N_+$. Here, we know the correlation coefficient is 1. Now, suppose $X \sim N(0,1)$. Here, we know $X, Y$ are not independent. Cov($X,Y$) = ...

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