# All Questions

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 ...
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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 $F$ ...
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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
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 ...
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. ...
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, ...
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: $$\frac{1}{(2n+1)!(2n+1)} \leq 5*10^{-9}$$ ...
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 ...
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 ...
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 ...
10 views

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) := ... 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 ... 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 ... 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) = ... 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 ... 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 ... 0answers 23 views ### Meaning of superscript following brackets in set definition What does the n mean in this set? U = \{0,1\}^n 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 ... 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=\{ ... 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 ... 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 ... 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 ... 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? 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 ... 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 ... 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 ... 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. ... 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::: ... 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 ... 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? 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 ... 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 ... 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 ... 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 ... 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 ... 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 ...
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?
### 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($X,Y$) = ...