0
votes
1answer
29 views

Vanishing of 1-form

If $\theta \in \frak{X}^* \mathrm{(M)}$ and $\theta (X) = 0$ $\forall X \in \frak{X} \mathrm{(M)}$ then $\theta = 0$. How do I prove this statement? Consider a manifold $M$ with chart $x^1, \dots, ...
1
vote
0answers
82 views

An attempt to a triple integral with spherical coordinates.

If $f(x,y,z)=xyz$ and $D=\{(x,y,z)\in\mathbb{R}^3:x\geq 0, y \geq 0, z \geq 0,\; x^2+y^2+z^2\leq 1\}$, find $$ \iiint_D f(x,y,z). $$ I tried to solve this with spherical coordinates: ...
1
vote
2answers
60 views

evaluating moment generating functions

Let $Z_1,Z_2,\ldots,Z_{14} $ be 14 independent N(0,1) variables, and let $Y=Z_1^2+Z_2^2+\cdots+Z_{14}^2$. Provide answers to the following to two decimal places. Evaluate the moment generating ...
2
votes
1answer
41 views

Upper bound on a sum similar to a telescoping sum: $|p_{n'}-p_0| \leq \sum_{i=1}^{n'}{|p_{i}-p_{i-1}|}$

Does someone know why the following is true: $$|p_{n'}-p_0| \leq \sum_{i=1}^{n'}{|p_{i}-p_{i-1}|}$$ If we did not have the "size of" operator, there would be an equality, but in this case its a ...
3
votes
0answers
65 views

Optimization problem (Sum of distances)

Given an ordered sequence $x_1 \leq x_2 \leq \cdots \leq x_n $ of length $n$ and a cost function $C(i) = \sum_{j}^{n}{\left|x_i-x_j\right|}$. The goal is to minimize the cost function. How do you ...
1
vote
1answer
68 views

edge coloring of a specific graph

For the graph $D_n$ created from complete graph $K_n$ by replacing one of edges by path on 3 vertices. For example, the graph attached is $D_4$. I can prove that the edge chromatic number is $n$. ...
1
vote
2answers
43 views

Is there any difference in meaning between these two statements

I'm reading text that states the vector $\ x \in \mathbb{R}^{n} $ is $\ s$-sparse if at most $\ s<n$ of its entries are non-zero Is that any different from stating: the vector $\ x$ is $\ ...
0
votes
2answers
670 views

Solving a Calculus velocity question using Precalc.

Decide whether on not Calculus is needed for the problem. If it is, use Pre-Calculus to approximate an answer. Find the distance traveled in 10 seconds by an object traveling at a velocity of v(t) = ...
2
votes
2answers
188 views

Which methods different than the natural one can one devise to confirm that the limit is $\;2/\pi\;$?

Good evening, I have found this exercise (Which methods different than the natural $\lim_{n\to\infty}\frac{|\cos{1}|+|\cos{2}|+|\cos{3}|+\cdots+|\cos{n}|}{n}$) What is the limit of: ...
1
vote
1answer
147 views

Mismatching Results - Keno and Probability

In Keno, a player picks from 1 to 70 (at least in this version), 20 of these numbers are drawn, and the payouts are based on the number of matches. What I have tried to do is to check that the Swedish ...
1
vote
5answers
101 views

Problem with understanding generating functions.

I am given generating functions $f(x)= \frac{x}{1-x}$ or $f(x)=\frac{1}{1+x^{2}}$ or $f(x)=\frac{1}{x^2-5x+6}$ and I am obliged to write sequence which are generated by this functions. What is the ...
1
vote
1answer
22 views

Confidence interval on the difference of probabilities

Suppose you have two coins 1 and 2 with unknown probabilities $p_1$ and $p_2$ of coming up heads. You then flip coin 1 $n_1$ times and coin 2 $n_2$ times. I would like to then be able to say that ...
36
votes
1answer
442 views

Can I solve an integral (or other tough problem) by playing with knots?

I've seen that in calculating things in knot theory that involves a lot of hard looking integrals and matrices, even though the knots themselves appear fairly simple. So is there some way in which ...
0
votes
1answer
48 views

divergence problem with polar coordinates

How can I show that $\nabla \cdot (\phi \nabla )=0$. when $\phi (r, \theta)=(r+\frac{R^2}{r})\cos\theta, r> R $?
4
votes
3answers
399 views

Empty functions are not injective?

Many sources say that empty functions such as $f:\emptyset \rightarrow S$ are injective because it is a vacuous truth. But currently I am reading a book on axiomatic set by Patrick Suppes, and he ...
1
vote
2answers
75 views

Variance deduction

the definition of variance is $V(X) = E((X-E(X))^2 )$ For a discrete random variable: if we have put $Y = g(X)$ , where $g$ is a real function $E(Y) = E(g(X)) = \sum\limits_{k} g(k)p_X(k)$ , ...
1
vote
2answers
187 views

Unitary similarity transformation

I have a matrix: $ A= \dfrac{i}{3} \begin{bmatrix} 1&-2&1\\-2&1&1\\1&1&-2\end{bmatrix} $ Could someone explain me how to find a corresponding diagonal matrix for a ...
13
votes
3answers
866 views

Why is an orthogonal matrix called orthogonal?

I know a square matrix is called orthogonal if its rows (and columns) are pairwise orthonormal But is there a deeper reason for this, or is it only an historical reason? I find it is very confusing ...
3
votes
2answers
105 views

If $x$ is integral over $A_m$ for all maximal ideals $m$, then $x$ is integral over $A$

I am going over an old exam, and there is this question that I am stuck: Given $A$ a commutative ring with unity, show that if $x\in\operatorname{Frac}(A)$ is integral over $A_m$ for all maximal ...
1
vote
1answer
66 views

Is there a name for this theorem?

Theorem: If you have a number $x$ with $n$ number of decimal places and another number $y$ with $l$ number of decimal places, then $x \cdot y$ will never have more than $n + l$ decimal places. For ...
3
votes
3answers
102 views

Is this function well-defined or just an abuse of notation?

Let's say we have a real-valued function $f$, perhaps something very simple. I define $g$ to be $g(x) = \int_0^xf(x)dx$. This looks like total nonsense to me. But I can't satisfactorily prove to ...
0
votes
2answers
92 views

Is this $\epsilon-\delta-$proof correct?

I have to Show that $$\mathbb{C} \rightarrow\mathbb{R}; z \rightarrow \Re z$$ is a continuous function using the $\epsilon-\delta-$criteria. So what I did is the following: I have to Show that ...
4
votes
3answers
207 views

Prove trigonometric relation

Recently, I found this identities in a sheet of paper I was given as studying material: $$\prod^n_{k=1}\sin\left(\frac{k\pi}{2n+1}\right)=\frac{\sqrt{2n+1}}{2^n}\tag1$$ ...
0
votes
2answers
155 views

Reducing quartic equations to quadratic

I'm trying to re-learn basic math/algebra, and I can't get passed one question concerned with reducing quartic equation to a quadratic: Find, correct to 3 significant figures, all the roots of the ...
1
vote
1answer
92 views

Prove that $f(n)$ is in $\Theta(g(n))$

Suppose $f(n) = 1^k + 2^k + \ldots + n^k \;$ and $\; g(n) = n^{k+1}.\;$ Prove that $\;f(n)\in \Theta(n^{k+1})$. My understanding is that we have to find $C_1, C_2 \gt 0$ such that: ...
3
votes
1answer
520 views

Trick to diagonalize symmetric matrices?

I will write an exam on Quantum Mechanics soon. I was wondering whether there is any smart and fast way to determine the eigenvalues/eigenvectors of a symmetric 3x3 matrix other than by calculating ...
3
votes
2answers
85 views

Let $N$ be the product of the first $m$ primes and $M$ be the product of the first $n-m$ primes.

Could someone tell me if this proof is correct? Suppose there are finitely many primes: $p_1,p_2,\cdots,p_n$ primes. Let $m<n$. Then let $N=p_1p_2p_3\cdots p_m$ and $M=p_{m+1}\cdots p_{n-1}p_n$. ...
0
votes
1answer
129 views

Axiom of Completeness for set of integers

If $A$ is a subset of the integers $\mathbb{Z}$, and is bounded above, then A has a supremum $\alpha$ that is an element of the integers $\mathbb{Z}$. Is this statement true?
0
votes
1answer
137 views

How many trailing zeroes does 1000! have in base 8

First I checked this for powers of two: $$\sum\limits_{i=1}^{9}\left\lfloor\frac{1000}{2^i}\right\rfloor=994$$ I was told the answer to this is $331$ since $994=331\cdot 3+1$. I'm wondering why its ...
1
vote
4answers
107 views

Why are these two series identical?

Could someone please show/explain to me explicitly why this is true (from wiki): $S = 1 - 1 + 1 - 1 + 1 - 1 + \cdots$ The series can be rearranged as: $S = 1 - (1 - 1 + 1 - 1 + 1 - 1 + \cdots)$ I ...
0
votes
1answer
57 views

directional derivatives at (0,0) vanish

Is the statment that all directional dervatives vanish at (0,0) really true, it seems to me the last equation states the opposite. The example is from: Mathematical Analysis: An Introduction to ...
1
vote
2answers
68 views

Finding general solution for a nonhomogeneous system of equations

I have a system of differential equations: $\begin{cases} x_1'=x_2+2e^t \\ x_2'=x_1+t^2 \end{cases}$ And I want to find the general solution for it. I started by finding the general solution for the ...
2
votes
3answers
53 views

Inverse Laplace of $F(s) = \frac{3s}{(s^2+9)^2}$

Can somebody please show how to go about answering the following; ${\scr L^{-1}}(F(s)) $ where $F(s) = \dfrac{3s}{(s^2+9)^2}$ I know the ${\scr L}\left(\dfrac{3}{s^2+9}\right)=\sin(3t)$ and that ...
0
votes
2answers
145 views

Solve the following differential equation: $y\frac{dx}{dy}-x=2y^2$, with the initial condition $y(1)=5$.

Solve the following differential equation: $y\dfrac{dx}{dy}-x=2y^2$, with the initial condition $y(1)=5$. The thing that is throwing me off (I think) is the $\dfrac{dx}{dy}$ instead of what I am ...
1
vote
1answer
90 views

almost sure convergence. martingale

I have a simple question regarding almost sure convergence. Assume a sequence $X_n = a_1 + ... + a_n$. and $P(a_n \neq -1 \space i.o) = 0$ This means $X_n/n \rightarrow -1 a.s $. Does it also mean ...
7
votes
1answer
512 views

What pure mathematics foundations should an applied mathematician have?

I'm studying mathematics, with some statistics also, and I've always chosen applied courses. I'm getting to the point where I'm studying 3rd year undergraduate to graduate level material. My first ...
7
votes
4answers
255 views

How do I convince someone that $\mathbb{R}^2$ and its copy inside $\mathbb{R}^3$ are different?

One of my friends is taking a first course in linear algebra now, and one of the problems on his latest homework was to explain why $\mathbb{R}^2$ and $\{(a_1,a_2,a_3) \in \mathbb{R}^3 \mid a_3 = 0\}$ ...
4
votes
3answers
181 views

Having trouble with a combinatorics question.

I'm not so good at combinatorics, but I want to know if my answer for this question is right. Originally this question is written in spanish and it says: Se dispone de una colección de 30 pelotas ...
2
votes
2answers
181 views

Does the distributive law for dot products go for both addition and subtraction?

I know that $\vec a\cdot(\vec b+\vec c)=\vec a\cdot\vec b+\vec a\cdot\vec c$, but is it also true that $\vec a\cdot(\vec b-\vec c)=\vec a\cdot\vec b-\vec a\cdot\vec c$?
0
votes
1answer
86 views

expectation conditioned on sigma fields

could anyone please explain to me a simple question regarding expectations conditioned on sigma fields? Consider a sample space {a, b, c} and $F_1$ = $\sigma$(a) and $F_2$ = $\sigma$(b) and a random ...
3
votes
2answers
257 views

Pushout from initial object isomorphic to coproduct

Let $C$ be a category with an initial object $0$, pushouts, and coproducts. If $G$ and $H$ are two objects, I want to know whether $G\sqcup H$ and $G\sqcup_0 H$ are isomorphic. Letting $i_1$ and $i_2$ ...
3
votes
1answer
196 views

Determine a sequence of random variables is a martingale

I'm trying to solve a problem from an old exam. This is an easy but a bit lengthy exercise, divided into subproblems. Since they are based on each other and probably are quite short, I was hoping that ...
3
votes
0answers
103 views

Cartan and Eilenberg Homological Algebra

OK, I am looking at Cartan and Eilenberg Homological Algebra book (1956, 1973 printing). Chapter V.9, p97 they define functors T(-,-) of type L$\Sigma$ and R$\prod$. T is of type L$\Sigma$, if T(A,C) ...
1
vote
1answer
61 views

can the projective dimension be read from any projective resolution?

Let $P_{\bullet}, P'_{\bullet}$ be two projective resolutions of an $R$-module $M$. Denote their differentials by $d,d'$ respectively. Define $M_i = \operatorname{ker} d_{i-1}, M'_i = ...
3
votes
2answers
70 views

$G$ is an infinite group and $G/Z(G)$ is finite. Show that every conjugacy class is finite [closed]

$G$ is an infinite group and $G/Z(G)$ is finite. Need to show that every conjugacy class is finite.
3
votes
2answers
211 views

pole, order and residue

I was practising for an exam and I had some trouble with the following excersice: $$f(z)= \frac{1}{z \sin z}$$ a. Find the pole and its order. $$\frac{1}{z(z-z^3/3!+ z^5/5! + \cdots)}= ...
1
vote
1answer
34 views

Small question on relative holology

if $Y\subset X$ , what is $\ker \delta$ such that $\delta: H_k(X,Y)\rightarrow H_k(Y)$ ? is it $\ker \delta = H_k(X,Y)$ ? $\delta$ is the usual connecting homomorphism from the long exact sequence ...
4
votes
2answers
211 views

Find the volume between two surfaces

Find the volume between $z=x^2$ and $z=4-x^2-y^2$ I made the plot and it looks like this: It seems that the projection over the $xy$-plane is an ellipse, because if $z=x^2$ and $z=4-x^2-y^2$ ...
1
vote
1answer
75 views

Avoiding collision with a plane

I am developing the software to control a common fly's flight in Opensim simulator, although I am facing some troubles determining when the fly must maneuver to avoid hitting a wall. Questions How ...
3
votes
1answer
106 views

$K_{1,3}$ packing in a triangulated planar graph

I am trying to show that every planar triangulated graph $G=(V,E)$ with $|V| \ge 5$ has an edge decomposition into $|V| - 2$ groups of $K_{1,3}$. In other words, that we can pack $|V| - 2$ instances ...

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