All Questions

2
votes
0answers
3 views

$\mathbf{H}^{3}$ is diffeomorphic to $\mathbf{SL}\left( 2,\mathbf{C}\right) \mathbf{/SU}\left( 2\right) $

I'm reading the book from Jensen's "Surfaces in Classical Geometries". Could anyone help me understand why $\mathbf{H}^{3}$ is diffeomorphic to $\mathbf{SL}\left( 2,\mathbf{C}\right) \mathbf{% /SU}\...
0
votes
0answers
9 views

Calculating homology of mapping torus (is this question incorrect?)

This question is on an old qualifying exam for my institution: Let $Y$ be a space and $f:Y\to Y$ a self-mapping. Let $X$ be the mapping torus of $f$ (i.e. the space obtained from $Y\times I$ by ...
0
votes
0answers
8 views

Closed geodesic in an orientable Manifold is homotopic to a curve of shorter length

Let M be an even dimensional Riemannian manifold with positive sectional curvature. Let $\gamma: S^{1} \rightarrow \mathbb{R}$ be a closed geodesic in M. Show that $\gamma$ is homotopic to a closed ...
1
vote
0answers
6 views

Proving $V(x) = \frac{|y|^2-|x|^2}{|y-x|^N}$ is harmonic in $\mathbb{R}^N-\{y\}$

$$V(x) = \frac{|y|^2-|x|^2}{|y-x|^N}$$ for fixed $y\in\mathbb{R}$, and $x\in\mathbb{R}^n-\{y\}$ I need to prove that this this is harmonic. That is, the sum of its second partial derivatives are $0$....
0
votes
1answer
31 views

How to evaluate $(\mathbf{a}\times\mathbf{b})\cdot (\mathbf{a} - \mathbf{b})$

The answer is $\mathbf 0$ but I don't understand how to get to this answer. If the answer is $\mathbf 0$ then that means $(\mathbf{a}-\mathbf{b})$ is perpendicular to $(\mathbf{a}\times\mathbf{b})$, ...
0
votes
0answers
14 views

Lost in a Forest Problem 3d Verision: Point and Plane

The lost in a forest problem is famous, specifically see this problem. The problem takes place in the plane. Consider the following 3d version. There is a point $p \in \mathbb{R}^3$ and a plane $P$ ...
0
votes
1answer
5 views

Exercise of position measurements and their values

I have this exercise of statistics: The attached table shows the approximate values of the distribution in quintiles of the family income per capita in Chile. Which of the following statements ...
1
vote
1answer
10 views

Proof of the number of degrees of freedom of a rigid body with more than 3 mass points

I know that the number of degrees of freedom of a system composed of M particles subject to j constraints is: $$ DOF = 3M - j$$. Now my book mentions that the number of $DOF$ of a rigid body with more ...
0
votes
2answers
20 views

How to solve for $x$: $[ (23 * 60 * 60 *1000)\mod x = 1195 ]?$

I want to solve $$(23 * 60 * 60 *1000)\mod x = 1195$$ for $x$. I understand there might be multiple solutions. I have tried plugging it into Wolfram Alpha and I get no output for $x$. How might I ...
1
vote
0answers
24 views

A question about Bochner's theorem.

I am studying Fourier analysis and finding two Bochner's theorem: Bochner's theorem VERSION $1$: In order that a function $f:\mathbb{R}^d\rightarrow \mathbb{R}$ be positive definite and continuous, ...
1
vote
0answers
17 views

Are derivations as general as derivatives in R^n?

Assuming we start with the derivations, $w_p(f)$ that are linear and satisfy the product rule. Is it possible to show that what $w(f) = D_v(f)$ without first defining what a directional derivative, ...
0
votes
1answer
13 views

product of two measurable spaces

If $X$ and $Y$ are two measurable spaces. Is it true that $X \times Y$ (cartesian product) is measurable? I can not understand how is the sigma-algebra of $X \times Y$ ? Could someone helo me to ...
0
votes
0answers
16 views

If $T\in M_{n\times n}(\mathbb R)$ invertible, why is $T^tT$ positive definite?

Let $T\in \mathcal M_{n\times n}(\mathbb R)$ invertible, why is $T^tT$ positive definite? My only idea is $$x^tT^tTx=(Tx)^t(Tx)=\|Tx\|^2,$$ and since $T$ invertible $\|Tx\|=0$ only if $x=0$, but I ...
1
vote
1answer
30 views

How tough is this graph?

In graph theory, toughness is a measure of the connectivity of a graph. A graph $G$ is said to be $t$-tough for a given real number $t$ if, for every integer $k > 1, G$ cannot be split into $k$ ...
-4
votes
0answers
18 views

A B C Truth Table [on hold]

For which line of the truth table do the following expressions differ assuming the truth table is constructed using A B C? Numerical Response F=A'B'+AB'+ABC G=B'C+B'C'+AB
1
vote
1answer
9 views

Existence of a spectral resolution

So, I'm proving that, given a spectral family $(E_t)_{t \in \mathbb{R}}$ in a Hilbert space $\mathscr{H}$ and a continuous function $f \in C(\mathbb{R}, \mathbb{C})$, the conditions below are ...
2
votes
2answers
21 views

$G$ acts on $\operatorname{Hom}(G,K)$ by conjugation

I am working on an example of the Drinfeld Double of the Group Algebra and stumbled upon the book On Characters of Finite Groups. My issue is with 8.1.1, page 192 of the PDF (relevant part here). It ...
-2
votes
0answers
15 views

Which is the function that gives you the nth digit of the champernowne constant? [duplicate]

Which is the function that gives you the nth digit of the champernowne constant? And how do you come up with such a function, proof. I have bruteforced with python to find the nth digit but I am ...
4
votes
0answers
22 views

A compact complex manifold admits an ample line bundle if and only if it is projective

Given a holomorphic line bundle $L$ on a complex manifold $X$, a point $x\in X$ is called a base point of $L$ if $s(x)=0$ for all $s\in H^0(X,L)$ (the space of global holomorphic sections of $L$). The ...
1
vote
1answer
13 views

Understanding supremum in this proof of weak maximum principle

I'm reading this proof of the weak maximum principle. The operator $L$ is this one: $$L = \sum a_{jk}(x)\partial_j\partial_k + \sum b_j\partial_j$$ but I don't think it's relevant for what I'll ask. ...
1
vote
1answer
15 views

Double integral that seems to go over undefined region

Evaluate $\int_R {8\over7}x^2y^{−3}dydx$ where R = {$(x, y) : 1 ≤ x, y ≤ 2, x ≥ y$}. I believe that the integral should be $$\int_1^{3}\int_0^{x} {8\over7}x^2y^{−3}dydx $$ However when I put this ...
0
votes
0answers
10 views

Expected ordering of random variables, given pairwise ordering probabilities

Suppose I have a collection of continuous random variables $V_1, V_2, ... V_n$ and for each $i, j <= n, i \neq j$, $P(V_i > V_j) = k_{i, j}$ is known. Is there an established algorithm for ...
0
votes
1answer
16 views

Equivalent representations for transformation Matrices

I'm having trouble creating a transformation matrix between two joints of a robot. Background: I have a robot and these two joints represented by two different coordinate systems, 4 and 5 (4th and ...
0
votes
2answers
11 views

Calculating Cumulative Binomial Probabilities

I was given the question to find $P(10\leq X\leq 12)$ with $n=15$ and $p=0.666$. Does this mean that $P(10\leq X\leq 12)=P(X\leq 12)-P(X\leq 10)$? If so, I got the answer $0.9206-0.5959=0.3247$ yet ...
1
vote
0answers
15 views

Integration of $\int_{0}^{R}\int_{-1}^{1} \mathrm{d}r\mathrm{d}zr^2\sin\bigg({\frac{1-3z^2}{r^3}}\bigg)$

I need to evaluate the following 2D integral: $$\int_{0}^{R}\int_{-1}^{1} \mathrm{d}r\; \mathrm{d}z\; r^2\sin\left(\frac{1-3z^2}{r^3}\right)$$ where $R$ tends to infinity. We have made several ...
0
votes
0answers
13 views

Does a flat limit of non-degenerate schemes become degenerate?

Say we have a flat family $ \{X_t \}_{t \in \mathbb{P}^1-{0}}$ with $X_i$ sub-schemes of $\mathbb{P}^n$. Assume that each $X_i$ does not lie in a hyerplane. Is it true that the limit $X_0$ also doesn'...
3
votes
1answer
22 views

How to correctly express a variable with two solutions?

Say we have $(x+1)(x+2) = 0$. So possible solutions of $x$ are $x=-1$, and $x=-2$; Would it be considered correct syntax to say "$x = -1, -2$" ?
0
votes
2answers
36 views

$\sum_{1\leq l \lt m\lt n} \dfrac{1}{5^l3^m2^n}$

I need a help to deal with the serie $\sum_{1\leq l \lt m\lt n} \dfrac{1}{5^l3^m2^n}$ (OBM level university, 2018). Some ideia? Thanks very much.
3
votes
1answer
14 views

Chern classes of tangent bundle over the Grassmannian G(2,4)

What are the Chern classes of the tangent bundle $\tau_G$ of the Grassmannian $G=G(2,4)$ of lines in $\mathbb{P}^3$? This is Exercise 5.37 on page 191 of 3264 & All That by Eisenbud and Harris. ...
2
votes
0answers
12 views

Construction G-Invariant Riemannian Metric

Let $M$ be a smooth manifold and $G$ be a lie group acting transitively on $M$. I know by Corollary 1.27 of these notes that there to exist a Riemannian metric $g_G$ on $M$ satisfying the in-variance ...
0
votes
0answers
10 views

Find Mean of a distribution (geometric) using R

I am currently struggling to find a way to calculate the mean of the geometric function (or any other function for that matter using R. So basically I want R to calculate for me $\frac{1-p}{p}$ for ...
1
vote
0answers
13 views

Exercise from Harris Algebraic Geometry: Homogeneity Implies Smoothness?

I'm rather confused by Exercise 14.13 in Harris's Algebraic Geometry, which asks the reader to prove that the Veronese and Segre varieties are smooth. The suggestion is that, 'given Exercise 14.3', ...
0
votes
1answer
24 views

Help proving a measure is not Borel regular.

In this question an example of a measure that is not Borel regular was given. However, I do not understand why this measure would not be Borel regular. I should think that what is said only works for ...
0
votes
0answers
21 views

Positivist polynom: Product of two non-constant polynoms with real coefficients $\geq 0$

I am dealing with the test of the OBM (Brasilian Math Olimpyad), University level, 2017, fase 2. I hope someone can help me discussing this test. The question 1 says: We say that a polynom is ...
4
votes
0answers
16 views

Where does the symplectic structure on coadjoint orbits of Lie groups on their Lie algebras come from?

I have read in several places that if $\Omega$ is the coadjoint orbit of $\zeta \in \mathfrak{g}^*$, the map from $G \to \Omega$ that sends $g \mapsto Ad^*(g)(\zeta)$ gives a surjection, and taking ...
3
votes
2answers
23 views

Number of vertices of degree n not connected to vertices of degree 1

Imagine a tree made up only of vertices of degree 1 and vertices of degree $n$. Let $m$ be the number of vertices of degree 1 and $\frac{m-2}{n-2}$ be the number of $n$-vertices of the tree. How can ...
0
votes
0answers
10 views

Fraction of Variation Explained by a Variable

I have perhaps a basic question, but I cannot seem to find an answer online. Essentially I would like to explain how much of the variation a change in one independent variable contributes to a ...
2
votes
1answer
13 views

Relative homology $H_n(T^2,S^1)$ (Hatcher, Exc 2.1,17.b)

I am going through Hatcher's Algebraic Topology and I am trying to solve exercise 17.b of part 2.1. The question asks to compute the relative homology $H_n(X,B)$. As far as I can understand , if on ...
0
votes
0answers
10 views

Row-sum condition for Runge-Kutta methods

Consider a general RK-method with weights $\vec{b}$ $(s\times 1)$, nodes $\vec{c}$ $(s\times 1)$ and matrix $\vec{A}$ $(s\times s)$. In the literature, there is a widely repeated minimum condition for ...
0
votes
0answers
9 views

Solving a non-homogeneous system of two equations and three variables where product of two of the variables are constant

Consider the following non-homogeneous system of equations where $x,y,z$ are variables and for a constant $\mathrm C$, $y \times z = \mathrm C \neq 0$ \begin{equation} \left\{ \begin{array}{lcl} a_1x ...
0
votes
0answers
9 views

Change of variables with spherical/cylindrical intersection

If $r$ is a positive real number and $W$ is the region of intersection of $\{x \in\mathbb R^3 \mid \|x\| ≤ r\}$ and $\{(x, y, z) \in \mathbb R^3\mid z^2 \ge x^2 + y^2\}$. What would this integral be? ...
2
votes
1answer
7 views

Prove that $\sum_{k=1}^{\infty} a_k g(x-r_k) \not\in L^2(\mathbb{R})$

Problem. Let $$g(x)=\begin{cases} \frac{1}{\sqrt{x}} &x\in(0,1)\\0 &\text{Otherwise} \end{cases}$$ And let $\{r_k, k\in \mathbb{N}\}$ be enumeration of rational numbers in $\mathbb{R}$ and $\{...
0
votes
1answer
15 views

solving pde with initial conditions

I know how to solve it when it is homogeneous and the initial conditions the constants are 0 .But how to solve it when there is some non-homogeneous part. Any help will be appreciated. Thanks in ...
1
vote
1answer
14 views

Sketching complex plane

Sketch all points in the complex plane such that $Re(\frac{1}{z})=1$ I managed to solve it $(\frac{1}{z})= \frac{|z|}{z|z|}=\frac{(x - iy)}{(x^2+y^2)}$ Meaning that $Re(\frac{1}{z})= \frac{x}{(x^2+y^2)...
1
vote
0answers
13 views

Integration of a complex exponential

I would like to know how to integrate a function that looks like $$ \int_a^b \frac{dx}{|x|^{-n-it}}, \quad x \in \mathbb{R}^n, \, 0<a<b. $$ Thank you.
0
votes
1answer
22 views

Radian Measure given the arc is 3/7 diameter

The question is what is the radian measure of a central angle of a circle if the subtended arc has a length that is $3/7$ of the length of the diameter. I reasoned as follows: if $1/2$ the diameter =...
1
vote
2answers
33 views

What does $f(x)=\dfrac{e^{(4x)\pi\cdot i}}{x}$ do as $x\to0$ from the right?

What does $f:\Bbb R\to\Bbb C:: f(x)=\dfrac{e^{(4x)\pi\cdot i}}{x}$ do as $x\to0$ from the right? This is a spiral parametrised by $x$. I can see the real part goes to infinity and it does so in the ...
0
votes
0answers
19 views

Tennis Group Optimization

I organize a group of players. The group is mixed, and there are 4 courts. There are 8 men and 8 women. The group plays 3 rounds, and I avoid repeated matches, so that: - No man or woman will play ...
-3
votes
1answer
13 views

ABCD pattern calculation?

I've been doing a programming code to draw ABCD pattern but I need help to calculate some value like the one on the picture 1 . I wont to find the expression for these values inside green rectangle ? ...
0
votes
1answer
25 views

Prove that $\inf\left\{\int_E f d\mu : E \in \mathcal{M}, \mu(E)\geq 0 \right\}>0$

Problem. Suppose that $(X,\mathcal{M},\mu)$ is a nonzero finite measure space and that $f$ is a positive measurable function on $X$. Let $\alpha\in (0,\infty)$ with $0<\alpha<\mu(X)<\infty$ ...

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