0
votes
0answers
2 views

Height function of a hypersurface

I was reading an article by do Carmo and Warner, which says: "By the height function for an oriented hypersurface at a point $p$ we shall mean the function defined on a neighborhood of the origin in ...
0
votes
0answers
2 views

divergence theorem with singularity at r = 0

I am trying to evaluate the volume integral given by \begin{align} \int_V [\nabla(\vec{x} \cdot \vec{u}) - \nabla \cdot (\vec{x}\vec{a})] dV \end{align} where $\vec{x}$ is the position vector and ...
0
votes
0answers
10 views

Folland 8.20 (Fourier Analysis)

I'm stuck a bit on this problem from Folland: The first part I can't figure out at all. The second part, I know: $\|Pf(x)\|_1 = |Pf(x)| = |\int f(x,y)dy| \leq \int |f(x,y)|dy$. If the last term is ...
0
votes
1answer
9 views

$X,Y$ are iid from distribution $F$, which is a continuous function, then is $P(X=Y)>0$?

Suppose $X,Y$ are iid random variables from a distribution function $F$, which is a continuous function. Then is it true that $P(X=Y)>0$? For me, the answer is trivially YES. Well, why not? We ...
0
votes
0answers
2 views

Linear Regression without X? : Using procedure to construct the GLS, derive the Best Linear Unbiased Estimator of α and compute its variance

(Have been working in matrix algebra) Given model: yi = a + ei ( y_i= α+ϵ_i ) That is y subset i and error term subset i Where the expected value of each error term for each entry is = 0 ...
0
votes
0answers
5 views

Can we exploit FFT for evaluating quadratic on regular, gridded data with stationary covariance?

I would like to evaluate the quadratic $\mathbf{y}^{T}K^{-1}\mathbf{y}$ with the following assumptions: $\mathbf{y}$ is on a regular grid, same spacing in all directions $K$ is separable and ...
1
vote
0answers
13 views

Is T an infinity spectrum whenever T is a spectrum?

Definitions: For a given first order sentence $\phi$ define $\text{spectrum}(\phi)$ to be the set of all cardinalities of the finite models of $\phi$. A set $S\subseteq\mathbb N_+$ is said to be a ...
0
votes
1answer
18 views

Express a relation as a function $f: A \to \mathcal{P} (A)$

Explicitly give the "$\leq$" relation shown in the following graph as a function from $A$ to $\mathcal{P} (A)$. Is this right? For $a,b \in A$ and $(a,b) \in \mathcal{P} (A)$ $$ f := \{ (a,b) ...
0
votes
1answer
10 views

Solving $\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$

If we have $$\arcsin\left(2x\sqrt{1-x^2}\right) = 2 \arcsin x$$ we have to find the set of $x$ for which this is true. I tried to solve it by putting $x = \sin a$ or $\cos a, but got no ...
0
votes
1answer
16 views

Nth Term of a series

Say I have a series like the following; $1 + \frac{1}{5}x + \frac{1 \times 6}{5 \times 10}x^2 + \frac{1 \times 6 \times 11}{5 \times 10 \times 15}x^3 + ...$ How do I find the sum of this? I'm trying ...
0
votes
0answers
17 views

$n(p(p(p(p(b))))$ Start from left to right?

$B$ has $1$ element. Which explains $2^1$ of power sets on the right, which one should I start to calculate first? A bit confused.
1
vote
0answers
6 views

What are those weighed graphs called?

Let $G = (V, E)$ be a directed graph, and define the weight function $f : V \sqcup E \to \mathbb{R}^+$ as follows: sum of weights of vertices is 1, if a vertex has edges coming out of it, their ...
1
vote
0answers
9 views

Rank of a locally free $\mathbb Z[G]$- module

Let $G$ be a finite group, $M$ a finitely generated $\mathbb Z[G]$-module so that the $\mathbb Z_p[G]$-module $M_p$ is free for all prime numbers $p$. Here $\mathbb Z_p[G]=\mathbb Z_p\otimes_\mathbb ...
0
votes
0answers
12 views

If $X = \{0,1\}$ is a discrete topological space, is $\{0,1\}^{|A|}$ a discrete topological space ($A$ uncountable)?

Take $X = \{0,1\}$ with the discrete topology. Then, let $A$ be an uncountable set, then $\{0,1\}^{|A|}$ is a discrete topological space? I am trying to prove that arbitrary product of connected ...
0
votes
1answer
18 views

Find $n$ such that the congruence $x^n\equiv 2\mod 13$ has a solution for $x$.

Find $n$ such that the congruence $x^n\equiv 2\mod 13$ has a solution for $x$. I am not getting any idea how to start this problem. Please give some hits
0
votes
0answers
8 views

Difference in real vs computer simulated probability distribution results.

If this is not the right place to ask this,please guide me. I had a thought about what if our basic laws are somehow flawed such that they work in the situations we have observed but not in some ...
0
votes
0answers
11 views

Uniform convergence of power series on closed unit disk implies holomorphy there?

I am having trouble proving that the space of holomorphic functions continuous till the closure in the unit open disk coincides with the power series whose coefficients form an absolute convergence of ...
1
vote
0answers
7 views

Poisson Equation: 'Dirichlet' type problem on all of $\mathbb{R}^N$

I've seen a great deal about solving the Dirichlet problem for Poisson's equation \begin{equation} \Delta u = f \quad \mbox{in} \quad \Omega \subset \mathbb{R}^N\\ u = 0 \quad \mbox{on} \quad ...
0
votes
0answers
10 views

Finding the image of the annulus $\{1 < |z| < e\}$ under the principal logarithm

What I did so far is test the intercepts to see where they get mapped and I got $0$ for the lower ones and $1 + i\pi$ and $1 \pm i\frac{\pi}{2}$ and $1$ for the others. It seems all over the place. ...
1
vote
0answers
12 views

inequality involving heights and bisectors

Let $a,b,c,a \le b \le c$ be the sides of the triangle $ABC$, $l_a,l_b,l_c$ the lengths of its bisectors and $h_a,h_b,h_c$ the lengths of its heights. Prove that: $$\frac {h_a+h_c} {h_b} \ge \frac ...
0
votes
0answers
5 views

Should The domain in Simple approximation theorem be measurable?

Here's Royden's version of The Simple Approximation Theorem, My question is do we really need domain $E$ be measurable? Or in other way, do we always define measurable function on a measurable ...
0
votes
0answers
4 views

Existence of Type I Self Dual Codes

How would I prove Type I self-dual codes (binary self-dual codes that are not necessarily doubly even) exist for every even length $n$?
1
vote
2answers
14 views

Laurent Series about $z=0$ of $f(z) = \frac{1}{z^3 - iz}$

So far: $$ \frac{1}{z^3 - iz} = \frac{1}{z(z^2 - i)} = \frac{i}{z} - \frac{iz}{z^2 - i} $$ Now I see that: $$ \frac{-iz}{z^2 - i} = z\left(\frac{i}{i - z^2}\right), $$ and this is where I get stuck. ...
1
vote
1answer
23 views

Does $\displaystyle \lim_{x \rightarrow 0^{+}}f(\ln x+ x)=-\infty$ imply $\displaystyle \lim_{x \rightarrow -\infty}f(x)=-\infty$?

Let $f$ be a function defined on $\mathbb{R}$ such that $\displaystyle \lim_{x \rightarrow 0^{+}}f(\ln x+ x)=-\infty$. True or false? $\displaystyle \lim_{x \rightarrow -\infty}f(x)=-\infty$. ...
0
votes
2answers
8 views

Isomorphism of a ring of matrices

Is it possible for a ring of matrices to isomorphic a ring of numbers? Suppose $$R = \begin{pmatrix} a & b \\ -3b & a \\ \end{pmatrix} a,b \in \mathbb Z $$ Can $R$ ...
0
votes
1answer
10 views

Show that the operator induced by $T$ on the quotient space $V/\operatorname{ker} (T-5I)$

A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)$. Show that the operator induced by $T$ on the quotient space ...
0
votes
1answer
18 views

Determine the maximum and the minimum of an expression

Let $x,y,z \in \Bbb R, x,y,z \gt 0$ such that $x^2+y^2+z^2=1$. Determine tha maximum and the minimum possible values of the expression $$\frac {x^3+y^3+z^3} {x+y+z}.$$
1
vote
0answers
19 views

Explaining the Ackermann function as A: $\mathbb N \times \mathbb N \rightarrow \mathbb N$

We have the following variation of the Ackermann function: $$A(0,m) = m+1$$ $$A(n,0) = \begin{cases}1, & \text{if } n=0 \\ 2, & \text{if } n=1 \\ 0, & \text{if }n=2 \\ 1, & ...
0
votes
1answer
5 views

characters in semi-direct product.

The character tables of the irreducible representations of $T_d$ and $C_{3v}$ are linked. In the notation on those pages, $A_1$ and $A_2$ are irreducible representations of degree 1, $E$ is degree 2 ...
0
votes
1answer
13 views

Arrangements of crew in two sides of a boat - permutations and combinations

A boat crew consist of 8 men, 3 of whom can row only on one side and 2 only on the other. The number of ways in which the crew can be arranged is This is a problem my math teacher has given ...
0
votes
1answer
18 views

Are Z and Z* (defined below) isomorphic as rings?

Define $\mathbb{Z}^*$ to be the set of integers but with the following operations: $a \circ b = a + b - 1$ and $a * b = a + b - ab$ where $a+b$ and $ab$ are the usual integer addition and ...
0
votes
0answers
5 views

Show that a connected $\alpha$-critical graph has-no cut vertices.

A graph is $\alpha$-critical if $\alpha$ (G - e) > $\alpha$ (G) for all e$\in$E. The number of vertices in a maximum independent set of G is called the independence number of G and is denoted by ...
1
vote
1answer
23 views

$[0,\infty]$ valued measurable function

I have a question about measure theory. Let $(X,\mathcal{M}, \mu)$ be a measure space and $f$ be a $[0,\infty]$ valued $\mathcal{M}$ measurable function on $X$. Is there $a_{k} \ge 0, A_{k} \in ...
1
vote
0answers
7 views

PDEs on higher genus Riemann surfaces, e.g. Klein Curve

I'm trying to solve a PDE on compact Riemann surfaces of genus g > 1. Since these can be obtained as quotients of the upper half plane $\mathbb{H}_2$ by some Fuchsian group $\Gamma$, I suppose it's ...
1
vote
0answers
28 views

Question on the hitting probability

I am confused on the relation between some basic operations with sets and probability. Consider a set $$ A=B\cup C\cup D $$ with $B,C,D$ disjoint sets. Take a random set $S$ almost surely non ...
0
votes
0answers
14 views

Groups of order 36 have a normal subgroup of order 3 or 9

I am trying to understand the proof from the first two paragraphs from https://math.berkeley.edu/~wodzicki/257/G36.pdf that groups of order 36 have a normal subgroup of order 3 or 9. This is what I ...
0
votes
0answers
13 views

Is there any proof of the criterion of determining maximal ideal in a commutative ring with unity by Third Isomorphism Theorem?

Theorem. Let $R$ be a commutative ring with unity $1$ and $M$ is an ideal of $R$. Show that $M$ is maximal iff $\dfrac{R}{M}$ is a field. In the proof of this theorem the methods so far I have ...
2
votes
3answers
32 views

Fundamental Period of $\tan x \cot x$

What is the period of $\tan x \cot x?$ I was given this question today. What I did was simplify the expression , and it reduced to a constant function. So it had no fundamental period. But my teacher ...
0
votes
0answers
6 views

Markovian Model: scheduling jobs to servers

I have the following problem. I tried to look at queuing theory, but it probably fits better as a scheduling problem. I have a set of $C$ servers: each one can perform 1 job. Processes arrive ...
1
vote
0answers
8 views

Fourier Sequence Converges Uniformly Implies Almost Everywhere Pointwise Convergence

I'm trying to understand this problem: Let $f$ be Riemann integrable on $[0,2\pi]$ Suppose that the Fourier Series of $f$, $S_{n}^{f}(x)$, converges uniformly on the interval. I want to show that ...
2
votes
2answers
14 views

Proportional Distribution

I have a problem regarding supply distribution. I distribute widgets on a monthly basis; I have many customers and each of them request a different quantity each month. My monthly supply is limited ...
-1
votes
0answers
17 views

PHI Conjecture: Fibonacci Number Line Segments & Simple Golden Ratio Puzzle?

This is a golden ratio conjecture. Does the golden ratio exist as conjectured below? The three line segments in the figure below have lengths of the Fibonacci numbers 2, 3, and 5. blue=2 green=3 ...
2
votes
3answers
46 views

If $f$ function then $f^{-1}$ function iff $f$ function injective (one-to-one).

During the lecture we learned this phrase: "If $f$ is a function then $f^{-1}$ is a function iff $f$ is injective (one-to-one)." But why? What with onto? $f$ doesn't need to be Surjective ...
0
votes
0answers
7 views

Finding a function within a dynamical system using a lyapunov function

Consider the system for $(x_1(t),x_2(t))$ \begin{align} \dot{x}_1 &= x_1^2+x_1^3+x_2\\ \dot{x}_2 &= x_1^2+u \end{align} Find a function $u=\phi(x_1(t),x_2(t))$ so that if $$V(x) = ...
0
votes
0answers
12 views

Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale?

Let $(x_n,\mathcal{F}_n, n\ge 1)$ be a martingale diference. Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale and why?? $a_n$ is a constant.
0
votes
0answers
19 views

Financial mathematics- finding yield rates for bonds

I'm not sure if its appropriate to post here but oh well QF put me on hold. Joe must pay liabilities of $1,000$ due $6$ months from now and another $1,000$ due one year from now. There are two ...
0
votes
0answers
5 views

Regularity connected union of smooth sets

Consider a countable family of bounded, open connected sets with smooth boundary $S_i \subset\mathbb{R}^n$. What can be said about the regularity of the boundary of $S = \bigcup_{i = 1}^{\infty} S_i$ ...
0
votes
1answer
10 views

Use of Bayes theorem in the Lovásk local lemma

Here's a line from the proof on Wiki I don't understand. $$\Pr(A\mid\bigwedge_{B\in S}\bar{B}) =\frac{\Pr(A\bigwedge_{B\in S_1}\bar{B} \mid \bigwedge_{B\in S_2}\bar{B})}{\Pr(\bigwedge_{B\in ...
1
vote
0answers
18 views

Laurent series for $f(z) = \exp(z+\frac{1}{z})$ around $0$

I need to find the Laurent series of the following function around $0$ - $$f(z) = \exp(z+\frac{1}{z})$$ Now by power series expansion, I got $$f(z) = \sum_{m=0}^{\infty} \frac{z^m}{m!} ...
1
vote
1answer
30 views

what is the CDF of $f(x)=\frac{3x^2}{2}$?

This is probably a dumb question but I just want to make sure. The pdf is $f(x)=\frac{3x^2}{2}$ if $-1 \leq 0 \leq 1$. The CDF is $F(x)=\frac{x^3}{2}$ but with what bounds? sorry if this is an easy ...

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