0
votes
0answers
7 views

Examples of compactly supported exact differential forms

I am having some trouble finding any examples of compactly supported exact differential forms on $\mathbb{R}^n$. I have found $e^{\frac{1}{x^2 -1}}$ when taken to be zero everywhere except on the ...
0
votes
0answers
8 views

$A\cup \text{Fr}A$ is a manifold

I'm dealing with the following exercise from Spivak's "Calculus on Manifolds": Let $A\subset \mathbb R^n$ be an open set such that the (topological) boundary of $A$, $\text{Fr}A$ is an $n-1$ ...
4
votes
1answer
11 views

Geometric progression question. Year less?

Here is the math question. A 100m cliff erodes by 2/7 of its height each year. (a) What will the height of the cliff be after 10 years? This is how I worked out the question. ...
0
votes
1answer
11 views

How to show two integers are coprime in a linear combination?

Show that $a$ and $b$ are relatively prime if and only if 1 is expressible as a linear combination of $a$ and $b$. Using that fact, show that any two consecutive integers are relatively prime. ...
0
votes
0answers
4 views

Show there is no intersection between sequence of balls on an open set.

Is there a way to show that a finite sequence of balls, in an open set $U ⊆ C$, with a small enough radius's such that there is no intersection between any of the balls? Intuitively it makes sense to ...
0
votes
0answers
5 views

Let $X$ and $Y$ be iid real-valued random variables. Show $P[|X-Y| \le 2] \le 3P[|X-Y| \le 1]$.

Found this question in The Probabilistic Method and tried for hours to prove it, but I'm not getting anywhere. Can anyone walk me through it?
1
vote
0answers
9 views

Convergence of $\sum_{i \leq n} X_i/n$

I have a question like this: Let $(X_n)$ be an i.i.d sequence of random variables with values in {-1,1}, and define $Y_n:= \sum_{i \leq n} X_i/n$. Show that $(Y_n)$ converges almost surely and in ...
0
votes
1answer
10 views

Fermat's Little theorem (Num Theory)

How can I compute 31^(1209)mod 101 using Fermat's Little Theorem?
0
votes
0answers
5 views

$G \subset G(F(x,y)/F)$ where$G = <\sigma>$, $\sigma(x) = y$, $\sigma(y)=x$. Describe the subfield of F(x,y) consisting of elements fixed by $\sigma$

Let $F$ be a field and let $E = F(x,y)$, where $x$ and $y$ are indeterminates. Let $G \subset G(E/F)$ be the subgroup defined by $G = <\sigma>$, where $\sigma(x) = y$ and ...
0
votes
0answers
8 views

The problem about the gödel proposition V in 1931 paper?

Proposition V says that every recursive function R can find a relation symbol in system P such that: R(v1, v2....,vn) -> prove(subst( r(u1,u2,...,u3), (z(v1),z(v2),...,z(vn))) ~R(v1, v2....,vn) ...
-3
votes
0answers
5 views

How can I calculate O - Notation of an algebraic expression?

Explain the process of obtaining O - Notation for a given algebraic expression. For example 5n^2+n^(3/2)
0
votes
0answers
3 views

Outer measure of a nested sequence of non-measurable sets

Let $\bigcup_{n=1}^\infty E_n=E$ and $ E_{n} \subseteq E_{n+1} $ then $\lim\limits_{n\mapsto \infty} \mu^*(E_n) = \mu^*(E) $ even if each $E_n$ is a non-measurable set, where $\mu^*$ is outer ...
0
votes
0answers
6 views

turning a certain chebychev polynomial-like expression into a hypergeometric form

Can the following expression be represented in terms of hypergeometric function $$\sqrt{3}\sin(\arcsin(7/25)/3)-\cos(\arcsin(7/25)/3)$$ It looks similar to the one presented on this site
0
votes
0answers
5 views

Find the most general bilinear transformation that maps $|z-1|=1$ to $\Re(f(z)) = 1$.

Find the most general bilinear transformation that maps $|z-1|=1$ to $\Re(f(z)) = 1$. Attempt Assuming that $$ 2 \mapsto 1 \\ 0 \mapsto \infty \\ 1+i \mapsto 1+i $$ we have the bilinear ...
0
votes
0answers
10 views

Still not clear why longest path problem is NP-hard but shortest path is not?

I've heard/read many times that shortest path problem is P, but longest path problem is NP-hard. But I have a problem with this: we say longest path problem is NP-hard because of graphs with positive ...
-1
votes
0answers
8 views

How do you isolate and solve for n in a Sigma notation equation?

"isolate and solve for n:" $$P(n)= \sum_{i=0}^n = (n^2+n)$$ If the above equation is a function of P, how would the equation be stated as a function of N? P.S. what should I be tagging this ...
0
votes
2answers
35 views

Why cannot the power of $i$ be negative?

In a question, The penultimate step of the result was $(i)^n = 1$, and it required to figure least value of n. I checked the $-4$ option, but it said the answer was $4$. Why so? The complete question ...
0
votes
0answers
5 views

Showing uniform convergence to origin in 3rd quadrant for $x(t)=\frac{1}{\frac{1}{x_0}-t}$ as $t\ \rightarrow \infty$

I want to show that for the system $\dot{x}=x^2, \dot{y}=y^2$,any solutions starting in the 3rd quadrant not including 0, converge uniformly to the origin. For an initial point $(x_0,y_0)$, (note both ...
0
votes
0answers
10 views

Converting higher-order ODE to first order ODE

given $y''' + 2y'' -5y' = 2y + 5y^3$ convert to a system of first order equations. My question is do we need to make substitutions for $y$ and $y^3$ or are we only concerned with the derivatives, if ...
0
votes
1answer
16 views

This seems simple but, how do i prove A isn't in C?

$A\in B$ and $B\in C$, Is it posible to prove $A$ isn't in $C$? Sorry for simplistic exercise, but been wondering this for days now... The question in the book says: Can you deduce that $A\in C$ ? ...
0
votes
1answer
15 views

Show that if $X$ is a bounded a subset of $\mathbb R$ then the closure of $\bar X$ is also bounded.

Show that if $X$ is a bounded subset of $\mathbb R$ then the closure of $\bar X$ is also bounded. My attempt I can show that if $A$ is bounded then so is $\bar A$. So if we have have limit point of ...
2
votes
3answers
12 views

Show that the $C_n \geq 4^{n-1}/2^{n}$ where $C_n$ is the Catalan number

I write $C_n=\frac{1}{n+1} {2n\choose n}$ and try to prove this claim by induction. But it didn't quite work out. Any idea how to do this without much computation?
0
votes
2answers
22 views

When a Rational function becomes a line?

Why is it that when $AD = BC$, this equation becomes a horizontal line? $$y = \frac {Ax+B}{Cx+D} $$ For any other values where $AD$ isn't equal to $BC$ it is a rational function.
0
votes
1answer
12 views

Cumulative distribution function (CDF) strictly less than

Suppose a distribution function for the random variable $X$ is given by $$F(x)=\left\{ \begin{array}{11} \hfill 0 \hfill & x \lt 0\\ \hfill \dfrac{x}{2} \hfill & 0 \leq x \lt 1\\ \hfill ...
0
votes
0answers
19 views

Need a reference book for stokes theorem

I am studying singular homology, I would like a good reference for the proof of stokes theorem for chains in manifolds. Thank you!
0
votes
0answers
10 views

Explain why the algorithm for modular exponentiation works

Can someone explain with a proof or similar why the algorithm for modular exposition works?
1
vote
2answers
22 views

Use induction to prove that $2^n \gt n^3$ for every integer $n \ge 10$.

Use induction to prove that $2^n \gt n^3$ for every integer $n \ge 10$. My method: If $n = 10$, $2^n \gt n^3$ where $2^{10} \gt 10^3$ which is equivalent to $1024 \gt 1000$, which holds for $n = ...
0
votes
0answers
5 views

Calculate a quadratic irrational from its periodic continued fraction

I have a periodic continued fraction [2; 1, 3] and I want to convert it into a quadratic irrational. Any helps?
0
votes
1answer
14 views

Lipschitz continuity of continuously differentiable function

Is it true that a continuously differentiable function in a Banach space $X$ is locally lipschitz in $X$?
0
votes
2answers
17 views

Basic Number Theory (Divisibility)

Not sure where to start. Thank you in advance! Find all positive integers $n$ such that $12$ divides $n$ and $n$ divides $816$.
0
votes
2answers
19 views

Differential Equation - Where does the solution end?

I was asked to solve the differential equation $y'+\frac{y}{x+1}=\frac{2y-1}{x}$, given the starting point y(0.5)=5/6. The equation meets the criteria for Existence and Uniqueness for every x>0 (as y' ...
0
votes
0answers
16 views

Poker Pot Odds EV Problem

My issue is with how EV is calculated. I think it's wrong in the approach how it treats the Pot. What's put in the Pot belongs to the Pot. Sure! But that's kinda stupid because the Pot will partially ...
0
votes
1answer
12 views

Function estimation

I have a list of chest measures and I need to find the size that fits as many measures that it can. If I need to find 1 size it's easy as I can minimize the function $(x−a)^2$ with $x$ as size ...
0
votes
0answers
9 views

Prove that a subset C of $\mathbb R^n$ is closed if and only if it contains all its limit points

Prove that a subset C of $\mathbb R^n$ is closed if and only if it contains all its limit points. A closed set is defined by a set of all boundary points. My professor said "We may prove that C is ...
1
vote
0answers
14 views

Taylor series roots at infinity

I started thinking about this after this MathSE thread. Take a sequence of Taylor polynomials $f_n$ that converge to $f$. Does $f_n$ always have a growing number or roots in $\mathbb{C}$ which grow ...
0
votes
1answer
11 views

Prove the general solution principle for linear algebra

Given that a vector v solves the homogeneous equation Av = 0, and that a particular solution y Ay = b has been found, prove that the general solution principle, stating that any solution can be given ...
2
votes
1answer
28 views

Why are function spaces generally infinite dimensional

The other day, I was trying to explain some concepts in Fourier analysis and wavelets to my girlfriend (an electrical engineering student) and obviously, the concept of Lebesgue integration came up in ...
0
votes
0answers
14 views

Fourier Transform of a radial function in $L^1(\mathbb{R}^2)$

Let $f \in L^1(\mathbb{R}^2)$ be radial, i.e. there exists $g: [0,\infty) \rightarrow \mathbb{R}$ such that $f(x) = g(|x|)$. Prove that $f$ is also radial. (Note that this result is true for ...
0
votes
0answers
25 views

Solving This Polynomial?

I have the following polynomial $$\frac{(ar-1)(ar-2)(ar-3)(ar-4)(ar-5)}{(r-1)(r-2)(r-3)(r-4)(r-5)} = P$$ where $r$ is my variable and $P$, $a$ are just real constants. I was wondering whether or not ...
0
votes
3answers
50 views

Integrating arctan

$x^2\over(1+x^2)$ Is there any other way to integrate this other than make the numerator $x^2 +1 -1$ and then split this into two integrals? My smartass friend told me that this is the only way but ...
2
votes
1answer
38 views

Show that $3\cdot |A|<9^{|A|}$ for every $A$

I am trying to prove this question which came up in my university's set theory exam last year. A few similar questions have been asked over the last few years and I cannot figure out the method to ...
0
votes
0answers
23 views

Biggest number of teams with 16 wins in a tournament

Here is a problem from a math competition - the solution of which requires the enumeration of combinations. I am asking for affirmation of my solution. Twenty teams are in a round-robin tournament; ...
0
votes
0answers
22 views

Central limit theorem and the sequence with general term $e^{-n} ( 1+n+ \cdots + n^n/n!)$ [Proof check]

As an exercise I need to find the limit of the said sequence $$e^{-n} ( 1+n+ \cdots + n^n/n!)$$ using the toolkit of probability theory. Since no solution (only hints) is provided, I would appreciate ...
0
votes
0answers
19 views

Quotient space, torus complex

Good evening! I'm reading Chapter 2 of the $Advances in Moduli Theory$ - Yuji, Kenji, my goal is the theorem of Torelli. So I find in the way of my reading the following: "...a point of the ...
0
votes
0answers
9 views

Solving solution given initial condition condition

Suppose we know that: $$u_t=ku_{xx},~~~~~~~~0<x<l,~~~t>0$$ and $$u(x,t)=\sum_{i=0}^\infty[C_n~cos(n\pi x/l) ~e^{-w_nkt}]$$ where $w_n=\frac{n\pi}{l} ~~~ for~~n=1,2,3,...$ What if the ...
0
votes
0answers
16 views

Would this be considered a valid proof for $\forall r \in R$ if $0 < r < 1 $, then $\frac{1}{r(1-r)}\geq 4$

I did a proof of the following $\forall r \in R$ if $0 < r < 1 $, then $\frac{1}{r(1-r)}\geq 4$ using a proof by contra-positive, which was different from the direct proof that the solutions ...
1
vote
2answers
19 views

How do I transform the matrix with similarity transformation to diagonal form and use this result to calculate $A^{10}$?

I have matrix $A=\left (\begin{matrix} -2 & -8 & -12\\ 1 & 4 & 4\\ 0& 0 &1 \end{matrix} \right )$ and I need to transform with similarity transformation to diagonal form. ...
1
vote
1answer
15 views

Judicious guess for the solution of differential equation $y''-6y'+9y= t^{3/2} e^3t$

$(a)$ Let $L[y]=y''-2r_1y'+r_1^2y.$ Show that $$L[e^{r_1t}v(t)]=e^{r_1t}v''(t).$$ $(b)$Find the general solution of the equation $$y''-6y'+9y= t^{3/2} e^3t.$$ I have problems only in part $(b)$.
1
vote
0answers
5 views

Projection of measure with bowen - walters metric.

Given $X$ a compact metric space, $f:X\to X$ be a homeomorphism and consider the quotient space $Y^{1,f}=(X\times [0,1])/\sim$, where $(x,1)\sim(f(x),0)$ for all $x\in X$. Let $d^{1,f}$ be the ...
-1
votes
0answers
34 views

Complex limit $\displaystyle \lim_{h \rightarrow 0} \lim_{z \rightarrow 0} \frac {h^2 \pm \sqrt{h^4 - 36(h - h^2\tan(z) + 9\tan(z))}}{2\tan(z)}$

I am trying to evaluate the following limit: $$\displaystyle \lim_{h \rightarrow 0} \lim_{z \rightarrow 0} \frac {h^2 \pm \sqrt{h^4 - 36(h - h^2\tan(z) + 9\tan(z))}}{2\tan(z)}$$ I thought I could ...

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