0
votes
0answers
67 views

Estimating the radius of a circle

I have a circle iwth radius $r$. I want to test the hypothesis that $r \leq 2$ vs. $r >2$ based on the posterior of $r$. $r$ follows the prior distribution: $f(r) = \frac{2}{r^{2}}$, $ r >0.5$. ...
2
votes
2answers
49 views

Inequality proof 2

How to prove the inequality : for real numbers $\alpha_1, \ldots \alpha_n, \beta_1, \ldots \beta_n$: $$\sqrt{(\alpha_1 + \beta_1)^2+\cdots+(\alpha_n + \beta_n)^2} \leq \sqrt{\alpha_1^2 + \cdots + ...
1
vote
3answers
402 views

Find integer solutions of $x^2 -px +q=0$, where $p$ and $q$ are prime

Quick number theory question that I have just come across, was wondering if anyone could shed some light on it. So $p$ and $q$ are given to be prime numbers, and we are told that the equation $x^2 ...
6
votes
3answers
540 views

First derivative of $\sqrt[\large 5]{\frac{t^3 + 1}{t + 1}}$

I have yet another derivative I need help with. I have to differentiate : $$\sqrt[\uproot{3}{\Large 5}]{\frac{t^3 + 1}{t + 1}}$$ with respect to $t$. I had two thoughts about this, use the chain ...
2
votes
1answer
514 views

Intuitive way to understand the square wave spectrum?

What is a intuitive way to understand that a transform of a square wave can result into something like this?
1
vote
1answer
480 views

Composition of Lebesgue measurable function $f$, with a continuous function $g$ having a certain property, is Lebesgue measurable

Suppose that $f$ is Lebesgue measurable and $g$ is real valued, continuous, and has the property that for any null set $N$, $g^{-1} (N)$ is measurable. Then $f \circ g$ is also Lebesgue measurable. ...
6
votes
3answers
2k views

Prove equality in triangle inequality for complex numbers

We need to show that $$ |z_{1}+z_{2}+\cdots+z_{n}|=|z_{1}|+|z_{2}|+\cdots+|z_{n}|$$ if and only if $z_{1},z_{2},\dots,z_{n}$ have the same argument (i.e. $z_{j}=r_{j}e^{i\theta}$ for $j=1,\dots,n$). ...
7
votes
2answers
227 views

What is the name of a game that cannot be won until it is over?

Consider the following game: The game is to keep a friend's secret. If you ever tell the secret, you lose. As long as you don't you are winning. Clearly, it's a game that takes a lifetime to win. ...
18
votes
3answers
1k views

For x < 5 what is the greatest value of x

It can't be $5$. And it can't be $4.\overline{9}$ because that equals $5$. It looks like there is no solution... but surely there must be?
1
vote
3answers
98 views

Trigonometric substitution integral

Trying to work around this with trig substitution, but end up with messy powers on sines and cosines... It should be simple use of trigonometric properties, but I seem to be tripping somewhere. ...
2
votes
3answers
111 views

Proof by induction that $\left(\sum_{k=1}^nk\right)^2\ge\sum_{k=1}^nk^2$

I am a bit new to logical induction, so I apologize if this question is a bit basic. I tried proving this by induction: $$\left(\sum_{k=1}^nk\right)^2\ge\sum_{k=1}^nk^2$$ Starting with the base ...
1
vote
3answers
9k views

How to calculate the median of a continuous random variable

$X$ is a continuous random variable with probability density function $f(x)= \dfrac{2x}{15}$ where $1≤x≤4$. What is the median of X?
3
votes
1answer
584 views

What is the order of convergence of Newtons root finding method? And when does it converge?

Given a function $f(x)$, we can approximate $x_r$ where $f(x_r)=0$ , by using Newton's method: $$x_{n+1}=x_n -\frac{f(x_n)}{f'(x_n)} $$ The method only works when 'you choose an $x_0$ near enough to ...
0
votes
2answers
147 views

What exactly is the complex plane, and how is it useful?

A lot of functions are defined on the complex plane, like the Gamma function: the Lambert W function, etc. But I have no idea about what the complex plane means and how it's useful, or just ...
1
vote
1answer
325 views

Geometric representation of product rule?

At time 1:06 of this video by minutephysics, there is a geometric representation of the product rule: However, I don't understand how the sums of the areas of those thin strips represent $d(u\cdot ...
3
votes
1answer
494 views

Lyapunov Stability of Non-autonomous Nonlinear Dynamical Systems

Let $\mathbf{F}:X\times\mathbb{R}^{+}\to X$ be a non-autonomous dynamical system, which is governed by $\dot{\mathbf{x}} = \mathbf{F}(\mathbf{x}, t, u)$, viz, \begin{equation} \begin{split} \dot{x}_1 ...
0
votes
0answers
78 views

conditions on coefficients of univariate polynomial so that it has only real roots

Consider a univariate polynomial of degree $n$ with real coefficients. Are there general equalities/inequalities on its coefficients, so that it has precisely $n$ real roots? For example for the case ...
1
vote
2answers
408 views

Combining two identical uniform distributions

Say two random variables, $X$ and $Y$, are such that $X$ ~ $U(0,a)$ and $Y$ ~ $U(0,a)$. What will the pdf be for $Z$, where $Z=X-Y$?
22
votes
3answers
425 views

How to solve $\int_0^\infty J_0(x)\ \text{sinc}(\pi\,x)\ e^{-x}\,\mathrm dx$?

I need some help with solving this integral involving Bessel function: $\hspace{2in}\displaystyle\int_0^\infty$$J_0(x)\ $$\text{sinc}(\pi\,x)\ $$e^{-x}\,\mathrm dx.$
4
votes
2answers
236 views

Rotation of a point in 3d space

I'm trying to rotate a point around a single axis of a 3D system. Given $P=\begin{pmatrix} 101 \\ 102 \\ 103 \end{pmatrix} $, And the rotation matrix formula for rotation around the X axis only, I ...
4
votes
2answers
60 views

Why the root of this tree has to be “1”?

Arrange $2^{n-1}-1$ zeroes and $2^{n-1}$ ones in a balanced full binary tree of depth $n$. If we want the number of edges that connect the same (and respectively different) digits are the same, then ...
1
vote
0answers
89 views

General solution for $M^{\circ -1 }(y)=x $ when $g(x)e^{f(x) }=y$

Reading this question $e^{C/x }-1=D/(x + a) $, i found my self completely unable to do anything. This is much more hard for me than my easy exercises about Lambert $W$-function. So I probably need ...
3
votes
1answer
61 views

Application of the Identity Theorem to $|x|^3$ for $-1<x<1$

Oxford Exam $2602$ $1997$ $Q3$ We want to show that there is no function $f$ which is holomorphic in $D(0;1)$ and such that $f(x)=|x|^3$ for $-1<x<1$. Here are my thoughts thus far: Suppose ...
0
votes
2answers
153 views

General Linear Groups with Homomorphisms [closed]

Let $G=\mathrm{GL}_n(\mathbb R)$ and $H=\mathbb R^*$. Let $\phi : G=\mathrm{GL}_n(\mathbb R) \rightarrow \mathbb R^*$ be the map defined by $\phi(A)=\det(A)$. Show that $\phi$ is a group ...
2
votes
1answer
85 views

Show a certain vector field $V$ only commutes with others that are collinear with $V$

I found this on an old qualifying exam. I started the problem, but I'm not sure what my next step should be: Let $T^2$ be the standard 2-dimensional torus with $\mathbb{Z}$-periodic coordinates ...
4
votes
0answers
108 views

how prove $\phi(n)\ge \frac{n}{6\log \log (n)} $ $\forall n\ge5 $ [duplicate]

How to prove$\forall n\ge5 $ $$\phi(n)\ge \frac{n}{6\log \log (n)} $$ $\phi$ is Euler function Thanks in advance
1
vote
1answer
34 views

Results following from Analyticity on a domain

This is part of an old Oxford exam paper (1997 2602 Q2) I'm working on for revision. Suppose we have a function $f$ which is holomorphic on the disc radius $R$ about $0$. We want to show that there ...
1
vote
1answer
131 views

Dimension of the set of self-adjoint operators

I'm trying to figure out what the dimension of the set of self-adjoint operators on V would be, or in more concrete terms: Let $dim V =n$. Let $S(V)$ denote the set of self-adjoint linear operators ...
1
vote
3answers
301 views

Confusing math problem

How would I solve this question? I came across it and is really confused. The payment of Jon was bigger by $960$ than the payment of David. After the payment of David got increased by $10\%$, Jon and ...
7
votes
1answer
163 views

Ulm and Frattini Subgroups

Let $A$ be an abelian group. We define $U(A)=\cap (nA), n\in \mathbb N$ be the Ulm subgroup of $A$. The Frattini subgroup of $A$ is $\Phi(A)=\cap(pA)$ ($p\in \mathbb P$). I was trying to show that ...
1
vote
0answers
254 views

Integral of gaussian and sine/cosine

I really need the solution of two integrals involving exponentials and sine/cosine. For $n\in \mathbf{N}$ even : $$\int_{-\infty}^{+\infty}\left(\frac{2\,ax\sin(\pi ...
2
votes
0answers
144 views

The partial sum and partial product of $\zeta$function

Taking the partial sum of the $\zeta $ function: $$\zeta^H(s,k)=\sum_{n=0}^k \frac{1}{n^s}$$ and the partial product f the $\zeta $ function: $$\zeta^P(s,j)=\prod_{i=0}^j \frac{1}{1-p_i^{-s}}$$ I ...
7
votes
1answer
226 views

Proving that the cohomology ring of $\mathbb{R}P^n$ is isomorphic to $\mathbb{Z}_{2}[x]/(x)^{n+1}$

We have to prove that $H^{*}(\mathbb{R}P^n, \mathbb{Z}_2) \simeq \mathbb{Z}_{2}[x]/(x)^{n+1}$ as ring. So we have to find an isomorphism $$ \phi: \mathbb{Z}_{2}[x]/(x)^{n+1} \rightarrow ...
2
votes
3answers
551 views

number of ways to make $2.00

How many different ways can you make $2.00 using only 1 cent, 5 cent, 10 cent, and 25 cent pieces, and 1 and 2 dollar bills (there are 100 cents in a dollar)? I have worked out an equation: $$p + 5n ...
2
votes
1answer
115 views

How to find the norm of the operator $(Ax)_n = \frac{1}{n} \sum_{k=1}^n \frac{x_k}{\sqrt{k}}$?

How to find the norm of the following operator $$ A:\ell_p\to\ell_p:(x_n)\mapsto\left(n^{-1}\sum\limits_{k=1}^n k^{-1/2} x_k\right) $$ Any help is welcome.
2
votes
2answers
68 views

Is the following version of the fundamental lemma of the calculus of variations valid?

Let $U$ be an open subset of $\mathbb{R}^n$ with smooth boundary $\partial U$. Consider a function $f$ in $L^2(u)$. Suppose that for every $h$ in the Sobolev space$ H^2_0(U)$ it holds that $$\int_U f ...
1
vote
0answers
70 views

Relation between $H^i_I(-)$ and $H^i_J(-)$ when $I\subset J$

What is the relation between $H^i_I(-)$ and $H^i_J(-)$ (cohomological functors) when $I\subset J$ are ideals of a (local) noetherian ring?
1
vote
2answers
55 views

determining the value of e by using mathematical induction

using the fact that $n!>2^{n}$ $\forall n\ge 4$ conclude that $e<\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\displaystyle\sum_{n=4}^{\infty}\frac{1}{2^{n}}$ where ...
0
votes
1answer
321 views

Find the first 5 terms of the expansion in a power series

Find the first 5 terms of the expansion in a power series $$y′=xe^{x}+2y^{2}$$ I've got a riccati equation $$ x e^{x}+2y^{2}, y(0)=0$$ After solving: $$y=e^{x}(x-1)+\frac{2}{3}y^{3} - 1$$ And I ...
4
votes
0answers
88 views

Stokes' Theorem and Measure Zero Sets

This is probably a very naive question but I am trying to connect two pieces of information in my head regarding integration of differential forms and integration with respect to a measure. The first ...
3
votes
0answers
103 views

Number of ways to partition n fixed points using cubic grids

What is the number of different ways to partition $n$ points in $\mathbb{R}^d$ using cubic grid partitions of given cube size h? Notation: $n$ is a positive integer. The class of cubic grid ...
0
votes
0answers
100 views

Prolate spheroidal coordinates

If $\alpha \in (0,\infty)$, $\beta \in (0,\pi)$ and $\theta \in (0,2\pi)$. $$\varphi (\alpha,\beta,\theta) =( \sinh(\alpha)\sin(\beta)\cos(\theta), \sinh(\alpha)\sin(\beta)\sin(\theta), ...
0
votes
1answer
118 views

Easy way to check for a valid solution in this triple equality?

Let's say I have the following equalities $a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 = b_1x_1 + b_2x_2 + b_3x_3 + b_4x_4 = c_1x_1 + c_2x_2 + c_3x_3 + c_4x_4$ Where the $a$'s, $b$'s, and $c$'s are known, ...
2
votes
3answers
1k views

Image of a union of collection of sets as union of the images

I am having problems with establishing the following basic result. Actually, I found a previous post that is close in nature (it is about inverse image), but I was interested in this specific one, ...
-1
votes
2answers
73 views

Two questions on finite abelian groups

Which of the following are true? 1.Every group of order $6$ abelian. 2.Two abelian groups of the same order are isomorphic
12
votes
4answers
2k views

Examples of nonlinear ordinary differential equations with elementary solutions.

I am looking for nice examples of nonlinear ordinary differential equations that have simple solutions in terms of elementary functions. (But are not trivial to find, like, for example, with ...
1
vote
1answer
66 views

What kind of functions can be moment-generating functions for a random variable?

Given an infinitely differentiable function $ g: \mathbb{R} \rightarrow \mathbb{R}$, can we always find a distribution function $f_X$ of some random variable $X$ so that $g(t) = \int_{-\infty}^\infty ...
4
votes
3answers
109 views

proving $n!>2^n\;\;\forall \;n≥4\;$ by mathematical induction

My teacher proved the following $n!>2^n\;\;\;\forall \;n≥4\;$ in the following way Basis step: $\;\;4!=24>16$ ok Induction hypothesis: $\;\;k!>2^k$ Induction step: ...
3
votes
2answers
374 views

Finite group is generated by a set of representatives of conjugacy classes.

Could you tell me how to prove that a finite group is generated by a set of representatives of conjugacy classes? I've read this ...
2
votes
1answer
77 views

there is no bounded linear functional on $ H$

let $ H= L^2[0,1]$ and $ C^1 $ be the set of all continuouse functions on $ [0,1] $ that have continuouse derivative.Let $ t \in [0,1] $ and define $ L: C^1 \longrightarrow F $ by $ L (h)= h'(t) $. ...

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