2
votes
1answer
70 views

Finding the average sum of 15 numbers out of 100 numbers

I've got a question. For the numbers 1,2,...,100 we choose with the same probability 15 numbers. 1) What is the average sum of those numbers? 2) What is the average sum of those numbers when the ...
1
vote
5answers
67 views

Limit with L'Hospital with infinite indeterminate formats

I'm trying to find the limit: $$\large \lim_{x\to0}(\sin x)^x$$ Whst I did was apply L'Hospital Rule: $$\large \text{let }y =(\sin x)^x\implies \ln y=x\ln\sin x$$ $$\large \lim_{x\to0}\ln y = \lim_{x\...
4
votes
1answer
96 views

Probability of drawing all “socks” of a given color from a drawer, given certain number of tries

Let's talk "socks." Say I have 7800 socks in a drawer (it's a big drawer), 800 of which are red and 7000 of which are black. If I randomly pull 1300 socks from the drawer, what is the probability ...
0
votes
1answer
27 views

Solve ODE of plane pendulum

I have seen in a book this system of ode's: $$ \begin{cases} \dot x =y\\ \dot y=-\sin x \end{cases} $$ and the say that the solution is $$ \begin{cases} x(t)=\pm 2\arctan(\sinh(t)) \\y(t)=\pm 2\,\...
1
vote
0answers
38 views

Radical Inequality related to three variables

Prove the following. $$\sum_\text{cyclic}\sqrt[4]{\dfrac{(a^{2}+b^{2})(a^{2}-ab+b^{2})}{2}}\leq\dfrac{2}{3}\left(\sum_\text{cyclic}\dfrac{1}{a+b}\right)\left(\sum_\text{cyclic}a^{2}\right)$$
0
votes
2answers
138 views

$\infty + \infty = \infty$?

(The context is a measure-theoretic one.) I know that $\infty - \infty$ is indeterminate, but what about $\infty + \infty = \infty$? It seems this statement is true and if I input it into Wolfram ...
0
votes
2answers
52 views

To what is $f^{-1}(G_2)$ equal??

If $\displaystyle{f: G_1 \rightarrow G_2}$, then $\displaystyle{f^{-1}: G_2 \rightarrow G_1}$, right?? Could you tell me which set is $$f^{-1}(G_2)$$?? Is it the set $G_1$??
4
votes
0answers
99 views

Blowing up a Singular Point More Than Once.

I am trying to understand how $I_n$-fibres appear in an elliptic surface by performing a sequence of blow-ups. To be concrete, I am looking at the following elliptic surface given in Weierstrass ...
2
votes
2answers
494 views

Forming a simple polygon from the extrusion of a polygonal chain

Let's say I have a set of vertices connected by edges to form a polygonal chain. Each vertex may be shared by a number of edges to form various sub-chains. An example is shown below. Each edge has ...
0
votes
2answers
53 views

Question regarding morphism of ringed spaces

I have recently started studying schemes, and I have encountered this passage from the book by Kenji Ueno: My questions: i) If $(X,O_X)$ is a local ringed space, why is $(X, i_*({O_X}_{|U}))$ also ...
0
votes
2answers
268 views

What is the difference of an n-tuple and a permutation of n elements

My understanding of n-tuple and a permutation of n elements is, that both are ordered sequences of n elements. Are there differences in the objects correlating to these two terms ? I guess it ...
1
vote
1answer
68 views

on automorphisms groups a finite 2-group

Let $G=\langle a,b\mid a^4=b^4=1, bab^{-1}=a^3\rangle$. Please prove that $Aut(G)$ is generated by the automorphisms $$a\mapsto ab,\hspace{10pt} a\mapsto a^3,\hspace{10pt} a\mapsto ab^2, \hspace{...
2
votes
0answers
36 views

VCG - plynomial time algorithm when bidders are unit demand

Is there a polynomial time algorithm to run a VCG when bidders are unit-demand? I though to look at the Bipartite graph when the left side is the bidders the right is the items and the edges are the ...
0
votes
2answers
74 views

What is the value of $E[|X|]$? [closed]

Let X be a zero mean unit variance Gaussian random variable.What is the value of $E[|X|]$?
4
votes
1answer
346 views

Infinite number of poles and residue theorem

I suppose a stupid question but I was wondering about it for a while: Can one apply the residue theorem to a function $f$ which is defined and holomorphic on $U-\{a_1,a_2,\dots\}$ where $U$ is simply ...
1
vote
1answer
274 views

Real part of Complex Function

I've this function $$f(k,\theta) = \frac{1}{k}\frac{1}{\cot\delta_0(k) -i }$$ and i know that $k\cot\delta_0(k) = -\frac{1}{a} + \frac{1}{2}r_ek^2 + \cdots$ it is an expansion. How can i get that ...
3
votes
2answers
95 views

Rotation matrices are similar if and only if their angles add up to 2 pi

Let $\theta_0, \theta_1 \in [0, 2\pi)$ and $\theta_0 \ne \theta_1$. Consider the rotation matrices $$M_0 = \left[ \begin{matrix}\cos(\theta_0) & -\sin(\theta_0) \\ \sin(\theta_0) & \cos(\...
1
vote
1answer
81 views

Neighbourhood of a matrix

Sometimes I find definitions which say that something happens in a neighbourhood of a matrix. For example a dynamical system generated by $x'=Ax, \ A \in \mathcal{M}(n)$ is structurally stable if ...
2
votes
1answer
147 views

What is a circle's area if its radius is $\pi$?

The area of a circle equals $\pi r^2$. If a circle's radius is $\pi$, what is its area? I believe the answer is $\pi^3$, right?
1
vote
2answers
128 views

Prove that the product satisfies the universal property for coproducts in Ab.

I refer to this question. The question is Is there an intuitive way to understand why finite products and coproducts in Ab coincide, while the same is not true in Grp? The author of the ...
0
votes
2answers
76 views

$m(E∩I)≥αm(I)$ for all intervals $I⊂[0,1].$ Prove that $m(E) = 1.$

Let $E$ be a measurable subset of $[0, 1].$ Assume there is a constant $α > 0$ such that $m(E∩I)≥αm(I)$ for all intervals $I⊂[0,1].$ (Here $m(·)$ denotes Lebesgue measure.) Prove that $m(E) = 1.$ ...
3
votes
3answers
811 views

Inverse of constant matrix plus diagonal matrix

Is there an efficient way to calculate the inverse of an NxN diagonal matrix plus a constant term? I am looking at N of around 40000. $\left[\begin{array}{cccc} a & b & \cdots & b\\ b &...
1
vote
0answers
40 views

Maximum of $P$ in the disk $|z|=1$ depending on co-efficients

Let $P(z)=a_nz^n+a_{n-1}z^{n-1}+\ldots+a_mz^m$ be a polynomial with complex coefficients such that $a_m\neq 0, a_n\neq 0$ and $n>m$. Prove that $$\max_{|z|=1}\{|P(z)|\}\ge\sqrt{2|a_ma_n|+\sum_{k=m}^...
1
vote
3answers
79 views

How to show $n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$ has no nonzero integer solutions?

How do we prove that $$n^2+m^2 = a^2+b^2 = (n-a)^2+(m-b)^2$$ has no nonzero integer solutions? I know two ways to prove this by taking a geometric interpretation but I don't want such a version. How ...
2
votes
1answer
61 views

Prove that $\int|f − g| = \int_{-\infty}^{\infty} μ(F_t △ G_t) dt.$

Let $f$ and $g$ be integrable functions on a measure space $(X,Σ,μ).$ For each $t ∈ \mathbb{R},$ consider the sets $F_t =\{x∈X :f(x)>t\}, G_t =\{x∈X :g(x)>t\}.$ Prove that $\int|f − g| = \int_{-\...
1
vote
2answers
62 views

“Simple” math question about length and rotation relations

I'm currently building a robot arm as a hobby, and I'm still in the planning phase. But I've encountered a small problem, where my knowledge doesn't suffice. This is what I am trying to achieve: I ...
1
vote
1answer
76 views

How to solve Bellman's optimal equation from the first principle

How to solve the following set (finite) of equations $$ v_*(s) = \max_{a\in A(s)} \sum_{s'} p(s'|s,a) [r(s,a,s') + \gamma v_*(s')]$$ $p$ and $r$ functions are given.
1
vote
1answer
49 views

Proof or disproof $Y = 2\sum_{i=1}^{n} a^{2}_{i} b^{2}_{i} + \sum_{i \not=j}^{n} a^{2}_{i}b^{2}_{j} - 2\sum_{i=1}^{n} a_{i} b_{i} \ge 0$

In my attempt to answer this question I came cross this question : if $$ \sum_{i=1}^{n}a^2_{i}\le 1,\sum_{i=1}^{n}b^2_{i}\le 1 $$ do we have ? $$Y = 2\sum_{i=1}^{n} a^{2}_{i} b^{2}_{i} + \sum_{i \...
1
vote
2answers
30 views

Having trouble understanding phrasing.

I am having a little trouble understanding the following: "If $p_1, \ldots, p_k$ be the list of distinct primes dividing the product $mn,$ then we can factor $m$ and $n$ as $m=p_1^{r_1} \cdots p_k^{...
2
votes
2answers
195 views

A question about 4 concyclic points

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles $...
0
votes
1answer
72 views

Proof of definite to indefinite integral transformation problem

I'd like to prove If $f(x)$ is continuous, then $$\int^a_b f(x) \mathrm{d}x=\int f(a) \mathrm{d}a - \int f(b) \mathrm{d}b$$ The problem is that my math teacher uses this "fact" in the proof of one-...
2
votes
1answer
78 views

Linear equation: $(A^\top A+B^\top B + D)x=c$ where $A,B$ are structured sparse and $D$ is diagonal.

Updated: the goal is to solve $(A^\top A+B^\top B + D)x=c$. Maybe it is not necessary to compute $(A^\top A+B^\top B + D)^{-1}$. Denote $e=(1,1,\ldots,1)^\top\in\mathbb{R}^n$ and $$A=\begin{bmatrix}...
1
vote
1answer
278 views

Prove the computational formula of Anderson-Darling test statistic.

The Anderson-Darling test statistic is defined as $$n\int_{-\infty}^\infty \frac{(F_n(x) - F(x))^2}{F(x)(1 - F(x))}dF(x)$$ and there is a computational formula $$A^2 = -n - S$$ where $$S = \sum_{...
2
votes
1answer
88 views

Clarification of an old question: Galois Groups of Finite Extensions of Fixed Fields

The question is: Let $\overline{K}$ be an algebraic closure of $K$, $\sigma\in G(\overline{K}:K)$ and $E=\{x\in\overline{K}:\sigma(x)=x\}$. Prove that every finite extension $L\mid E$ is a Galois ...
0
votes
0answers
89 views

Simple groups of order 7920

I would like to prove that every finite simple group of order 7920 is isomorphic to $M_{11}$. I would be appreciated if someone can give me some points.
1
vote
0answers
54 views

Change of variable for Lebesegue Integral

Let $G$ be an absolutely continuous function, $G:[a,b] \rightarrow [c,d]$ and $f \geq 0$ a Lebesegue measurable function in $[c,d]$. I managed to prove that if $f$ is just Borel measurable it holds $$\...
0
votes
1answer
84 views

Why is the product of these projection matrices a zero matrix

2 vectors are given: $$ \vec{a_1}=\begin{bmatrix}1 && 2 && -2\end{bmatrix}^T \\ \vec{a_2}=\begin{bmatrix}-2 && 2 && 1\end{bmatrix}^T $$ To calculate their projection ...
1
vote
1answer
37 views

A relationship among multiple periodic arrays

There are N periodic arrays ai[n] with period Ti, respectively, where i=1, 2, … , N. Each array has a property that a[n]=1 when n=k*T where k is integer, otherwise a[n]=0. Then a new array is created ...
2
votes
2answers
82 views

Why are these lines tangent?

I was trying the problems at http://euclidthegame.org and for level 20, ending up using, but couldn't see the reason behind the following: We have a circle centred on B and a point A outside the ...
1
vote
2answers
109 views

Is it true that $\frac{A\cup B}{C\cup D}=\frac{A}{C}\cup \frac{B}{D}$?

Suppose $A, B$ are sets and $C\subset A$, $D\subset B$ subsets such that $C\cap D=\phi$. Is it true that $$\frac{A\cup B}{C\cup D}=\frac{A}{C}\cup \frac{B}{D}.$$ Recall that if $X$ is a set and $A\...
1
vote
1answer
31 views

Proof of Cauchy's functional equation for rational arguments

We have thesis that for every $c\in\mathbb{Q}$ every additive function has form of $f(x)=cx$. In the proof we're showing that $f(nx)=nf(x)$. Then we're supposed to replace $nx$ by $\frac{1}{n}x$. Why ...
3
votes
1answer
230 views

Orientation double cover

Let $M$ be a manifold and let $\bigwedge^\text{top}TM$ be the top exterior product of the tangent bundle. Then this becomes a line bundle. Let $g$ be any metric on $\bigwedge^\text{top}TM$ and define $...
1
vote
3answers
78 views

Extrema of $f(x)=\frac{\sin (5x)} 5 - \frac{2\sin(3x)} {3} + \sin (x)$.

(a) I need help in finding maxima and minima of the following funcion: $$f(x)=\frac{\sin (5x)} 5 - \frac{2\sin(3x)} {3} + \sin (x)$$ therefore I need to find the roots of $f'(x)=\cos(5x)-2\cos(3x)+\...
1
vote
0answers
118 views

Find the value of $\sqrt{1+2\sqrt{1+3\sqrt{1+…}}}$ [duplicate]

Find the value of $$\sqrt{1+2\sqrt{1+3\sqrt{1+....}}}$$ I have done a similar question before in which only a single number is involved, for example $\sqrt{2+2\sqrt{2+2\sqrt{2+....}}}$ which ...
1
vote
1answer
33 views

Balanced Core: Explicit Expression?

Denote the collection of all balanced subsets by: $\mathcal{B}:=\{B\subseteq X: B\text{ balanced}\}$ Since the union of arbitrary balanced sets is balanced we can form the balanced core of arbitrary ...
8
votes
4answers
488 views

The standard role of intuitive numbers in the foundations of mathematics

In my career I've been formed mostly in the formal side of mathematics, that is, standard set theory and every classical branch of mathematics that uses set theory. However, I am not quite sure about ...
3
votes
1answer
256 views

uniform continuity on $(a, b]$ implies limit at $a^+$ exists and finite

Let a uniformly continuous function $f$ on $(a, b]$. Prove that $\lim_{x\rightarrow a^+} f(x)$ exists and finite. What I did so far: from the definition of uniform continuity: $$\forall\...
1
vote
0answers
26 views

$\gamma(t)$ is not asymptotically stable unless $\int_0^T \nabla \cdot f(\gamma(t))dt \leq 0$

Let $f \in C^1(E)$ where E is an open subset of $\mathbb{R^n}$ containing a periodic orbit $\gamma(t)$ of $x'=f(x)$ of period $T$. Then $\gamma(t)$ is not asymptotically stable unless $$\int_0^T \...
2
votes
1answer
480 views

Generating Matlab Code using Maple

When exporting Matlab Code from Maple using for example the command fprintf(FileName, %s, CodeGeneration[Matlab](VariableName, output = string)) I get the text <...
7
votes
0answers
138 views

When is separation of variables possible?

In classical PDE courses it is common to learn to perform a change of variables without really learning how to find the adequate equations of the change (polar, cylindrical or spherical coordinates ...

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