All Questions
0
votes
0answers
9 views
Does Euler totient function gives exactly one value(answer) or LEAST calculated value(answer is NOT below this value)?
I was studying RSA when came across Euler totient function. The definition states that- it gives the number of positive values less than n which are relatively prime to n.
I thought I had it, untill I ...
0
votes
0answers
7 views
1 complex addition = 2 real additions?
Is it true that one complex addition requires 2 real additions. Could you show a proof that one complex addition requires 2 real additions?
-1
votes
0answers
18 views
$f_n(z)={z^n\over n}$, $z\in D$ open unit disk then
$f_n(z)={z^n\over n}$, $z\in D$ open unit disk then
1.$\sum f_n$ converges uniformly on $D$?
2.$f_n$ and $f'_n$ converges uniformly on $D$?
3.$\sum f'_n$ converges on $D$ pointwise?
4.$f_n''(z)$ ...
0
votes
1answer
14 views
Statistics Question
This is probably super simple to most of you on here, but I was chatted by a friend earlier with a question. It reads just like this:
...
-4
votes
0answers
21 views
0
votes
0answers
12 views
FFT: Why does 4log(4)/log(2) result into 8 complex adds and 4 complex multiplications?
FFT supposed to take O(N log N) operations. Take an radix-2 FFT with N=4.
Does 4log(4)/log(2) really result into 8 complex adds and 4 complex multiplications?
If so, why doesn't it result into 8 ...
2
votes
0answers
22 views
Can an odd perfect number be divisible by $165$?
I know that an odd perfect number cannot be divisible by $105$ or $825$. I wonder if that's also the case for $165$ (this is actually a stronger statement).
0
votes
2answers
28 views
Does $P(A\cap B) + P(A\cap B^c) = P(A)$?
Based purely on intuition, it would seem that the following statement is true, when thinking of the events as sets:
$$P(A\cap B) + P(A\cap B^c) = P(A)$$
However, I am not sure if this is true, and ...
2
votes
0answers
19 views
Analogy between prime numbers and singleton sets?
While trying -- in vain -- to write an alternative answer for another question (If $\cup \mathcal{F}=A$ then $A \in \mathcal{F}$. Prove that $A$ has exactly one element.), I discovered the following ...
1
vote
1answer
7 views
Prove that if $n\geq\text{lcm}(a,b)$ and $\gcd(a,b)|n$ then $n=xa+yb$ for some integers $x,y\geq 0$
I thought I had it, but then I realized I didn't. Even just a hint—am I going in the right direction or should I try something completely different?
We know that $\gcd(a,b)=wa+zb$ for some integers ...
0
votes
0answers
6 views
Help with school solid geometry
In the triangular pyramid $MABC$ all side edges equals $1$, $\angle AMB = \angle BMC = 60 ^\circ$, $\angle AMC = 45 ^\circ$.
Find:
1) square of the $\triangle ABC$;
2) dihedral angle on the $AB$ ...
1
vote
0answers
8 views
Prove that $[L,Rad(L)] \subseteq N$ for finite-dimensional Lie algebra $L$
I need to prove the following fact: if $L$ is a finite-dimensional Lie algebra over field of characteristic $0$, $Rad(L)$ is its radical, and $N$ is the maximal nilpotent ideal in $L$, then ...
0
votes
2answers
22 views
If $d(x,A)=0\forall x\in X$ for some subset $A$ of $X$, does it follow that $A$ is dense?
If $d(x,A)=0\forall x\in X$ for some subset $A$ of $X$ then $A$ is dense in $X$, right? Once I did one problem which says $d(x,A)=0\Leftrightarrow x\in \bar{A}$ so by the condition here we get ...
0
votes
0answers
5 views
probability to cover straight line with circles
suppose sensors of homogeneous radius r are dropped by Poisson distribution on a straight line of length L. how to calculate that the straight line is covered by sensors with probability P
1
vote
0answers
17 views
$\int_{0}^{\pi/2} \text{arctanh}(\sin x) \text{arctan}(a \tan(x)) \cos(x) \ dx$
Are you kind to let me know the way? By the way, don't you have a "curiosity" tag?
$$\int_{0}^{\pi/2} \text{arctanh}(\sin x) \text{arctan}(a \tan(x)) \cos(x) \ dx, \quad a>0$$
0
votes
2answers
17 views
Finding Area of a shape
I'm doing some revision and I'm a bit stuck. How do you find out the area of this shape?
I know you have to do 10cm x 2cm = 20cm devided by 2 = 10cm - 2cm x ? (something) = area.
I'm not sure what ...
1
vote
0answers
21 views
Is there a continuous version of $tan^{-1}(\frac{y}{x})$ for the entire unit circle?
The fact that $tan^{-1}(\frac{y}{x})$ only "works" for the upper-right quadrant makes some calculations (for a physics simulator) impossible. I of course use $atan2(y,x)$ in the code, that's not what ...
0
votes
0answers
14 views
Size and Order of a Line Graph of Another Graph G
This is an extension of the problem at this thread: Graphs, line graph and complement of graph.
Just to reiterate definitions:
The line graph $L(G)$ of a graph $G$ is defined in the following ...
1
vote
2answers
44 views
A is a matrix of integers , prove that A+I is invertible
Question:
$2 \le d \in \Bbb Z$
Let $A \in M_n(\Bbb Q) s.t$ All of it's elements are integers divisible by d.
Prove that $I+A$ is invertible.
What I thought:
I thought of using the determinant of ...
0
votes
0answers
10 views
Proving an identity regarding the Cauchy problem (using convolutions)
Given $u_0 \in C_c(\mathbb{R}^n)$, consider the solution of the Cauchy problem $$u(x,t) = \int_{\mathbb{R}^n} \Gamma (x - y,t)u_0(y) dy \qquad x \in \mathbb{R}^n,t>0\, \, .$$ Given $0<s<t$ , ...
-1
votes
0answers
15 views
$p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$
$p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$ where $a_i(z)$ are non constant poly in complex variable, $k\ge 1$, I need know $$\{(z,w):p(z,w)=0\}$$ is
1.bounded with empty interior
2.unbounded with ...
1
vote
0answers
21 views
numerical approximation to logarithm
we know that $$ \ln(x)=\frac{1}{x} \int_{0}^{1}dt \frac{1}{1+xt} $$
then given a cuadrature formula inside $(0,1)$ is that true
$$ \ln(x)= \frac{1}{x}\sum_{i}\frac{w_{i}}{1+xt_{i}} $$
wht other ...
2
votes
1answer
58 views
Find the limit without use of L'Hôpital or Taylor series: $\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2 x}\right)$
Find the limit without the use of L'Hôpital's rule or Taylor series
$$\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2x}\right)$$
0
votes
2answers
21 views
On a question chosen at random, what is the probability that the student answers it correctly?
I'm really confused about this question. I appreciate your help.
A student takes a multiple choice exam where each question has five possible answers. He answers correctly if he knows the answer, ...
3
votes
4answers
90 views
Can we get just $3$ from $\pi$?
Today, a friend and I solved a question and one point came up where we were discussing whether we should write $(-1)^{n-1}$ or $(-1)^{n+1}$ and quickly we remembered that it was the same thing (for ...
1
vote
1answer
25 views
Questions about epimorphisms and projectives in functor categories
Suppose $I$ is a small category, $R$ is a ring and $_R\mathrm{Mod}$ is the category of left $R$-modules. How do I show that the category $[I,~_R\mathrm{Mod}]$ of all functors from $I$ to ...
4
votes
1answer
25 views
Unity in the rings of matrices
Suppose we are given an arbitrary ring $R$. Then the set $M_n(R)$ of all square matrices with elements from $R$, together with usual matrix addition and multiplication forms a ring. If R is a unitary ...
2
votes
1answer
13 views
Finding the MLE of $\theta$ where $\theta \leq x$
consider the following PDF
$$
\begin{eqnarray}
f(x;\theta) &=&
\left\{\begin{array}{ll}
2\frac{\theta^2}{x^3} & \theta \leqslant x\\
0 & x< \theta; 0 < \theta
...
4
votes
2answers
27 views
2
votes
0answers
23 views
Differential 2-form has a primitive…
Can anybody show me a proof of this theorem? : If a differential form of degree $2$ in $R^3$ has a primitive form of degree $1$ of class $C^2$, it also has a primitive form: $u \, dx + v \, dy$ where ...
2
votes
1answer
18 views
On Lévy collapsing the reals
Consider the Lévy forcing notion. Let $M$ be a transitive standard model of $\mathsf{ZFC}$. Let $\aleph_n$ be the cardinality of the real numbers $2^\omega$ in $M$. Now collapse $\aleph_n$ to ...
1
vote
0answers
20 views
Enumerate partitions of identical objects
I have a problem concerning enumerating partitions of a set of identical objects. I know, now someone is going to talk about Stirling number of second kind, but I'm quite sure this is not the answer.
...
2
votes
2answers
28 views
combinations and permutation questions
I have a number of dogs and monkeys living with me. If I decide to take out for a walk a monkey AND a dog, I find I have fewer options than if I decide to take out a monkey OR a dog.
What can you say ...
0
votes
0answers
28 views
1 complex addition = 2 real additions, 1 complex multiplication = 4 multiplications + 2 additions.
Could you show that, when talking about Fourier transforms, that one complex addition requires 2 real additions, and one complex multiplication requires 4 multiplications and 2 additions.
0
votes
1answer
14 views
Minimal Red-Black tree with depth 3
I'd like to ask what is minimal RBT with black depth 3. Is this following RBT ? Values are not important. And that tree can't have depth 2 or 1.
1
vote
1answer
42 views
Name for three-valued sign $+, -, 0$
Is there an accepted term (an adjective or prefix) like strict, trichotomous, strong or definite sign to indicate the three-valued sign whose values are $+$, $-$, and $0$?
Are there words reserved ...
3
votes
1answer
16 views
Square of sum of matrices
I'm trying to follow these lecture notes on Linear Discriminant Analysis (LDA) but I can't seem to figure out how the author gets from:
$$ \Sigma_{x\epsilon\omega_{i}} (w^{T}x - w^{T}\mu_{i})^2$$
to
...
0
votes
1answer
22 views
Using the Chebyshev Inequality
This is the Q:
A 20 fair coins tosses, (f means the "H" of the coin).
I have to block the probability that I will get n/2+n/100 "H"-s by Chebyshev Inequality. [n=20 in this case...], so:
n/2+n/100 = ...
0
votes
0answers
17 views
Stochastik and Bernoulli experiment
My Task is:
You got N coins, where N is distributet by a Poisson-distribution with the Parameter lambda.
Now we got the likelihood for head of 0 < p < 1. Where K are the heads that you got ...
1
vote
1answer
18 views
Subalgebras of certain C*-algebras
Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and ...
0
votes
0answers
10 views
test of sheffer polynomials
Are the any tools available for testing if a polynomial sequence corresponds to a generating function (e.g Sheffer sequence GF)? The Coxeter configuration is a configuration whose incidence graph is ...
1
vote
1answer
16 views
Solve this PDE using method of characteristics
I am not sure how to begin this problem, I have looked up how to use the method of characteristics but can find no example where p^2 so I am unsure of how one would approach this.
0
votes
1answer
26 views
I don't know where to begin with this functions question
a) Suppose that $f:\Bbb Z\to \Bbb Z$ is a one-to-one function. Define a function $g:\Bbb Z\to \Bbb Z$ by: for all integers $x$, $g(x)= -f(x)$. Prove that $g$ is also one-to-one.
b) Suppose $f:\Bbb ...
2
votes
1answer
24 views
Matrices manipulation
I am having difficulty with the following question
I have to determine if the following claim is true or not.
If it is true I have to proof it else I need to give an example
I believe it is not ...
2
votes
2answers
36 views
Commutator property
Can you please show me how to prove this? If H,K,L are normal subgroups of G then $$[[H,K],L]\subseteq [[K,L],H][[L,H],K]$$ Thanks in advance!
4
votes
1answer
55 views
Continuity of $d(x,A)$
I am doing a head-check here. I keep seeing this theorem quoted as requiring $A$ to be closed (as in Is the function distance continuous?), but I don't think that it is needed.
Theorem. Let ...
1
vote
0answers
35 views
How to solve it in radicals?
How to solve the equation $x^5+10x^3+20x-18=0$ in radicals? One of its roots is
$$\frac 1 5\, \left( -\frac1 4- \frac 1 4\,\sqrt {5}+\frac 1 4\,\sqrt {-10+2\,\sqrt {5}}
\right) \sqrt ...
2
votes
2answers
32 views
Simplifying a Product of Summations
I have, for a fixed and positive even integer $n$, the following product of summations:
$\left ( \sum_{i = n-1}^{n-1}i \right )\cdot \left ( \sum_{i = n-3}^{n-1} i \right )\cdot \left ( \sum_{i = ...
2
votes
1answer
34 views
Question about Euler numbers
How to prove that
where $\ E_{2n}$ is the $2n^{th}$ euler number
and $$\frac{1}{\cosh(x)}=\sum_{n=0}^{\infty }\frac{E_n}{n!}x^n$$
is there any help?
thank for all
1
vote
1answer
14 views
find $\theta_{MLE}$ for a function
For
$$
f(x;\theta)=(\theta+1)x^{-\theta-2}
$$
find the maxmimum likelihood estimators (MLEs) for $\theta$ based on a random sample of size $n$.
My work so far:
$$
\begin{align}
\prod_{i=1}^n ...
