2
votes
1answer
44 views

What is the meaning of $1_{a>b}$?

What would this mean: $1_{a>b}$ .. Based on the context, it could mean "$1$ if $a>b$ else $0$", but it's the first time I see it so help would be appreciated.
1
vote
3answers
105 views

$Z$ score probability

I was given a question where I was supposed to find the probability of obtaining $y$ between two scores, however when I input my answer it tells me that I'm wrong, the question is given below along ...
0
votes
2answers
72 views

Proof a function is continous.

Is the function $f: \mathbb{R}^2 \rightarrow \mathbb{R}$, where $f(x_1,x_2) = x_1^2 + x_2^2$, a continuous function? My attempt: Suppose that $\forall \varepsilon > 0$ $\exists \delta >0$ such ...
1
vote
1answer
82 views

Integral curve for vector field tangent to sphere

Let $S^1$ be the unit sphere $x_1^2+x_2^2=1$ in $\mathbb{R}^2$ and let $X=S^1\times S^1\in\mathbb{R}^4$ with defining equations $f_1=x_1^2+x_2^2-1=0, f_2=x_3^2+x_4^2-1=0$. The vector field ...
3
votes
1answer
85 views

Asymptotic behaviour of e * !n - n! , n tends to infinity

What is the asymptotic behaviour of the function $e !n-n!$ , where $!n = n! \sum_{k=0}^n \frac{(-1)^k}{k!}$ is the subfactorial of $n$. I tried Wolfram Alpha but the series for n=$\infty$ is ...
0
votes
2answers
44 views

Fourier transform of a function over finite group

Let $G$ be finite abelian group and $\hat G$ be its character group. The Fourier transform of a function $f:G \to \mathbb C$, is the function $\hat{f}:\hat{G}\to \mathbb C$ defined by ...
1
vote
2answers
127 views

Books about Turing machines and undecidability

I need help with finding literature about Turing machine and undecidability. First book I was suggested is Introduction to Automata Theory, Languages, and Computation by Hopcroft, Motwani and Ullman. ...
2
votes
2answers
55 views

Validity of a Limit Proof

I am trying to show that if $\displaystyle{\lim_{s\to\infty}} s_n = s$, $\displaystyle{\lim_{s\to\infty}} \sqrt{s_n} = \sqrt{s}$ for some sequence $s_n$. We must note that $s_n > 0$ for all $n$ ...
0
votes
1answer
31 views

Solve NC + BN =F

I asked this in the computer section; someone suggested asking in the maths section: Is there a simple way to solve the following matrix equation for N: NC + BN = F The matrices B, C, and F are ...
1
vote
2answers
63 views

Doubts about graded algebra.

In the last days I've been studying the tensor algebra $T(V)$ of a vector space $V$ over the field $K$ and I've realised that what I'm not understanding hasn't to do with tensor products, but rather ...
2
votes
1answer
113 views

Homology and cohomology are basically the same

Is my following understanding correct: A chain complex $(C,\partial)$ is a family $\{C_i\}_{i\geq 0}$ of $R$-modules ($R$ is a given ring) together with a family of $R$-module homomorphisms ...
1
vote
1answer
56 views

Proof of the properties of limits of CDFs

The cumulative distribution function is defined as $F(a) = \mu((-\infty,a])$ where $\mu$ is a probability measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. Given this definition, it is easy to prove ...
1
vote
2answers
156 views

Finding $\lim_{x \to 0} x^x$ without l'Hôpital [closed]

I have to find the limit of $x^x$ as $x$ approaches $0$ without derivatives.
0
votes
1answer
54 views

Prove that the eigenvectors are independent.

Given two vectors $\boldsymbol\alpha=\left(\alpha_1,...,\alpha_N\right)^{T}$ and $\boldsymbol\beta=\left(\beta_1,...,\beta_N\right)^{T}$, let $M$ be the $N\times N$ matrix whose entries are expressed ...
0
votes
4answers
219 views

Test for convergence/divergence of $\sum_{n=1}^{\infty}(-1)^n\sin\left(\frac{n}{\pi}\right)$

Given the series $$\sum_{n=1}^{\infty}(-1)^n\sin\left(\frac{n}{\pi}\right)$$ I need to test for convergence/divergence. I think the divergent test might work here. I could see that the ...
2
votes
2answers
267 views

What is the meaning of “integral point”?

While reading this paper (http://cowles.econ.yale.edu/P/cd/d04b/d0473.pdf) I encountered the concept of "integral point", used first in definition 5.1, on page 34. Does anybody know more details about ...
1
vote
1answer
571 views

Solving the differential equation $y'' + 2y' + 2y = 0$ given constraints

How can I solve this initial value problem? $$ y'' + 2y' + 2y = 0,$$ given $y\,(\pi/4)=2$ and $y'(\pi/4)=0$. I've found $y(t)=e^{-t} \left(C_1\cos t + C_2\sin t \right)$ but I wasn't able to find ...
1
vote
1answer
110 views

Properties of Lie derivative

Let's have Lie derivative: $$ L_{V}\varphi = V^{\mu}\partial_{\nu}\varphi , \quad L_{V}A_{\mu} = V^{\nu}\partial_{\nu}A_{\mu} + (\partial_{\mu}V^{\nu})A_{\nu}. $$ How to show that for scalar and ...
0
votes
1answer
54 views

Continuity of functions from complex numbers

i have a question about continuity. Suppose i have a function from $\Bbb{C}$ into a Banachalgebra $A$ for example $r\mapsto exp(ra)$ for a fixed $a\in A$. Do we have to prove continuity by ...
1
vote
1answer
69 views

Find the sum of the series

For any integer $n$ define $k(n) = \frac{n^7}{7} + \frac{n^3}{3} + \frac{11n}{21} + 1$ and $$f(n) = 0 \text{if $k(n)$ is an integer ; $\frac{1}{n^2}$ if $k(n)$ is not an integer } $$ Find $\sum_{n = ...
3
votes
2answers
63 views

Poles of abelian differentials

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$. As a corollary of the Riemann-Roch theorem we know that for every abelian differential $\omega$ on $X$ we have ...
3
votes
0answers
88 views

In which space does my (weak) solution exist?

I am reading through some numerical analysis papers, and was wondering what characteristics of a problem define which function space the weak solution of a problem belongs to. For example, for the ...
0
votes
2answers
68 views

Is the boundary of this set compact?

Let $X$ be a topological space, $Y$ a subspace of $X$ and $A\subseteq Y $ such that $\partial(A)$ is compact in $X$. Is $\partial(A)$ compact in $Y$?
1
vote
1answer
72 views

Real analysis: prove $\;\lim_{x\to -1} x^3 + x -2 = -4$

Prove that $\lim_{x\to -1} x^3 + x -2 = -4$ My try is the following: To begin, note that if x is a real number, satisfies (sth that I need help with), then (sth that I need help with). Now, for ε>0 ...
1
vote
2answers
66 views

Counting permutations

how many words, without making any reference to their meaning can be written from the letters: $ a,a,a,b,b,c,c,d$ ? what is the best approach to solve this kind of problem ?
1
vote
0answers
135 views

what is the use of the hyperboloid model for hyperbolic geometry?

I am quite new to hyperbolic geometry so even an answer that this question doesn't make any sense can be very helpful. As far as i understand: There are different models of a plane where hyperbolic ...
0
votes
1answer
33 views

Subset of non-units of germs of smooth functions at $x$ is an ideal

For a manifold $X$ with a point $x \in X$ define the ring of germs of the smooth functions at $x$ to be $C^{\infty}_x(X)=C^{\infty}(X)/\sim$ where $f_1 \sim f_2 \iff \exists U \in \tau_X.f_1|U=f_2|U$ ...
0
votes
3answers
141 views

“closure preserves homeomorphism”

Let me explain the title of the problem and the problem very clearly : If $X$ and $Y$ are subsets of a topological spaces $A$ and $B$ respectively, which are homeomorphic in the respective subspace ...
7
votes
2answers
148 views

chameleon puzzle- modified one

In one island there are 3 colors of chameleon 12 blue,15 green and 7 red. When two different color’s chameleon meet together , they convert into third color. What is the number of minimum no. of ...
1
vote
2answers
5k views

How to prove that inverse Fourier transform of “1” is delta funstion?

$\mathscr{F}\{\delta(t)\}=1$, so this means inverse fourier transform of 1 is dirac delta function so I tried to prove it by solving the integral but I got something which doesn't converge.
7
votes
1answer
140 views

Why this two spaces do not homeomorphic?

Consider $\Bbb Q$ with subspace topology and $\Bbb Q\times \Bbb Q$ with product topology. Why this two spaces are not homeomorphic?($\Bbb Q$ is the rational numbers)
2
votes
4answers
101 views

Sum of the series $\frac{(-3)^{n-1}}{8^n}$

It might looks obvious to you but I don't manage to find the sum: $$\sum_{n=1}^{\infty} \frac{(-3)^{n-1}}{8^n}$$ Can anyone help me ? Thanks
2
votes
2answers
70 views

How do you prove that $\lim f(x) = 0$, when $f$ is rapidly decreasing?

Let $f: \Bbb{R} \to \Bbb{R}$ be rapidly decreasing in the sense than $\sup_{x \in \Bbb{R}} |x|^k |f^{(\ell)}(x)| \lt \infty$ for all $k, \ell \geq 0$, where $f^{(\ell)}$ is the $\ell$th derivative. ...
3
votes
1answer
186 views

show that 210 is a triangular number

Show that 210 is a triangular number. Would it suffice to solve the equation 210=((n)(n+1))/2 ? Then n is equal to 20 and -21but n in this case must be positive, so 210 would be the 20th triangular ...
1
vote
1answer
97 views

How to solve this system of equations.

Solve the system of equations: $$\begin{cases}\dfrac{x^2+xy+y^2}{x^2+y^2+1}=\dfrac{1}{xy} \\\left(\sqrt{3}+xy\right)^{\log_2x}+\dfrac{x}{\left(\sqrt{3}-xy\right)^{\log_2y}}= ...
0
votes
3answers
77 views

Prove: $\binom{n}{r-1} + 2\binom{n}{r} + \binom{n}{r+1} = \binom{n+2}{r+1}$

Prove: $\binom{n}{r-1} + 2\binom{n}{r} + \binom{n}{r+1} = \binom{n+2}{r+1}$ This was one of the questions that my professor gave me as extra practise however I want to know a more efficient way of ...
1
vote
1answer
46 views

What is this set?

I have the following set, ${F}' = \{ A \subset \Omega :^\exists B,N \in {F}$ s.t. $\mu(N) = 0$ and $A \Delta B \subset N\}$ Here, $F$ is a collection of the subsets of $\Omega$, precisely ...
2
votes
1answer
117 views

Integral from $0$ to $\infty$ of $\frac{1}{3}\ln\left(\frac{x+1}{\sqrt{x^2-x+1}}\right)+\frac{1}{\sqrt 3}\arctan \left(\frac{2x-1}{\sqrt 3} \right)$

Evaluate the integral $$ \int_0^\infty \left( \frac{1}{3}\ln\left(\frac{x+1}{\sqrt{x^2-x+1}}\right)+\frac{1}{\sqrt 3}\arctan \left(\frac{2x-1}{\sqrt 3} \right) \right) dx $$ I have read about ...
0
votes
1answer
20 views

Show $\iint_S f(ax+by+cz) dS= \int_{-1}^1 f(t\|v\|)dt$ by cylindrical shells

Show $\iint_S f(ax+by+cz)dS = \int_{-1}^1 f(t\|v\|)dt$ by cylindrical shells. Here $S$ is the 2-sphere and $f$ is assumed continuous. Here there are two solutions: the first of which was the one I ...
0
votes
1answer
46 views

is the set of all sequences that converge to zero first category

the metric space is: the set of all convergent sequences with metric supremum. the subset is: all sequences that converge to zero. is the subset first category in this metric space? we have figured ...
0
votes
2answers
179 views

How to solve a congruence of a polynomial $x^3+2x^2+x+2\equiv 0 \mod 45$?

$x^3+2x^2+x+2\equiv 0 \mod 45$ $f(x)=(x^2+1)(x+2)$ by inspection $\fbox{1}$$x=7$ is a possible solution $\mod 9$ ,since $45=5\times 3^2$ $\fbox{2} $ $x= 2 $ is a solution $\mod 5$ but i want to ...
2
votes
0answers
78 views

Finding limit of a sequence [duplicate]

Hi I'm having a lot of trouble with this: I need to find the limit of the following sequence: There's a hint that I should use the following formula: Any help is much appreciated!
1
vote
4answers
104 views

Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $

Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $ Thanks in advance, my professor asked us to this a couple weeks ago, but I was enable to get to the right answer. Good luck! Here is ...
1
vote
1answer
136 views

Equivalent Definition of the Closure of a Set

I want to prove the following Lemma: Let $(X,\tau)$ be a topological space and $B\subseteq X$ be a subset. Then holds $$\overline{B}=\bigcap_{X\setminus A\in\tau \atop B\subseteq A}A.$$ I have ...
2
votes
1answer
145 views

Conformal map between annulii

Is there any conformal map between $D_1= \lbrace z \in \mathcal{C} \; ; \; 1 \leq |z| \leq 2 \rbrace$ and $D_2 = \lbrace z \in \mathcal{C} \; ; \; 1 \leq |z| \leq 3 \rbrace$. By the Schottky theorem ...
1
vote
1answer
97 views

Find the general formula of a the series.

If we are given that the first 4 terms of a series are 1,2,4, and 8. And the rest of the terms are summation of the previous 4 terms i.e. 5th term is 8+4+2+1 and so on. So how can we find the general ...
4
votes
2answers
68 views

estimation of a parameter

The question is: $x_i = \alpha + \omega_i, $ for $i = 1, \ldots, n.$ where $\alpha$ is a non-zero constant, but unknown, parameter to be estimated, and $\omega_i$ are uncorrelated, zero_mean, ...
2
votes
1answer
64 views

Let $\Omega$ be a countable set $\mathcal A=2^{\Omega}$ be the collection of all subsets of $\Omega$.Then $\dots$

Let $\Omega$ be a countable set $\mathcal A=2^{\Omega}$ be the collection of all subsets of $\Omega$.Then If $\mu:A\rightarrow [0,\infty]$ is defined by $\mu(E)=|E|$ that $|E|$ is number of ...
1
vote
2answers
471 views

Minimum number of iterations in Newton's method to find a square root

I am writing an algorithm that evaluates the square root of a positive real number $y$. To do this I am using the Newton-Raphton method to approximate the roots to $f(x)=x^2-y$. The $n^{th}$ iteration ...
1
vote
3answers
127 views

combinatorics implementation in real life problems

How many ways there are to organize $7$ men in a row, if two insist on not standing next to each other? How do I approach this?

15 30 50 per page