All Questions

0
votes
0answers
10 views

Images of some regions of the complex plane by given function?

I'm trying the draw the image of $A=\lbrace z\in \mathbb{C}:-1<Im((1+i)z)<1\rbrace$ by $f(z)=1/z$ and the one of $B=\lbrace z\in\mathbb{C}:|z|<1\rbrace$ by $f(z)=(z-1)^{-1}$. I've managed to ...
1
vote
1answer
27 views

Family of “something very close to be a curve” over a curve $C$

Hartshorne (IMHO restrictive) definition of a curve: Definition of (complex) curve: A curve is an integral separated scheme of finite type over $\mathbb C$ of dimension $1$. (The definition of a ...
0
votes
1answer
30 views

Boundedness of sequence of functions

Consider a sequence of continuous integrable functions $\{f_n(t)\}_n$ such that $\ast\ \displaystyle\lim_{n\rightarrow\infty}f_n(t)=0$ for all $t>0$ $\ast\ \{f_n(t)\}_n$ is such that ...
2
votes
3answers
44 views

Probability of $2$ boys in a family.

In a family there are 3 children with minimum $1$ boy.What is the probability there are exactly $2$ boys in the family? I think I have to use combinatorics to solve this problem. I have solved some ...
-1
votes
0answers
10 views

Finding SD given mean, median, 25th and 75th percentile [on hold]

I have the data from 2009 till 2013. I would like to plot a graph using these figures but i lack the SD. so is there a way to find it? Mean: 3048,3292,3369,3348,3520 Median: 2800,3999,3020,3000,3200 ...
0
votes
1answer
31 views

Efficient ways to evaluate an integral with a logarithm

Is the approximation in terms of series for the logarithm $$\log(z)= \sum_{n=0}^{\infty}\frac{2}{2n+1}\Bigl(\frac{z-1}{z+1}\Bigr)^{2n+1} $$ a good approximation if I replace this series inside the ...
1
vote
1answer
24 views

Trigonometric Functions on a unit circle

I have to find all solutions for $\theta$ in the given range: \begin{equation} tan (\theta) = \frac {-1}{\sqrt3}, -\pi \le \theta \lt 2\pi \end{equation} I said that if $(x,y)$ is on the unit circle ...
1
vote
3answers
29 views

Proving a mod b < a/2 when a > b > 0

Suppose that $a \gt b \gt 0$. How can one prove that $a$ mod $b \lt a/2$? I understand why is that happening: if $a$ mod $b \gt a/2$ that means that $a/b \lt a/2$ and $a/b$ has enough "space" to ...
1
vote
0answers
10 views

Theory of Interest - Inflation rate

A newly retired employee has a retirement benefit of \$ 1,000,000 that he can obtain in 2 ways; 1. A lump sum paid immediately, or 2. A 10-year annuity-due with monthly payments starting from ...
2
votes
2answers
19 views

Solving Lagrange equation systems?

Given an equation system when using Lagrange multipliers to find maxima and minima, how does one solve it will all these variables that I cannot isolate because I don't know if they are 0 or not, so I ...
0
votes
0answers
26 views

What is explicit form of this kernel?

Let $G$ be a group and $N$ be a normal subgroup of $G$. Let $F$ and $S$ be a free group such that $F/R=G$ and $S/R=N$ for some normal subgroup $R$ of $F$. The map from $N \rtimes G$ to $G$ given by ...
0
votes
0answers
18 views

Find a bijection between cosets

Let $G \supseteq H \supseteq K$, G Group with subgroups H and K. I want to show that $$\phi : R_{G/H} \times R_{H/K} \rightarrow G/K, (x,y) \mapsto x y K$$ is a bijection, where $R_X$ is set that ...
0
votes
2answers
21 views

Find the matrix $A$ with this condition…

If $\theta \in\mathbb{R}\setminus\{k\pi, k\in\mathbb{Z}\}$ and $A\in M_{2\times 2}(\mathbb{C})$ such that $$A^{-1} \begin{pmatrix} \cos \theta & -\sin\theta \\ \sin \theta & ...
2
votes
1answer
35 views

Best score in this puzzle

I want to maximise the score of the following table, choosing one item from each column/row, so no two items are on the same row or column. Score to maximise is just adding all the choices together. ...
0
votes
0answers
11 views

Can you give a example about of curvature tensor

Can your give a Riemann manifold $(M^n,g)$,let $R(X,Y,Z,W)=g(R(Z,W)X,Y)$,and under some coframe $w^1,...w^n$, $$R=R_{ijkl}w^i \bigotimes w^j\bigotimes w^k \bigotimes w^l$$ such that,$\forall i,j ...
3
votes
1answer
36 views

Wedge product = set intersection?

In a research article [1] I found the following formulation: The wedge product may be considered as set intersection. For example, surfaces of constant $f(x,y,z)$ and surface of constant ...
0
votes
1answer
16 views

Subset notation with the bar crossed

Reading the book 'An Introduction To Continuous Optimization', I ran across the $\subseteq$ notation, but with the little bar crossed over with a small $45^o$ dash - only the bar, not the whole ...
0
votes
1answer
23 views

proving limits with Epsilon-Delta definition

Say I've got $\lim_{x\to a} f(x)= A$ and $\lim_{x\to a} g(x)= B$ How do I do the following: prove that if A < B, then there is the existence of a $\delta$ such that when $0 < |x-a| < ...
2
votes
1answer
34 views

Probability conundrum

Good morning, wondered if you could help me please? I would like to work out the probability of and event happening 5 times out of 6. all 6 events have a 1 in 60 chance of a particular outcome. I ...
3
votes
2answers
28 views

Problem to understand a recurrence relation

In Norris, Markov chains, I found the following: [...] a recurrence relation of the form $$ ax_{n+1}+bx_n+cx_{n-1}=0 $$ where $a$ and $c$ were both non-zero. Let us try a solution of ...
0
votes
0answers
17 views

Asymptotic expansion at infinity of integral function

Given $q\in(0,1)$ find $z$ such that $$ F(z)\equiv\int_{-\infty}^{z}\frac{e^{-\frac{y^2}{2 \sigma _{22}^2}} \text{erfc}\left(\frac{\rho \sigma _{11} y-\sigma _{22} V}{\sqrt{2-2 \rho ^2} \sigma ...
3
votes
1answer
25 views

Find the number of possible 4x4 matrices such that :

Find the number of possible 4x4 matrices such that : 1) each row has two 0's and two 1's 2) each column has two 0's and two 1's example : $$\large \begin{pmatrix} ...
0
votes
0answers
6 views

Characterise expression involving white noise

I would like to characterise an expression, for example by finding its spectral density. The function is $\int_{-t_0}^{t}\mathbf{C}_s^t(\tau)\mathbf{q}_{\omega}(\tau)d\tau\cdot\mathbf{q}_{a}(t)$ ...
0
votes
1answer
15 views

Can the b-adic representation of rational numbers (by quote notation) be extended to non-terminating expansions?

The wikipedia article on p-adic numbers warns about $b$-adic expansions where $b$ is not a prime: Although for p-adic numbers p should be a prime, base 10 was chosen to highlight the analogy with ...
0
votes
0answers
20 views

Examples of compact subsets in topological spaces [duplicate]

I'm trying to find a topological space and a compact subset whose closure is not compact. It is an exercise in the text book Armstrong, Basic Topology, which I can not figure out. Any hint or answer ...
0
votes
0answers
8 views

There is a question about Digamma function

what's the definition of this Digamma function? Just like Ψ(α,α+1-ρ;pz),where α,ρ,z are independent variable,p is coefficient. I'm really not sure what to do with this question.How can I solve ...
1
vote
1answer
19 views

Bijection of homotopy classes

I want to prove the following: given the (already proven) fact that the we have a bijection between (continuous maps) $f:X\rightarrow Y^K$ and $g:X\wedge K\rightarrow Y$ for pointed spaces $X$,$Y$ and ...
1
vote
4answers
25 views

How do I know if two vectors with $n$ components are parallel?

How do I know if two vectors with $n$ components are parallel? For example $$\begin{pmatrix}5\\2\\1\\3\\4\end{pmatrix} \text{, and } \begin{pmatrix}4\\1\\2\\3\\6\end{pmatrix}.$$
1
vote
0answers
17 views

What is the structure of the group with at most five elements of composite order?

What is the structure of the group with at most five elements of composite order? For instance can we say any thing about the order of such a group?
1
vote
2answers
35 views

meaning of integration

I read that integration is the opposite of differentiation AND at the same time is a summation process to find the area under a curve. But I can't understand how these things combine together and ...
0
votes
0answers
16 views

An inverse Fourier transform of Riemann $\Xi(z)$ function

Riemann $\Xi(z)$ function is related to Riemann $\zeta(s)$ function via ($s=1/2+i z$): $$\Xi(z)=\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)\tag{1}$$ The functional equation for $\zeta(s)$ is ...
-1
votes
0answers
15 views

surface and cone integrals [on hold]

can someone take me through these two questions, I have the answers but not the steps and I have no idea how to even get started, thanks!
1
vote
1answer
21 views

Can You Pass Nonlinear Functions of Conditioned Variable Through Conditional Expectation?

In general, nonlinear functions cannot pass through the expectation operator. For example, it is not true that $E\left(e^X\right)=e^{E(X)}$. However, when one conditions on $X$, is this true? Does it ...
0
votes
1answer
18 views

polynomial modulo polynomial

If $h(x) = x^2 + 1$, $g(x) = x^2 + x + 1$ and $f(x) = x^3 + x + 1$, then $$ \begin{align} g(x)h(x) \mod f(x) &\equiv (x^2 + x + 1)(x^2 +1) \mod x^3 + x + 1 \\ &\equiv x^4 + x^3 + 2x^2 + x + ...
1
vote
1answer
16 views

Secret sharing: modular arithmetic

I have this problem of sharing a secret code $n\in\mathbb{Z}$ such that $0\le n\le250$ among five people. There are 5 people, each one of whom receives a secret number $s_i$, $1\le i\le 5$ such that ...
-1
votes
0answers
33 views

How to prove the left limit and the right limit? [on hold]

The followings lines are on the real axis. What I am interested in to prove that the left limit of the red interval is to the left of the left blue one and correspondingly for the right end. Also ...
0
votes
1answer
10 views

Finding the max distance between two arrays

I have the solution of an ode as an array $x(t_k)$ where $k=1,...K$. I have another array which is an approximation to the solution of the ...
0
votes
1answer
8 views

3D Vector Equation

Consider the points $A (1, 5, 4)$, $B (3, 1, 2)$ and $D (3, k, 2)$, with $\overline{AD}$ being perpendicular to $\overline{AB}$. (i) Find $AB$ (ii) $AD$ , give the answer terms of $k$. Show that ...
0
votes
0answers
20 views

nice name for the image of multivariable function

Consider a differentiable function $f:D\subset\mathbb R^m\mapsto \mathbb R^n$ with $m\le n$. I know if $m=1$ then $f(D)$ is called by "path", if $m=2$ then $f(D)$ is called by "surface" and if $m=3$ ...
0
votes
1answer
21 views

$\epsilon -n_0$ argument for convergence of $(3n^2+5)/(2n^2-1)$ to $3/2$

I was doing the proof to show that given $\epsilon > 0$, there exist an $n_ 0 \in \mathbb{N}$ such that etc.. And I chose my $n_0 > \max\lbrace2,\sqrt{(13+ \epsilon)/\epsilon} \rbrace$. But, is ...
2
votes
3answers
43 views

$\varphi(n) \leq 5$, where $\varphi$ is the Eulerian function

If $\varphi(n) \leq 5$, then can we find a bound for $n$ itself, where $\varphi$ is the Eulerian function?
6
votes
1answer
46 views

A function is smooth at a point and not smooth in any neighbourhood of it, exist or not?

Suppose that a function $f$ defined in an open set $U \subseteq \mathbb{R}^m$ is smooth at a point $p \in U$. Then we have that there exists an open set $U_n \subseteq U$ $($ say $U_{n+1} \subseteq ...
3
votes
1answer
23 views

Existence of unique solution on $(-\delta,\delta)$ for $f(x)=1+x+\displaystyle\int^x_0\sin(tf(t))dt$

The following was a question previously given in a test at my university: Show that there exists some $\delta>0$ for which there is a unique continuous function ...
3
votes
0answers
22 views

A topic for seminar in finite group theory.

I am doing a course on "Structure of finite groups" and we have a choice of giving a presentation on topics related to the course, Course outline is as follows- I was thinking of O-Nan Scott ...
1
vote
3answers
44 views

A question about the proof of a theorem in Representation theory of groups

My Question is about one part of the proof of theorem in the book "A Course in the Theory of Groups" by Derek J.S. Robinson. I highlight the part that my question is about. We know that if $G$ is a ...
1
vote
0answers
28 views

Probability Distribution for a Weird Card Game

I promise this is not for a homework problem, even though this sounds like only something a professor would dream up. Here is the game: I begin with a deck of 13 cards: 1 through 10, Jack, Queen, and ...
0
votes
1answer
16 views

Harnack's inequality

Let $u$ be harmonic on $\{|z|<1+\epsilon\}$ for some $\epsilon>0$ and $u \geq 0$ on $\{|z|=1\}.$ Could anyone advise me how to show $\dfrac{1-|z|}{1+|z|}u(0) \leq u(z) \leq ...
2
votes
0answers
17 views

Nondimensionization of a simple system.

A damped spring mass system is modelled below: $$m\frac{d^2y}{dt^2}=F_s+F_d\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space t>0$$ ...
2
votes
1answer
16 views

How to prove that a set of connectives is adequate

I have this Table: $$\begin{array} {|c|} \hline A & B & A*B\\ \hline 1 & 1 & 0\\ \hline 1 & 0 & 1\\ \hline 0 & 1 & 1\\ \hline 0 & 0 & 0\\ \hline \end{array}$$ ...
1
vote
0answers
14 views

How to evaluate a line integral over the lemniscate of Bernoulli

Evaluate the integral over the $\int_c$ |y| ds where curve c is given by the equation $(x^2+y^2)^2=40^2(x^2-y^2)$ I used polar coordinates but keep getting 0 as a answer. Help?

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