0
votes
0answers
7 views

Sketch a figure which has a group of symmetries of order 5.

I am trying to draw a shape which has only 5 symmetries I know Square has 8 Rectangle/parallelogram has 4 Triangle has 6 Circle has infinite how do i know which shape has only 5 I know that ...
0
votes
0answers
6 views

Infinite sets having no RE subsets

I'm back trying to learn recursion theory on my own. I'd like to prove the following result: There exists an infinite set having no infinite R.E. subset. Constructive comments are appreciated. Proof: ...
0
votes
0answers
11 views

Efficient way of checking linear independence

Suppose I have a $4 \times 4$ matrix $A$ whose columns represent vectors $v_1,v_2,v_3,v_4$ in $\mathbb{R}^4$. Now, given that $\det{A} = 0$ (i.e. the vectors are linearly dependent), I want to make ...
1
vote
4answers
11 views

Prove $4^k - 1$ is divisible by $3$ for $k = 1, 2, 3, \dots$

For example: $$\begin{align} 4^{1} - 1 \mod 3 &= \\ 4 -1 \mod 3 &= \\ 3 \mod 3 &= \\3*1 \mod 3 &=0 \\ \\ 4^{2} - 1 \mod 3 &= \\ 16 -1 \mod 3 &= \\ 15 \mod 3 &= \\3*5 ...
0
votes
0answers
12 views

Domain of convergence of series

Could you help me to find the domain of convergence of series : $$\sum\limits_{n,m=1}\frac{n}{m!}z_1^nz_2^m$$ in $\mathbb{C}^2$.
0
votes
0answers
4 views

Question about upcrossings of Brownian motion

I am very stuck on this problem: Given a Brownian motion $(B_s)$, write $S_t = \sup_{0 \leq s \leq t} B_s$, for each $t>0$. All stopping times and martingales are considered w.r.t the filtration ...
0
votes
0answers
5 views

How to find the wave equation for a given dispersion relation?

Let's assume I have general wave function written like this: $$ f(x,t) = \displaystyle\int_{-\infty}^{\infty}A(k)e^{i(kx-\omega(k) t)}dk $$ So it's composited from lots of plane waves and each of ...
2
votes
2answers
16 views

If consecutive elements commute each other, does it mean that all of them commutes with each other?

Let $x_1,x_2,...x_k$ be $k$ different elements of a group $G$ and $k\geq4$. if we know that $x_i$ commutes with $x_{i+1}$ and $x_k$ commutes with $x_1$, can we say that all $x_i$ commutes with each ...
0
votes
3answers
21 views

Proving by Cauchy's definition $\lim_{x\to -1} x^2+3x-5=-7$

Prove by Cauchy's definition $\displaystyle\lim_{x\to -1} x^2+3x-5=-7$ From definition: $|x+1|<\delta\Rightarrow |x^2+3x+2|<\epsilon \iff |x+1||x+2|<\epsilon$. Now I'm not really sure ...
0
votes
0answers
8 views

$L^2$ regularity of a convolution with newtonian potential.

I am reading Bertozzi, Majda Vorticity and incompressible flow and in page 71 72, we are concerned with recovering the velocity field of a flow from its vorticity. At some point we need to have the ...
-1
votes
3answers
25 views

Combinatorics Question (discrete math)

In how many ways can one mark 6 blocks on a grid of 5 columns and 3 rows such that in every row at least one block will be marked? An explanation will be appreciated! Thanks a lot
1
vote
0answers
13 views

Problems about generators for Sylow p-subgroups

There are several problems I met asking to find the generators for some different Sylow $p$-subgroup. $(i)$ a Sylow 2-subgroup in $S_{8}$; $(ii)$ a Sylow 3-subgroup in $S_{9}$; $(iii)$ a Sylow ...
3
votes
1answer
34 views

Show that series converge or diverge

If $\displaystyle \sum_{n=1}^{\infty} a_n$ converge and has positive terms then decide if following series converge or diverge : a) $\displaystyle \sum_{n=1}^{\infty} a_n \cdot \sin{a_n}$ I think it ...
2
votes
7answers
31 views

Solution of an exponential equation

Probably very simple question. Why the solution of $$1=n(1-a)^{t}$$ in terms of $t$ is equal to: $$t=\frac{\ln n}{\ln \frac{1}{1-a}}$$
1
vote
4answers
30 views

Quotient spaces in linear algebra

I'm having a bit of difficulty understanding what a quotient space is to a vector space $V$. I will present the part I'm finding trouble with below. Let $V$ be a vector space and let $U$ be a sub ...
0
votes
0answers
8 views

Showing that the Brownian Bridge is Gaussian

Take $X_t = (1-t)B_{t/(1-t)}$ for $t\in[0, 1)$ where $B_t$ is a $1$-dimensional Brownian motion. I want to show that $X_t$ is Gaussian. I have actually never been able to find a precise definition ...
0
votes
1answer
12 views

Computationally Plotting an Implicit Equation

I have the following equation: $$F(x,y) = \sqrt{ (x^2 y)^{-0.2}}$$ and, $w/o$ a specific question, is there an advisable computational technique for plotting the above $(x,y>0)$? I am ...
0
votes
0answers
12 views

Compact open topology and nets

If $X$ and $Y$ are topological spaces we can form a topology on $Y^X$, which has as subbasis sets of the form $B(T,U) := \{f \in Y^X : f(T) \subset U \}$ where $T$ compact and $U$ open. Is there a ...
2
votes
0answers
43 views

difficult complex integral $\int_\gamma \frac{1}{z^2+i}dz$

We are asked to calculate $\int_\gamma \frac{1}{z^2+i}dz$ where $\gamma$ is the straight line from $i$ to $-i$ in that direction. My parametrization is simple, I chose $z(t)=i-2it$. Notice that ...
0
votes
0answers
12 views

Lebesgue intergral over $\mathbb R$

Let $f:\mathbb R^2\to \mathbb R$ be given bij $$f(x,y)=\begin{cases}2(x-y)e^{-(x-y)^2}& \text{ if }x>0 \\0&\text{ otherwise}\end{cases}$$ Given that $\int^\infty_{-\infty} e^{-z^2} dz=\sqrt ...
1
vote
1answer
20 views

Proof regarding notations

I tried to solve the following question: Let $f,g$ be non-negative functions such that $f(n)=g(n)\left[1+o(1)\right]$. Prove that $f(n)=\Theta(g(n))$. I looked on two cases: ...
-1
votes
2answers
36 views

Set theory: Why are these two sets different?

I'm currently working through a set theory book and one of the exercises is to explain why $\{z|z\subseteq \{\emptyset\}\}$ and $\{x|x\in \mathbb{Z}, 0<x<1\}$ are different. I'm just completely ...
2
votes
0answers
19 views

Clarification on the definition of $X^{\omega}$

I have never seen this notation before (graduated with a math degree a few months ago; not in school currently). Here's what I gather from Munkres' Topology: Given a set $X$, an $\mathbf{\omega}$ ...
2
votes
4answers
53 views

Abelian group of order 99 has a subgroup of order 9

Prove that an abelian group $G$ of order 99 has a subgroup of order 9. I have to prove this, without using Cauchy theorem. I know every basic fact about the order of a group. I've distinguished ...
1
vote
1answer
22 views

A non-UFD where prime=irreducible

It is easy to see that in an atomic domain (where every element factors into irreducibles), we have that all irreducibles are prime iff the domain in question is an UFD. I think it is not true for a ...
0
votes
0answers
14 views

The equivalence of homogenous systems of linear equations in two unknowns that have the same solutions

I am self-studying Linear Algebra by Hoffman & Kunze. Exercise 6 in Section 1.2: "Prove that if two homogenous systems of linear equations in two unknowns have the same solutions, then they are ...
-6
votes
0answers
19 views

Find the area of the shaded portion in the circle. [on hold]

AC=28 cm Find the area of the shaded portion. EDIT: I have found out the area of the lower semi-circle I'm not being able to find the area of the upper half.
1
vote
1answer
18 views

Asymptotic sequence of tan(z)

I have a question about the asymptotic sequence of $\tan(z)$: I have a problem with the second sequence in $(\sin^n(z))_n$. The problem is that I don't see a priori that an asymptotic sequence of ...
2
votes
1answer
31 views

20th derivative of a rational function

I could not find the 20th derivative of the function below : $$f(x) = \frac{2x}{x^2 - 4}$$ I have taken 1st and 2nd derivatives but I could not succeed at generalizing the derivative function.
1
vote
2answers
19 views

Deciding $\displaystyle o,\omega,\Theta$ notations

I have a question which I couldn't solve for about two hours. It goes like this: Let $\displaystyle f(n)=\left(\frac{n+3\ln(n)}{n}\right)^n \ ; \ g(n)=27^{\ln(n)}$. Fill the blank box with ...
-1
votes
0answers
30 views

Interstellar's Gargantua (Black Hole) Equation

I don't really know if this is a physics, computer science or mathematics question so kindly excuse me if I posted it at the wrong place. I want to know what was maths/equations that were used to ...
-1
votes
0answers
24 views

Why is this function an embedding?

We have the canonical function $\epsilon_p: \mathbb{Z} \to \mathbb{Z}_p, x \mapsto (\overline{x})_{k \in \mathbb{N}_0}=(\overline{x}, \overline{x}, \overline{x}, \dots )$. The function $\epsilon_p: ...
0
votes
3answers
41 views

What is the determinant of matrix?

Find determinant of the $n \times n$ permutation matrix $$ M= \left[ {\begin{array}{cccc} 0 & 0 & \ldots & 0 & 1\\ 0 & 0 & \ldots & 1 & 0\\ \vdots & ...
2
votes
1answer
26 views

Proving by Cauchy's definition $\lim_{x\to 0} x^2\cos x=0$

Prove by definition that $$\displaystyle\lim_{x\to 0} x^2\cos x=0$$ So take $\delta=\sqrt\epsilon$, and from definition we have: $|x|<\delta\Rightarrow|x^2|<\delta^2\Rightarrow|x^2\cos ...
-3
votes
0answers
23 views

Boolean simplification of $(ABC)' + (AB)'C + A'BC' +A (BC)' + AB'C$? [on hold]

What will be the answer for this? $$(ABC)' + (AB)'C + A'BC' +A (BC)' + AB'C$$ I tried solving it and ended up having answer 1 by using De'Morgan's laws
3
votes
3answers
77 views

Proving convergence of a series. Is my proof correct?

Prove that if $\sum_{n=0}^{\infty}{a_{2n}}$ and $\sum_{n=0}^{\infty}{a_{2n+1}}$ are convergent series then $\sum_{n=0}^{\infty}{a_{n}}$ is also convergent From the assumption we know that ...
-1
votes
0answers
16 views

Solving eqn. of the form K = AGL + BGT, where A,B,L,T are invertible matrices.

I am obtaining the following equation in a regression problem: \begin{eqnarray} Z'_1Y_1\Omega^{-1}_{1}A+Z'_2Y_2\Omega^{-1}_{2}A = Z_{1}'Z_1\Pi A'\Omega^{-1}_1A + Z_{2}'Z_2\Pi A'\Omega^{-1}_2A ...
1
vote
2answers
29 views

Find $x$ as the given $n$th term in the Fibonacci sequence?

With a given $n$ and I am trying to find the value of $x$, as in: $$Fib(x)=n$$ Using the formula for Fibonacci sequence, where $\varphi$ is the Golden Ration ($\approx1.61803399\ldots$) $$Fib(z) = ...
2
votes
3answers
97 views

Proving no rational satisfy $p^2 = 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 ...
0
votes
2answers
48 views

What does it mean to “cut-off 10% your weakest exam”?

Say that I have two exam grades: $$e_1$$ and $$e_2$$ and that exam $e_1$ is worth $p_1$ of your grade and $p_2$ percent of your grade ($p_1 + p_2 = 1 $). If on top of that weighting I promise to ...
-2
votes
1answer
64 views

Is $d(x,y)=|x-y|^2$ a distance on $\mathbb{R}$?

Please how to prove that $d(x,y)=|x-y|^2$ is a distance on $\mathbb{R}$, I don't know how to solve the triangular inequality. Thank you.
1
vote
3answers
62 views

Why does $1+p+p^2+\dotsb+p^{n-1}=\frac{1-p^n}{1-p}$ [duplicate]

$$y_n=\rho^ny_0+(1+\rho+\rho^2+\cdots+\rho^{n-1})b.$$ If $\rho \not=1$, we can write this solution in the more compact form $$y_n=\rho^ny_0+\frac{1-\rho^n}{1-\rho}b.$$ This is from Elem. Diff. ...
0
votes
1answer
24 views

How does the author apply the implicit function theorem?

Let $m$ and $k$ be integers with $0\leq m\leq k$. Let $M\subset \mathbb R^k$ and $p\in M$. Suppose there is an open set $U\subset \mathbb R^k$ and a smooth map $f:U\to \mathbb R^{k-m}$ such that ...
1
vote
1answer
19 views

Let X be Hypergeometric, Find $E\left(\binom{X}{2}\right)$

Let X be Hypergeometric: $X \sim \operatorname{HGeom}(w,b,n)$, so that $X$ is the number of white balls in a sample of size $n$ out of a population of $w+b$ white and black balls. Find ...
0
votes
1answer
30 views

Consider the function, f and its second derivative:

$$f(x)=\frac{4x^2}{x^2+3} $$ $$f'(x)=\frac{24x}{(x^2+3)^2} $$ $$f''(x)=\frac{72(1-x^2)}{(x^2+3)}$$ a)What are the critical numbers(if any)? b)On what intervals is the function increasing and on ...
2
votes
2answers
45 views

Proving $\lim_{x\to9}\sqrt x=3$ using Cauchy's definition

Prove: $\displaystyle\lim_{x\to9}\sqrt x=3$ using Cauchy's definition for a limit. After doing the scratch work I get that: $\delta=\epsilon^2+6\epsilon$, so going back, I have to show that ...
2
votes
1answer
23 views

How to find LCM of this equation?

Can $(x+1)(2x-1)$ be the LCM of this biquadratic equation $$\frac{5x-1}{x+7}=\frac{3x+1}{x+5}$$
0
votes
0answers
9 views

Show that $\max_{\bar{U}} |u|\le C(\max_{\partial U} |g|+\max_{\bar{U}} |f|)$

Let $U$ be a bounded, open subset of $\mathbb{R}^n$. Prove that there exists a constant $C$, depending on only $U$, such that $$\max_{\bar{U}} |u|\le C(\max_{\partial U} |g|+\max_{\bar{U}} |f|)$$ ...
0
votes
1answer
15 views

consider a base-16 adder. explain how to modify the adder so that it can perform a base-10 addition

consider a base-16 adder. explain how to modify the adder so that it can perform a base-10 addition I found this when I searched in Google but not understand please guide me to understand this ...
1
vote
1answer
52 views

Proof, that the floor and ceiling functions exist

I want to proof the following, with elementary properties of the integers and reals: Let $x\in \mathbb{R}$. Then there are unique $p,q\in \mathbb{Z}$, such that: $$p\leq x < p+1\text{ and ...

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