1
vote
0answers
12 views

How can we extend $\operatorname{div}:C_c^\infty(\Omega)^d\to L^p(\Omega)$ to $W_0^{1,\:p}(\Omega)^d$?

Let $d\in\mathbb N$ and $p\ge 1$ $\lambda$ be the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be open and $$W_0^{1,\:p}(\Omega):=\overline{C_c^\infty(\Omega)}^{\left\|\;\cdot\;\...
0
votes
1answer
8 views

How can I find a DNF and Minimal Form for this boolean expression?

$Q(x,y,z)=(y′\vee z′ \vee 0\vee x′)\wedge1\wedge(z\vee x′\vee 0\vee y\vee z)′\wedge(z′\vee x\vee y\vee z′)$ I'm not supposed to use tables but only proprieties like De Morgan ecc.
0
votes
2answers
31 views

What does $u^T$ mean? How to compute this?

if $u = \begin{bmatrix}7\\7\\5\end{bmatrix}$ then $uu^T = \begin{bmatrix}\\\\\\\end{bmatrix}$ and $u^Tu = \begin{bmatrix}\\\\\end{bmatrix}$ What in the world does the $T$ mean? Also, are they ...
-1
votes
1answer
16 views

Given $g(x,y)$ with continous and differnatiable derivations for all $(x,y)$ points

Given $g(x,y)$ with continous and differnatiable derivations for all $(x,y)$ points(I mean by that: $f'_x$ has a value, continous and differantiable, same for $f'_y$), and given $g'_x(0,0) = 1 , g'_y(...
-1
votes
0answers
11 views

Proof that SSR = SST when they have the same degree of freedom? [on hold]

Intuitively, this makes sense, but is there any way to prove it?
1
vote
2answers
38 views

Find the Maclaurin series of f(x)=(arctan(x)-x)/x^3

What I think I need to to do is find a general series expansion of the function and then derive term by term to get the Macaurin series...but I'm not quite sure how to expand this function. Any help ...
1
vote
2answers
20 views

Is there a faster way to determine partial orderings of basic finite sets?

For example, consider the set $S = \{ 0, 1, 2, 3 \}$, and the following relation on $S$: $$ R = \{(0,0), (1,1), (1,2), (1,3), (2,0), (2,2), (2,3), (3,0), (3,3) \}. $$ Obviously, I can go through ...
2
votes
3answers
32 views

Proving $-1$ is the infimum of $E=\{\frac{2}{n}+(-1)^n\mid n\in \mathbb{N}\}$

Prove: $\inf\{E\}=-1$ for $E=\{\frac{2}{n}+(-1)^n\mid n\in \mathbb{N}\}$ Let assume the contrary , there is $x\in E$ such that $\frac{2}{n}+(-1)^n\leq-1$. Because we are looking at negative ...
0
votes
0answers
6 views

Print the image of a map (morphism) in Singular

I try to learn some basics in SINGULAR. I just wonder, how to get the image of a morphism printed. Here's a short example: (for those, who don't know Singular, there is a small description added) <...
0
votes
0answers
11 views

Is there any example of a Markov chain (discrete) with limit distribution (discrete) of heavy tail?

Is there any example of a Markov chain with limit distribution (discrete) of heavy tail? In other words, a Markov chain whose limit distribution has infinite second moment?Already, thanks for the help!...
0
votes
2answers
18 views

Given $h(x,y) = f(x^2y + y^3)$, and $f(t)$ is differentiable $\forall t$ and $f'(t) = \frac{1}{2}$, then

Given $h(x,y) = f(x^2y + y^3)$, and $f(t)$ is differentiable $\forall t$ and $f'(t) = \frac{1}{2}$, then compute $h'_x(1,1) + h'_y(1,1)$. Well, I'm having difficulties using the chain rule to solve ...
-1
votes
0answers
15 views

Prove for formula Tñ=a(n-1)f

The formula for finding the nth term of an A.P. is Tñ=a+(n-1)f. Where a= 1st term of A.P. n=number of terms. f=common difference between two consecutive terms of an A.P.
1
vote
0answers
11 views

Variant of local inversion theorem in special case

Let $F:\mathbb{R}^2\to\mathbb{R}^2$ be defined by $F(x,y)=(x+2y+x^2\ ,\ y-x^3+y^2)$. Then show that for $p_0=(4,1)$ and $p_1=(1,1)$ there exists $\delta>0$ such that for every $\vec y\in B(p_0,\...
1
vote
0answers
10 views

Interpolating polynomial such that it is convex in specified region

The problem I have is that I have data at two points $x_1,x_2$ and $x_2>x_1>0$. At these two points, I know that the function $f$ has values $f(x_1)$ and $f(x_2)$ respectively. It is also ...
-1
votes
2answers
60 views

Given the matrix $A$, find $A^3$

Given the matrix $A = \begin{bmatrix}-2&1\\0&-1\end{bmatrix}$, find $A^3$ I don't understand what $A^3$ is supposed to represent? What are they asking me to find exactly??
0
votes
5answers
56 views

Limit of this sequence [duplicate]

So guys I need to find the limit of: $\displaystyle\lim_{n \to \infty}\left(\sqrt{n^2+2n+5}-n\right)$ The quadratic equation is hard to factorise and I really struggle to answer these questions. ...
0
votes
1answer
17 views

Rotation of 3d vector alone a plane?

I have vector PQ which lies on plane Ax+By+Cz+D=0, now after i rotate this vector in this plane with angle t,about the point P what will be the new position of Q ? Here the position of new Q is ...
1
vote
3answers
49 views

Every subspace is the kernel of a linear map

I know that every kernel of a linear map from $\mathbb R^n$ to $\mathbb R^m$ is a subspace of $\mathbb R^n$. I am wondering if the converse is true, i.e. every subspace of $\mathbb R^n$ is the kernel ...
0
votes
1answer
10 views

Summation with Floor and Square Root functions + Tight Bounds

I was applying a methodology that allows to come up with iterative algorithms time-complexity function's closed-form. I ran into a particular where I ended up with the result below. I wouldn't have ...
0
votes
0answers
13 views

Expected value of stochastic process

How can i calculate the expected value of $$S_t= S_0 \exp\left( mt-\frac12\int_{0}^{t}e^{2Y_s}ds+\int_{0}^{t}e^{Y_s}dB_s\right)\quad $$ where $${Y_t}$$ is the solution of a sde and follows tha normal ...
4
votes
0answers
80 views

$\displaystyle\int_1^2\sqrt\frac{x^6+4x^4-2x^3+1}{x^4}\ \mathrm dx$ [on hold]

Find the value of: $\displaystyle\int_1^2\sqrt\frac{x^6+4x^4-2x^3+1}{x^4}\ \mathrm dx$ I do not really know where to start, so please forigve me for not showing my attempt. Wolfram alpha gives $2....
0
votes
1answer
19 views

If a normal series has maximal lengh in $G$, then it is a composition series.

A normal series of subgroups of $G$ is a decreasing sequence of subgroups $... \subset G_1 \subset G_0 = G$ where $\subset$ denoted strict inclusion and $\forall i \ \ G_i$ is normal in $G_{...
0
votes
0answers
15 views

Exam Recurrence and Complexity [on hold]

I have an exam coming up in a few days and my prof gave us a couple questions we should know as he will make new questions based on these topics the explanations must be in-depth because it will be a ...
0
votes
1answer
28 views

Calculate velocity $\nu$.

A lorry of mass $3.5\times10^4\text{ kg}$ attains a steady speed $\nu$ while climbing an incline of $1$ in $10$ with its engine operating at $175$ kW. Find $\nu$. $g=10ms^{-1}$. Neglect friction. The ...
1
vote
0answers
17 views

Proof of an irrationality criterion

I have attached a proposition whose proof I don't understand at two points. Here are my questions: Why do we have $|a_{0n}+\theta_{1}a_{1n}+\dots+\theta_{k}a_{kn}|<(\rho-\varepsilon)^{-n}$ for ...
2
votes
0answers
16 views

Estimator bias and consistency

Let $x_1, x_2, \ldots,x_n$ be a simple random sample from a random variable $X$ with support $\{0,1,2,3,4\}$ and probability function $p(0)=\frac{5}{12}(1-\lambda)^2$, $p(1)=\lambda$, $p(2)=\lambda(1-\...
-7
votes
1answer
50 views

Help Me Solve A Few Problems [on hold]

Here is it $$2y^2y'-x^2=0,\quad y=\text{?}$$ For example $$y^2y'-5x=0$$ $$y^2 \,dy -5x \,dx = 0$$ $$y^2\,dy=5x\, dx$$ $$\int y^2\, dy = \int 5x\,dx$$
2
votes
1answer
25 views

Finding arc length of the curve $6xy=x^4+3$ from $x=1$ to $x=2$

Looking at this as a graph of a function of $y$ is more convenient $$ y=\frac{x^4+3}{6x}\Rightarrow \frac{dy}{dx}=\frac{x^3-1}{2x^2}\Rightarrow \left( \frac{dy}{dx} \right)^2=\frac{x^6-2x^3+1}{4x^4} ...
0
votes
0answers
6 views

What different between huffman and EBCOT?

I have a question about entropy coding and Image compression. Huffman coding and EBCOT coding are entropy coding. correct or not? I knew about huffman coding but I don't know EBCOT, How it working?...
1
vote
0answers
26 views

Transformations of two Laplace distributions resulting in a Laplace distribution

Suppose we have two independent identical random variables $X_1$ and $X_2$ with Laplace distribution \begin{align} f_X(x)=\frac{1}{2b}e^{-\frac{|x|}{b}} \end{align} I am looking for a non-...
0
votes
1answer
19 views

if the metric $d_1$ is complete, and $\lim_{n \to \infty} d_1(x_n,x)=0$ iff $\lim_{n \to \infty} d_2(x_n,x)=0$, is $d_2$ complete?

two metrics $d_1, d_2$ on $X$, For all $x_n$ and $x$ from $X$ it holds : $\lim_{n \to \infty} d_1(x_n,x)=0$ iff $\lim_{n \to \infty} d_2(x_n,x)=0$ Is it true that $(X, d_1)$ complete implies that $...
0
votes
0answers
5 views

local sections transformation formula

I am trying to prove a formula which states the relations between two local sections in a principal bundle: Let $P(M,G)$ be a principal bundle let $\{ U_\alpha \}_{\alpha \in I}$ be an open cover for ...
1
vote
1answer
15 views

Periodic Function with Integral

Problem: $f(x)$ is a continuous function, and it is periodic with period $T$. For any $a<b$, prove that $$\lim_{n\to\infty}\int_a^bf(nx)dx=\frac{b-a}{T}\int_0^Tf(x)dx$$ I tried substituting ...
0
votes
1answer
14 views

Diffuse-like decomposition of the segment $[0,1]$

Consider the segment $[0,1]\subset\mathbb{R}$ and the standard Lebesgue measure $\mu$ on $\mathbb{R}$. I wonder if we can find such decomposition $A\sqcup B=[0,1]$, that for any $x\in[0,1]$ we have $\...
0
votes
0answers
8 views

Determine SE(3) transform between pair of sensors producing 2d line segments

Say we have sensorA and sensorB with an unknown transform between them, and that we have n ...
1
vote
1answer
22 views

Why does MLE work for continuous distributions?

In the attachment below you can see the definition of the likelihood function. Likelihood 1) Whilst the explanation of why the whole max likelihood method is viable for discrete distributions is ...
0
votes
2answers
46 views

What is embedding?

I am new to this so do I need to learn topology in order to understand this? Cause I come across this which says that unlike the 2D sphere, 2d saddle surface cannot be embedded in 3D Euclidean space(...
2
votes
2answers
40 views

Prove: Let $a,b\in R$ such that $a\lt b$, and $f:[a,b]\rightarrow R$ be monotonic, then $\frac{1}{f}$ is also monotonic

Prove or disprove: Let $f:[a,b]\rightarrow \mathbb{R}$ be monotone. If $f(x)\ne 0$ for all [a,b], then $1/f$ is also monotone on [a,b]. I've been sitting on this for quite a while trying to find a ...
0
votes
1answer
19 views

Inverse Matrix=echolon form of $(M|E_n)$?

Why is it, that if i want to calculate the inverse of a matrix, the echelon form of $(M|E_n)$ will give it to me? For example: $\{\{1,-2,0,1,0,0\},\{0,2,1,0,1,0\},\{-1,1,2,0,0,1\}\}$ in echelon form ...
3
votes
2answers
72 views

A convergent series: $\sum_{n=0}^\infty 3^{n-1}\sin^3\left(\frac{\pi}{3^{n+1}}\right)$

I would like to find the value of: $$\sum_{n=0}^\infty 3^{n-1}\sin^3\left(\frac{\pi}{3^{n+1}}\right)$$ I could only see that the ratio of two consecutive terms is $\dfrac{1}{27\cos(2\theta)}$.
6
votes
0answers
65 views

Generators of so(7)

Short version: Let $V$ be a 7-dimensional linear space of (real) square matrices. Suppose further that $[V,V]$ (the linear space spanned $[X,Y]$, $X,Y\in V$) is isomorphic to $\mathfrak{so}(7)$. Can ...
0
votes
1answer
18 views

How to prove this function is upper semicontinuous?

From Rudin's RCA: If $f:\Bbb R\to\Bbb C$ is an arbitrary function, and $$\phi(x,\delta):=\sup\{|f(x_1)-f(x_2)|\mid x_{1,2}\in(x-\delta,x)\cup(x,x+\delta)\},\,\phi(x):=\lim_{\delta\to 0^+}\phi(x,...
0
votes
3answers
18 views

How do I find the the third vertex of an isosceles triangle, if the vertex is on the y-axis?

$ABC$ is an isosceles triangle ($AB=AC$). We know that $A(5;9)$, $B(4;2)$ and $C$ lies on the $y$-axis. How to find $C$?
-6
votes
0answers
21 views

Permutation, Arranging letters [on hold]

Please help me! I am in a hurry! The six letters of the word “MOTHER” are rearranged in all possible orders and the words so formed are listed in alphabetical order
0
votes
3answers
41 views

Prove or disprove $A$ compact/closed $\implies$ $\mathcal{P}(A)$ compact/closed

For every $A \subset \mathbb R^3$ we define $\mathcal{P}(A)\subset \mathbb R^2$ by $$ \mathcal{P}(A) := \{ (x,y) \mid \exists_{z \in \mathbb R}:(x,y,z) \in A \} \,. $$ Prove or disprove that $A$ ...
0
votes
2answers
18 views

Automorphism of Upper half plane

Let $M=\left\{\left.\displaystyle z\mapsto\frac{az+b}{cz+d}\ \right|\ \ ad-bc\not =0\right\}$,$$p:GL(2,\mathbb C)\to M, \begin{bmatrix}a & b \\ c & d \end{bmatrix}\mapsto\frac{az+b}{cz+d}.$$ ...
1
vote
1answer
21 views

Is a Blaschke product/rational function a covering map for a $n$-sheeted covering of $S^{1}$?

We have a Blaschke product $B(z)$ of order $n$ (you can think of it as a rational function with $n$ zeros and $n$ poles), the zeros are obviously inside $\mathbb{D}$. Why is $B(z) \colon S^{1} \to S^{...

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