All Questions

1
vote
0answers
12 views

The equality condition of $||A||_p \leq \left[ \sum_{j=1}^m || A_{j\cdot}^T ||_q^p \right]^{1/p}$

I want to show that the inequality below gets the equality $$||A||_p \leq \left[ \sum_{j=1}^m || A_{j\cdot}^T ||_q^p \right]^{1/p}$$ when $A$ is a rank 1 matrix. Here $A\in\mathcal M(m,n)$ and $p, q$...
-1
votes
2answers
26 views

How can I prove that $10=2^{a}*3^{b}*7^{c}$ has infinite solutions?

Both in a unrescrited case and with the following restriction: $a+b+c=1$
0
votes
0answers
9 views

gradient of a transformation that uses an orthogonal matrix

If I have Y=QX, where Y and X are vectors of dimension n and belong to $R^{n}$, and Q is an orthogonal matrix. Then, why do we have $\nabla_{Y}f=Q\nabla_{X}f$? I know that orthogonal matrices have ...
0
votes
0answers
22 views

Prove that $ C^m_{n+k+1}= $ $\sum_{i=0}^{m}$ $C^{i}_{k+i} $ $\times$ $C^{m-i}_{n-i} $ [duplicate]

Prove that $ C^m_{n+k+1}= $ $\sum_{i=0}^{m}$ $C^{i}_{k+i} $ $\times$ $C^{m-i}_{n-i} $. I tried using double counting and Newton binom. Any idea? I don't now many identities......
0
votes
1answer
24 views

Inverse Laplace Transform - Pulling out the constant

If you refer to my picture: https://i.stack.imgur.com/lVsU1.png I'm having a hard time understanding why in the 2nd step the fraction is split up in two terms when 2 is a constant. I get why you ...
1
vote
2answers
53 views

How reliable is the linear problems like $ \min \|Ax - b\|^2$?

The following linear optimization is common used $$\min_x \|Ax - b\|^2$$ Here, $A$ is the matrix; $x,b$ are the vectors. I am curious about how reliable is the solution $x$ by solving above ...
0
votes
1answer
12 views

Sobolev-Gagliardo-Nirenberg: Why is $|f|^q$ continously differentiable?

I wanna understand a proof of the Sobolev-Gagliardo-Nirenberg inequality. Therefore, I need to know why $|f|^q \in C_c^1(\mathbb{R}^n)$ for $f \in C_c^1(\mathbb{R}^n)$ and $q>1$. Can eventually ...
0
votes
0answers
10 views

Subderivative of a product and of a quotient

Let $r$ be one subderivative of $f(x)$ at $x_0$ and $s$ be one subderivative of $g(x)$ at $x_0$. Does there exist a formula for a subderivative at $x_0$ of $h(x)=f(x)g(x)$ and of $i(x)=\frac{f(x)}{g(...
-1
votes
1answer
34 views

How to discretize and normalize an $n*n$ Gaussian kernel?

A 3x3 Gaussian kernel is usually shown as $$\frac{1}{16} \begin{bmatrix}1 & 2 & 1\\ 2 & 4 & 2\\ 1 & 2 & 1\end{bmatrix}$$ But where does that actually come from?
2
votes
1answer
75 views

Prove that $\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$ when $f(f(x))=x^2$

Let $f:[0, \infty) \to [0,\infty)$ be a differentiable function with $f'$ continuous. If $f(f(x))=x^2$, prove that $$\int_0^1 (f'(x))^2dx \geq \frac{30}{31}$$ without explicitly finding $f.$ Since we ...
0
votes
1answer
33 views

How does distinguishability of boxes change the number of ways to distribute n objects into separate boxes [duplicate]

I was going through problems related to distributing n distinguishable objects all into boxes, a in the first, b in the second, c in the third where a+b+c = n. The solutions I've seen usually comes to ...
0
votes
0answers
17 views

How to solve this equation which is of Cauchy-Euler form?

I have a second order ode in the Cauchy-Euler form. The problem is, the coefficients of differentials in the equation are arbitrary and are not known numbers. This leaves me with a problem of ...
0
votes
1answer
53 views

Not Sure Why Limit Is In Book

I am looking at logarithms and derivatives. In my books, Bostock and Chandler, it saying: $\frac{dy}{dx} = \lim_{\delta x \to 0} \frac{\delta y}{\delta x} = \lim_{\delta x \to 0}(\frac{1}{\frac{\...
0
votes
2answers
26 views

What is $m_n$, when $(m_1,m_2,m_3) = (1,2,3)$ and $m_n =\sum_{i=1}^{3}(4-i)m_{n-i}$?

If $m_1, m_2, m_3$ are given then what will be the nth term of the series if $$m_{n}=\sum_{i=1}^{3}(4-i)m_{n-i}$$ ex if $m_1 ,m_2, m_3$ are $1, 2 ,3$ then $m_4 = 14$ $m_5= 15$ So what will be the ...
0
votes
0answers
14 views

Power Series Solution for deriving a recursion relation

Unfortunately, I'm not familiar with Power series solution method. Also reading some guide files did not help me much. I hope you offer me some hints. I have an equation as follows $$ [- \frac{d^2}{dr^...
0
votes
1answer
49 views

what is the applications of the gamma function? [on hold]

Can I get perfect confident sources for applications of the gamma function? Sits or books or notes ,i get some sources and find 'incomplete gamma and distribution of gamma"like \begin{eqnarray*} \...
1
vote
1answer
35 views

Is $A+(A^{−1})^{*}$ invertible?

Let $A$ be an $n\times n$ invertible matrix. I think this is true because I have tried a few different real and complex matrices and they satisfy this. The trouble I'm having is showing it is true. ...
3
votes
1answer
32 views

Simple probability and statistics problem

If there are 12 football teams in a league, how many different bets can you make if you bet on 3 first teams and 2 that will get kicked out of the league? The solutions says its (12*11*10*9*8)/2. ...
0
votes
1answer
32 views

Functional Analysis- invertible operators

Let ${c_n}$ $\in \ell^{\infty}$, and let $T_{c_n}$$\in$ B$(\ell^2)$, $\:$ be defined by $\\$ $T_{c_n}$ ({$x_n$}) = {$c_nx_n$}. $\\$ If inf {$\vert c_n \vert$: n $\in\mathbb{N}$} $\gt$ 0 , and $d_n$ =...
2
votes
1answer
71 views
+50

How to see if a subgroup is normal from Cayley graph

Let be a Cayley diagram of group $G$. Let $H$ be the orbit of element p. Is $H$ a normal subgroup of $G$? Is there a simple way to check that because going by definition seems complicated. I tried ...
0
votes
0answers
20 views

Clarification for inverse trigonometric fuctions. [duplicate]

Since sin⁻¹(x) means the inverse of sin(x), also written as arcsin(x), how would you write 1/sin(x)? Since sin²(x) actually does mean sin(x) * sin(x), can you write 1/(sinx *sinx) as sin⁻²(x)?
1
vote
2answers
41 views

Homework question using Liouville theorem and Cauchy estimate.

The statement is if $f$ is an entire function that satisfies $$ |f(z)| \leq \pi e^{2Rez} $$ Then $\forall z \in \mathbb{C}$ , there is a complex number $a \in \mathbb{C}$ such that: $$f(z) = ae^{2z}$...
0
votes
0answers
24 views

Need help with the integral: $\int_{0}^{\infty}\prod_{i}^{N/2}(e^{-\lambda_{2i-1}t}+e^{-\lambda_{2i}t}-e^{-(\lambda_{2i-1}+\lambda_{2i})t})dt$

I am trying to understand the behavior of the following integral $$\int_{0}^{\infty}\prod_{i=1}^{N/2}\left( e^{-\lambda_{2i-1}t}+e^{-\lambda_{2i}t}-e^{-(\lambda_{2i-1}+\lambda_{2i})t} \right)dt$$ ...
0
votes
2answers
18 views

On the signature of a quadratic form

Prove that the determinant in $M_2(\Bbb R)$ is a quadratic form of signature $(2,2)$. I found the first part: the symmetric bilinear form $$B(M,N)=\frac{1}{2} ( \det(M+N) - \det (M) -\det (N) )$$ ...
0
votes
0answers
36 views

Is this only definition?

If $u$ is twice differentiable and $\Omega$ a domain and $\Delta$ the Laplace operator then $$ -\Delta u \in C(\Omega) \Rightarrow u \in C^2(\Omega) $$ I think it is only definition . Thanks ...
0
votes
0answers
11 views

Linear approximation non-zero implies surjectivity?

Consider $M$ a smooth manifold, and $g:M\to\mathbb{R}$. I want to prove the following equivalence: $dg_x:T_xM\to\mathbb{R}$ is surjective, if and only if $dg_x\neq 0$. That surjective implies $...
2
votes
1answer
34 views

how to check if there is an automorphism mapping between two conjugacy class

Let $G\le S_n$ be a permutation group and suppose that $C_1,C_2$ are two distinct conjugacy classes that have the same cardinality and is represented by a permutation of the same cycle-type. My ...
1
vote
1answer
30 views

Existence of periodic solution for non-autonomous system

"Consider the n-dimensional system $x^{'}=f(x)+g(t)$ where $x^{T}f(x)\leq -k|x|^{2}, k>0$, for all $x$ and $|g(t)|\leq M$ for all t. Show that this equation has a w-periodic solution if $g$ is w-...
0
votes
0answers
20 views

How do I evaluate the following combination of random variables? Is it martingale?

I'm about to analyse the following expression $$Z_n:=\prod_{k=1}^n \left(\frac{\frac{Y_k}{\prod_{i=1}^k X_i}}{\sum_{j=1}^k \frac{Y_j}{\prod_{i=1}^j X_i}} \right),$$ where $Y_j$ for all $j\in \mathbb{...
0
votes
2answers
37 views

Find distance between a line and the origin [on hold]

The given line is defined as following: $$\text{plane 1: } x+y+z = 6$$ $$\text{plane 2: } 2x - y - 5z = -5$$ What is the distance between the line and the origin?
1
vote
2answers
36 views

Could someone explain explain to what correspond the moments of a random variable?

Could someone explain what are (at least the four first) moments ? (normalized moment to be more precise) Let $X$ a r.v. So the first moment is the expectation. This will correspond to $\mathbb E[X]$...
0
votes
0answers
17 views

Why is true this equality? Afinne conection

Let $\gamma : [0,l] \to M$ be the only geodesic joining $p$ and $q$ in a complete riemannian manifold, where $q$ is out of cut locus of $p$.Let $E_1,...,E_n = \gamma'$ a orthonormal parallel field ...
2
votes
1answer
62 views
+50

Inference regarding the mean lifetime of a bulb using a new technique

The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean $\lambda$. We need to test the null hypothesis $H_0: \lambda=1000$ ...
0
votes
1answer
13 views

Deriving a demand function from a specific utility function

Let's say I have a utility function of the form $Ax^b + Cx^d$. Now I would like to find the consumption depending on the price for one unit of good $x$. This means for any given $p$ I would maximize $...
1
vote
1answer
41 views

Is the normal bundle construction idempotent?

Let $X$ be a submanifold of $M$. Inductively, let $N_0$ be the normal bundle of $X$ in $M$, and $N_{k+1}$ the normal bundle of $X$ in $N_k$. (Identify $X$ with the zero section of $N_k$, of course.) ...
0
votes
0answers
7 views

Lipschitz Mapping of $\mathbb{R}_+$ to $[0,1)$

During my research, I encountered a problem where I have some points in $\mathbb{R}^2_+$ (actually, there are in $\mathbb{N}^2$), and I would like to map them into the two-dimensional square $[0,1)^2$ ...
0
votes
1answer
17 views

How to use mean value theorem to get the inequality $|\frac{1}{n^s}-\frac{1}{x^s}|\le \frac{|s|}{n^{\sigma+1}}$?

How to use the mean-value theorem to get the inequality $|\frac{1}{n^s}-\frac{1}{x^s}|\le \frac{|s|}{n^{\sigma+1}}$?
-2
votes
1answer
30 views

Limit of $\sqrt[n]{n^k}$ for k $\in \mathbb{N}$ for $n \rightarrow \infty$

So the title basically says what my question is I'm looking for a proof for the limit of the given sequence (title).
0
votes
2answers
32 views

Why do we have to check ker(T)=0 for isomorphism when I know that dim (V)=dim(W)?

Suppose I want to prove that $P_2$ is isomorphic to $R^3$. Why is it not sufficient to prove dim($P_2$)=dim($R^3$) and apply Thm part (b)? Why do I have to show that the ker($T$)=$0$?
1
vote
0answers
51 views

How to evaluate the line integral (checking Stokes Theorem)

Consider the vector field: $$\vec F = ye^x \hat i + (x^2 + e^x) \hat j + z^2e^z \hat k$$ A closed curve $C$ lies in the plane $x + y + z = 3$, oriented counterclockwise. The parametric ...
3
votes
3answers
38 views

Complex Number: $\frac{(3+i)^2}{(1+2i)^2}$ - cannot arrive at textbook solution

I have a complex quotient $\frac{(3+i)^2}{(1+2i)^2}$ The solution provided in my textbook is $-2i$. I arrived at different solutions and I'd like to know where I went wrong. Till now in my textbook ...
2
votes
1answer
44 views

How many different ways can $\mathbb{Z}_3$ act on the set $\{1, 2, 3, 4\}$

How many different ways can $\mathbb{Z}_3$ act on the set $\{1, 2, 3, 4\}$ This is my attempted proof. Proof: Any action of $\mathbb{Z}_3$ on the set $\{1, 2, 3, 4\}$ is equivalent to a homomorphism ...
3
votes
0answers
39 views

Hungerford Chapter 2 Section 2 Problem 2 WITHOUT using the structure theorem of finite abelian groups

Let $G$ be a finite abelian group and $x$ an element of maximal order. Show that$\langle x \rangle$is a direct summand of $G$. Use this to obtain another proof of Theorem 2.1. Theorem 2.1: Every ...
0
votes
0answers
36 views

When do you learn calculus 1, calculus 2 and calculus 3?

As an English student who is using an American resource for learning material, could someone explain at what age you would learn calculus 1, calculus 2 and calculus 3?
0
votes
0answers
37 views

For a field $K$ of characteristics $p>0$, when is a finite purely inseparable extension $F/K$ (with $[F:K]=p^n>1$) such that $F\cong K$?

An example that illustrates the question is: $F=\mathbb{F}_p(t)$ and $K=\mathbb{F}_p(t^p)$, for which $F\cong K$ by Luroth's theorem. Also, for $p>3$, consider $F=\overline{\mathbb{F}}_p(x,y)...
2
votes
1answer
72 views

How to prove that the derivative of a homogeneous equation is $y'=\frac{y}{x}$

I was solving a differential equation problem which required me to find out the derivative of. For the curve $x^2y^3=(2x+3y)^5$, $\frac{dy}{dx}=\frac{-y}{g(x)} $ The author simply stated that ...
2
votes
3answers
46 views

Showing $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial.

I encountered this problem in Sims' "Computation with Finitely Presented Groups". Show that $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial. The book uses coset enumeration or ...
1
vote
1answer
42 views

10th Grade Algebraic Rate of Bacteria Growth Problem

"A certain type of bacteria doubles every 6.5 hours. If there were 60 bacteria to start with, what is the hourly growth rate of the bacteria? How many bacteria will there be after a day and a half? ...
0
votes
0answers
10 views

Deriving the residual error for finite elements.

I am using the following set of notes for adaptive finite elements (https://www.ruhr-uni-bochum.de/num1/files/lectures/AdaptiveFEM.pdf) and am trying to go through the error calculations on page 29&...
0
votes
0answers
9 views

Normal Equations (OLS) : Summation notation to system of equation form

I am struggling to "visualize" the following equation. I would like to know how the following equation can be written in system-of-equation form in order to see that there are $K$ unknowns. The ...

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