0
votes
0answers
10 views

How to solve an equation involving euclidean norm operation?

On page 3 of Scalable, Versatile and Simple Constrained Graph Layout it describes the equation: ||(p-r)-(q+r)||=d Where p and q are points q, r is a vector and d is a scalar. It then goes on to ...
1
vote
0answers
23 views

$f$ is monotone on D and $f(D)$ is an interval

$f$ is monotone on D and $f(D)$ is an interval then $f$ is continuous Is my proof right? pf) First, suppose it is monotone increasing Since $f(D)$ is an interval there is $[c,d]$ such that ...
0
votes
0answers
8 views

hypothesis testing ; F-testing

lnQ=1.37+0.632lnK+0.452lnL (0.257). (0.219) cov(bk,bl)=0.055 R^2=0.98 H0: bk+bl=1 F=(Rb(hat)-r)'[R(X'X)^-1R']^-1(Rb(hat)-r)/e'e/n-k-1 I found the numerator value but i ...
0
votes
3answers
18 views

Radius of Convergence

Is the radius of convergence of $$\frac{n(x+3)^n}{4^n}$$ equals 4? I got $|x+3|\lt 4$ as the final result. How do you know, what is the radius from here?
5
votes
4answers
69 views

Interesting summation question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to?

Question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to? $$\sum_{n=1}^{1729} \left[(-1)^n\cdot V(n)\right]$$ Where $$V(n)=a^n+b^n$$ My effort: I think I ...
0
votes
0answers
10 views

What is the difference between single and double layer potential

I want to know the difference between $\textbf{single-layer}$ and $\textbf{double layer potentials}$ . so is it exist a $\underline{link}$ between the choice of $\underline{single/double~ layer~ ...
0
votes
0answers
11 views

Inverse Laplace operator $\Delta^{-1}$ in $H^1_0, \ H^{−1}$.

Let open domain $\Omega \subset \mathbb R^n$, $u : \Omega \rightarrow \mathbb R$, $u \in H^1_0 (\Omega)$ and $f \in H^{−1}(\Omega)$. I'm looking for some known expressions of the inverse Laplace ...
2
votes
2answers
13 views

Correct term for percentage in decimal form

I have 35% of something, but when I calculate how much that is I multiply the total by 0.35 Is there a unambiguous word for the decimal form of a percent? "Decimal" is too broad because it can refer ...
0
votes
0answers
12 views

Tetrahedron symmetries

What are the order of lines of symmetry, plane of symmetry and rotational symmetry in a tetrahedron with its base is an equilateral triangle and other sides are all isosceles triangles?
1
vote
1answer
26 views

Determining the center of the p-Sylow subgroup of $S_p $

My Algebra book says without proof that the center of the p-Sylow subgroup (we will call it P) of $S_p$ is the subgroup itself. Now I can understand that P will be a subset of its center as it is of ...
1
vote
3answers
25 views

Should I use the comparison test for the following series?

given the following series $\sum_{k=0}^\infty \frac{\sin(2k)}{1+2^k}$ I'm supposed to determine whether it converges or diverges. Am I supposed to use the comparison test for this? My guess would ...
0
votes
1answer
22 views

Find the smallest prime number that doesn't divide any 5-digit number whose digits are in strictly increasing order.

Find the smallest prime number that doesn't divide any 5-digit number whose digits are in strictly increasing order. I have posted an answer of my own below; any alternative solutions will also be ...
0
votes
0answers
28 views

Semplify $\det\left(D+M+A\right)$.

Let $D$, $M$, $A$, $n\times n$ matrices, with $n\in\mathbb N$. $D$ is a diagonal matrix, $M$ with elements all equal to $k\in\mathbb R$, $A$ is an antisymmetric matrix. Is possible to calculate ...
0
votes
0answers
13 views

Calculating probability of an event using mean and stnd deviation

A factory produces electrical devices. It is known from experience, that the life-time of this devices will be normal distributed. With:  Mean = 1100 hours  standard deviation = 50 hours. ...
1
vote
3answers
153 views

Symmetric matrices and eigenvalues

If the eigenvalues of a symmetric matrix $A$ are greater than 0, show that $v^{\top}Av > 0$ for every $v \ne 0$ I am trying to prove this as follows: If $v$ is an eigenvector of $A$, then $Av ...
0
votes
1answer
15 views

Finding this Probability Density Function

I would much appreciate if you help me out with this problem Let $X \sim Unif(0,1)$ Find the density of $Y = -\lambda^{-1} \log(1-X)$ with $\lambda > 0$ Then calculate $P(Y>t+s|Y>t)$ for ...
-1
votes
0answers
14 views

Find the population

Every year, the emigration rate from country A to B is 𝛼 (0 < 𝛼 < 1), whereas the emigration rate from country B to A is 𝛽 (0 < 𝛽 < 1). Note that the fluctuation in population of both ...
0
votes
0answers
9 views

Wave Operators: Cook

Problem Given Hilbert spaces $\mathcal{H}_0$ and $\mathcal{H}$. Consider Hamiltonians: $$H_\#:\mathcal{D}H_\#\to\mathcal{H}_\#:\quad H_\#=H_\#^*$$ Denote for shorthand: ...
-5
votes
3answers
29 views

The sum of the multiples of 2 and 17 under 767 [on hold]

What is the sum of the multiples of 2 or 17 under 767
0
votes
0answers
22 views

Why does symmetry happen in reset-based random walks?

I am studying the basic concepts about random walks / brownian motion, and based on the idea of a Möbius-based walk in Wolfram's website, I wanted to try my own version of it in Python to compare it ...
1
vote
3answers
26 views

Show that the operator sequence $ A_n = 1/2(A_{n-1} + A^{-1}_{n-1})$ converges strongly, $A_0 = I+T$, where $T$ is compact and $||T|| \le 1/2$.

I'm studying for an analysis prelim and am stumped on an old exam problem for which there are no solutions given. The full question is as follows: Let $X$ denote a Hilbert space, and $T$ a ...
0
votes
2answers
18 views

Find isomorphism for an operation

I was trying to solve this problem, but am having trouble seeing why it is an isomorphism. To map from R* to G, I think that the phi function would be Phi(x)=x/2 but that doesn't work. This phi ...
1
vote
2answers
39 views

Proving a partial derivative identity

I'm currently studying for a resit and I've been faced with this partial differentiation question: If $z = f(y/x)$ show that $$x^2\frac{\partial^2 z}{\partial x^2}+2xy\frac{\partial^2 z}{\partial ...
1
vote
3answers
32 views

What does it mean to evaluate a polynomial at a linear map, e.g. $p(T)$ where $p(x)$ is a polynomial?

I'm learning about the Cayley-Hamilton Theorem and the Primary Decomposition Theorem, and have trouble understanding what the notations mean. What does $\chi_T(T)$ mean? ($\chi_T(x)$ is the ...
4
votes
5answers
198 views

Showing a function is injective using that $f'(x)\ne0$

Given a differentiable function $f\colon \mathbb R\to\mathbb R$ which we must prove to be injective, does it suffice to show $f'(x)≠0$ for all $x$ (for which the function is defined)? It makes sense, ...
3
votes
2answers
55 views

Prove for any set $E\subset R$ with lebesgue measure 1 there exists a subset with lebesgue measure 1/2.

Prove for any set $E\subset R$ with Lebesgue measure 1 there exists a subset with Lebesgue measure 1/2. It looks easy but I have tried for an hour and could not find a way to prove it. Can anyone ...
2
votes
3answers
203 views

Finding the roots of a different Quadratic equation from the roots of a Given Quadratic equation

The Question: If $\alpha$ and $\beta$ are the roots of the equation $ax^2+bx+c=0$... Then find the roots of the equation $ax^2-bx(x-1)+c(x-1)^2=0$ My Attempt: The new equation can be ...
0
votes
1answer
15 views

What is the summatory function of the number of (not necessarily distinct) prime factors?

In the Math World article on Merten's Constant, a related constant $B_2$ is mentioned which "appears in the summatory function of the number of (not necessarily distinct) prime factors." I am very ...
1
vote
0answers
25 views

Direct sum notation

I was reading the direct sum of the groups and the index notation looks little bit strange for me. Group $G = \bigoplus_{\alpha < \beta}\mathbb Zx_{\alpha},$ where $\beta$ be an ordinal. Is this ...
3
votes
2answers
118 views

Convergent by Ratio test?

I am lost with this problem: $$\sum_{n=1}^\infty \frac{n^n}{2^n n!}.$$ I am suppose to find if it is convergent or divergent. I have the correct set up. After cancelling everything I am left with ...
0
votes
3answers
34 views

Evaluating the closed integral of an elliptical path

I've been working on a problem that states: Evaluate $\int F*dr $ where $F(x,y,z) = x\,i+xy\,j+x^2yz\,k $ and C is the elliptical path given by $$ x^2+4y^2-8y+3=0 $$ in the xy-plane, traversed ...
4
votes
3answers
51 views

Square roots equations

I had to solve this problem: $$\sqrt{x} + \sqrt{x-36} = 2$$ So I rearranged the equation this way: $$\sqrt{x-36} = 2 - \sqrt{x}$$ Then I squared both sides to get: $$x-36 = 4 - 4\sqrt{x} + x$$ Then I ...
-1
votes
1answer
12 views

Area of a quarter circle C1 equals the area of an inner circle C2 where C2.diameter = C1.radius

Say we have a circle C1 with radius 2. Inside of that we draw circle C2 going from the centre point of C1 to the perimeter of C1 (making it diameter = 2) ...
2
votes
2answers
55 views

Show that $\frac{n}{n^2-3}$ converges

Hi I need help with this epsilon delta proof. The subtraction in the denominator as well as being left with $n$ in multiple places is causing problems.
2
votes
1answer
28 views

Does the $5x + 1$ sequence for 7 reach a power of 2 or does it get stuck in a period?

This is much like $3x + 1$, except that if $x$ is odd, you do $5x + 1$. If $x = 7$, then we have 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, 458, 229, ... I've iterated this twenty ...
9
votes
1answer
236 views

Sum of roots of cubic = -coefficient of quadratic term?

Working through Ian Stewart's "Galois Theory, Third Edition," he states at the end of the second paragraph on page 13: "Because we know that $\alpha_1+\alpha_2+\alpha_3$ is minus the coefficient of ...
1
vote
4answers
196 views

Is the following Alternating Series Absolutely Convergent?

$$\sum_{n=1}^\infty\frac{(-1)^n}{2n+1}$$ I think it is Absolutely Convergent because it converges by direct comparison to Harmonic series? Am I right or wrong?
2
votes
1answer
27 views

Jacobi Elliptic Functions built from Jacobi theta functions

I believe I understand the general theory of elliptic functions to an extent. What I can't seem to find is the distinct method which was used to show that a particular combination of Jacobi Theta ...
0
votes
2answers
33 views

State a reason the given function is not a homomorphism

$f:\Bbb R \rightarrow \Bbb R$ and $f(x)=\sqrt x$ For $\forall x\lt0\in\Bbb R$, $f(x)=\sqrt x\in\Bbb C\notin\Bbb R$ Does my answer make sense, or should I elaborate with words?
7
votes
4answers
72 views

Showing a function $f: \mathbb{N} \times\mathbb{N} \to \mathbb{N}$ is injective

Let $f: \mathbb{N} \times\mathbb{N} \to \mathbb{N}$ with $$ f(i,j) = \frac{(i+j-2)(i+j-1)}{2}+j. $$ I want to show $f$ is an injection. This is how I approached the problem: I tried to show ...
2
votes
1answer
32 views

Prove that the space of function $C^{k,\gamma}(\bar{\Omega})$ is a Banach space

Hi this is problem 1 in Chapter 5 Evans PDE. I have trouble showing the second part, i.e. completeness of $C^{k,\gamma}$. I know there is some previous posts but I did not quite get the answers. And I ...
0
votes
1answer
18 views

What can I assume, when given a matrix with information about its eigenvalues but not its action?

Basically, I've had to use linearity a couple of times yesterday and today, in order to write up a few proofs. But I notice that I am only given information such as positivity conditions and ...
2
votes
0answers
34 views

How could one invert this sum of Stirling numbers?

In this paper, under Stirling Numbers and their Asymptotics, the author takes equation (3.1): ...
17
votes
1answer
99 views

$f$ entire, $f$ satisfies $|f(x+iy)|\leq\frac{1}{|y|}$ for all $x,y\in\mathbb{R}$. Prove that $f\equiv 0$.

Let $f$ be an entire function. Suppose that $f$ satisfies $$ |f(x+iy)|\leq\frac{1}{|y|}. $$ for all $x,y\in\mathbb{R}$. Prove that $f$ is identically zero. I'm having some trouble with this, ...
3
votes
5answers
49 views

Why do we use base $e$ in population growth questions?

I know that we need base e to differentiate but I don't see what makes this formula work. $$ P = P_0 e^{rt} $$ where the 0 refers to initial population, $r$ the rate, and $t$ the time. Changing ...
0
votes
1answer
17 views

Show that an operator of rank $n$ can have at most $n$ nonzero eigenvalues.

Show that an operator of rank $n$ can have at most $n$ nonzero eigenvalues. I'm not sure how to proceed. I think induction will be the best. Any solutions or hints are greatly appreciated.
1
vote
1answer
12 views

Negative binomial distribution problem!

Every year for Halloween, it is estimated that $5$ out of every $8$ houses give away candy. (a) If we want to receive candy from $5$ different houses, what is the probability that we will need to ...
0
votes
0answers
7 views

Prove the properties of penalized likelihood estimator in Fan and Li (2001) paper

I'm reading through Fan and Li's paper "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties". In Page 2 near bottom right corner, they proposed three properties that a ...
3
votes
1answer
27 views

Find the inverse of the function given:

$$f(x)= \frac{5x}{(x − 2)}$$ My work: $$y=\frac{5x}{(x-2)}$$ $$x=\frac{5y}{(y-2)}$$ $$x(y-2)=5y$$ $$xy-2x=5y$$ $$\frac{xy-2x}{5}=y$$ $$f(x)=\frac{(xy-2x)}{5}$$ Any help is appreciated.
-1
votes
0answers
27 views

Hamiltonian: Compactness

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Denote the resolvent: $$R(z):=(z-H)^{-1}\in\mathcal{B}(\mathcal{H})$$ Denote compact ...

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