All Questions

123 views

Prove that gradient of $\operatorname{tr}(A \cdot B \cdot A^{T} \cdot C$) with respect to $A = C \cdot A \cdot B$ +$C^{T} \cdot A \cdot B^{T}$

I've been poking through SEE's Machine learning notes and am having difficulty proving the relationship: Gradient, $\nabla$, of the trace $\operatorname{tr}(A \cdot B \cdot A^{T} \cdot C$) with ...
107 views

150 views

Are waiting times always going to be exponentially distributed?

I'm studying for CAS/SOA Exam P/1 and a question I have here is: We have a portfolio of $20$ insurance policies. The number of claims per policy in a $3$-month period has a Poisson distribution ...
163 views

If a continuous real function is additive, then it is linear

I have to prove the following problem Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x + y) = f(x) + f(y),\ \forall x,y \in \mathbb{Z}$. Then $f$ is a linear ...
73 views

Show by example that this need not be true if we do not assume that the groups are finitely generated

Let $G, H,$ and $K$ be finitely generated abelian groups. If $G \times K \cong H \times K$, show that $G \cong H$. Show by example that this need not be true if we do not assume that the groups ...
377 views

Involutions and Abelian Groups

Suppose that $G$ is a finite group where at least three-fourths of the elements are involutions, i.e., $$|I(G)| \geq \frac{3}{4} |G|.$$ (Here, $I(G)$ denotes the set of all involutions of $G$, ...
2k views

Software to Plot 3D Vector Fields

I don't have programs like MAthematica or Maple that plot vector fields out of the box, the ones I use are Maxima and Scilab for simbolic/numeric, and none of them can easly plot 3d vector fields, so ...
367 views

Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$

The following is a historical question, but first some background: Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be ...
872 views

Mixed Strategy Nash Equilibrium of Rock Paper Scissors with 3 players?

It seems like most game theory tutorials focus on 2-player games and often algorithms for finding Nash equilibria break down with 3+ players. So here is a simple question: Is ...
68 views

806 views

What is gained by computing additional digits of $\pi$? [duplicate]

Possible Duplicate: Do We Need the Digits of $\pi$? Given that at 39 digits, we have enough of $\pi$ to calculate the volume of the known universe with (literally!) atomic precision, what ...
115 views

Modular form weight 0

Why is an entire modular form of weight 0 must be a constant? In particular, does a function defined on the upper half plane that is analytic everywhere, including i/infty, imply boundedness? Can we ...
65 views

On functions in $L^p$

I am trying to solve the following problem. It seems easy, but for some reasons I am confused by the hint. Suppose $0<p_0<p_1\leq\infty$. I need to find an example of functions $f$ on ...
154 views

how to show $f(1/n)$ is convergent?

Let $f:(0,\infty)\rightarrow \mathbb{R}$ be differentiable, $\lvert f'(x)\rvert<1 \forall x$. We need to show that $a_n=f(1/n)$ is convergent. Well, it just converges to $f(0)$ as ...
2k views

Is the difference between consecutive prime numbers always an even number?

If we look at the difference between consecutive prime numbers, $p \gt 2$, it always appears to be an even number. For example, here are the seven consecutive primes starting at the $10^{10th}$ ...
65 views

hopf bifurcation for an ode

I understand how to analyse a system of equations like $x'(t) = f(x,y)$ $y'(t) = g(x,y)$ set $x'$ and $y'$ to zero and find the fixed points etc, and find the stability. What Im am not sure of ...
110 views

Hello :) i want to give a answer op the following question: For which prime number $p$ can we give a solution of the diophantic equation given by $x^2-65y^2=p$. I want to solve the question without ...
112 views

215 views

349 views

Let $\mathrm{M}_{d\times d}\left(\mathbb{C}\right)$ denote the set of all $d\times d$-matrices with complex entries. My goal is to show that the set $\mathcal{M}:= \left\{ \rho\in \mathrm{M}_{d\times ... 2answers 10k views Trapezoid area proof by dividing it into two triangles? I am trying to figure out how the formula for the area of a trapezoid with exactly two parallel sides is deduced. In my textbook it says that the formula for the area of a trapezoid is deduced by ... 1answer 115 views Domain of bijectivity of function$f:\mathbb{C}\rightarrow\mathbb{C}$There is a type of problems in my course in Complex analysis that I don't fully understand them. Given function$f:\mathbb{C}\rightarrow\mathbb{C}$,$f(z)=z^2$. You must specify the analytic and ... 1answer 127 views Which of the following improper integrals are convergent? Which of the following improper integrals are convergent? a.$\quad\displaystyle \int_{1}^{\infty} \frac{dx}{\sqrt{x^3+ 2x + 2}}$b.$\quad\displaystyle \int_{0}^{5}\frac{dx}{(x^2− 5x + 6)}$... 3answers 373 views Free group n contains subgroup of index 2 My problem is to show that any free group$F_{n}$has a normal subgroup of index 2. I know that any subgroup of index 2 is normal. But how do I find a subgroup of index 2? The subgroup needs to have ... 5answers 550 views Successful approaches to the modelization of ''randomness'' If you pick a number$x$randomly from$[0,100]$, we would naturally say that the probability of$x>50$is$1/2\$, right? This is because we assumed that randomly meant that the experiment was to ...

15 30 50 per page