I am solving a certain kind of integral equations using iteration and Volterra series. Now I get a formal solution and in order to prove convergence I need to estimate the $L^1$ an …

I came across the following differential equation when considering some direct scattering problems: $$ N'_x(x,z)=G(x,z)N(x,z) $$ where $N(x,z)$ is a $2\times2$ complex matrix wit …

The claim that appears in the title of this question is mentioned in the paper "On Shelah's amalgamation" by Judah and Roslanowski. I'd really like to see a proof of this fact, but …

Let $f = \sum_n a_n q^n$ be a cuspidal newform of weight $k$ on $\Gamma_0(N)$ for some $N$. Let $K_f$ be the number field generated by the $a_q$ as $q$ runs over all primes. My q …

For a given set $S$ of complex square matrices $M1,M2\cdots,Mn$, one can obtain a matrix group $G$ generated by matrx multiplication. For any $i$, we can define a matrix space $Gi$ …

Any closed form for series like $$F(x)=\Sigma_{i=p}^{\infty}x^p,\text{p is prime}$$ or $$F(x)=\Sigma_{i=0}^{\infty}x^{i!}$$? More generally,we can obtain a power series from decim …

I wanted to introduce and develop the complex logarithm from scratch. As the result I've arrived a couple of months ago at the following identity after which the road to complex lo …

I'm working through Local Fields by Serre and am stumped by something that he thinks should be obvious. Let $A$ Be a complete D.V.R with uniformizer $\pi$ and $\overline{K}$ be i …

Given an even integer N, what is the minimum set of primes such that any even number $x \leq N$ can be expressed as the sum of two primes in the set? Goldbach's conjecture said Ev …

Suppose $X$ and $Y$ are two Brownian motions such that $|X|$ and $|Y|$ are independent. Then it is easy to show that $\langle X,Y \rangle =0$ using the Tanaka formula, for example, …

Let $$J_1=\int_0^1\frac{1}{\sqrt{1-t_2}}dt_2,$$ $$J_2=\int_0^1 \int_0^{t_2}\frac{1}{\sqrt{1-t_2}}(\frac{1}{\sqrt{1-t_3}}+\frac{1}{\sqrt{t_2-t_3}})dt_3dt_2,$$ $J_3=\int_0^1 \int_0^{ …

I'm searching for a proof that a finitely generated projective module over a Von Neumann Regular ring is free. I know that this result is true, because a friend of mine have proved …

I am an amateur mathematician, and certainly not a set theorist, but there seems to me to be an easy way around the reflexive paradox: Add to set theory the primitive $A(x,y)$, whi …

Hello, I am currently studying Extension 1 Mathematics. I missed two classes and I figured out that tomorrow I will have a quiz. Can you help me to solve this permutation and combi …

Fix a positive integer $d \geq 2$, and let $n,k$ be natural numbers with $k \leq n$. Let $b(n,k)$ denote the number of monomials of degree $kd-(n+1)$ in $n+1$ variables $x_0,\ldo …