Intereting Posts

What is a formal polynomial?
Looking for examples of first countable, compact spaces which is not separable
Legendre polynomials, Laguerre polynomials: Basic concept
Today a student asked me $\int \ln (\sin x) \, dx.$
Special linear group as a submanifold of $M(n, \mathbb R)$
system of ode with non-constant coefficient matrix
Crossing the road
Showing that $\sum\limits_{n \text{ odd}}\frac{1}{n\sinh\pi n}=\frac{\ln 2}{8}$
Intuition of Addition Formula for Sine and Cosine
What is the surface by identifying antipodal points of a 2-torus embedded in $\mathbb{R}^3$?
possible signatures of bilinear form on subspaces
Prove that a function whose derivative is bounded is uniformly continuous.
Problem about right triangles.
Best approximation in a Hilbert space
Let$\ p_n$ be the$\ n$-th prime. Is$\ \lim_{n\to\infty} \log \log n \prod_{i=1}^{\lfloor \log n \rfloor} \frac{p_i-1}{p_i}>0$?

Prove that if $f\in \hbox{Hom}_{\mathbb{Z}}(\prod_{i=1}^{\infty }\mathbb{Z},\mathbb{Z})$ and $f\mid_{\bigoplus_{i=1}^{\infty } \mathbb{Z}}=0$ then $f=0$.

I took an element of $\prod_{i=1}^{\infty }\mathbb{Z}$, that is, $(m_{1},m_{2},…)$, and because $f$ is homomorphism we have $f(m_{1},m_{2},…)=f(m_{1},0,0,…)+f(0,m_{2},0,…)+\cdots$ and because $f$ is zero on $\bigoplus_{i=1}^{\infty } \mathbb{Z}$ so $f(m_{1},m_{2},…)=0$, but I am not sure that my solution is right. Please tell me if it is right or wrong, and if it is wrong, please help me to make it right. Thank you.

- Betti numbers and Fundamental theorem of finitely generated abelian groups
- Check if $(\mathbb Z_7, \odot)$ is an abelian group, issue in finding inverse element
- In a group we have $abc=cba$. Is it abelian?
- For which $n$, $G$ is abelian?
- Problem on abelian group
- If a group mod its commutator subgroup is cyclic, then the group is abelian?

- Sums and products of algebraic numbers
- Every group of order 203 with a normal subgroup of order 7 is abelian
- Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
- Are these two definitions of a semimodule basis equivalent?
- The center of a non-Abelian group of order 8
- Applications of the Jordan-Hölder Theorem.
- Showing associativity of (x*y) = (xy)/(x+y+1)
- Zero divisors in $A$
- Find an integer $n$ such that $\mathbb{Z}=\mathbb{Z}$.
- Let $F$ be a field and $f: \mathbb{Z} \to F$ be a ring epimorphism.

Let $f:\Bbb Z^{\Bbb N}\to\Bbb Z$ such that $f$ is zero on $\Bbb Z^{(\Bbb N)}$ (the direct sum of countable copies of $\Bbb Z$), and let $x=(x_n)_{n\ge 0}\in\Bbb Z^{\Bbb N}$. We want to prove that $f(x)=0$. Write $x_n=2^nu_n+3^nv_n$ with $u_n,v_n\in\Bbb Z$. (We can do this since $\gcd(2^n,3^n)=1$ for all $n\ge 0$.) Set $u=(2^nu_n)_{n\ge 0}$ and $v=(2^nv_n)_{n\ge 0}$. Then $u,v\in \Bbb Z^{\Bbb N}$ and $f(x)=f(u)+f(v)$. But $$f(u)=f(u_0,2u_1,\dots,2^{n-1}u_{n-1},0,\dots)+f(0,\dots,0,2^nu_n,\dots).$$ Since $f$ is zero on $\Bbb Z^{(\Bbb N)}$ we get $f(u_0,2u_1,\dots,2^{n-1}u_{n-1},0,\dots)=0$, so $f(u)=f(0,\dots,0,2^nu_n,\dots)$. But $f(0,\dots,0,2^nu_n,\dots)=2^nf(0,\dots,0,u_n,2u_{n+1},\dots)$ and therefore $2^n\mid f(u)$ for all $n\ge 0$, so $f(u)=0$. Analogously $f(v)=0$.

- A differentiable manifold of class $\mathcal{C}^{r}$ tangent to $E^{\pm}$ and representable as graph
- How to find nth moment?
- Interpreting Line Integrals with Respect to $x$ or $y$
- Error estimation for spline interpolation
- Prove that a complex valued polynomial over two variables has infinitely many zeroes
- $\lfloor a n\rfloor \lfloor b n\rfloor \lfloor c n\rfloor = \lfloor d n\rfloor \lfloor e n\rfloor \lfloor f n\rfloor$ for all $n$
- Why is the fixed field of this automorphism $\mathbb Q(\pi^2)$?
- Find the particular solution of $u_x+2u_y-4u=e^{x+y}$ satisfying the following side condition $u(x,-x) = x$
- When does gradient flow not converge?
- $\frac{\partial T}{\partial t} = \alpha \nabla ^2r$ for spherically symmetric problems
- Extension of bounded convex function to boundary
- Why isn't NP = coNP?
- Proving two entire functions are constant.
- Is it a new type of induction? (Infinitesimal induction) Is this even true?
- Limit of $\left(1-\frac{1}{n^2}\right)^n$