Intereting Posts

How do you find the solutions for $(4n+3)^2-48y^2=1$?
Example of uncomputable but definable number
Number of binary strings containing at least n consecutive 1
What is a degree $2$ covering of $S$ by a surface $S'$ of genus $3$?
Prove that $\sin(\sqrt{x})$ is not periodic, without using derivation
Show that $\sum\limits_{n=1}^{\infty}\frac{x}{1+n^2x^2}$ is not uniformly convergent in $$
Help explain a new theory on small sines
Some equivalent formulations of compactness of a metric space
Is $SL_n(\mathbb{R})$ a normal subgroup in $GL_n(\mathbb{R})$?
$\gcd(ab,c)$ equals $\gcd(a,c)$ times $\gcd( b, c)$
$\ell_p$ is Hilbert space if and only if $p=2$
The probability that two vectors are linearly independent.
A nice pattern for the regularized beta function $I(\alpha^2,\frac{1}{4},\frac{1}{2})=\frac{1}{2^n}\ $?
Calculate $\dim W+V$ and $W\cap V$
How to count the number of solutions for this expression modulo a prime number $p$?

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.

- Prove that the group $G$ is abelian if $a^2 b^2 = b^2 a^2$ and $a^3 b^3 = b^3 a^3$
- Quotient group $\mathbb Z^n/\ \text{im}(A)$
- Isomorphic abelian groups which are not isomorphic R-modules
- On the Definition of multiplication in an abelian group
- A nonsplit short exact sequence of abelian groups with $B \cong A \oplus C$
- If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$

- Prove that $f(x)$ is irreducible iff its reciprocal polynomial $f^*(x)$ is irreducible.
- A problem with tensor products
- What does it mean to represent a number in term of a $2\times2$ matrix?
- Irreducible polynomial over $\mathbb{Q}$ implies polynomial is irreducible over $\mathbb{Z}$
- Abelianization of free group is the free abelian group
- Proving that a ring is not a Principal Ideal Domain
- Fixed Field of Automorphisms of $k(x)$
- For $f\in\mathbb{Q}$, Gal($f)\subset S_n$ is a subset of $A_n$ iff $\Delta(f)$ is a square in $\mathbb{Q}^*$
- If $|a| = 12, |b| = 22$ and $\langle a \rangle\cap \langle b\rangle \ne e$, prove that $a^6 = b^{11}$
- Show group of order $4n + 2$ has a subgroup of index 2.

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 strange “pattern” in the continued fraction convergents of pi?
- Constructing a set with exactly three limit points
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- Necessary and sufficient conditions for differentiability.
- Maps in Opposite Categories
- Proof for exact differential equations shortcut?
- Finding $\binom{999}{0}-\binom{999}{2}+\binom{999}{4}-\binom{999}{6}+\cdots +\binom{999}{996}-\binom{999}{998}$
- Pointwise estimate for a sequence of mollified functions
- A characterisation of quadratic extensions contained in cyclic extensions of degree 4
- Can $R \times R$ be isomorphic to $R$ as rings?
- function whose limit does not exist but the integral of that function equals 1
- Are all equiangular odd polygons also equilateral?
- Units in quotient ring of $\mathbb Z$
- How would you explain why $e^{i\pi}+1=0$ to a middle school student?
- $T(n) = 4T({n/2}) + \theta(n\log{n})$ using Master Theorem