# All Questions

35 views

### Is this homomorphism in general surjective?

Let $R$ be a commutative ring and $I$ an ideal of $R$. Pick a fix $0 \not= a \in I$ and consider the map $\phi: R \to I$ given by $r \mapsto ra$. Is this map surjective?
25 views

### To test following system of linear equation for equivalency

Let F be field of complex numbers I have two system of equations $x_1 - x_2 =0$ $2x_1 + x_2 =0$ And $3x_1 + x_2 -0$ $x_1 + x_2 =0$ The definition says that each if equation in first system ...
82 views

119 views

### Proving set is dense

Let $A$ be a dense set of real numbers in $[0,1]$. I need to prove that $B=\{na : a \in A, n \in \mathbb{N}\}$ is dense in $[0,\infty)$. This is very intuitive but I fail to prove it. Any tips?
245 views

### To prove in a Group Left identity and left inverse implies right identity and right inverse

Let $G$ be a nonempty set closed under an associative product, which in addition satisfies : A. There exists an $e$ in $G$ such that $a \cdot e=a$ for all $a \in G$. B. Given $a \in G$, ...
152 views

### Determine if the following series is convergent or divergent:

$S=\sum_{k=1}^\infty (-1)^{k+1} \frac{k}{k^{2}+1}$ Now, I started by saying: consider, $\sum_{k=1}^\infty \left\lvert \frac{k}{k^{2}+1} \right\rvert$ , if this converges, that means S ...
65 views

### Paths and connectivity of graphs

I am trying to show that for a graph on $n\ge 3$ vertices with minimum degree of all vertices $\ge k/2$, G connected that G has a path of length k. I know if n is greater than k but n/2 is less than ...
47 views

### Logarithmic question

In the following question I fail to understand why the A option is correct. I understand that D is wrong, and that B and C are correct, but why is A correct? If $3^x=4^{x-1}$, then $x$cannot be ...
45 views

### For what value of $k$ is the vector field solenoidal

Problem: Let $\mathbf{r} = x \hat{i} + y \hat{j} + z \hat{k}$ be the position vector of a general point in $3$-space, and let $s= |\mathbf{r}|$ be the length of $\mathbf{r}$. For what value of the ...
178 views

### Prove that the set of all integers $>0$ is the smallest inductive set

An inductive set is a set $I$ such that $1 \in I$ and if $x \in I$ then $x+1 \in I.$ Some authors define the set of all integers $>0$ as the smallest inductive set, say Apostol's Analysis. But I ...
105 views

### Choosing new teammates

My sister gave me a combinatorical riddle. It doesn't appear to be hard, but I ask you if my thoughts are right, just for certainty. Here it is. Assume you belong to a group of $100$ people, and ...
55 views

### Non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i,$ for two consecutive integers.

Given an example of a non-abelian group $G$ satisfying $(a \cdot b)^i=a^i \cdot b^i, \forall a, b \in G$ for two consecutive integers. This is question 5 from Herstein Page 35. I have proved that ...
126 views

### Pseudo-Surreal numbers are analogous to?

I've been exploring surreal numbers. Real equivalent of the surreal number {0.5|} I see that pseudo-surreal numbers seem to have an interesting branch of game theory. Still having a form of {x|y}, ...
59 views

### Integration of Exponential and Logarithms, $\int_{z-1}^z \log(\frac{1}{z-y}) \exp (-| y| ^{3}) \, dy$

The integral I am dealing with is: $$\frac{3}{2 \Gamma \left(\frac{1}{3}\right)}\int_{z-1}^z \log \left(\frac{1}{z-y}\right) \exp \left(-\left| y\right| ^{3}\right) \, dy$$ where $z\in \mathbb{R}$ ...
15 views

### can we say in finite measure that $f_n \to f$ in measure iff every subsequence of $f_n$ coverge almost every where to f

Assume $(X, \mu )$ be finite measure space then can we say that $f_n \to f$ in measure iff every subsequence of $f_n$ coverge almost every where to f . My tries : I know a theorem that states ...
81 views

145 views

62 views

471 views

### How to prove that the exponential function is the inverse of the natural logarithm by power series definition alone

The exponential function has the well-known power series representation/definition: $e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$ And the natural logarithm has the less well-known power series ...
368 views

### Ball and urn method (counting problems)

How many ordered triples $(a, b, c)$ of positive integers exist with the property that $abc = 500$? Since, $500 = 2^2 5^3$ I believe this can be solved using Ball and Urn let $a = 2^{x_1}5^{y_1}$ ...
24 views

### On Procedure Constrained Matrix Factorization

Given rank $r$ $A\in\Bbb R^{n\times m}$, is there a procedure to find $XY=A$ such that $X\in\Bbb R^{n\times r}$, $Y\in\Bbb R^{r\times m}$ with property that $\max_{i,j}|x_{i,j}|$, $\max_{i,j}|y_{i,j}|$...
62 views

### Distribution equation (x-a)T=0

I have to solve the equation $(x-a)T=0$ , T is a distribution. By definition : $(x-a)\int T(x)\varnothing (x)=0$ I know if I pose $X=x-a$ I find $XT(X)=0$ and $T(X)=\delta(X)$. But I stuck to find ...
50 views

367 views

### What is the intuition behind / How can we interpret the eigenvalues and eigenvectors of Euclidean Distance Matrices?

Given a set of points $x_1,x_2,\dots,x_m$ in the euclidean space $\mathbb{R}^n$, we can form a $m\times m$ Euclidean Distance Matrix $D$ where $D_{ij}={\|x_i-x_j\|}^2$. We know a little bit about ...
48 views

### Cauchy sequence under a uniform continuous function

Let , $f:(1,4)\to \mathbb R$ be uniformly continuous and $\{a_n\}$ be a Cauchy sequence in $(1,2)$. Consider: $x_n=a_n^2f(a_n^2)$ and $y_n=\frac{1}{1+a_n^2}f(a_n^2)$ Then which is correct ?...
I am solving warm up problem 1.2 from Concrete Mathematics book. I've got the right answer by induction: $$f(0) = 0\\ f(n) = 3f(n-1) + 2,$$ But I can not figure how to simplify it to the closed ...
Let $G$ be a finie group, and $H$ be a core-free subgroup of $G$ (that is to say, there is no nontrivial normal subgroup of $G$ contained in $H$). Denote by $\Omega$ the set of right cosets of $H$ in \$...