# All Questions

158 views

### Cute coloring problem on a board

Suppose we color an $n\times n$ square board using $n$ colors exactly $n$ times each. Prove that there is either a column or a row containing at least $\lceil \sqrt n \rceil$ different colors. A ...
190 views

### Union of Chain of Ideals

I'm writing a project in a "Rings and Modules" course, and I've come across the following proposition, stated without proof: Proposition 1.2. In a commutative ring R , the product of ideals is ...
76 views

79 views

42 views

### function with two parameters

We have function $f(x)=x^{2014}-ax^2-bx$ determine how many roots have this function. We have for sure one root $x=0$ becouse we can rewrite $f(x)=x(x^{2013}-ax-b)$ but how I can determine the ...
758 views

### Why doesn't this work in Geogebra

I've got a really simple equation that I want GeoGebra to plot: $\sqrt {2x}-\sqrt {3y} =2$ It says it's an illegal operation so I try: $3y=2x-4\sqrt{2x}+4$ When this doesn't try, I try changing ...
62 views

### How to prove this splitting field isomorphy criterium?

Given two irreducible polynomials $f(x) = x^2+a, g(x) = x^2+b \in \Bbb Q [x]$, the task is to prove that splitting fields of $f$ and $g$ ($\Bbb Q[\sqrt{-a}]$ and $\Bbb Q[\sqrt{-b}]$ actually) are ...
205 views

### define a cost function

I would like to define a cost function that penalies when the amount of a variable is out of a base. I mean assume that the value of $x$ should be $a\le x\le b$ now how can I define a single cost ...
171 views

### Rings that are isomorphic to the endomorphism ring of their additive group.

Every ring is isomorphic to a subring of the endomorphism ring of it's underlying group. That's Cayley's theorem for rings. What can we say about rings that are isomorphic to the endomorphism ring of ...
42 views

### notation for equations solving

I have some notation questions on the following equation solving: 2+x=5$\iff$x=5-2$\iff$x=3 Would you read the above as "two plus x equals five if and only if x equals five minus two if and only if ...
59 views

284 views

### Text on Probability Theory applied to Actuarial Science

I am a senior undergraduate who has passed the first three actuarial exams on probability (P), financial mathematics (FM), and models for financial economics (MFE). I am working on passing the life ...
152 views

### Newton-Raphson's method

Hello MathExchange community ! I am working on some "simple" numerical methods to solve 4th degrees and below equations. To make it easier I am working on the $[0, 1]$ interval and I know for sure ...
235 views

### Chess Piece Combinations

I came across a question yesterday about combinations, and I wanted to know what the correct answer was. The question states as follows: There are 8 spaces that are alternately black and white. There ...
242 views

### Finding intersection with newton's method for $\cos(x) = 2x$

I have spent the last 30 minutes to figure out what I am doing wrong. Maybe someone can spot the error: I have to find the intersection using newton's method for $$\cos(x) = 2x$$ Newton's Method ...
390 views

Compute the volume of $$S_n=\{(x_1,x_2,...,x_n)\in\mathbb{R^n},x_i\geq 0,\displaystyle\sum_{k=0}^{n} x_i<1\}$$ I don't really have an idea how to solve it. My 'work': Perhaps I could use $$... 1answer 102 views ### \mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q}) ?? Let p,q be primes, p≠q, then I have to show that \mathbb{Q}(\sqrt{p},\sqrt[3]{q})= \mathbb{Q}(\sqrt{p}\cdot \sqrt[3]{q}) So far I've tried a lot of things with minimal polynomials and bases, ... 2answers 1k views ### Why can a 1 element set be a member of another set but not a subset of it? I have came across this in a textbook: \{2\}\nsubseteq\{\{2\}\} but \{2\}\in\{\{2\}\} I understand that \{2\} is an element (member) of the other set but considering \{2\} is a set ... 2answers 131 views ### when is a ring a free module over a subring? Let S \subset R be rings, S not necessarily an ideal of R, and S \neq R. Is there anything that can be said about when R is free as an S-module? 1answer 23 views ### Given point A(-4,2,3) and B(4,0,1) what conditions is the line: [x,y,z] = [4,0,1] + t[m,n,1] perpendicular to AB? Then determine a vector equation either in terms of m or n, of the line that satisfies the condition. Attempt: AB = [8,-2,-2] Therefore, the dot product of [8,-2,-2] and [m,n,1] must be zero. ... 3answers 72 views ### \mathbb{Z}_m is homomorphic image of \mathbb{Z}_n Doesn't this always work as long as n\geq m? Can't we get rid of the condition that n is a multiple of m? If n is a multiple of m, show that \mathbb{Z}_m is homomorphic image of ... 2answers 54 views ### Recurrence relation - Show that a sum of a sequence is zero We are given the following sequence: f(n)=4f(n-1)-5f(n-2), f(0)=f(1)=a where a is some value in \mathbb C. We are asked to show that$$\sum_{n=0}^{\infty}\frac{f(n)}{3^n}=0 First thing I ...
For a coin, there is no information whether it is fair or not. The following two hypothesis are supposed for getting tail : $H_0 : p = 0,5$ and $H_1 : p = 0,7$. This coin is tossed $10^4$ times and if ...
### Show that if a prime number $p|a^n$ then $p|a$ [duplicate]
The title says it all, how can I prove the following: Show that if a prime number $p|a^n$ then $p|a$