5
votes
2answers
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 ...
1
vote
1answer
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 ...
2
votes
1answer
76 views

Weak convergence in $L^1$ toward initial data of a pde

Let $\varphi:[0,1]\to\mathbb{R}$ be nondecreasing and continuous, and let $s\in L^\infty(\Omega\times[0,T],[0,1])$ be such that $\varphi(s)\in L^1(0,T;H^1(\Omega))$ be a solution of $$ \partial_t s ...
0
votes
2answers
27 views

Need to Rearrange for P

$$E=2.5\cdot P\cdot V\cdot \left(1-\left(\frac xP\right)^{.286}\right)$$ Missed a variable last time. Still can not isolate for P. Last time the formula was edited incorrectly.
2
votes
0answers
126 views

Pseudo-inverse of an underdetermined Toeplitz matrix

I have an undetermined Toeplitz matrix (more columns than rows). For example: \begin{equation*} T = \begin{pmatrix} 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 ...
0
votes
1answer
82 views

Is this equation separable?

I have the equation: $dy/dx=4y^2-3y+1$ From what I can tell, this is NOT separable because there is no g(x) on the right side. Does that look right? It doesn't factor either. Thanks!
2
votes
1answer
44 views

Simple inequation

for some values $\lambda_1, \ldots, \lambda_{n-1} \in \left[-1, 1\right] \subset \mathbb{R}$ and a constant $\delta \in \mathbb{R}$ with $\delta \geq 0$ the following properties hold: $ 1 \geq ...
0
votes
3answers
67 views

Is the iterative method suitable??

We have the equation $g(x)=x^{2}-3x-4=0$ that has the roots $-1$ and $4$ and we are looking for a suitable iterative method $x_{n+1}=\varphi(x_{n}),n=0,1,2$ so that the sequence $(x_{n})$ converges ...
2
votes
3answers
211 views

The (ε, δ) definition of limit , is it essential ? What is its importance ? and the difference between it and normal limit calculation?

As the title says , why should we learn (ε, δ) definition of limit ? is it essential ? What is its importance ? and the difference between it and normal limit evaluation?
0
votes
0answers
67 views

Proving that a cyclic permutation has subgroups

This is my own work, I'm "self-learning" and trying to give Algebra some much needed attention. I've defined a permutation as follows: Let X be the set {1,2,3,...,n} Let f be a permutation on X. ...
0
votes
1answer
60 views

How is it possible that a p-series converges when p>1

I have just studied p-series and I was taught that when p<=1 the series diverges and that when p>1 the series converges. It seems that both serieses converge only that when p>1 it happens faster. ...
57
votes
2answers
6k views

Can 18 consecutive integers be separated into two groups,such that their product is equal?

Can $18$ consecutive positive integers be separated into two groups, such that their product is equal? We cannot leave out any number and neither we can take any number more than once. My work: ...
5
votes
3answers
977 views

Compact operator with closed range has finite dimensional range

Let $X,Y$ be Banach Spaces, and let $T\in K(X,Y)$ be a compact operator from $X$ to $Y$. I have to prove that $T(X)$ is closed in Y if, and only if, $\dim(T(X))<\infty$. Can anybody help me with ...
3
votes
1answer
98 views

$\mathrm{Pol}_m(\mathbb{A})$ viewed as a relation pp-definable from $\mathbb{A}$

First let me recall some (abbreviated, and possibly simplified to suit my situation) definitions: Let $A$ be a finite set and $\mathbb{A}$ some set of relations on $A$. Let $m, n$ be positive ...
2
votes
1answer
107 views

Conditional probability with independent conditions

Given two independent events A and B, what can we say about $P(X\mid A,B)$ ? Is the following correct ? $$P(X\mid A,B) = \frac{P(A,B\mid X)P(X)}{P(A,B)}\tag{Bayes}$$ $$P(X\mid A,B) = \frac{P(A\mid ...
1
vote
2answers
79 views

Complete System Residue problem

I have to proof that given a complete system residue modulo $k$ ( $k$ is prime ) { $a_1, a_2, a_3, \ldots a_k$ } that, for every integer $n$ there exists s such that: $n \equiv \sum\limits_{i=0}^s ...
8
votes
3answers
433 views

$\exp(A+B)$ and Baker-Campbell-Hausdorff

A few years ago, I did research in quantum mechanics, specifically dealing with generalized displacement operators. In such musings, BCH lights (or gets in, depending on your viewpoint) the way. A ...
4
votes
2answers
92 views

Is this a helpful way of thinking about modular arithmetic?

Could one consider the structure of $\mathbb{Z_{n}}$ under addition and multiplication to simply be that of the ordinary integers, with one additional axiom - namely that $n=0$? So we would have ...
0
votes
2answers
113 views

Proof that $2\sin(1.5x)\cos(1.5x)=3\sin(x)-4\sin^3(x)$

I was looking at the double angle formula which is: $2\sin(x)\cos(x)$ and the triple angle formula which is: $3\sin(x)-4\sin^3(x)$ I was wondering if there is a "easy" proof, something that a pre-calc ...
1
vote
1answer
519 views

How does the Jacobian relate 3D to 2D?

May be it is simple, but I'm on Google for hours without finding a clue. I'm reading an article in computer vision where the optical flow equation is $\nabla I\cdot v + {dI \over dt} = 0 $ and for the ...
2
votes
1answer
94 views

Can one visualise the dual groups to Cantor groups?

My question is very simple, and, probably an answer can be found in any harmonic analysis textbook, but it seems I have failed that task. It occurred to me that I don't understand the structure of the ...
0
votes
1answer
251 views

Little-Oh and Big-Oh to prove property for function product

I have to prove that: If $f = O(|x-x_0|^k)$ and $g = o(|x-x_0|^j)$ , then $f.g(x) = O(|x-x_0|^{k+j})$, where $f.g$ means the product of functions $f$ and $g$. Also, notice the difference between ...
1
vote
2answers
100 views

Can you say that $\mathbb{R}$ is Dense in $\mathbb{R}$?

Denote the set of irrationals by $\mathbb{I}$. We know that $\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$, and we know that $\mathbb{Q}$ is dense in $\mathbb{R}$, and $\mathbb{I}$ is dense in ...
3
votes
1answer
228 views

The proof that every bounded linear operator generates an unique uniformly continuous semigroup.

Let $X$ be a Banach space and $A: X \to X$ a bounded linear operator. So, $A$ is the infinitesimal generator of an uniformly continuous semigroup $\{T(t)\}_{t\geq 0}$ on $X$. The proof, as presented ...
4
votes
1answer
93 views

How general are $\mathrm{Aut}$ groups and $\mathrm{End}$ rings?

For any locally small category $X$ and object $A\in X$ the set $\mathrm{Aut}_X(A)$ is a group w.r.t. composition $\circ$. For any locally small abelian category $X$ and object $A\in X$ the set ...
2
votes
2answers
1k views

Any bipartite graph has a matching that covers each vertex of maximum degree

I need to prove following lemma: Any bipartite graph has a matching that covers each vertex of maximum degree Any help will be appreciated.
0
votes
1answer
223 views

What does “Ord” mean

I have a question as follows: Solve for x: Ord29(x)=7. I have never seen Ord before I have an exam and something like this will come up so can someone just tell me how I would go about doing this?
0
votes
1answer
25 views

How do i prove that a $m(TE)=0$ and $TE$ is measurable when $T$ is a singular operator on $\mathbb{R}^n$?

Let $m$ be the $n$-dimensional Lebesgue measure and $m^*$ denote the Lebesgue outer measure. Let $T$ be a linear operator on $\mathbb{R}^n$ such that $\text{rank}(T)<n$ and $E\subset ...
1
vote
1answer
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 ...
1
vote
3answers
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 ...
1
vote
2answers
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 ...
-2
votes
2answers
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 ...
7
votes
1answer
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 ...
1
vote
1answer
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 ...
0
votes
2answers
59 views

Euclidean lemma proof [duplicate]

According to Euclidean lemma it is defined that if $p$ is prime then $$p|ab\Rightarrow p|a\lor p|b$$ How to prove by descending induction that if $$p|a^n \Rightarrow p|a $$ knowing that $a^n = a ...
0
votes
2answers
97 views

The space $\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}$

For a open Lipschitz domain $\Omega$, consider the space $$A =\{ u \in H^1(\Omega) \mid \Delta u \in L^2(\Omega)\}.$$ Now I heard somewhere that all the second derivatives of a function $u$ are ...
5
votes
1answer
273 views

how to show that integral depending on a parameter are continuously differentiable

I'm trying to solve this exercise Let $f:[0,1]\to \mathbb{R} \space$ an integrable function, show: $$g(z):=\int_{0}^1\frac{f(x)}{x-z}dx$$ is a continuous differentiable function on $\mathbb R ...
3
votes
1answer
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 ...
3
votes
1answer
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 ...
0
votes
2answers
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 ...
0
votes
1answer
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 ...
2
votes
1answer
390 views

Volume of an n-simplex (Without Probabilities) [duplicate]

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 $$ ...
4
votes
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, ...
0
votes
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 ...
2
votes
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?
0
votes
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. ...
3
votes
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 ...
0
votes
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 ...
0
votes
1answer
89 views

Hyphothesis test for a coin tossed 10000 times

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 ...
0
votes
2answers
48 views

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 $p|a$

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