# All Questions

343 views

### Diophantine equations: ternary forms

Thue proved that all Diophantine equations consisting of an irreducible binary form (cubic or higher) equal to a constant, i.e., $$c_nx^n+c_{n-1}x^{n-1}y+\cdots+c_oy^n=k$$ ($n,k$ fixed) have finitely ...
888 views

### Are all vectors matrices?

It's pretty clear to me that all vectors are matrices (either 1 x n or n x 1). But there is some discussion about this at work. We're wondering whether the expression "Vectors/Matrices" in a ...
103 views

### first partial derivative of $F(x,y)=\int_a^x f(t,y)dt$

According to the fundamental theorem of calculus, the first partial derivative is f(x,y). I'm wondering why I can't apply L'Hopital's rule in the following reasoning: ...
204 views

### Koszul algebra of a ring

I'm studying on Cohen-Macaulay Rings of Bruns-Herzog. Let $(R,\mathfrak{m},k)$ be a Noetherian local ring and $H_{\bullet}(R)$ its Koszul algebra. I found on the book (page 75) that "since ...
200 views

### Auslander-Buchsbaum and Ferrand-Vasconcelos

I'm studying on "Cohen-Macaulay rings" of Bruns-Herzog, here a link: http://books.google.co.in/books?id=ouCysVw20GAC&printsec=frontcover&hl=it#v=onepage&q&f=false At page 65 there is ...
112 views

### Simple algebra question

$$\dfrac{\qquad\dfrac{5p+10}{p^2-4}\qquad}{\dfrac{3p-6}{(p-2)^2}}$$ Im all confused about this question. Can someone go through it step by step. Can you please list when I can cancel numbers. ...
1k views

### Minimal free resolution

I'm studying on the book "Cohen-Macaulay rings" of Bruns-Herzog (Here's a link and an image of the page in question for those unable to use Google Books.) At page 17 it talks about minimal free ...
1k views

### Why is 33 1/3% of 240 = 79.992 wrong?

I'm embarrassed to ask this fundamental question among questions of particle filters but : my daughter just had this marked wrong on a test. The teacher said the answer was 1/3 of 240 = 80. On ...
139 views

### A question about an example on flat families from Hartshorne. In particular, is this local ring reduced?

Is the local ring $R_p$ reduced, where $p=(a,x,y,z)$ and $R=k[a,x,y,z]/I$ and $I=(a^2(x+1)-z^2,ax(x+1)-yz,xz-ay,y^2-x^2(x+1))$ ? This comes from example III.9.8.4 in Hartshorne's algebraic geometry. ...
201 views

### Help on a proof of a Theorem of Rees

I'm studying on this book http://books.google.co.in/books?id=ouCysVw20GAC&printsec=frontcover&hl=it#v=onepage&q&f=false on page 10 there is a Rees Theorem. I'd like to know why the ...
1k views

### Relation between the tangent to a curve and the first derivative of a function

There is a relation between the tangent to a curve of a function and the first derivative of that function. However, how do I show that connection? How can you explain it to someone so that it becomes ...
71 views

2k views

628 views

### [Dummit and Foote : Algebra] 2.1 Subgroups Question 10(b) Some help/hint please

So, the question goes like this "Prove that the intersection of an arbitrary nonempty collection of subgroups of $G$ is again a subgroup of $G$ (do not assume the collection is countable)." At first, ...
779 views

### Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$

How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and $$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$ for all real numbers $x$ and $y$ with $y\neq0$? PS. This is ...
163 views

### A notion of topology for computability

A topology on a space $X$ is defined as a subset of the power-set of X, that is closed under arbitrary unions, finite intersections and includes the empty set and the full space. Is anybody aware of ...
548 views

### Intuition behind using dot product to find component of vector in direction of another

I'm reading Chris Hecker's third article on rigid body dynamics http://chrishecker.com/Rigid_body_dynamics Quoting... "More importantly, if our collision detector supplies us with a 'normal vector' ...
179 views

### Formula to generate a score from 1 to 100 based on 2 percentages?

I am trying to come up with a formula that will result in a score of 1 to 100 (never anything lower or higher). I have two numbers that I can use to come up with this score, a specific percent and an ...
508 views

### Maximum number of mutually orthogonal latin square pairs (definition provided)

An $n\times n$ matrix is defined to be a "latin square" if each row and column is a permutation of the first $n$ natural numbers. Two squares of same order are orthogonal if the $n^2$ pairs ...
257 views

### Moving variables around

Sorry for the easy question in advance, I'm trying to remember my algebra and I've come across a problem I don't quite remember the answer to: I've gotten the original problem (trying to get the $y$ ...
212 views

### discontinuous optimization

I'm solving the following problem: $$\max_\rho \;\; \rho \; \min\left[\left( \frac{bn}{an-bm} \right)[(a-m)-\rho], \frac{b}{a}[a-(p+\rho)]\right]$$ where all constants and variables are defined ...
217 views

### MCMC Metropolis Hastings

Does anyone know a webpage or a document where I can find a practical example of implementation of the Metropolis-Hastings algorithm, with some thoughts about burn-in time and how to construct the ...
387 views

### elliptic generalizations of Euler's trick

So Euler employed the following identity $$\sin(z) = z \prod_{n=1}^{\infty} \left[1-\left(\frac{z}{n\pi}\right)^{2}\right]$$ to evaluate $\zeta(2n)$, for $n\in\mathbb{N}$ I'm curious if there's been ...
399 views

### Computing centralizers

How does GAP (or Magma, Maple, ...) compute the centralizer of an element in a group? What about the center of a group? I'm interested in semidirect products, so probably there are simpler answers ...
We say that an ideal $\alpha$ of $A$ is finitely generated if $\alpha =(x_1,\cdots,x_n)=\sum_{i=1}^{n} Ax_i$, i.e. finitely generated as an $A$-module. Then how we call if $\alpha$ is generated by ...
Consider the recurrence given by $(n+1)^2 a_{n+1} = (9n^2+9n+3)a_n-27n^2 a_{n-1}$ $a_0 = 1, a_1 = 3$. Clearly, $a_n$ is rational, but unexpectedly, the recurrence seems to output only integral ...