1
vote
3answers
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 ...
2
votes
4answers
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 ...
1
vote
1answer
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: ...
3
votes
1answer
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 ...
2
votes
2answers
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 ...
3
votes
2answers
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. ...
4
votes
2answers
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 ...
2
votes
5answers
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 ...
4
votes
0answers
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. ...
2
votes
1answer
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 ...
1
vote
3answers
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 ...
1
vote
1answer
71 views

Is the co-domain of a Hilbert transform of a function the same as the function itself?

Let $f:\mathcal{D}\to\mathcal{D}$ be a function whose domain and co-domain are $\mathcal{D}$. Let $\hat{f}$ be its Hilbert transform, which is defined as $$\hat{f}(t)=\mathcal{H}(f(t))=\frac{1}{\pi} ...
7
votes
6answers
1k views

Why is $\infty^0$ indeterminate?

In a recent test question I was required to us L'Hopital's rule to evaluate: $$\lim_{x\to 0^+} x\ln{(e^{2x}-1)}$$ I assumed that anything multiplied by 0 would give an answer of 0. This turns out ...
0
votes
2answers
106 views

Touch up on Trig

I have forgotten a few things about trigonometry and angles. I have this trig equation, $\sin \theta = \frac{200\text{ dyn}}{224\text{ dyn}}$ What exactly are the steps of getting the angle, ...
2
votes
1answer
245 views

Filtrations and right continuity

Here is a question that has been posted on another forum (in french) that I couldn't answer and about which I'm really curious. So here it is, let's be given as a state space $\Omega=\{f\in ...
0
votes
0answers
55 views

Prove that single machine can reduce from 2-partition problem

I'm given the following problem: Prove that single machine \sum U_i (number of tardy job) with release date constraints problem can reduce from 2-partition problem ...
0
votes
1answer
64 views

“full coverage” in hyperspherical space

I'm working on a computer algorithm that considers the relationships between data points in a theoretical n-dimensional space. I am "looking" from the origin in all directions in a programmatic way ...
5
votes
0answers
45 views

How to construct certain cover given in Mumford's Abelian Varities book

In chapter I, appendix to section 2 of the book "abelian varieties" by Mumford, we consider a discrete group $G$ acting freely and discontinuously on a good topological space $X$ (i.e., $\forall x \in ...
1
vote
3answers
1k views

Smooth transition between two lines (2d)

I have function that is defined as $$ Y = \frac{1}{15} x \longrightarrow {\rm if}\qquad 0 \leq x \leq 30 $$ $$ Y = \frac{1}{70} x + \frac{11}{7} \longrightarrow {\rm if}\qquad x > 30 $$ The ...
1
vote
1answer
160 views

Why does $q \equiv (r-1)/2 \mod r$ mean that $2q \equiv -1 \mod r$?

In the paper Safe Prime Generation with a Combined Sieve by Michael J. Wiener, the author states: For any small odd prime $r$, we can eliminate candidates for $q$ that are congruent to $(r − ...
2
votes
1answer
63 views

Is such a cryptographic system possible?

Suppose you want to design a program that plays a card game (like Skat) for three players over the Internet. At the beginning of the game it is needed to deal the cards in a way, that each player gets ...
2
votes
1answer
396 views

quaternion representation of the rotation of a sphere into plane displacement

I do have a sphere of known radius which does have a coordinate frame rigidly attached to it. Let's call the coordinate frame attached to the sphere XYZs. The sphere can be rotated and displaced ...
1
vote
1answer
380 views

How can I solve a vector equation in Z2?

I have a equation with 256 variables * 256-dimensional vectors in $\mathbb{Z}_2 $: $$ x_1 \cdot \left(\begin{array}{c} 1\\ 1\\ 1\\ \vdots\\ 0 \end{array}\right) + x_2 \cdot ...
7
votes
2answers
2k views

Change of order of double limit of function sequence

The more general quesion is under what conditions the folloing equality will hold (all functions are real valued): $$\lim_{x \rightarrow a} \ \lim_{j \rightarrow \infty} f_j(x) = \lim_{j \rightarrow ...
3
votes
2answers
283 views

How to get 'rectangular size' of arbitrary circular sector?

Given a circular sector defined by sweeping from a 'start' to a 'stop' angle (see diagram below) and a radius, how do you compute the bounds of the rectangle that fits to the edges of the sector? ...
1
vote
2answers
736 views

Raising a square matrix to the k'th power: From real through complex to real again - how does the last step work?

I am reading Applied linear algebra: the decoupling principle by Lorenzo Adlai Sadun (btw very recommendable!) On page 69 it gives an example where a real, square matrix $A=[(a,-b),(b,a)]$ is raised ...
2
votes
2answers
680 views

How to get the cardinal direction from one location to another?

Given are two geo locations, each with latitude and longitude. One is the current location, the other is a target location. is there a formula for calculating the target's cardinal direction for 0 ...
6
votes
3answers
475 views

how to compute the Euler characters of a Grassmannian?

Let $G(n,m)$ be the Grassmannian of all n-dim subspaces of an m-dim vector space over $\mathbb{C}$. How to compute the Euler characters of $G(n,m)$? For example, $G(1, 2)$ is $\mathbb{C}P^1$ which is ...
1
vote
2answers
719 views

Non repeating random number generation with x(i+1) = x(i) + increment mod m

I have to generate millions of non-repeating random numbers and came across this equation: $x_{i+1} = x_i+c \space(mod \ m)$, where c and m are relative primes and $m \geq total\ to\ be\ generated$. ...
15
votes
2answers
524 views

What is Riemann-Roch in arithmetic all about?

I learn number theory recently and I could not understand what Riemann-Roch was all about in arithmetic; could someone give me a bit hint? What is the advantage of viewing all this stuff geometrically ...
39
votes
2answers
3k views

Compute $ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$

I need your help with evaluating this limit: $$ \lim_{n \to \infty }\underbrace{\sin \sin \dots\sin}_{\text{$n$ compositions}} n,$$ i.e. we apply the $\sin$ function $n$ times. Thank you.
4
votes
3answers
187 views

Anyone know if this integral has an analytic solution?

I've come across the following integral: $$\int_{-\pi}^{\pi}\left[\frac{1}{A-R \cos(2\theta-\phi)}\right]^{\frac{N-1}{2}}d\theta$$ I know how to approximate this integral using the Laplace method, ...
2
votes
1answer
128 views

Continuous functional calculus question

I have defined a continuous functional calculus on the bounded self-adjoint linear operators for functions continuous on the spectrum, and are defined on an interval. How do I deal with wanting to ...
2
votes
1answer
444 views

How to find the eigenvalues/eigenvectors of a non-triangular matrix?

http://en.wikipedia.org/wiki/Characteristic_polynomial#Characteristic_equation According to the above Wiki, the characteristic equation is very easy to solve if the matrix is a triangular matrix, or ...
6
votes
3answers
2k views

Secretary problem - why is the optimal solution optimal?

I have read about this problem: http://en.wikipedia.org/wiki/Secretary_problem But I want to see how it is proven that the "optimal" solution is indeed optimal. I understand how to prove that if the ...
10
votes
2answers
749 views

Algebraic proof of a trig matrix identity?

I'll put the question first, and then the background, because I'm not sure that the background is necessary to answer the question: I have a geometric proof, but is there an elegant algebraic proof ...
4
votes
1answer
159 views

Show the correctness of a logarithmic inequality

Let $p_1>p_2$ and $n_1>n_2$ be positive numbers. I want to show that, $$ \frac{\log \left(\frac{p_1}{n_1}+1\right)}{\log \left(\frac{p_2}{n_2}+1\right)}\leq \frac{\log ...
3
votes
2answers
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, ...
13
votes
4answers
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 ...
2
votes
3answers
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 ...
0
votes
1answer
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' ...
2
votes
2answers
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 ...
1
vote
2answers
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 ...
1
vote
4answers
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 $y$ ...
4
votes
2answers
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 ...
1
vote
1answer
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 ...
11
votes
1answer
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 ...
4
votes
1answer
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 ...
1
vote
1answer
461 views

Finitely generated ideal

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 ...
8
votes
1answer
127 views

Prove that a holonomic (p-recursive) difference equation returns only integral values

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 ...

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