4
votes
1answer
111 views

Is there any relation between a group being unimodular and having equivalent uniform structures?

Recall: A topological group is said to have equivalent uniform structures if its left and right uniform structures coincide. A locally compact group is said to be unimodular if left Haar measures and ...
1
vote
3answers
2k views

Solving an integral when using a dummy variable

I'm trying to solve an integral and I don't remember how to when a variable is in the integral symbol, if that makes sense. I'm not sure what the correct terminology is. So day I have an integral ...
14
votes
1answer
872 views

Proving that the genus of a nonsingular plane curve is $\frac{(d-1)(d-2)}{2}$

I'm studying from Joseph Silverman's book The Arithmetic Of Elliptic Curves and I'm trying to do as many exercises as I can. Right now I'm trying to do Exercise 2.7 from chapter II which reads as ...
5
votes
2answers
217 views

Excision via simplicial sets

Let $X$ be a topological space, covered by a collection of open sets $\{U_\alpha\}$ (or more generally a cover by subspaces whose interiors cover $X$). Consider the singular simplicial set ...
2
votes
2answers
282 views

About Cardinality in Hausdorff Spaces

I have two problems: 1.- Let $X$ be a compact Hausdorff space, then $X$ has a basis with cardinality less than or equal to $|X|$. 2.- Let $X$ be a Hausdorff space and $D$ a dense subset in $X$, ...
6
votes
2answers
556 views

Cohomology ring of a product

I am trying to calculate $H^*(\mathbb{R}P^3 \times \mathbb{C}P^5,\mathbb{Z})$ as a cohomology ring. I know that $$H^*(\mathbb{R}P^3,\mathbb{Z}) = \frac{\mathbb{Z}[\alpha,\beta]}{(2 \alpha, ...
3
votes
3answers
116 views

Solving $ \left| \frac{-2x-6}{4} \right| \le 5$ for $x$

Say I have a statement like: $$ \left| \frac{-2x-6}{4} \right| \le 5. $$ And I want to find the closed interval form of $x$. i.e. I want to know what the maximum and minimum $x$ can be. How do I ...
2
votes
1answer
220 views

Kruskal and Prim

Can someone explain why Kruskal's algorithm and Prim's algorithm are each a special case of the general algorithm for minimum spanning trees? The general algorithm: Let A be the set of selected ...
1
vote
3answers
932 views

Isosceles trapezoid with semicircle

I have an isosceles trapezoid, with a semicircle in the middle. I need to know the difference in area of the two shapes. I Radius of the semicircle is 6cm, and the longest base is 14cm.
1
vote
2answers
688 views

Prime factorization rules

When you have a number like 81. Is it safe to assume that if the number can't be divided by 2 or 3 that it's prime if it ends with a 1?
0
votes
1answer
189 views

Linear versus non-linear integral equations

I'm having trouble solving an integral equation. It appears to me to be a homogeneous Fredholm equation of the second kind. However, I'm being told that this can't be a Fredholm equation, because it ...
3
votes
1answer
177 views

Weierstrass Equation and K3 Surfaces

Let $a_{i}(t) \in \mathbb{Z}[t]$. We shall denote these by $a_{i}$. The equation $y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}$ is the affine equation for the Weierstrass form of a ...
11
votes
4answers
351 views

Roadmap to SPDEs

I'm trying to learn about the Kushner-Stratonovich-Pardoux equations in filtering theory. I'm familiar with Itô calculus at the level of Øksendal's book (but struggle with much of Karatzas and ...
4
votes
1answer
227 views

Why is $a\equiv b \pmod n$ equivalent to the congruences $a\equiv b,b+n,b+2n,\dots,b+(c-1)n\pmod {cn}$?

I learned the following proposition (in which there is no proof) in a GRE math preparation book. I don't understand what it means and I am not able to find any theorem about this statement in Hardy's ...
1
vote
1answer
469 views

Why can't you integrate all power functions without a log function?

You need a logarithm function to solve all power functions. That's a fact. Power functions look like this: $f\colon x \mapsto a x^r \qquad a,r \in \mathbb{R}$ But why would you need a logarithm ...
12
votes
3answers
291 views

Differential equations that are also functional

I was toying with equations of the type $f(x+\alpha)=f'(x)$ where $f$ is a real function. For example if $\alpha=\frac{\pi}{2}$ then the solutions include the function $f_{\lambda,\mu}(x)=\lambda ...
4
votes
0answers
298 views

The cohomology of finite $G$-modules

This is to some extent a continuation of an earlier question of mine. Now that I'm all cleared up on what it means for a finite group to have periodic cohomology, I have another question; first I will ...
6
votes
2answers
499 views

Brouwer's Fan Theorem

I am not a mathematician, but I really would like to understand/know the following: What is so special about Brouwer's "Fan Theorem"? Is there an easy to undestand proof somewhere? Why was Brouwer ...
2
votes
1answer
316 views

Monodromy correspondence

Lately I've been studying monodromy and covering maps (in particular ramified covering mapos of Riemann surfaces), and I came across something I didn't fully understand. Let $V$ be a connected real ...
7
votes
2answers
294 views

Continuity of semigroups on $L^2$ and $L^1$: Is this simple proof correct?

Let $(X, \mu)$ be a $\sigma$-finite measure space, and $P_t$ a symmetric, Markovian, strongly continuous contraction semigroup on $L^2(X,\mu)$. (Markovian means that if $f \in L^2$ with $0 \le f \le ...
3
votes
2answers
129 views

How can I solve this intial value problem?

How can I solve this IVP, 1st order differntial equation. $$\frac{dy}{dt}= \frac {1}{e^y-t}$$ with initial value $y(1)=0?$ any help will be apperciated.
2
votes
3answers
283 views

Why Do Structured Sets Often Get Referred to Only by the Set?

Why do structured sets, like (N, +) often get referred to just by their set? Under this way of speaking, where N denotes the natural numbers, + addition, and * multiplication, (N, +, *) and (N, +) ...
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 ...

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