instruction stringlengths 12 30k |
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It is a local constant sheaf, but why the local system on topological space $X$ is one-to-one correspondes to $\tilde{X}\times_G V$, where $G$is fundamental group of $X$ and $\tilde{X}$ is the universal space of $X$, and $V$ is a $G$-module, I am wondering why $\tilde{X}\times_G V$ is a local system (local constant sh... |
I have a helix in parametric equations that wraps around the Z axis and a point in space. I want to determine the shortest distance between this helix and the point, how would i go about doing that?
I've tried using Pythagorean theorem to get the distance and then taking the derivative of the distance function to fi... |
You are on the right track by applying: $\rm\ A/I\ $ is a domain iff $\rm\ I\ $ is prime. For $\rm\ A = R[x],\ \ I = (x)\ $ the problem reduces to showing that $\rm R[x]/(x)\cong R\:.\:$ As you surmise, this is easily achieved by exhibiting an evaluation map with kernel $\rm\ (x)\:,\:$ i.e. an evaluation which maps $\r... |
I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\frac{\pi}{2}.$$
What do you think?
It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} dx$$ is also $\frac{\pi}{2}.$ |
You are on the right track by applying: $\rm\ A/I\ $ is a domain iff $\rm\ I\ $ is prime. For $\rm\ A = R[x],\ \ I = (x)\ $ the problem reduces to showing that $\rm R[x]/(x)\cong R\:.\:$ As you surmise, this is easily achieved by exhibiting an evaluation map with kernel $\rm\ (x)\:,\:$ i.e. an evaluation which maps $\r... |
Let $X$ be a Polish space. Let $d,d'$ be two complete compatible metrics. Is $1:(X,d)\to(X,d')$ uniformly continuous.
What I really want to know is: can we unambiguously speak of $f:X\to K$ from one Polish space to a compact Polish space as being uniformly continuous? Clearly it won't depend on the compatible metric... |
Is uniform continuity well defined on Polish spaces? |
I am looking for a short proof that $$\int_0^\infty \left(\frac{\sin x}{x}\right)^2 dx=\frac{\pi}{2}.$$
What do you think?
It is kind of amazing that $$\int_0^\infty \frac{\sin x}{x} dx$$ is also $\frac{\pi}{2}.$ Many proofs of this latter one are already in this [post][1]
[1]:http://math.stackexchange.com/quest... |
Hmm, yes that is. Also $X=[0,\infty)$ with $d(x,y)=|y-x|$ $d'(x,y)=|y^{2}-x^{2}|$ would work too. Thanks for that.
Okay then, scrapping that too general a question\ldots can my other concern be addressed, *i.e.* can we unambiguously call a map from a Polish space to a compact Polish space uniformly continuous withou... |
??????????????????????????????????? |
Given the following formula:
$$\bigg[\forall x P(x,x) \wedge \forall x \forall y \forall z\bigg( P(x,y) \wedge P(y,z) \Rightarrow P(x,z)\bigg) \wedge \forall x \forall y (P(x,y) \vee P(y,x)) \bigg] \Longrightarrow \exists y \forall x P(y,x)
$$
It is known that this formula is always true if x,y,z are in any domain... |
I have some problems understanding a question in a math class test for 10 year olds.
The question is:
>Someone pays in 200€ to a bank. The interest rate is 3.75%. Some time later, he looks at the account balance and he sees that there are 235€ now. How much time passed?
I would calculate this using logarithms,... |
I don't have enough reputation to post a comment to your question (I think), but it would seem to me that this follows from the last point. Hints: What special property does 1 have in V_0? Have you tried expressing an arbitrary element of V_0 in the basis given by integer translations of the scaling function? |
What does complete in algebra category mean?Thanks! |
the question is how to calculate for the general problem.
the specific problem is this.
Given a door code of n numbers on a 10 digit panel how many combinations do you have to try?
the catch is that the mechanism that senses whether the correct code has been entered has no concept of the start and end of the co... |
The following is a screenshot plot of comparison between
Exp[Derivative[1,0][Zeta][0,x]+1/2Log[2 Pi]] and Gamma function:
![alt text][1]
In theory the blue and the red graphs should be identical: the Gamma function. But in reality we have the blue plot representing the absolute value of the Gamma function instea... |
the question is how to calculate for the general problem.
the specific problem is this.
Given a door code of n numbers on a 10 digit panel how many combinations do you have to try?
the catch is that the mechanism that senses whether the correct code has been entered has no concept of the start and end of the co... |
**Problem:**
***In a school competition,the probability of hitting the target bu Dick is $\frac{1}{2}$,by Betty is $\frac{1}{3}$ and by Joe is $\frac{3}{5}$.If all of them fire independently,calculate the probability that the target will be hit.***
This general approach for solving is to **find the complement of ... |
Intersection of neighborhoods of 0. Subgroup? |
Is the formula A∨ (B∧C) = (A ∨ B) ∧ (A ∨ C) correct ? |
A friend of mine introduced me to the following question:
Does there exist a smooth function $f: \mathbb{R} \to \mathbb{R}$ ($f \in C^{\infty}$), such that $f$ maps rationals to rationals and irrationals to irrationals and is nonlinear?
He has been able to prove that such a polynomial (with degree at least 2) doesn... |
I'm having trouble understanding this proof:
> **Lemma:** Let $\lambda$ be the Lebesgue measure restricted to the Borel sets of $[0,1]$. Then there exists a decreasing sequence $(E_n)_{n\ge0}$ of subsets of $[0,1]$ such that each of them is $\lambda$-thick and $\bigcap E_n = \emptyset$.
> **Proof:** Let $E_n = [0... |
(**Disclaimer:** I'm an engineer)
Hi everybody, I found this “riddle” posted on the internet:
![Portal Riddle][1]
It's meant as a joke, but I do think it deserves an answer :)
A bit of background: the orange and blue ellipses are a citation from the videogame Portal. They are "portals" connected so that eve... |
Have I made a straight line, or a circle? |
As we saw [here][1], the minimum of two quantities can be written using elementary functions and the absolute value function.
$\min(a,b)=\frac{a+b}{2} - \frac{|a-b|}{2}$
There's even a nice intuitive explanation to go along with this: If we go to the point half way between two numbers, then going down by half the... |
Let $(X,d),(Y,d)$ be polish spaces and $Y$ compact. Consider the spaces $C_{u}(X,Y)$ of uniformly continuous functions, and $C(X,Y)$ of continuous functions. We can endow these with the uniform-metric.
Can $C(X,Y)$ be generated from $C_{u}(X,Y)$ using only point-wise limits of $\omega$-sequences? *I.e.* Define $\mat... |
I read that any function can be represented as a sum of convex and concave function, meaning for all f(x), f(x) = g(x) + h(x) where g is convex and h is concave.
There could be infinitely many decompositions of that sort.
Anyone knows where I can see a proof or this, or knows of a proof of this?
Thanks.
|
can any function be represented as a sum of convex and concave function? |
Since you wanted a different proof approach, you can try using [Abel's Identity][1], which has turned out to be quite useful in analytic number theory.
For instance see this: [An estimate for sum of reciprocals of square roots of prime numbers][2].
To apply this to your question:
Since we know that
$\disp... |
What is the best program to factor large arbitrary-form integers on a single computer, or on a few disjointed computers? "Best" is obviously subjective, but what do you recommend?
I'm working on a project to factor general-form numbers that I know are composite, with a scale of 100 - 1000 digits. I have a few comput... |
The question is how to calculate for the general problem.
The specific problem is this:
> Given a door code of $n$ numbers on a 10 digit panel, how many combinations do you have to try?
The catch is that the mechanism that senses whether the correct code has been entered has no concept of the start and end of ... |
Though I (slightly) disagree with Nate's characterization of geometers in his comments, I think there is actually some deep notions worth discussing contained in it.
The heart of the matter is, **how do you define a circle**, and **how do you define a line**? And in the apparent paradox of the original question, th... |
I believe theorem 1.2 [here][1]
[1]: http://books.google.com/books?id=ZhzXJHUgcRUC&pg=PA28 |
I read that any continuous function can be represented as a sum of convex and concave function, meaning for all f(x), f(x) = g(x) + h(x) where g is convex and h is concave.
There could be infinitely many decompositions of that sort.
Anyone knows where I can see a proof or this, or knows of a proof of this?
Tha... |
can any continuous function be represented as a sum of convex and concave function? |
I read that any continuous function can be represented as a sum of convex and concave function, meaning for all $f(x)$, $f(x) = g(x) + h(x)$ where $g$ is convex and $h$ is concave.
There could be infinitely many decompositions of that sort.
Anyone knows where I can see a proof or this, or knows of a proof of this... |
Lets say I take a computation that involves only addition and multiplication:
(a+b)*(c+d)
which can be done in many other ways, eg.
a*(c+d) + b*(c+d)
a*c + a*d + b*c + b*d
In terms of additions and multiplications the number of operations required for each of the three examples shown are (2,1) ... |
What is an Integral Domain? |
So I have this problem to solve...
----------
Let X denote the number of paint defects found in a square yard section of a car body painted by a robot.
These data are obtained:
8, 5, 0, 10, 0, 3, 1, 12, 2, 7, 9, 6
*Assume that X has a Poisson distribution with parameter λs.*
a) Find an unbiased esti... |
A $C^1$ function $f$ on a compact interval is convex (or is it concave?)
if and only if $f'$ is increasing and is concave (or is it convex?)
if and only if $f'$ is decreasing. If a function on a compact interval
is a sum of a convex and a concave function, each of which is $C^1$, then
$f'$ is the sum of an increasi... |
I believe the first theorem in article 1.2 [here][1]
[1]: http://books.google.com/books?id=ZhzXJHUgcRUC&pg=PA28 |
This is an answer to the question in one of your answers: is the Stone-Cech Compactification of a Polish space necessarily Polish?
The answer is a resounding no: not even close, in some sense. Let $X$ be a countably
infinite set equipped with the discrete metric (and thus the discrete topology). Its Stone-Cech c... |
Alright, so I got two points in 3d space, so they have a x,y, and z.
Now if the line's y - which I get like so:
Vector3 v = new Vector3();
v = a.subtract(b, v);
v.normalizeLocal();
float cosine = (float) v.dot(v);
float angle = (float) Math.toDegrees(Math.acos( cosine ));
Now I have... |
I have a camera looking at a computer monitor from varying angles. Since the camera is a grid of pixels, I can define the bounds of the monitor in the camera image as:
![alt text][1]
I hope that makes sense. What I want to do is come up with an algorithm to translate points within this shape to this:
![alt text]... |
Let $(X,M,\mu)$ be a measure space. Why are the following statements equivalent?
i) There exists a sequence of pairwise disjoint measurable sets $\{A_{n}\}$ in $M$ , each of finite measure, such that $\mu(A) = \sum_{n=1}^{\infty} \mu(A \cap A_{n})$ for every measurable set $A \in M$.
ii) $\mu$ is a countable sum ... |
Given the following expression:
$$(7y-1)(y-7) \le 0$$
To me this inequality implies $y \le 7$ and $y \le \frac{1}{7}$ but the correct expression (from my module) happens to be $\frac{1}{7} \le y \le 7$
Where exactly I am wrong? |
I want to find a nice simple expression for the definite integral
$$\int_0^\infty \frac{x^2\,dx}{(x^2-a^2)^2 + x^2}$$
Now, I can numerically compute this integral, and it seems to converge to $\pi/2$ for all real values of $a$. Is this integral actually always equal to $\pi/2$? How can I show this?
Also, why d... |
Is this definite integral really independent of a parameter? How can it be shown? |
Hint: If the product of two numbers is nonpostive, one of them must be nonpositive and one of them must be nonnegative.
Edit: $(7y-1)(y-7)\leq 0$. So either, $7y-1 \leq 0$ and $y-7 \geq 0$ or $7y-1 \geq 0$ and $y-7 \leq 0$. Now, can you eliminate one of these two cases? |
Well, my answer to 1. is that you can convienently do standard analysis in a chart. Did you ever try working on more complicated (singular) objects? It's a pain in the neck! (2. and 3. have already been answered) |
Let R be a commutative ring and I and J be two ideals of R, how to show that IJ= the intersection of I and J? |
By definition, the weak law states that for a specified large $n$, the average is likely to be near $\mu$. Thus, it leaves open the possibility that $|\bar{X_n}-\mu| \gt \eta$ happens an infinite number of times, although at infrequent intervals.
The strong law shows that this almost surely will not occur. In parti... |
Let R be a commutative ring and I and J be two ideals of R, how to show that IJ= the intersection of I and J if R=I+J? |
How would you expand the moment generating function (1-2*t)^(-n/2) as a power series in t??
What method would you be able to use? |
Is there solutions in $Q-Z$ to $X^2+2y^2=1$?
Im pretty sure that there isnt but Im not sure how to show this. I dont have much experience (yet) with P adics. |
Does there exist $X$, $Y \in Q-Z$ such that $X^2+2Y^2=1$? |
I came across an interesting problem in Jacod's probability book. But have no idea how to approach it. Should I approach it using induction? Any ideas?
Let $X_1, X_2, \cdots$ be an infinite sequence of iid sequence of integrable random variables
and let $N$ be a positive, integer-valued integrable random variable w... |
I need to find the two roots (x-intercepts) from the equation:
(x + 7)(x - 3) = (x + 7)(5 - x)
How would I go about this? I would greatly appreciate a step-by-step explanation. |
I need to find the two roots (x-intercepts) from the equation:
(x + 7)(x - 3) = (x + 7)(5 - x)
How would I go about this? **_I would greatly appreciate a step-by-step explanation._** |
The book I am studying uses the notation $w_k$ to denote the number of roots of unity contained in $K$, an imaginary quadratic number field.
Supposedly, this is the same as the size of the unit group in $O$, the ring of integers in $K$. I am well familiar with the standard arguments for finding the units in $O$.
... |
I need to find the two roots (x-intercepts) from the equation:
$$(x + 7)(x - 3) = (x + 7)(5 - x)$$
How would I go about this? **_I would greatly appreciate a step-by-step explanation._** |
Are there solutions in $\mathbb{Q}-\mathbb{Z}$ to $X^2+2y^2=1$?
Im pretty sure that there isnt but Im not sure how to show this. I dont have much experience (yet) with $p$-adics. |
Does there exist $X$, $Y \in\mathbb{Q}-\mathbb{Z}$ such that $X^2+2Y^2=1$? |
How would I solve $(x + 7)(x - 3) = (x + 7)(5 - x)$? |
I suppose a question to ask here is : what kinds of applications?
One of the areas where symplectic geometry is used is Hamiltonian dynamics, where the geometric framework allows one to generalize away from cotangent bundles, but this direction is directly related to the origins of the subject.
Another direction... |
Let $K$ be an imaginary quadratic field and $U$ denote the unit group in the ring of integers in $K$. Are there $\alpha \in K-U$ with finite multiplicative order? That is, is there $n \in N$ such that $\alpha^n=1$? |
Hint: If $x+7\neq 0$, your equation is equivalent to $x-3=5-x$, because you can divide both sides by a quantity different from zero. And what does happen to your equation when $x+7=0$? |
I want to solve this equation:
$y''+2y'-3y=0$
I did this:
$y' = z$
$y'' = z\dfrac{dz}{dy}$
$z\dfrac{dz}{dy}+2z-3y=0$
$zdz+2ydy-3ydy=0$
$zdz=(3y-2z)dy$
$z=3y-2z$
$z=y$
$y=y'=y''$
???
now, I'm pretty sure I did something wrong. could you please correct. |
how do you solve $y''+2y'-3y=0$? |
The product of two univariate generating functions is simply given by the Cauchy product.
$$ A(x) = \sum_{n=0} a_n x^n $$
$$ B(x) = \sum_{n=0} b_n x^n $$
$$ A(x)B(x) = C(x) = \sum_{n=0} x^n c_n $$
with $c_n = \sum_{k=0}^n a_k b_{n-k}$.
What is the resulting generating function for the bivariate case?
$$ ... |
$\displaystyle\sum_{n=0}^{\infty}\sqrt{n}(\sqrt{n^4+1}-n^2)$
Ok I was really hesitant to post this because it's such an elementary question, and I really should be able to do this. However, I've been looking at this for over half an hour and I've tried many methods, but I'm not able to get the series into a nice for... |
Let $\rho$ be a representation of $G$ on $V$. Why are its eigenvalues roots of unity? |
how to read a mathematical paper? |
I want to solve this equation:
$y''+2y'-3y=0$
I did this:
$y' = z$
$y'' = z\dfrac{dz}{dy}$
$z\dfrac{dz}{dy}+2z-3y=0$
$zdz+2zdy-3ydy=0$
$zdz=(3y-2z)dy$
$z=3y-2z$
$z=y$
$y=y'=y''$
???
now, I'm pretty sure I did something wrong. could you please correct. |
Given two random samples a and b of a normal distribution, how can I calculate the probability that their difference |a-b| is bigger (or lesser) than c? |
what is the formula for P(|a-b| > c) in a normal distribution? |
I seem to remember reading somewhere an "abstract nonsense" proof of the well-known fact that the fundamental group of a connected topological group is abelian. By "abstract nonsense" I mean that the proof used little more than the fact that topological groups are the group objects in the category of topological space... |
Sum[(i + 1) (n - i), {i, 0, n - 1}]
that is a sum of ( i+1)(n-1) with bounds from i=0 to n-1.
is that O(n^2) or O(n^3)? and can you explain me how you found it? thanks.
|
is this expression O(n^2) or O(n^3)? |
$$\sum_{i=0}^{n-1} (i+1)(n-1)$$
Is that $O(n^2)$ or $O(n^3)$? Can you explain me how you found it? Thanks.
|
A version of the Riesz Representation Theorem says that a continuous linear functional on the space of continuous real-valued mappings on a compact metric space, $C(X)$, can be identified with a signed Borel measure on the set $X$. Are there any similar results when we replace $C(X)$ by the space of continuous function... |
Is there a good explanation of a manifold on the web somewhere? The wikipedia article isn't really working for me. I was actually hoping for a whiteboard lecture on youtube, but can't find one. |
![diagram][1]
Given a circle, a point $H$ outside the circle, segments $\overline{HE}$ and $\overline{HT}$ tangent to the circle at $E$ and $T$, respectively, and points $I$ and $G$ on the circle such that $I$, $G$, and $H$ are collinear (all as shown above), knowing the measures of $\angle EHG$ and $\angle GHT$ (ca... |
I am having issues solving the following equation:
x/(2x-3) - 1/(2x) = 3/(4x-6)
The resolution of this is 1.
It is in the section of parabolas and it should be pretty easy to solve.
My steps are as follows:
x/(2x-3) - 1/(2x) = 3/(4x-6) multiply by 2x(4x-6)
2x^2 -1(4x-6) = 6x gets me here, remove 6x
... |
I am having issues solving the following equation:
$$\frac{x}{2x-3} - \frac{1}{2x} = \frac{3}{4x-6}$$
The resolution of this is 1.
It is in the section of parabolas and it should be pretty easy to solve.
My steps are as follows:
$$\frac{x}{2x-3} - \frac{1}{2x} = \frac{3}{4x-6}$$
Multiply by $2x(4x-6)$
... |
Are there solutions in $\mathbb{Q}-\mathbb{Z}$ to $X^2+2y^2=1$?
I'm pretty sure that there isnt but Im not sure how to show this. I don't have much experience (yet) with $p$-adics. |
Is there a good explanation of a manifold on the web somewhere? The wikipedia article isn't really working for me. I was actually hoping for a whiteboard lecture on youtube, but can't find one.
My math experience is calculus through differential equations, twenty five years ago. I also have some computer science ... |
ok .. so from what you guys say: that will be: n-1*Σ from 0 to n-1 of (n/2).
the inner bracket is O(n^2), and the outer is N, so this is the O(n^3) complexity
right? |
Consider the set $A$ of natural numbers which are of the form $k^3-p$, for $k$ a positive integer and $p$ a positive prime. Does $A$ have a density (of any of the usual kinds for sets of natural numbers) and if so, what is it?
$A$ contains a lot of numbers, and it seems somewhat difficult to me to prove that a parti... |
This isn't in any way a thorough or rigorous introduction to manifolds, but as far as visualization it's quite nice: Weeks' <a href="http://books.google.com/books?id=Lurp6nB4LtQC&printsec=frontcover&dq=shape+of+space&source=bl">The Shape of Space</a>. |
Here is an exercise, on analysis which i am stuck.
- How do i prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on $\mathbb{R}$? |
Here is an exercise, on analysis which i am stuck.
- How do I prove that if $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$, then the sequence $\{F_{n}(x)\}$ is boundedly convergent on $\mathbb{R}$? |
I generally have a "top down approach" to reading papers. More often than not, I am driven to a particular paper looking for a particular result. So I would start by looking at the statement of the result and try to see what background is needed in order to parse the statements which appear there. Once I have understoo... |
There is a problem in Griffiths Introduction to Electrodynamics as follows.
Check the divergence theorem for the function $\mathbf{v} = r^2\mathbf{\hat{r}}$, using as your volume, the sphere of radius R, centered at the origin?
Here is my calculation for the surface intergral part, please help me finding the erro... |
What's wrong with my calculation for checking the divergence law? |
There is a problem in Griffiths Introduction to Electrodynamics as follows.
Check the divergence theorem for the function $\mathbf{v} = r^2\mathbf{\hat{r}}$, using as your volume, the sphere of radius R, centered at the origin?
Here is my calculation for the surface intergral part, please help me finding the erro... |
Here is a problem in Griffiths Introduction to Electrodynamics as follows.
Check the divergence theorem for the function $\mathbf{v} = r^2\mathbf{\hat{r}}$, using as your volume, the sphere of radius R, centered at the origin?
Here is my calculation for the surface intergral part, please help me finding the error... |
I was Looking at another questions title, and given the tag of DG, I thought it would read a little more like this one. Or at least that answers to this question would be answers to that question.
There are many different techniques one might use to show a manifold is simply conencted, I am only interested in a spec... |
How to prove a manifold is simply connected?... using geometry |
Continuous and 1-1 $\Longrightarrow$ Borel Sets $\longrightarrow$ Borel sets? |
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