instruction stringlengths 12 30k |
|---|
Cohomological decomposition of tensor sheaves? |
I'm reading [High Performance JavaScript](http://www.amazon.com/Performance-JavaScript-Faster-Application-Interfaces/dp/059680279X), and I think the graphs in one chapter are just plain wrong. [Here is one on Google Books](http://books.google.co.uk/books?id=ED6ph4WEIoQC&lpg=PA48&ots=2Twc2DWbgH&dq=high%20performance%20j... |
What does -1.13 times faster mean? |
Let $a_n$ be a sequence in $\mathbb{R}$, $\phi: \mathbb{N} \rightarrow \mathbb{N}$ a monotonically increasing function with $\phi (0)=0$. Show that
$\sum _{n=0}^{\infty } a_n$ converges $\Rightarrow \sum _{i=0}^{\infty } {(\sum _{j=\phi (i)}^{\phi (i+1)-1 }a_{j})}$ converges.
My notes: Can you use that ${(\sum _... |
I'm given the following exercise:
$\iint\limits_D \exp(x^{2}+y^{2})dA$
And I dont even know where to start, any chance someone could give me a hint? |
I'm reading [High Performance JavaScript](http://www.amazon.com/dp/059680279X), and I think the graphs in one chapter are just plain wrong. [Here is one on Google Books](http://books.google.com/books?id=ED6ph4WEIoQC&pg=PA41).
The y axis is "Times faster", and it runs from -1.5 to +4.0. Now, I would have thought that... |
I'm reading [High Performance JavaScript](http://www.amazon.com/dp/059680279X), and I think the graphs in one chapter are just plain wrong. [Here is one on Google Books](http://books.google.com/books?id=ED6ph4WEIoQC&pg=PA41).
The y axis is "Times faster", and it runs from -1.5 to +4.0. Now, I would have thought that... |
I'm given the following exercise:
$\iint\limits_D \exp(x^{2}+y^{2})dA$
And I dont even know where to start, any chance someone could give me a hint?
D is a half circle, given by:
$9\le x^{2}+y^{2}\le 16, y\ge 0$
Wheres A, I don't know what it is, its not given in the exercise. - Thats partly whats buggin... |
How can I compute the following limit?
$\displaystyle \lim_{a \to 0^+} \left(a \int_{1}^{\infty} e^{-ax}\cos \left(\frac{2\pi}{1+x^{2}} \right)\mathrm dx\right)$
Any hints you can please give?
Cheers |
-5 = 2x + 5
I want to "re-start" my mathematical education. I am needing it more and more at work. A co-worker asked me to break that down (he couldn't remember either).
Where should I start? What books?
Thanks! |
20 years since I was in high school. How do I break this down? |
First, what's $\lim_{a \to 0^+} \int_1^\infty ae^{-ax} \: dx$? Second, can you show that your integral is not so far from $\int_1^\infty ae^{-ax} \: dx$? |
What is the probability of getting 3 heads in a row? Would it be 1/8? |
There is a slight mistake in your argument: a priori, you don't know if the order $bN$ in $G/N$ is exactly $k$, but you do know that it is of order *dividing* $k$, hence relatively prime to $p$. In order to show that the order cannot be $1$, you need to use the fact that $N$ itself is a $p$-group (did you notice that y... |
"$-x$ times faster than" just means "$1/x$ times the speed of".
Suppose browser $A$ loads a page in 5 seconds which browser $B$ loads in half the time.
How many times faster is $B$ than $A$? Well, twice, of course. But what about the inverse? How many "times faster" is $B$ than $A$?
Well, first of all, it's no... |
Assuming a fair coin, how many sequences of three throws are there? Are they equally likely? How many of those are "3 heads in a row"? |
There is a slight mistake in your argument: a priori, you don't know if the order $bN$ in $G/N$ is exactly $k$, but you do know that it is of order *dividing* $k$, hence relatively prime to $p$. In order to show that the order cannot be $1$, you need to use the fact that $N$ itself is a $p$-group (did you notice that y... |
Wwhat are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some important consequences of a strong formulation of the axiom of choice?
[1]: http://math.stackexchange.com/questions/648... |
Motivating implications of the axiom of choice? |
What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some important consequences of a strong formulation of the axiom of choice?
[1]: http://math.stackexchange.com/questions/6489... |
a simple argument? |
It is proved in *Advanced Calculus* by Angus Taylor, § 20.8, that
$$\log n!=\log \left( \left( \frac{n}{e}\right) ^{n}\sqrt{2\pi n}\right)
+r_{n},$$
where
$$r_{n}=\sum_{k=1}^{\infty }S_{k}$$
with
$$S_{k}=\sum_{p=n+1}^{\infty }\frac{k}{2(k+1)(k+2)p^{k+1}}.$$
This formula for $r_{n}$ provides a method ... |
Starting with the famous Basel problem, Euler evaluated the Riemann zeta function for all even positive integers and the result is a compact expression involving Bernoulli numbers. However, the evaluation of the zeta function at odd positive integers (in terms of getting a closed form sum) is still open. There has been... |
There are many abuses like this in the world. I have seen things on sale that were 400% OFF!!! Would they give me 3x the original price if I took it? Maybe -1.13 times faster means that you send them 113% as many bits as you received in the base case. |
Given the following formula:
\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)) \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 with finite size,... |
Given the following formula:
$$\forall x P(x,x) \wedge \forall x \forall y \forall z\bigg[ P(x,y) \wedge P(y,z) \Rightarrow P(x,z)\bigg] \wedge \forall x \forall y (P(x,y) \vee P(y,x)) \Longrightarrow \exists y \forall x P(y,x)$$
It is known that this formula is always true if $x,y,z$ are in any domain with finite... |
Given the following formula:
$\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)) \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 with finite si... |
The premises tell you that $P$ is reflexive and transitive, as well as total; this means that it is a total pre-order on the domain. The conclusion is that the pre-order has a smallest element.
How would you go about doing it by induction? Well, suppose you know the result holds in any domain with fewer than $k$ ele... |
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 w... |
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 w... |
Motivated by my ignorance [here][1], if $X$ is a projective toric variety, is
$$H^m(X, \mathcal O_X) =
\begin{cases}
0 & m > 0 \\
1 & m = 1
\end{cases}
$$
as for $\mathbb P^n$?
[1]: http://math.stackexchange.com/questions/12767/cohomological-decomposition-of-tensor-sheaves/12774#12774 |
If $F_{1}$ is a subfield of $F_{2}$, is $GL(N,F_{1})$ contained in $GL(N,F_{2})$? I am particular interested in the case of when $F_{1} =\mathbb{Q}(\omega)$ and $F_{2} = \mathbb{C}$. Here $\omega$ is a complex root of unity. |
If $F_{1}$ is a subfield of $F_{2}$, is $GL(N,F_{1]}$ contained in $GL(N,F_{2})$? |
So one way to define a group presentation is to say, well, let's generate the free group with some number of generators, and then quotient by saying certain elements (relators) cancel just as $aa^{-1}$ does (and other things, you have to take the normal subgroup generated by the elements, but the basic idea works, I th... |
Do people study "ring presentations"? Is this a dumb question? |
If $X$ is a locally compact and metrizable space such that its Alexandroff compactification is _not_ first countable. Does this imply that no other compactification of $X$ can be first countable? Why? |
Motivated by my ignorance [here][1], if $X$ is a projective toric variety, is
$$H^m(X, \mathcal O_X) \cong
\begin{cases}
0 & m > 0 \\
\mathbb C & m = 1
\end{cases}
$$
as for $\mathbb P^n$?
[1]: http://math.stackexchange.com/questions/12767/cohomological-decomposition-of-tensor-sheaves/12774#12774 |
"$-x$ times faster than" just means "$1/x$ times the speed of".
Suppose browser $A$ finishes loading a given page in five seconds, while browser $B$ completes in half the time.
How many times faster is $B$ than $A$? Well, twice, of course. But what about the inverse? How many "times faster" is $B$ than $A$?
We... |
Suppose browser $A$ finishes loading a given page in five seconds, while browser $B$ completes in half the time.
How many times faster is $B$ than $A$? Well, twice, of course. But what about the inverse? How many "times faster" is $B$ than $A$?
One way to answer this is to assume that "$-x$ times faster than" jus... |
Strictly speaking, $-1.13$ times faster would mean that the program is doing things in reverse order. And Opera would completely stand still. Since that can't be what they really mean, it must be an abuse of language as has already been pointed out.
Could it be though that the original scale was logarithmic and tha... |
> There are ten people in a room:
>
> Person 1 knows 9 people. Person 2
> knows 8 people. etc. Person 9 knows 1
> person.
>
> How many people does person 10 know?
What is the answer and what is the principal behind this question? |
(Assume all unit speed)
Prove that an admissable curve c:(a,b) -> R^3
(Hence c' X c'' is never zero)
with a zero torsion is contained in a plane.
I understand why this should be true, if there is no torsion element to the curve then c can't twist into the third dimension.
T = c'
N = c''
B = T X... |
> There are ten people in a room:
>
> Person 1 knows 9 people. Person 2
> knows 8 people. etc. Person 9 knows 1
> person.
>
> How many people does person 10 know?
What is the answer and what is the principle behind this question? |
The following is a multiplication problem in traditional base ten notation. Each letter rep-resents a different digit. What digit does each letter represent? How did you get your foot in the door?
XY*YX=XY
YZ/WVY |
The following is a multiplication problem in traditional base ten notation. Each letter represents a different digit. What digit does each letter represent? How did you get your foot in the door?
XY*YX=XY
YZ/WVY |
> A newspaper launches a competition. It said that readers should submit one number between 1 and 1000. A £2000 prize would be awarded to the person that got the closest to 2/3 of the mean of all the entries.
What would a smart entry be? What is the underlying principle behind this type of question. I know it invol... |
The following is a multiplication problem in traditional base ten notation. Each letter represents a different digit. What digit does each letter represent? How did you get your foot in the door?
XY*YX=XY
YZ/WVY
![i hate this course][1]
[1]: https://i.stack.imgur.com/cXC4Y.png |
(Assume unit speed parametrization)
Prove that an admissible curve $c:(a,b) \to \mathbb R^3$ (hence $c^{\prime} \times c^{\prime\prime}$ is never zero) with a zero torsion is contained in a plane.
I understand why this should be true, if there is no torsion element to the curve then $c$ can't twist into the thi... |
(Assume a unit speed parametrization)
Prove that an admissible curve $c:(a,b) \to \mathbb R^3$ (hence $c^{\prime} \times c^{\prime\prime}$ is never zero) with a zero torsion is contained in a plane.
I understand why this should be true, if there is no torsion element to the curve then $c$ can't twist into the t... |
I'm currently looking at the equation for a multiclass classifier of the form
$$\sum\limits_{i=1}^{N_I} \sum\limits_{k=1,\atop k \neq y_i}^{N_K} L(1+ \mathbf{w_k}\cdot\mathbf{x_i}-\mathbf{w_{y_i}}\cdot\mathbf{x_i})$$
where <br/>
$X$ is a $N_K \times N_F$ data matrix<br/>
$y$ is a vector of class labels <br/>
... |
I have this question:
a ≡ 12^772 (mod 71) when 0 ≤ a < 71
and am having troubles getting it started.
772 / 71 is 10 with a remainder of 62;
How do I do this question? |
I have this question:
$a \equiv 12^{772} (\bmod{71})$, when $0 \leq a <71$
and I am having troubles getting it started.
$\frac{772}{71}$ is $10$ with a remainder of $62$;
How do I do this question? |
Is there an infinite set of strings whose kolmogorov complexities are computable? |
Is there an infinite set of strings whose kolmogorov complexities are computable? |
I have this question:
$a \equiv 12^{772} \pmod{71}$, when $0 \leq a < 71$
and I am having troubles getting it started.
$\frac{772}{71}$ is $10$ with a remainder of $62$;
How do I do this question? |
Is there an infinite set of strings whose Kolmogorov complexities are computable? |
Is there an infinite set of strings whose Kolmogorov complexities are computable? |
I'm not very expert in math, but I have a curiosity that could help me many times ..
Is there any way to find a function, even just similar, from a set of values?
example:
I have some values (x, y):
{(200, 11), (72, 5), (36.3) (28.3) (18.2) (12.2)}
what is the function that can return these values and al... |
I'm not very expert in math, but I have a curiosity that could help me many times ..
Is there any way to find a function, even just similar, from a set of values?
example:
I have some values (x, y):
{(200, 11), (72, 5), (36,3), (28,3), (18,2), (12,2), ...}
what is the function that can return these value... |
I'm not very expert in math, but I have a curiosity that could help me many times ..
Is there any way to find a function, even just similar, from a set of values?
example:
I have some values (x, y):
{(200, 11), (72, 5), (36,3), (28,3), (18,2), (12,2), ...}
what is the function that can return these value... |
I was just reading a proof of the dimension theorem in Steven Roman's Advanced Linear Algebra. In addressing the cases of infinite bases, Roman proceeds to show that if $\mathcal{B}$ and $\mathcal{C}$ are bases of a space $V$, then $|\mathcal{B}|\leq |\mathcal{C}|$, working up to an application BSC-Theorem. Anyway, he ... |
I was just reading a proof of the dimension theorem in Steven Roman's Advanced Linear Algebra. In addressing the cases of infinite bases, Roman proceeds to show that if $\mathcal{B}$ and $\mathcal{C}$ are bases of a space $V$, then $|\mathcal{B}|\leq |\mathcal{C}|$, working up to an application of the BSC-Theorem. Anyw... |
Suppose I have a function f that is infinitely differentiable on some interval I.
When I construct a taylor series P for it, using some point a in I, does f(x) = P(x) for all x in I?
I'm confused as to whether taylor series approximate functions on one point, or on an interval. |
On what interval does a taylor series approximate its function? |
On an infinite grid of ideal one-ohm resistors, what's the equivalant resistance between two nodes a knights move away?
<sub>*(ple... |
The premises tell you that $P$ is reflexive and transitive, as well as total; this means that it is a total pre-order on the domain. The conclusion is that the pre-order has a smallest element.
How would you go about doing it by induction? Well, suppose you know the result holds in any domain with fewer than $k$ ele... |
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... |
If $F_{1}$ is a subfield of $F_{2}$, is $GL(N,F_{1})$ contained in $GL(N,F_{2})$? |
Suppose I have a function f that is infinitely differentiable on some interval I.
When I construct a taylor series P for it, using some point a in I, does f(x) = P(x) for all x in I?
I'm confused as to whether taylor series approximate (or equal - I'm not about that either) functions on one point, or on an interv... |
On what interval does a taylor series approximate (or equal?) its function? |
The following is a multiplication problem in traditional base ten notation. Each letter represents a different digit. What digit does each letter represent? How did you get your foot in the door?
![i hate this course][1]
[1]: https://i.stack.imgur.com/cXC4Y.png |
I have been interested in learning formal math formally enough so that I could write a proof assistant using some simple parsing tools, and explain to someone else with little math knowledge how to write one. Naturally, it would be a very clunky and tedious assistant, but this is the standard I have set for myself for ... |
The usual statement of the Lebesgue Decomposition Theorem says that given two $\sigma$-finite measures $\mu$ and $\nu$ on a measure space, we can decompose $\nu = \nu_1 + \nu_2$, where $\nu_1$ is absolutely continuous with respect to $\mu$ and $\nu_2$ and $\mu$ are mutually singular.
Wikipedia ([link text][1]) says... |
I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious what types of techniques are used and just how difficult of a problem it is. Would this result be a reasonable capstone... |
How hard is the proof of $\pi$ or $e$ being transcendental? |
Suppose I have a function f that is infinitely differentiable on some interval I.
When I construct a Taylor series P for it, using some point a in I, does f(x) = P(x) for all x in I?
I'm confused as to whether Taylor series approximate (or equal - I'm not about that either) functions on one point, or on an interv... |
On what interval does a Taylor series approximate (or equal?) its function? |
I understand that $\pi$ and e are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious what types of techniques are used and just how difficult of a problem it is. Would this result be a reasonable capstone t... |
How hard is the proof of $\pi$ or e being transcendental? |
I understand that $\pi$ and e are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious what types of techniques are used and just how difficult of a problem it is. Would this result be a reasonable capstone t... |
this is a question I have on a step in the proof that's confusing me.
\sum_{n\eq1} ({\frac{4^{n}}{3^{n-1}}}) ---> 4(4/3)^(n-1)
with limit from n=1 --> infinity
please explain how/why this happened?
cheers,
gregg
|
this is a question I have on a step in the proof that's confusing me.
\begin{align}
\sum_{n\eq1} ({\frac{4^{n}}{3^{n-1}}}) \end{align} ---> $4(4/3)^(n-1)$
with limit from n=1 --> \infty
please explain how/why this happened?
cheers,
gregg
|
This is an answer elaborating on my comment above, and largely addressing the
edits to the original question:
I don't think there were any "old techniques" before Hermite proved
transcendence of $e$ in the early 1870s. As far as I know, the only transcendental numbers known before then were Liouville's interestin... |
this is a question I have on a step in the proof that's confusing me.
\begin{align}
\sum_{n\eq1} ({\frac{4^{n}}{3^{n-1}}}) \end{align} ---> $4\frac{4/3}^{n-1}$
with limit from n=1 --> $\infty$
please explain how/why this happened?
cheers,
gregg
|
this is a question I have on a step in the proof that's confusing me.
$$\sum_{n \eq 1} ({\frac{4^{n}}{3^{n-1}}}) ---> 4\frac{4}{3}^{n-1}$$
with limit from $n=1 --> \infty$
please explain how/why this happened?
cheers,
gregg
|
this is a question I have on a step in the proof that's confusing me.
\begin{align}
\sum_{n\eq1} ({\frac{4^{n}}{3^{n-1}}}) \end{align} ---> $4\frac{4}{3}^{n-1}$
with limit from n=1 --> $\infty$
please explain how/why this happened?
cheers,
gregg
|
this is a question I have on a step in the proof that's confusing me.
$\begin{align}
\sum_{n=1} ({\frac{4^{n}}{3^{n-1}}}) \end{align}$ ---> $4\frac{4}{3}^{n-1}$
with limit from n=1 --> $\infty$
please explain how/why this happened?
cheers,
gregg
|
I'm trying to figure out whether my hardware function for mapping operating system pages to DDR memory controllers is [injective][1]:
F(x) = a * x ( mod b ) | x in [0,b), a and b are co-prime
My hunch is that it is injective and I think it has to do with the [Chinese Remainder Theorem][2], though it may be so... |
When `a` and `b` are co-prime, is `F(x) = a * x ( mod b) | x in [0,b) , a and b are co-prime` an injective function? |
this step in the proof is confusing me:
$\sum_1^\infty {\frac{4^{n}}{3^{n-1}}}$ ---> $\sum_1^\infty 4(\frac{4}{3})^{n-1}$
please explain how/why this happened?
cheers,
gregg
|
I'm currently trying to do some Fourier transformations, or at least trying to understand them. The only thing I'm worried about is the complex part of the function. All I have is some basic, self thought, understanding about complex numbers.
As far as I'm concerned, a complex number exists of a real part and a comp... |
I am not getting this theorem:
> ***Angle between chord AB and tangent at A is the same as subtended by segment AB at any point on the circumference.***
How to prove this theorem? |
I'm currently trying to do some Fourier transformations, or at least trying to understand them. The only thing I'm worried about is the complex part of the function. All I have is some basic, self thought, understanding about complex numbers.
As far as I'm concerned, a complex number exists of a real part and a comp... |
This is an answer elaborating on my comment above, and largely addressing the
edits to the original question:
I don't think there were any "old techniques" before Hermite proved
transcendence of $e$ in the early 1870s. As far as I know, the only transcendental numbers known before then were Liouville's interestin... |
in functional analysis, you encounter the terms 'adjoint' and 'formal adjoint'.
What does 'formal' in that case mean? It Sounds like a hint that 'formal adjoints' lack a certain property to make them a 'true' adjoint.
I have nowhere found a definition, and would be eager to know. |
And what about the other direction? If $(u_{n})$ is increasing, what about $(a_{n})$?
I'm *guessing* the former is true since we know that if $(a_{n})$ converges, $(u_{n})$ converges to the same limit. That tells me that around infinity the sequences behave roughly the same. I tried proof by induction but got stuck. |
If $(a_{n})$ is increasing, is $u_{n}=\frac{a_{1}+\cdots+a_{n}}{n}$ increasing as well? |
Guys is my logic correct? If Earth travels around the Sun in 29.78 km/s and I have to calculate how long will it take to travel 1km I did following:
1s / 29.78 = 0.03358 (to 5 d.p)
I think it's ok but I would like to get confirmation on this.
Thank you. |
Are you talking about differential operators on functions over a domain in $\mathbb{R}^d$? (This is the context in which the phrase "formal adjoint" usually comes up.)
The idea is that working with, say, smooth functions with compact support, we have the integration by parts formula
$$ \int Du\cdot v ~dx + \int u... |
Are you talking about differential operators on functions over a domain in $\mathbb{R}^d$? (This is the context in which the phrase "formal adjoint" usually comes up.)
The idea is that working with, say, smooth functions with compact support, we have the integration by parts formula
$$ \int Du\cdot v ~dx + \int u... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.