instruction stringlengths 12 30k |
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which is a good introductory book book for Markov chains and Markov processes?
Thank you. |
Worth having a look at is "Harmonic analysis on finite groups" for applications to probability theory (one of your stated strengths). That way you'll be going from what you already know to the new stuff. [Here][1] is the Google Books link.
[1]: http://books.google.com/books?id=3trud4weQ3AC |
http://i51.tinypic.com/2rfx91v.png
(sorry for the draw)
I try to get the blue angle , i tough it was easy but...
the curve is a bezier curve.
the black dot shows the tangent of D
I know position of A B C D.
I tried to get the angle by addition of angle a and angle b
the point F is found by multiplying A... |
(sorry for the draw)
I try to get the blue angle , i thought it was easy but...
the curve is a Bézier curve.
the black dot shows the tangent of D
I know position of A B C D.
I tried to get the angle by addition of angle a and angle b
the point F is found by ... |
Prove that there is a largest integer $n$ such that $n \le x$ for any rational $x$.
What about any real number $x$? |
I want to make an origami of a sphere, so I planned to print some net of a Pentakis icosahedron, but I have a image of another sphere with more polygones:
http://www.webkinesia.com/online/graphics/images/icos.gif
I would like to find the net of such model (I know it will be very fun to cut).
Do you know if it ha... |
How to make a sphere-ish shape with triangle faces ? |
I want to make an origami of a sphere, so I planned to print some net of a pentakis icosahedron, but I have a image of another sphere with more polygons:
I would like to find the net of such model (I know it will be very fun to cut).
Do you kno... |
Prove that there is a largest integer $n$ such that $n \le x$ for any fixed rational $x$.
What about any fixed real number $x$? |
Lets say I have a 2 x 2 Matrix (Actually [Structure Tensor][1] of a Discrete Image - I):
$$ \begin{bmatrix}
\frac{\partial I}{\partial x}\frac{\partial I}{\partial x} & \frac{\partial I}{\partial x}\frac{\partial I}{\partial y} \\\
\frac{\partial I}{\partial y}\frac{\partial I}{\partial x} & \frac{\partial I}{... |
This is a very basic question that I feel that I should know the answer to but haven't been able to think through clearly.
In linear algebra, we learn that the basis of a finite dimensional vector space can be thought of as 'co-ordinates' of that space. And, we model what we intuitively understand as the euclidean p... |
What are polar co-ordinates? |
[This][1] has a proof that for any ordered space separable implies hereditarily separable.
And the Sorgenfrey line is a subspace of an ordered space (e.g. the double arrow).
[1]: http://at.yorku.ca/cgi-bin/bbqa?forum=ask_a_topologist_2005;task=show_msg;msg=2905.0001.0002.0001 |
Given the semimajor axis and a flattening factor, is it possible to calculate the semiminor axis? |
How do you calculate the semiminor axis for an ellipsoid? |
Where is the Taylor series for (0,1500) centered? If at zero, it is the same Taylor series as for (0,0.5). But you really don't want a Taylor series for (0,1500)-it would take an enormous number of terms for any reasonable accuracy. [Numerical Recipes][1] (chapters 5 and 6) and all other numerical analysis books hav... |
I was reading a comment on MathOverflow and did not understand what was being suggested. I also don't understand Anton's response:
http://mathoverflow.net/questions/17732/difference-between-measures-and-distributions/18062#18062
I initially thought Regenbogen's comment was being applied to the derivative of the d... |
Dinesh, you seem to have the answer (for integer distributions) - need to show
$3(q_0+q_1)\geq q_0+q_1+q_2$, ie, $2(q_0+q_1)\geq q_2$, which is true since $2q_0\geq q_i$ for any $i$.
Also, I can't comment - anyone know why? |
I think the key point is understanding the notion of "howto parametrize points" in a manifold, and in particular the *big question* is "How can I pass from delightful, amenable coordinates in $\mathbb{R}^n$ to those fancy 'things' with a lot of cosines, logarithms and so on?".
Well, the answer (IMO) is "Because *I kn... |
What does the $\ni$ (backwards element of) symbol mean? It doesn't appear in the [Wikipedia list of mathematical symbols][1], and a Google search for "backwards element of" or "backwards epsilon" turns up contradictory (or unreliable) information.
It seems it can mean both "such that", or "contains as an element". ... |
In <a href="http://math.stackexchange.com/questions/12488/for-a-planar-graph-find-the-algorithm-that-constructs-a-cycle-basis-with-each-e"> your previous question</a> better use was made of graph theory terminology: you refer to the graph as planar and to a cycle basis.
Assuming you understand those terms, perhaps ... |
Situation A: Once only, I toss 2 identical fair coins and don't look at the outcomes. A truthful observer looks at one of the coins and tells me that at least one of the coins is a head.
Situation B: Once only, I toss 2 identical fair coins and don't look at the outcomes. A truthful observer looks at both of the co... |
I'm stuck on this problem:
I have a "truth-table" (well, I don't know if it can be called truth table, if there aren't true/false values only):
string | a | b
---------------------
x | 1 | 0
z | 1 | 0
xx | 1 | 1
xz | 1 | 1
zx | 1 | -1
... |
###Introduction###
The following response attempts to address two aspects I perceive to be of interest in this question: one concerns whether there are some relatively rote or standard procedures for evaluating the convergence of certain kinds of improper multivariate integrals; the other is whether there is some si... |
I'm stuck on this problem:
I have a "truth-table" (well, I don't know if it can be called truth table, if there aren't true/false values only):
string | a | b
---------------------
x | 1 | 0
z | 1 | 0
xx | 1 | 1
xz | 1 | 1
zx | 1 | -1
... |
I'm stuck on this problem:
I have a "truth-table" (well, I don't know if it can be called truth table, if there aren't true/false values only):
string | a | b
---------------------
x | 1 | 0
z | 1 | 0
xx | 1 | 1
xz | 1 | 1
zx | 1 | -1
... |
Does anyone know a combinatorial proof of the following identity, where $F_n$ is the $n$th Fibonacci number?
$$n F_1 + (n-1)F_2 + \cdots + F_n = F_{n+4} - n - 3$$
It's not in the place I thought it most likely to appear: Benjamin and Quinn's *[Proofs That Really Count][1]*. In fact, this may be a hard problem, a... |
Let $f: X \to Y$ be a proper morphism of noetherian schemes. If the natural map $\mathcal{O}_Y \to f_*(\mathcal{O}_X)$ is an isomorphism, then a version of Zariski's main theorem states that the fibers $X_y, y \in Y$ are all connected. (The case listed as "Zariski's main theorem" in Hartshorne is the case of $f$ birat... |
How to get the connectedness theorem from the quasi-finite version of ZMT? |
Sum of finite number of rational numbers is of course a rational number, but sum of infinite number of rational numbers might be an irrational number. Can someone give me some intuisions why this sum might be irrational? I just "dont feel it". |
Let $0<a\le 1$ be fixed. Is the sequence $$a_N=\int_1^N x^{-a}e^{ix} dx$$ bounded? |
Is this a bounded sequence? |
**Question:** What are some good resources for a student who has taken algebraic and point-set topology and who wishes to learn how filters and ultrafilters are applied in topology?
**Motivation:** I've seen in a number of papers where ultrafilters are briefly mentioned as tools which can be used to generalize an ar... |
Edit: Here is a more verbose proof. The original proof is retained below.
<h3>Introduction</h3>
Let $\Sigma$ be some alphabet with $n$ symbols. Let $L$ be the language of words in which each symbol appears an even number of times. Suppose $G$ is a grammar for $L$. Our goal is to given an exponential lower bound on ... |
This question is vaguely related to: http://math.stackexchange.com/questions/14915/different-formulations-of-class-field-theory
As I said there, I'm currently learning class field theory. For some motivation, I've also read a little about Langlands. Everything I've read about that program seems to be a generalizatio... |
Does the Langland's program preserve CFT's distinction between local and global theories? |
Edit: Here is a more verbose proof. The original proof is retained below.
<h3>Introduction</h3>
Let $\Sigma$ be some alphabet with $n$ symbols. Let $L$ be the language of words in which each symbol appears an even number of times. Suppose $G$ is a grammar for $L$. Our goal is to given an exponential lower bound on ... |
Edit: I was initially assuming that $E$ is the closed linear span of the basic sequence, but then I realized that from your question there is no justification for me having done so. Since my initial argument applied only to that case and I didn't know how to extend it, I did a Google search to find that one way to red... |
How to understand the compactness in topology space in intuitive way?
|
How to understand compactness? |
While combing around my notes looking for other possible examples for [this question][1], I chanced upon another one of my unsolved problems:
Cycloidal curves are curves generated by a circle rolling upon a plane or space curve. It's not too hard to derive the required parametric equations if the fixed curve is a pl... |
I recently read that the group of diffeomorphisms of a smooth manifold which preserve some sort of geometric structure (e.g. Riemannian structure, conformal structure, etc.) frequently turn out to be a Lie group. What are some examples of this?
I've read about the euclidean group $E(n)$ which are the isometries of ... |
Lets say I have a 2 x 2 Matrix (Actually [Structure Tensor][1] of a Discrete Image - I):
$$ \begin{bmatrix}
\frac{\partial I}{\partial x}\frac{\partial I}{\partial x} & \frac{\partial I}{\partial x}\frac{\partial I}{\partial y} \\\
\frac{\partial I}{\partial y}\frac{\partial I}{\partial x} & \frac{\partial I}{... |
This is a very basic question that I feel that I should know the answer to but haven't been able to think through clearly.
In linear algebra, we learn that the basis of a finite dimensional vector space can be thought of as 'co-ordinates' of that space. And, we model what we intuitively understand as the euclidean p... |
I have found the following ellipse representation $(x,y)=(x_0\cos(\theta+d/2),y_0\cos(\theta-d/2))$, $\theta \in [0,2\pi]$. This is a contour of bivariate normal distribution with uneven variances and correlation $\rho=\cos(d)$. I know that this is a rotated ellipse with centre $(0,0)$. How to find the lengths of major... |
I have a functor $F\colon \mathbf{Rng}\to\mathbf{Grp}$, and a correspondence on objects which assigns to every group $F(R)$ a suitable subgroup $G_o(R)\subseteq F(R)$. Is there a way to turn $G$ into a functor, defining $G_o(R)\to G_o(S)$ via the maps I have between $F(R)$ and $F(S)$?
$$
\begin{array}{ccc}
F(R) &\to... |
In my course there's a paragraph: Taylor polynom with Lagrange remainder,
The paragraph starts with a theorem (i left out the constraints):
http://img801.imageshack.us/i/screenshot2hf.png/
(sorry, spam prevention won't let me post an image)
With *f* an *R-R* function and *a* and *a + h* defined over an i... |
Show that $(n-r) \binom{n+r-1}{r} \binom{n}{r} = n \binom{n+r-1}{2r} \binom{2r}{r}$.
In the LHS $\binom{n+r-1}{r}$ counts the number of ways of selecting $r$ objects from a set of size $n$ where order is not significant and repetitions are allowed. So you have $n$ people you form $r$ teams and select $r$ captains a... |
In my course there's a paragraph: Taylor polynom with Lagrange remainder,
The paragraph starts with a theorem (i left out the constraints):
$ ( \exists \theta \in ]0,1[)(f(a +h) = T_{f,a,n}(a + h) + \frac{h^{n+1}}{(n+1)!}D^{n+1}f(a + \theta h)) $
With $f$ an $R-R$ function and $a$ and $a + h$ defined over an... |
In my course there's a paragraph: Taylor polynomial with Lagrange remainder,
The paragraph starts with a theorem (I left out the constraints):
$ ( \exists \theta \in ]0,1[)(f(a +h) = T_{f,a,n}(a + h) + \frac{h^{n+1}}{(n+1)!}D^{n+1}f(a + \theta h)) $
With $f$ an $R-R$ function and $a$ and $a + h$ defined over... |
Elementary proof that 2x^2+xy+3y^2 represents infinitely many primes? |
I got a quick question on riemannian geometry. I'm not quite sure whether this is the right place to ask this question, since it might be a rather elementary one from a research point of view. Nontheless I think it might be too advanced for a typical homework question (which by the way it isn't)
So let $M^m$ be a m... |
What is the shortest string S over a alphabet of size n, such that every permutation of the alphabet is a substring of S?
I thought of this problem while reading a open problem on [shortest supersequence of all permutations][1].
[1]: http://garden.irmacs.sfu.ca/?q=op/smallest_universal_supersequence |
What is the shortest string that contain all permutation of an alphabet? |
My question is ; How can I solve the following derivative question?
$ y = sin (arc tan x) + tan (arc sin x) $
Thanks in advance,
|
How can I find $ y = sin (arc tan x) + tan (arc sin x) $ ? |
Show that $\binom{2n}{2} = 2 \binom{n}{2}+n^2$.
LHS: This is the number of pairs of $2n$ distinct elements.
RHS: We can rewrite this as $2 \binom{n}{2}+ \binom{n}{1} \binom{n}{1}$. So you can divide up the $2n$ distinct elements into two subsets of $n$ elements. Then pick an element from each of these subsets to... |
What's your intended result?
A derivative, i.e. $y'$? Use the chain rule.
Solve for $x$? Use $cos^2(x) = 1 - sin^2(x)$ and $tan(x) = sin(x)/cos(x)$ to simplify |
What is the shortest string $S$ over an alphabet of size $n$, such that every permutation of the alphabet is a substring of $S$?
I thought of this problem while reading a open problem on [shortest supersequence of all permutations][1].
[1]: http://garden.irmacs.sfu.ca/?q=op/smallest_universal_supersequence |
What is the shortest string that contains all permutations of an alphabet? |
I'm having trouble finishing this integral:
\begin{align*}
\newcommand{\mba}{\mathbf{a}}
\newcommand{\mbb}{\mathbf{b}}
\newcommand{\mbl}{\mathbf{l}}
\newcommand{\mbp}{\mathbf{p}}
\newcommand{\mbv}{\mathbf{v}}
\newcommand{\mbx}{\mathbf{x}}
f(\mbl \mid \mba, \mbb)
&= \int_{l_1}^0 \dotsi \int_{l_k}^0 f(\mbx \mi... |
The standard way (other than generating up to $N$) is to check if $(5*N*N + 4)$ or $(5*N*N - 4)$ is a perfect square. What is the mathematical logic behind this? Also, is there any other way for checking the same? |
Checking if a number is a Fibonacci or not? |
First, define a probability distribution with density:
\begin{align*}
\newcommand\bm\boldsymbol
\newcommand{\mbl}{\mathbf{l}}
f(\mbl \mid \bm \alpha)
= \frac{1}{B(\bm \alpha + \bm 1)}\exp\left(\mbl \cdot \bm\alpha\right)
\end{align*}
where
$\mathbf l \in (-\infty,0]^k, \sum_i\exp(l_i) = 1$
I'm having tro... |
I'm not sure how well known this "riddle" is but here it goes.
3 people go to a restaurant, each buy food worth 10.00. When they're done, they give 30.00 to the waitress. She gives the money to the manager, manager says they paid too much, gives 5.00 back to the waitress. She comes back to the table, gives a doll... |
This is a very basic question that I feel that I should know the answer to but haven't been able to think through clearly.
In linear algebra, we learn that the basis of a finite dimensional vector space can be thought of as 'co-ordinates' of that space. And, we model what we intuitively understand as the euclidean p... |
I am not sure how 'mathematical' the riddle part of this story is. :) Each guest paid 9 because together they paid 30, and got back 27. Of those 27, the manager got 25, and the waitress got the other 2 -- there is no sense/reason in adding 27 to 2 -- though you might have a gut feeling that you are headed in the 'right... |
list = [random, random, random, ..., random]
largest := (list[0] > list[1]) ? list[0] : list[1]
secondlargest := (list[0] <= list[1]) ? list[0] : list[1]
for each num in list:
if num >= largest:
secondlargest := largest
largest := num
elseif num >= secondla... |
**Question:** What is the definition of a small class? Is there no such thing as a set which contains other sets such as {{1,2},{1},{2}, 1, 2} (ie, is this really called a "class")? What are some prototypical examples of small classes? What are some prototypical examples of large classes?
**Motivation:** A few th... |
My question is, how can I solve the following derivative question?
$$y = \sin(\arctan x) + \tan(\arcsin x)$$
Thanks in advance
|
How can I find the derivative of $y = \sin(\arctan x) + \tan(\arcsin x)$? |
How to input such system
![alt text][1]
into maple so that it would solve it?
[1]: https://i.stack.imgur.com/yFGkB.jpg |
Maple: How to input such system so that it would be solved? |
On page 18 of Rudin's *Real and Complex* analysis he defines $0 \cdot \infty = 0$ and says that "with this definition the commutative, associative, and distributive laws hold in $[0,\infty]$ without any restriction".
What is not clear to me is whether the quoted statement is a justification of the definition or jus... |
Arithmetic on $[0,\infty]$: is $0 \cdot \infty = 0$ the only reasonable choice? |
Dylan's right that diffeomorphism groups are usually infinite dimensional. There's a theory of infinite dimensional Lie groups, I've heard, but I think it's a lot harder to work out things like convergence. Worth noting, though, is that in this analogy, if $Diff(M)$ is a Lie group then its Lie algebra should be $Vect... |
what is the lim x-> 0 [x]/x ? for left limit and right limit in x->0
Thanks in Advance . |
what is the lim x-> 0 [x]/x ? |
Some books say even numbers start from two but if you consider the number line concept, I think zero should be even because it is in between -1 and +1 (i.e in between 2 odd numbers). What is the real answer? |
Is zero odd or even? |
How to input such system
![alt text][1]
into maple so that it would solve it?
BTW we some part of system will be given to us,, which we do not know certently - may be some Ai's may be some Bi's as constants. we do not know how much constants we will get. And we would need from given constants get a soltion for o... |
Assuming that [x] is the floor of x, then look at [this graph][1].
Assuming that [x] is the ceiling of x, then look at [this similar looking graph][2].
Assuming that [x] is the "nearest integer" function, then consider what the nearest integer is on the interval [-0.5, 0.5].
If this is something else, please... |
From wikipedia,
http://en.wikipedia.org/wiki/Riemann_zeta_function
"Furthermore, the fact that $\zeta(s) = \zeta(s*)*$ for all complex s ≠ 1 (* indicating complex conjugation) implies that the zeros of the Riemann zeta function are symmetric about the real axis."
I know the zeros are symmetrical. But what about ... |
Is Riemann Zeta Function symmetrical about the real axis? |
So I had a system
#for given koefs
k:=3; n:=3;
#let us solve system:
koefSolution:= solve({
sum(a[i], i = 0 .. k) = 0,
sum(a[i], i = 0 .. k)-(sum(b[i], i = 0 .. k)) = 0,
sum(i^n*a[i], i = 0 .. k)-(sum(i^(n-1)*b[i], i = 0 .. k)) = 0
});
So I have a vector like
... |
How to solve such system with given parts of it? (maple) |
Which numbers are you using? Positive integer? In this case, you don't have to consider zero (that's why, perhaps, some books says that even numbers start from 2). If 0 is in your number set, then yes, it is divisible by 2. |
Last month, I was introduced to the card game SET. The game raises several interesting questions eg what's the probability that n randomly drawn cards contain k sets, is it possible to end with 12 (or 15) cards which contain no sets... I would like to present the solution to a question like this to undergrads, and thro... |
EDIT: Bounty of 50 pts announced for the **poly**-size part. **You** can win the bounty for a single comment.
1. Can a CFG over large alphabet describe a language, where each terminal appears even number of times?
2. If yes, would the Chomsky Normal Form be polynomial in |Σ| ?
EDIT: What about a language where... |
Last month, I was introduced to the card game SET. The game raises several interesting questions eg what's the probability that n randomly drawn cards contain k sets, is it possible to end with 12 (or 15) cards which contain no sets... I would like to present the solution to a question like this to undergrads, and thro... |
**HINT**: Find out what points $(x|y)$ with $x = 1$ also have distance $2$ to the origin. What would these points represent? |
Note that you can write down the equation of a circle if you know the co-ordinates of the center $(h,k)$ and its radius $r$. You already know that the two circles in question have radius $2$. It remains to figure out where their centers lie. You are told that their centers lie on the line $x=1$ which is a line parallel... |
Assuming that [x] is the floor of x, then look at [this graph][1].
Assuming that [x] is the ceiling of x, then look at [this similar looking graph][2].
Assuming that [x] is the "nearest integer" function, then consider what the nearest integer is on the interval [-0.49, 0.49].
If this is something else, plea... |
From the first day which I went to University until now, I'm always thinking that the whole math science is based on lies. Why ?
**For example:** ***infinity*** !
What is infinity ? Is that real ? Is infinity concept real ? your answer is "Yes" ? What about [Zeno's paradox][1] ? I know that there are different so... |
Is mathematics based on lies ? |
[Update]: I know mathematicians(!) don't like these kind of questions. But please add comments and answer to me if you want.
From the first day which I went to University until now, I'm always thinking that the whole math science is based on lies. Why ?
**For example:** ***infinity*** !
What is infinity ? Is t... |
It's not mathematics but rather physics which is based on these "lies". In mathematics, we assume (if we're Platonists) that objects like the real numbers "really exist", just not in the physical world, and then everything makes sense. Some people pretend that when they're doing mathematics, they're just combining axio... |
[Update]: I know mathematicians(!) don't like these kind of questions. But please add comments and answer to me if you want.
From the first day which I went to University until now, I'm always thinking that the whole math science is based on lies. Why ?
**For example:** ***infinity*** !
What is infinity ? Is t... |
It seems like your premise for mathematics being a 'lie' is that it's theoretical background cannot be completely applied to the real world (e.g. having the smallest distance in the real world, but not in mathematics). If that's how you define a lie, sure, but mathematics is an abstract science and I am not sure why yo... |
I'm not going to prove something, this is just a question. From the first day which I went to University until now, I'm always thinking that the whole math science is based on lies. Why ?
**For example:** ***infinity*** !
What is infinity ? Is that real ? Is infinity concept real ? your answer is "Yes" ? What abo... |
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