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Winding yarn into a ball suggests some mathematical questions:
![Yarn][1]
1. Under some natural model, what paths should the yarn follow to achieve the most dense ball?
One model is that used by Henryk Gerlach and Heiko von der Mosel in their paper
"On sphere-filling ropes" [arXiv:1005.4609v1 (math.GT)][2].
(T... |
How to find a bijection between R-valued points of X and local ring homomorphisms? |
I need to write a coupon code system but I do not want to save each coupon code in the database. (For performance and design reasons.) Rather I would like to generate codes subsequent that are watermarked with another code.
They should like kind of fancy and random. Currently they look like this:
1: AKFCU2, 2: AK... |
We know that 2^4 = 4^2 and (-2)^(-4) = (-4)^(-2). Is there another pair of integers (x, y), with different x and y, which satisfies the equality **x^y = y^x**?
|
We know that $2^4 = 4^2$ and $(-2)^{-4}$ = $(-4)^{-2}$. Is there another pair of integers $(x, y)$, with different $x$ and $y$, which satisfies the equality $x^y = y^x$?
|
Is there a way of enumerating necklaces, in the spirit in which the factorial number system ( http://en.wikipedia.org/wiki/Factorial_number_system ) can be used to enumerate permutations? |
Is this a know special function? |
Is this a known special function? |
Given a row of 16 houses where 10 are red and 6 are blue, what is the expected number of neigbors of a different color? |
Is this a known special function:
$$\int_0^1 a^p(1-a)^{1-p}\\,b^{1-p}\\,(1-b)^p dp\qquad ?$$
I am really only interested in maximizing this over $(a,b)$ in $[0,1] \times [0,1]$, so a pointer to a nice numerical evaluation is appreciated as much or more so than an unstable exact formula.
Thanks for any help
|
My text gives a much more complicated proof of this result, which makes me wonder if the argument I have in my head for this has something wrong with it. Does this work, or have I made a bad assumption somewhere along the line?
Let $U \subseteq \mathbb{C}$ be open, $\overline{D}(z_0, r) \subset U$, and $f : U \righ... |
What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course?
My students in this course are generally from biochemistry, computer science, economics, business, and physics (with a few humanities folks taking the course for fun) - not just math ... |
My text gives a much more complicated proof of this result, which makes me wonder if the argument I have in my head for this has something wrong with it. Does this work, or have I made a bad assumption somewhere along the line?
Let $U \subseteq \mathbb{C}$ be open, $\overline{D}(z_0, r) \subset U$, and $f : U \righ... |
arising out of comment made by Yuval Filmus in http://math.stackexchange.com/questions/9395/what-is-the-cardinality-of-set-of-all-smooth-functions-in-l1 I got this idea (forgive me for my ignorance for if it is nothing but an elementary definition/result in real analysis). The idea is like this *... |
arising out of comment made by Yuval Filmus in http://math.stackexchange.com/questions/9395/what-is-the-cardinality-of-set-of-all-smooth-functions-in-l1 I got this idea (forgive me for my ignorance for if it is nothing but an elementary definition/result in real analysis). The idea is like this *... |
My text gives a much more complicated proof of this result, which makes me wonder if the argument I have in my head for this has something wrong with it. Does this work, or have I made a bad assumption somewhere along the line?
Let $U \subseteq \mathbb{C}$ be open, $\overline{D}(z_0, r) \subset U$, and $f : U \righ... |
arising out of comment made by Yuval Filmus in http://math.stackexchange.com/questions/9395/what-is-the-cardinality-of-set-of-all-smooth-functions-in-l1 I got this idea (forgive me for my ignorance for if it is nothing but an elementary definition/result in real analysis). The idea is like this *... |
arising out of comment made by Yuval Filmus in http://math.stackexchange.com/questions/9395/what-is-the-cardinality-of-set-of-all-smooth-functions-in-l1 I got this idea (forgive me for my ignorance for if it is nothing but an elementary definition/result in real analysis). The idea is like this *... |
Suppose I have two tables, each of unknown size, and I'd like to estimate the average of their true sizes. I hire 2 contractors: one guarantees good precision (i.e., her measurement normally-distributed about the true value, with a standard deviation of 5mm), while the other is a dimwit (i.e., his measurement is also u... |
I can't resist giving a couple of examples of (contrived) numerical coincidences (both are from [*Experimentation in Mathematics*][1] by J.M. Borwein, D.H. Bailey and R. Girgensohn).
Example 1.
$$\int_{0}^{\infty}\cos(2x)\prod_{k=1}^{\infty}\cos(x/k)\ dx=\frac{\pi}{8}-\epsilon,$$
where
$0<\epsilon<10^{-41}.$
... |
My text gives a much more complicated proof of this result, which makes me wonder if the argument I have in my head for this has something wrong with it. Does this work, or have I made a bad assumption somewhere along the line?
Let $U \subseteq \mathbb{C}$ be open, $\overline{D}(z_0, r) \subset U$, and $f : U \righ... |
This question was firstly posted on [Mathoverflow][1] Two answers are pretty interesting. The potential counter-example given in the second answer is really interesting, but it is not surely a counter-example.
Consider two partially ordered sets $A = \{a< b,a< c\}$, $B=\{x< z,y< z\}$.
Their linear extensions (her... |
Does order isomorphism of linear extensions of two partially ordered sets imply order isomorphism of themselves? |
This question was firstly posted on [Mathoverflow][1]. Two answers are pretty interesting. The potential counter-example given in the second answer is really interesting, but it is not surely a counter-example.
Consider two partially ordered sets $A = \{a< b,a< c\}$, $B=\{x< z,y< z\}$.
Their linear extensions (he... |
Given $xy=C$ where $x, y$ are integer variables and $C$ is integer constant.
> What is the most efficient algorithm that finds $x,y$ such that $x+y$ is minimum?
Providing references is highly appreciated. |
Hardness of perimeter minimization? |
Given $xy=C$ where $x, y$ are integer variables and $C$ is integer constant.
> What is the most efficient algorithm that finds $x,y$ such that $x+y$ is minimum?
Providing references is highly appreciated.
Edit: Input integers are reasonably encoded. |
I began reading Paul Halmos' "Naive Set Theory", and encountered the "Axiom of Specification".
<Blockquote> To every set $A$ and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly those elements $x$ of $A$ for which $S(x)$ holds.
</Blockquote>
Earlier in the same section, I learned ... |
How can I write the Axiom of Specification as a sentence? |
you could make sth inspired by RSA, where you do not need two keys. like this:
calculate everything modulo $p$, where $p$ is prim. that will then be the GF[$p$] [(GaloisField)][1].
now instead of finding a generator $g$ for that GF[$p$] and calculating anything in powers of $g$, you only need two large numbers $m$... |
Let $\Sigma$ be an alphabet of size $s$, with which we build strings of length $n$.
The **Hamming ball** of radius $d$ centered at $x\in\Sigma^n$ is the set $B(x, d)$ of words in $\Sigma^n$ that differ from $x$ in *at most* $d$ positions.
Similarly, the **Hamming circle** of radius $d$ centered at $x\in\Sigma^n... |
Let $A \in \mathbb{R}^{n \times n}$ be a positive definite matrix and let $a > 0$ be some constant.
I am interested in the following quantity
$$
\hat x = {\rm arg} \min_{x \in R^n}\ x' \left( \frac{1}{a + \|x\|^2_2} A \right) x
$$
Is it possible to relate $\|\hat x\|_2^2$ to the eigenvalue and $\hat x$ to the e... |
How to relate the following problem to known results on the eigenvectors? |
Finding Range of: $y=\log_e(3-2x)/\sqrt{x+1}$? |
f(a,b) = f(a-1, b) + f(a-1, b-1) + f(a, b-1), ab != 0 <br>
f(a,b) = 1, ab = 0 <br>
So what is f(a, b)?
|
General expression of f(a, b) if f(a, b)=f(a-1,b) + f(a, b-1) + f(a-1, b-1)? |
So, [wikipedia](http://en.wikipedia.org/wiki/Affine_space) has a page about "affine space", where such a space is an object (set, topological space, blah) with a free transitive (blah, continuous, blah) action by the additive group of a free module.
At least when the base ring is an algebraically closed field, we ca... |
Axiomatic affine n-space and the (classical) affine variety $A^n_k$ only agree for K an alg. closed field? |
Let $A \in \mathbb{R}^{n \times n}$ be a positive definite matrix and let $a > 0$ be some constant.
I am interested in the following quantity
$$
\hat x = {\rm arg} \max_{x \in R^n}\ x' \left( \frac{1}{a + \|x\|^2_2} A \right) x
$$
Is it possible to relate $\|\hat x\|_2^2$ to the eigenvalue and $\hat x$ to the e... |
I'd heard of propositional logic for years, but until I came across [this question][1], I'd never heard of predicate logic. Moreover, the fact that *Introduction to Logic: Predicate Logic* and *Introduction to Logic: Propositional Logic* (both by Howard Pospesel) are distinct books leads me to believe there are signif... |
What's the difference between predicate and propositional logic? |
What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course?
My students in this course are generally from biochemistry, computer science, economics, business, and physics (with a few humanities folks taking the course for fun) - not just math ... |
How can I prove that the Cartier dual of $\alpha_p$ is again $\alpha_p$ (using the Yoneda lemma)? It should be something like $\alpha_p(R) \to (\alpha_p(R) \to \mu_p(R)), x \mapsto (y \mapsto exp_{p-1}(x + y)$, where $exp_{p-1}$ is the truncated exponential sequence. My problem is that this isn't a homomorphism. |
Ok. That is a fact. There is a theorem that proof that. As it says in [this question][1].
[1]: http://math.stackexchange.com/questions/6148/if-a-graph-can-be-colored-with-max-4-colors-is-it-planar
I just cant understand how maths can proof such a thing, what does exactly the theorem says?? how mathematically ... |
From Wikipedia's triangle center article: "Thus every point is potentially a triangle center. However the vast majority of triangle centers are of little interest, just as most continuous functions are of little interest. The Encyclopedia of Triangle Centers is an ever-expanding list of interesting ones."
Hyperbolic... |
Reading the Wikipedia page for [Random Walk][1], I was wondering what is the definition for a general random walk as a random process, so that all the concepts such as [random walk on $\mathbb{Z}^d$][2], [Gaussian random walk][3] and [random walk on a graph][4] can fall into this general definition?
(1) In Ross's ... |
During my studies of real analysis I've come across the definition of a field (or algebra) of sets. My question is: What is the connection between this structure and the structure of a field or algebra known from abstract algebra? |
If there is none, why?
And for the other side, what about open set (0, 1) to closed set [0, 1] with a continuous function?
Thanks |
Is there a continuous function from [0, 1] onto (0, 1)? |
if $X_n$ converges to $X$ and $Y_n$ converges to $Y$ in distribution, what about $X_n + Y_n $ would that converge to $X+Y$ in distribution ?
any ideas how i could prove or disprove this |
I have an interesting math question I've developed while playing cards with my family and wonder if anyone has a way to approach the problem.
Say I have n people who each have a deck of cards. Each deck has a different number of cards in it. We are shuffling the cards for a game and in the end we all want to have th... |
I am writing a program to solve a rubiks cube, and would like to know the answer to this question.
There are 12 ways to make one move on a rubik's cube. How many ways are there to make a sequence of six moves? |
How many ways can I make six moves on a rubiks cube? |
I have an interesting math question I've developed while playing cards with my family and wonder if anyone has a way to approach the problem.
Say I have n people who each have a deck of cards. Each deck has a different number of cards in it. We are shuffling the cards for a game and in the end we all want to have th... |
I'm considering a parametrized sigmoid function such as the following logistic:
$f(x)=\frac{e^{a+bx}}{1+e^{a+bx}}$
And I'm interested only in the interval $x >= 0$ and $x < x_{max}$ (with a given $x_{max}$).
Two tiny questions regarding its properties:
1) Is the **sum of sigmoid functions always a sigmoid... |
I have always taken for granted that expected value is a linear operator. For *any* random variables $X$ and $Y$: $E(aX + bY) = aE(X) + bE(Y)$. Can anyone point me to a rigorous proof of this?
Also, I know that median $Med()$ is not a linear operator, meaning $Med(aX + bY)$ might not be equal to $a Med(X) + b Med(Y)... |
A proof that expected value is a linear operator? What about median? |
I'm considering a parametrized sigmoid function such as the following logistic:
$$f(x)=\frac{e^{a+bx}}{1+e^{a+bx}}$$
And I'm interested only in the interval $\displaystyle x >= 0$ and $\displaystyle x < x_{max}$ (with a given $x_{max}$).
Two tiny questions regarding its properties:
1) Is the **sum of sigm... |
There is a common problem (I often assign it in linear algebra, and I was assigned the problem in representation theory); seems like it might be related.
Suppose you have $n$ people sitting in a circular table, each of them with two neighbors. They each have a bowl of porridge in front of them, and a spoon in each h... |
What are the open sets of a Stone space of a Boolean algebra B?
[According to Wikipedia][1], they are the ultrafilters on B that contain an element of B. However, this cannot be the case since an ultrafilter is a set of subsets of B and contains no elements of B. Using the alternate definition of the Stone space as ... |
I have an interesting math question I've developed while playing cards with my family and wonder if anyone has a way to approach the problem.
Say I have n people who each have a deck of cards. Each deck has a different number of cards in it. We are shuffling the cards for a game and in the end we all want to have th... |
As an exercise in Python programming I've written a script that finds amicable pairs, and I've had fun doing so. But I'm wondering if amicable pairs have a use, in mathematics or elsewhere?
As a reminder:
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal t... |
Amicable pairs: any use for them yet? |
I have a set of functions that map square matrices with $n$ rows and columns to square matrices with $k < n$ rows and columns. Is there a name for this property? I know that 'projection' would be the wrong word. Also I don't mean that the function *only* changes the size, it might also do other things. |
What is the name for a function of a matrix that changes the matrix size? |
There is a common problem (I often assign it in linear algebra, and I was assigned the problem in representation theory); seems like it might be related.
Suppose you have $n$ people sitting in a circular table, each of them with two neighbors. They each have a bowl of porridge in front of them, and a spoon in each h... |
How to solve the recursion:
p[n,m] = p[n-1,m-1] + p[n+1,m-1] + p[n,m-1]
Ideally in general, but if you need base cases:
p[n,0] = 0 (for n!=0)
p[0,0] = 1
I've asked a similar question previously and believe n^(m-2) is
somehow involved.
Would turning this into a DFQ help?
|
There is a common problem (I often assign it in linear algebra, and I was assigned the problem in representation theory); seems like it might be related.
Suppose you have $n$ people sitting in a circular table, each of them with two neighbors. They each have a bowl of porridge in front of them, and a spoon in each h... |
I'm considering a parametrized [sigmoid][1] function such as the following logistic:
$$f(x)=\frac{e^{a+bx}}{1+e^{a+bx}}$$
And I'm interested only in the interval $\displaystyle x >= 0$ and $\displaystyle x < x_{max}$ (with a given $x_{max}$).
Two tiny questions regarding its properties:
1) Is the **sum of... |
Take a Multiplicative group G, defined by a prime p. Now take an element t from it. Its inverse is t<sup>-1</sup>. To calculate the inverse you can use the extended Euclidian algorithm or this trick, [which I found on wikipedia](http://en.wikipedia.org/wiki/Modular_multiplicative_inverse#Direct_Modular_Exponentiatio... |
Why does this trick to find the inverse of an element in a Multiplicative Cyclic Group work? |
There is a standard problem (I often assign it in linear algebra, and I was assigned the problem in representation theory); seems like it might be related.
Suppose you have $n$ people sitting in a circular table, each of them with two neighbors. They each have a bowl of porridge in front of them, and a spoon in each... |
There is a standard problem (I often assign it in linear algebra, and I was assigned the problem in representation theory); seems like it might be related.
Suppose you have $n$ people sitting in a circular table, each of them with two neighbors. They each have a bowl of porridge in front of them, and a spoon in each... |
I get confused in the following problem. Actually I don't know how to derive the derivative of the Jacobian. Could anybody help me?
Given a smooth vector field $\vec{b}$ on $R^{n}$, let $\vec{X}(s)=\vec{X}(s,x,t)$solves the ODE:
$\dot{\vec{X}}=\vec{b}(\vec{X})(s \in R)$,
$\vec{X}(t)=\vec{X}$
Define the... |
Firstly, I apologise for not being familiar with TeX.
It is a rather surprising fact (to me, at least) that $\binom{14}{4} = 1001$; $\binom{14}{5} = 2002$; $\binom{14}{6} = 3003$. Actually, this is the only instance where three consecutive binomial coefficients are in the ratio 1 : 2 : 3. I found it quite interestin... |
Reading a book on fractional calculus reminded me that I'd like to know more on the following method/idea.
Given an integral: $\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{e^{i\cdot c_j \cdot t}}}{\displaystyle\sum_{k=a}^b{e^{i \cdot d_k \cdot t}}} dt}$
I can use recurrences and generating functi... |
I know the general Black-Scholes formula for Option pricing theory (for calls and puts), however I want to know the other solutions to the Black-Scholes PDE and its various boundary conditions. Can someone start from the B-S PDE and derive its various solutions based on different boundary conditions? Even if you could ... |
There is a standard problem (I often assign it in linear algebra, and I was assigned the problem in representation theory); seems like it might be related.
Suppose you have $n$ people sitting in a circular table, each of them with two neighbors. They each have a bowl of porridge in front of them, and a spoon in each... |
There is a standard problem (I often assign it in linear algebra, and I was assigned the problem in representation theory); seems like it might be related.
Suppose you have $n$ people sitting in a circular table, each of them with two neighbors. They each have a bowl of porridge in front of them, and a spoon in each... |
I have always taken for granted that expected value is a linear operator. For *any* random variables $X$ and $Y$: $E(aX + bY) = aE(X) + bE(Y)$. Can anyone point me to a rigorous proof of this?
Also, I know that generally median $Med()$ is not a linear operator, meaning $Med(aX + bY)$ might not be equal to $a Med(X) ... |
Expected value is a linear operator? Under what conditions is median also a linear operator? |
Say I have two locally free sheaves $F,G$ on projective variety $X$. I know the cohomology groups $H^i(X,F)$ and $H^i(X,G)$. Is this enough to give me information about $H^i(X,F\otimes G)$? In particular, if $H^i(X,F)=0$, what conditions on $G$ guarantee that also $H^i(X,F\otimes G)=0$? |
Reading a book on fractional calculus reminded me that I'd like to know more on the following method/idea.
Given an integral: $\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{e^{i\cdot c_j \cdot t}}}{\displaystyle\sum_{k=a}^b{e^{i \cdot d_k \cdot t}}} dt}$
I can use recurrences and generating functi... |
Given your boundary conditions, if we open up the recurrence relation then we discover an $m$-step process, where at each step we change $n$ by $0$ or $\pm 1$. For every walk that ends up at $0$, we "score" one point.
How many such walks exist? There is an equal number $k$ of $+1$s and $-1$s, and so we get
$\disp... |
Let $L^{\infty}$ denote the set of all essentially bounded functions and suppose that
$f \in L^{\infty}$ and $g \in L^{\infty}$. Then the product $fg$ is in $L^{\infty}$. Now the question is: prove or disprove: $||fg||_{\infty} <= ||f||_{\infty} ||g||_{\infty}$.
Any hints?
|
Let $L^{\infty}$ denote the set of all essentially bounded functions and suppose that
$f \in L^{\infty}$ and $g \in L^{\infty}$. Then the product $fg$ is in $L^{\infty}$. Now the question is: prove or disprove: $\|fg\|\_\infty \leq \|f\|\_\infty \|g\|\_\infty$.
Any hints?
|
Wolfram|Alpha shows an [expansion][1].
[1]: http://www.wolframalpha.com/input/?i=limit+as+n+goes+to+infinity+of+%28+%28power%282%2C2n-1%29%29+%28sqrt%28n%29%29+%28power%28gamma%28n%29%2C2%29+%2F+gamma%282n%29%29+%29 |
Our math prof gave us this and we are not sure what to make of the notation, can someone give us a hand?
https://i.stack.imgur.com/6UcRj.png |
Our math prof gave us this and we are not sure what to make of the notation, can someone give us a hand?
![alt text][1]
[1]: https://i.stack.imgur.com/r5Fvf.png |
you could make sth inspired by RSA, where you do not need two keys. like this:
calculate everything modulo $p$, where $p$ is prim. that will then be the GF[$p$] [(GaloisField)][1].
now instead of finding a generator $g$ for that GF[$p$] and calculating anything in powers of $g$, you could use any large numbers for... |
you could make sth inspired by RSA, where you do not need two keys. like this:
calculate everything modulo $p$, where $p$ is prim. that will then be the GF[$p$] [(GaloisField)][1].
now instead of finding a generator $g$ for that GF[$p$] and calculating anything in powers of $g$, you could use any large numbers for... |
When I first learned linear algebra, both the professor and the book used brackets like `[` and `]` to enclose matrices. However, in my current differential equations textbook, matrices are enclosed by parenthesis, and I suddenly realize everybody else are using them too.
So are brackets/parenthesis for enclosing ma... |
Notation help. What does $\mathcal{P}(A)$ and $\chi_{-}$ mean here? |
Notation help. What do $\mathcal{P}(A)$ and $\chi_{-}$ mean here? |
To motivate my question, recall the following well-known fact: Suppose that $p\equiv 1\pmod 4$ is a prime number. Then the equation $x^2\equiv -1\pmod p$ has a solution.
One can show this as follows: Consider the following polynomial in ${\mathbb Z}_p[x]$: $x^{4k}-1$, where $p=4k+1$. The roots of this polynomial are... |
Since the $b_n$ are bounded, there is a constant $C$ such that $-C\lt b_n\lt C$ for all $n$. Fix $n$. Suppose first $a_n\gt 0$. Then $-C a_n< a_nb_n < Ca_n$. Similarly, if $a_n\lt 0$ then $Ca_n < a_n b_n < -C a_n$. In short: $|a_n b_n|\lt C|a_n|$.
Now, for any sequence $c_n$, $c_n\to 0$ iff $|c_n|\to 0$. Right?
... |
You write *... but since $a_n$ is not bounded...*
Presumably you mean that you have not been explicitly told that the sequence $a_n$ is bounded. That is true; you have not.
However, as it happens, the following holds:
**Theorem.** Let $c_n$ be any sequence of real numbers. If $c_n$ converges, then it is bounde... |
Suppose $\mu(X) < \infty$ and $0 < p< q \leq \infty$. I'm reading the following proof:
If $q=\infty$ then $L^{q}(\mu) \subseteq L^{p}(\mu)$.
Proof:
If $q=\infty$ then $\int |f|^{p} \leq (||f||_{\infty})^{p} \int 1$.
Why we have the above inequality??
|
Let $M$ be a compact, connected smooth manifold. If $p, q$ are points in $M$, is there always a geodesic that goes from $p$ to $q$?
I know that this is certainly not true if $M$ is not compact, but I couldn't find a counterexample for the compact case.
Can anybody help me out?
Thanks,
S. |
I am a bit confused about the concept of proper subsets,precisely whether to include one or both of the **void set** and **the set itself**.
An extract from my module goes like this :
***Obviously,every set is the subset of itself and the void set ($\phi$) is the subset of every set. These two subsets are called ... |
Subsets and Splits
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