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Suppose $A$ is a set of $2n$ positive integers { $a_{i}$ } with $a_{i} \geq 2$ which can be partitioned into at most $r \leq n$ coprime pairs. The following fact is true: For all $A$ and every $n \geq 1$,
$$
\prod_{i = 1}^{2n} (a_{i} - 1) \equiv 0 \pmod{2^{r}}.
$$
For example, the set {$2,3,5,6$} can be partitione... |
I am teaching a Calc II course and came across the following series when finding the interval of convergence for the Taylor series of $f(x)=\sqrt{x}$ centered at $x=1$:
$$
\sum_{n=2}^\infty \frac{1\cdot 3\cdot 5\cdot 7\cdots (2n-3)}{2^nn!}.
$$
I know that it doesn't converge, but how do you show it? I would li... |
Why Doesn't This Series Converge? |
Suppose $A$ is a set of $2n$ positive integers { $a_{i}$ } with $a_{i} \geq 2$ which can be partitioned into at most $r \leq n$ coprime pairs. The following fact is true: For all $A$ and every $n \geq 1$,
$$
\prod_{i = 1}^{2n} (a_{i} - 1) \equiv 0 \pmod{2^{r}}.
$$
For example, the set {$2,3,5,6$} can be partitione... |
Suppose a set S has 11 elements. How many subsets of S have an even
number of elements? Express your answer as a summation.
I'm not really sure how to approach this. I think the general formula for this type of question is n!/k!(n-k)! where n is 11 and k is the number we're choosing. So would we just take the summa... |
Where, or do there exist, good visualizations of sheaves, stalks, stacks, and/or schemes? I'm a better visual thinker than I am a symbolic thinker, and it would be easier for me to follow some of the algebraic geometry papers that have no pictures whatsoever if I had some kind of visual intuition of what these objects ... |
Suppose $A$ is a set of $2n$ positive integers { $a_{i}$ } with $a_{i} \geq 2$ which can be partitioned into at most $r \leq n$ coprime pairs. The following fact is true: For all $A$ and every $n \geq 1$,
$$
\prod_{i = 1}^{2n} (a_{i} - 1) \equiv 0 \pmod{2^{r}}.
$$
For example, the set {$2,3,5,6$} can be partitione... |
I don't think the series diverges.
$$ \frac{1 \cdot 3 \cdot 7 \cdots (2n-3)}{2 \cdot 4 \cdot 6 \cdots 2n}$$
$$ = \frac{{2n \choose n}}{(2n-1)4^n} = \frac{1}{(2n-1)\sqrt{\pi n}}(1 + O(\frac{1}{n}))$$
using the approximation
$$\frac{ {2n \choose n}}{4^n} = \frac{1}{\sqrt{\pi n}}(1 + O(\frac{1}{n}))$$
and... |
I don't think the series diverges.
$$ \frac{1 \cdot 3 \cdot 7 \cdots (2n-3)}{2 \cdot 4 \cdot 6 \cdots 2n}$$
$$ = \frac{{2n \choose n}}{(2n-1)4^n} = \frac{1}{(2n-1)\sqrt{\pi n}}(1 + O(\frac{1}{n}))$$
using the approximation
$$\frac{ {2n \choose n}}{4^n} = \frac{1}{\sqrt{\pi n}}(1 + O(\frac{1}{n}))$$
and... |
If x,y,z are natural numbers, how many solutions are there to x+y+z=25?
How would I figure this out? I cant even begin dissecting this problem. Where do I begin? |
let $f$:**R**->**N** such that given any $x_1$,$x_2$ belong to **R** such that $x_1 < x_2$ then there exists a $x_3$ belongs to $(x_1,x_2)$ such that $f(x_3)> max(f(x_1),f(x_2))$. How many such mappings are possible ? |
According to [Wikipedia][1], the number of terms in (x+y+z)^30 is 512. I'm assuming this is before like terms are added up. How many terms would there be if like terms were combined? How would I go about figuring that out?
[1]: http://en.wikipedia.org/wiki/Trinomial_expansion |
According to [Wikipedia][1], the number of terms in (x+y+z)^30 is 496. I'm assuming this is before like terms are added up. How many terms would there be if like terms were combined? How would I go about figuring that out?
[1]: http://en.wikipedia.org/wiki/Trinomial_expansion |
What is the cardinality of set of all smooth functions belonging to $L^1$ or $L^2$ ? What is that of set of all integrable or square integrable functions ? |
what is the cardinality of set of all smooth functions in $L^1$ ? |
I have a function given implicitly, you know. X og Y on both sides. Than it says, assume y=y(x). That's fine. And find y'(0), that I should be able to do. But what about y''(0)? How do you treat the dy/dx parts when taking the second derivative?
Edit: I would also like to follow the tip in the book, that says when ... |
Find the second derivative of some implicit function? |
Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form?
$$\int_0^\infty\frac{cos\;x\}{1+x^2}\mathrm{d}x$$ |
Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form?
$$\int_0^\infty\frac{\cos\;x\}{1+x^2}\mathrm{d}x$$ |
I don't think the series diverges.
$$ \frac{1 \cdot 3 \cdot 7 \cdots (2n-3)}{2 \cdot 4 \cdot 6 \cdots 2n}$$
$$ = \frac{{2n \choose n}}{(2n-1)4^n} = \frac{1}{(2n-1)\sqrt{\pi n}}(1 + O(\frac{1}{n}))$$
using the approximation
$$\frac{ {2n \choose n}}{4^n} = \frac{1}{\sqrt{\pi n}}(1 + O(\frac{1}{n}))$$
and... |
Let $f \colon (\mathbb{C}^{2},\mathbf{0}) \to (\mathbb{C},0)$ be a Brieskorn-Pham singularity of the form $f = x^{p} + y^{q}$ with coprime $p,q > 1$. Define the singular hypersurface $V_{f} = f^{-1}(0)$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f}$, and let $p_{g} = \dim H^{0}(V, \mathcal{O})$ denote... |
Let G be the group U(24) and H the cyclic subgroup generated by the element 7.
Exhibit all of the distinct cosets of H in G.
I am coming up with this.
(I created the Cayley table for U(24) - using the elements 1,5,7,11,13,17,19,23.)
<1> = {1}
<5> = (5*5 = 1) {5}
<7> = (7*7 = 1) {7}
and so forth since e... |
Exhibiting the distinct cosets of a cyclic subgroup H in group G? |
Let $f \colon (\mathbb{C}^{2},\mathbf{0}) \to (\mathbb{C},0)$ be a Brieskorn-Pham singularity of the form $f = x^{p} + y^{q}$ with coprime $p,q > 1$. Define the singular hypersurface $V_{f} = f^{-1}(0)$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f}$, and let $p_{g} = \dim H^{0}(V, \mathcal{O})$ denote... |
what are some *natural* examples of integral domains $A$ together with sequences of elements satisfying $... | f_2 | f_1 | f_0$, but so that $f_i$ divides no power of $f_{i+1}$? So in the spectrum there will be an ascending chain of basic-open subsets.
Of course we could just take the union of the $k[y^{-n} x,y]$ i... |
Let $f \colon (\mathbb{C}^{2},\mathbf{0}) \to (\mathbb{C},0)$ be a Brieskorn-Pham singularity of the form $f = x^{p} + y^{q}$ with coprime $p,q > 1$. Define the singular hypersurface $V_{f} = f^{-1}(0)$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f}$, and let $p_{g} = \dim H^{0}(V, \mathcal{O})$ denote... |
In a calculus course I just completed I struggled to understand the Taylor and Maclaurin series. I got the basics but not some of the harder concepts. Could anyone point me in the right direction? Books, tutorials, etc?
Many thanks. |
Is there any known bound on sum of independent but not identically distributed geometric random variables?
I have to show that the tail of the sum drops exponentially (like in the Chernoff bounds for the sum of iid geom. variables).
Formally, if $X_i ~ Geom(p_i)$, $X = \sum_{i=1}^n X_i$, and $E[X]=\Theta(n)$,
Is... |
Is there any known bound on sum of independent but not identically distributed geometric random variables?
I have to show that the tail of the sum drops exponentially (like in the Chernoff bounds for the sum of iid geom. variables).
Formally, if $X_i \sim Geom(p_i)$, $X = \sum_{i=1}^n X_i$, and $E[X]=\Theta(n)$,
... |
I have posted about induction on it's own so I think I understand that part. But I'm slightly confused when it comes to using induction to show that n+1 is also the derivative. The base case if fine, I just set n=1. And show that is equal to the derivative not using the formula. But how about k+1? Can I just choose k=t... |
Induction to prove a simple formula for calculating the n-th derivative? |
Say I pass 512 samples into my FFT
My microphone spits out data at 10KHz, so this represents 1/20s.
(So the lowest frequency FFT would pick up would be 40Hz).
The FFT will return an array of 512 frequency bins
- bin 0: [0 - 40Hz)
- bin 1: [40 - 80Hz)
etc
So if my original sound contained energy at say... |
Say I pass 512 samples into my FFT
My microphone spits out data at 10KHz, so this represents 1/20s.
(So the lowest frequency FFT would pick up would be 40Hz).
The FFT will return an array of 512 frequency bins
- bin 0: [0 - 40Hz)
- bin 1: [40 - 80Hz)
etc
So if my original sound contained energy at say... |
I am CS major, please be patient if my question is not well-stated.
The dataset is quantitative mass spectrometry (MS) data. By labeling proteins of two different samples A and B, we get the relative abundance of 100 to thousands of proteins in A/B. Alongside with this ratio, we can estimate its variance based on t... |
How to do hypothesis testing on Gaussian mixture model? |
I think it's great that you're seeking a deeper understanding of an important topic-- even after the class is over!
If you haven't done so, go back to your calculus text and carefully read the sections on Taylor and MacLaurin series. (I know math textbooks often seem obtuse and unhelpful, but it's a valuable skil... |
Suppose $h$ is a hash function, $h$ : \{ 0, 1 \} <sup>*</sup> $\rightarrow$ \{ 0, 1 \} <sup>n</sup> and for all $w_1$, $w_2$ it holds: $h(w_1 \oplus w_2) =
h(w_1) \oplus h(w_2)$.
*$\oplus$ is the XOR operation.*
Why isn't $h$ a cryptographically good hash function? |
Why is hash function $h$ ($h(w_1 \oplus w_2) = h(w_1) \oplus h(w_2)$) not good? |
Could you provide a geometric explanation of taylor expansion? |
Intuition explanation of taylor expansion? |
I read a little remark in a topology text that the _direct_ image of an open set under a continuous mapping is not necessarily open.
What is the definition of _direct_ image in this case? I tried googling to no avail, and only found references to sheaf theory on wikipedia, which I don't believe to be what I'm looki... |
Definition of direct image? |
Given an $R$-algebra $S$, is $Hom_{R-Alg}(\prod_{i=1}^n R, S) = Hom_{S-Alg}(\prod_{i=1}^n S, S)$? |
When is $5y - 2$ a square? Modulo 5 this is $5y - 2 = 3$, but squares mod 5 are always 0, 1 or 4 so the equation has no solutions. |
David Einsenbud and Joe Harris' book [The Geometry of Schemes](http://www.amazon.com/Geometry-Schemes-Graduate-Texts-Mathematics/dp/0387986375/ref=sr_1_1?s=books&ie=UTF8&qid=1289223071&sr=1-1) has a large number of really nice pictures which could help you gain a little bit of geometric intuition. That being said, it i... |
The Clifford product of a pair of vectors $a,b$ is an associative operation defined by
$$ ab = a \cdot b + a \wedge b.$$
In sufficiently low dimensions I am used to being able to define the Clifford product on arbitrary $k$-vectors by repeatedly applying the vector definition. For instance, suppose that I build ... |
How do I evaluate the Clifford product in dimensions greater than 3? |
Hint: When is $5y - 2$ a square? Modulo 5 this is $5y - 2 = 3$, but squares mod 5 are always 0, 1 or 4 so the equation has no solutions. |
We know that the higher the degree an equation has the more "turning points" it may have. For example a parabola has one "turning point."
![alt text][1]
(A parabola has an equation of the form $y=ax^2 + bx +c$.)
A cubic of the form $y=ax^3 + bx^2 +cx +d$ can have up to two "turning points," though it may have fe... |
We know that the higher the degree of an equation, the more "turning points" it *may* have. For example, a parabola has one "turning point."
![alt text][1]
(A parabola has an equation of the form $y=ax^2 + bx +c$.)
A cubic of the form $y=ax^3 + bx^2 +cx +d$ can have up to two "turning points," though it may have... |
Why is that true? |
We know that the higher the degree of an equation, the more "turning points" it *may* have. For example, a parabola has one "turning point."
![alt text][1]
(A parabola has an equation of the form $y=ax^2 + bx +c$.)
A cubic of the form $y=ax^3 + bx^2 +cx +d$ can have up to two "turning points," though it may have... |
Why is every group the quotient of a free group with a normal subgroup? |
It's not easy to give a proof without knowing exactly what your
definitions of the Clifford and exterior algebras are, so the
following argument is still a little handwaving. But let's say that we
somehow have defined what the exterior algebra of a vector space $V$
is; we know that its elements are multivectors, an... |
I don't think the series diverges.
$$ \frac{1 \cdot 3 \cdot 7 \cdots (2n-3)}{2 \cdot 4 \cdot 6 \cdots 2n}$$
$$ = \frac{{2n \choose n}}{(2n-1)4^n} = \frac{1}{(2n-1)\sqrt{\pi n}}(1 + O(\frac{1}{n}))$$
using the approximation
$$\frac{ {2n \choose n}}{4^n} = \frac{1}{\sqrt{\pi n}}(1 + O(\frac{1}{n}))$$
and... |
Why is every group the quotient of a free group by a normal subgroup? |
From *[Proofs without Words][1]*.
![alt text][2]
[1]: http://books.google.co.uk/books?id=Kx2cjyzTIYkC&printsec=frontcover&dq=proofs+without+words&hl=en&ei=7RnYTLrRHZnc4wbR-on9Bw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDEQ6AEwAA#v=onepage&q&f=false
[2]: https://i.stack.imgur.com/uW7pG.png |
From *[Proofs without Words][1]*.
![alt text][2]
[1]: http://books.google.co.uk/books?id=Kx2cjyzTIYkC&printsec=frontcover&dq=proofs+without+words&hl=en&ei=7RnYTLrRHZnc4wbR-on9Bw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDEQ6AEwAA#v=onepage&q&f=false
[2]: https://i.stack.imgur.com/uAyOX.png
|
It's not easy to give a proof without knowing exactly what your
definitions of the Clifford and exterior algebras are, so the
following argument is still a little handwaving. But let's say that we
somehow have defined what the exterior algebra of a vector space $V$
is; we know that its elements are multivectors, an... |
Further to the other answers posted, not only does the sum converge, but you can conclude that it converges precisely because of the way it was derived. As you state, the terms $C_n=\frac{1\cdot3\cdot5\cdot7\cdots(2n-3)}{2^nn!}$ arise as the Taylor expansion of $f(x)=\sqrt{x}$ about $x=1$,
$$
\sqrt{1-x}=1-\frac{x}{2}... |
Let $f \colon (\mathbb{C}^{2},\mathbf{0}) \to (\mathbb{C},0)$ be a Brieskorn-Pham singularity of the form $f = x^{p} + y^{q}$ with coprime $p,q > 1$. Define the singular hypersurface $V_{f} = f^{-1}(0)$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f}$, and let $p_{g} = \dim H^{0}(V, \mathcal{O})$ denote... |
From *[Proofs without Words][1]*.
![alt text][2]
[1]: http://books.google.co.uk/books?id=Kx2cjyzTIYkC&printsec=frontcover&dq=proofs+without+words&hl=en&ei=7RnYTLrRHZnc4wbR-on9Bw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDEQ6AEwAA#v=onepage&q&f=false
[2]: https://i.stack.imgur.com/0SBs6.png
[3]: h... |
Let $f \colon (\mathbb{C}^{2},\mathbf{0}) \to (\mathbb{C},0)$ be a Brieskorn-Pham singularity of the form $f = x^{p} + y^{q}$ with coprime $p,q > 1$. Define the singular hypersurface $V_{f} = f^{-1}(0)$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f}$, and let $p_{g} = \dim H^{0}(V, \mathcal{O})$ denote... |
Let $f \colon (\mathbb{C}^{2},\mathbf{0}) \to (\mathbb{C},0)$ be a Brieskorn-Pham singularity of the form $f = x^{p} + y^{q}$ with coprime $p,q > 1$. Define the singular hypersurface $V_{f} = f^{-1}(0)$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f}$, and let $p_{g} = \dim H^{0}(V, \mathcal{O})$ denote... |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n... |
Let $g: \mathbb{C}\times[a,b]\to\mathbb{R}$ a continuous function and
$$g(t)=\lim_{h\to h_0} g(h,t) \ \forall t\in \mathbb{R}$$
Is the following result/reasoning correct ? If we have a sequence of functions, we need uniform convergence to pass the limit inside the integral, but it seems it works everytime here. O... |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n... |
A set M which is an element of the Integers^2 (pair) is defined by
(Rule 1) (0, 1) $\in$ M
(Rule 2) If (x, y) $\in$ M, then (x + 1, y + 2x + 3) $\in$ M
(a) (2 points) Starting with (0, 1), write out the six pairs with the smallest first coordinates.
(b) (2 points) State the (simple) relationship that holds ... |
A set M $\in$ $\mathbb{Z}^2$ is defined by
(Rule 1) $(0, 1)$ $\in$ M
(Rule 2) If $(x, y)$ $\in$ M, then $(x + 1, y + 2x + 3)$ $\in$ M
(a) (2 points) Starting with $(0, 1)$, write out the six pairs with the smallest first coordinates.
(b) (2 points) State the (simple) relationship that holds between the firs... |
This is one of the most intuitive observations in all of group theory, and it illustrates the quotient operation in the most fundamental way.
I'll provide two separate answers. The first is fully intuitive; the second is a formalized version of the first.
**First answer:** Take a group $G$. A relation on $G$ is a... |
A set M $\in$ $\mathbb{Z}^2$ is defined by
(Rule 1) $(0, 1)$ $\in$ M
(Rule 2) If $(x, y)$ $\in$ M, then $(x + 1, y + 2x + 3)$ $\in$ M
(a) (2 points) Starting with $(0, 1)$, write out the six pairs with the smallest first coordinates.
(b) (2 points) State the (simple) relationship that holds between the firs... |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n... |
A set M $\in$ $\mathbb{Z}^2$ is defined by
(Rule 1) $(0, 1)$ $\in$ M
(Rule 2) If $(x, y)$ $\in$ M, then $(x + 1, y + 2x + 3)$ $\in$ M
(a) (2 points) Starting with $(0, 1)$, write out the six pairs with the smallest first coordinates.
(b) (2 points) State the (simple) relationship that holds between the firs... |
A set M $\in$ $\mathbb{Z}^2$ is defined by
(Rule 1) $(0, 1)$ $\in$ M
(Rule 2) If $(x, y)$ $\in$ M, then $(x + 1, y + 2x + 3)$ $\in$ M
(a) (2 points) Starting with $(0, 1)$, write out the six pairs with the smallest first coordinates.
(b) (2 points) State the (simple) relationship that holds between the firs... |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n... |
This is one of the most intuitive observations in all of group theory, and it illustrates the quotient operation in the most fundamental way.
I'll provide two separate answers. The first is fully intuitive; the second is a formalized version of the first.
**First answer:** Take a group $G$. A relation on $G$ is a... |
<a href="http://mathworld.wolfram.com/CatalansConstant.html">Quite</a> a <a href="http://mathworld.wolfram.com/MertensConstant.html">few</a> mathematical constants are known to arise in several branches of mathematics (more <a href="http://en.wikipedia.org/wiki/Euler_product">here</a>). I have no doubt that they are u... |
Suggesting closed-form representations of mathematical constants by means of experimental mathematics? |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n... |
Have you tried the [Inverse Symbolic Calculator][1]?
[1]: http://oldweb.cecm.sfu.ca/projects/ISC/ISCmain.html |
<a href="http://mathworld.wolfram.com/CatalansConstant.html">Quite</a> a <a href="http://mathworld.wolfram.com/MertensConstant.html">few</a> mathematical constants are known to arise in several branches of mathematics (more <a href="http://en.wikipedia.org/wiki/Euler_product">here</a>). I have no doubt that they are u... |
<a href="http://mathworld.wolfram.com/CatalansConstant.html">Quite</a> a <a href="http://mathworld.wolfram.com/MertensConstant.html">few</a> mathematical constants are known to arise in several branches of mathematics (more <a href="http://en.wikipedia.org/wiki/Euler_product">here</a>). I have no doubt that they are u... |
How to prove $\limsup(\{A_n\} \cup \{B_n\}) = \limsup(\{A_n\}) \cup \limsup(\{B_n\})$? Thanks!
|
Let $g: \mathbb{C}\times[a,b]\to\mathbb{R}$ a continuous function and
$$g(t)=g(h_0,t)=\lim_{h\to h_0} g(h,t) \ \forall t\in \mathbb{R}$$
Is the following result/reasoning correct ? If we have a sequence of functions, we need uniform convergence to pass the limit inside the integral, but it seems it works everytim... |
How to prove $\limsup(\\{A_n \cup B_n\\}) = \limsup(\\{A_n\\}) \cup \limsup(\\{B_n\\})$? Thanks!
|
The usual symbol for the cardinality of the continuum (i.e. the real numbers) is Fraktur $\mathfrak{c}$. However, I recall some sources also using $\aleph$ (with no subscript). This usage is not mentioned in Wikipedia or Mathworld, but I found some support for it over Google.
Is the $\aleph$ notation standard? |
Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. I want to show that if $X$ is isomorphic to a subset of Y then $X$ is isomorphic with an initial segment of $Y$. (The other direction is of course true).
I would appreciate some hints. Does this require the recursion theorem to construct the isomorphism? |
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n... |
Writing $p=a+b$ you have the cubic equation,
$$
p^3+p-2b=0.
$$
This is already in "depressed cubic" form (no $p^2$ term), so it can be solved directly by [standard methods][1]. The coefficient of p is positive, so it is strictly increasing and there will be a single real root.
$$
p = \sqrt[3]{\sqrt{b^2+1/27}+b}... |
Writing $p=a+b$ you have the cubic equation,
$$
p^3+p-2b=0.
$$
This is already in "depressed cubic" form (no $p^2$ term), so it can be solved directly by [standard methods][1]. The coefficient of p is positive, so it is strictly increasing and there will be a single real root.
$$
p = \sqrt[3]{\sqrt{b^2+1/27}+b}-\... |
If $a$ and $b$ are positive real numbers, and if $f(x)$ has the following asymptotic property
$f(x) = \frac{a + O(\frac{1}{\sqrt{x}})}{b + O(\frac{1}{\sqrt{x}})}$
then is the following true?
$f(x) = \frac{a}{b} + O(\frac{1}{\sqrt{x}})$
This might look like homework but it isn't. |
How to calculate daily user churn rate? |
We know that 2^4 = 4^2. Is there another pair of integers (x, y), which satisfies the equality **x^y = y^x**?
|
I can't resist giving a couple of examples of (contrived) numerical coincidences (both are from [*Experimentation in Mathematics*][1] by J.M. Borwein, D.H. Bailey and R. Girgensohn).
Example 1.
$$\int_{0}^{\infty}\cos(2x)\prod_{k=1}^{\infty}\cos(x/k)\ dx=\frac{\pi}{8}+\epsilon,$$
where
$0<\epsilon<10^{-41}.$
... |
I can't resist giving a couple of examples of (contrived) numerical coincidences (both are from [*Experimentation in Mathematics*][1] by J.M. Borwein, D.H. Bailey and R. Girgensohn).
Example 1.
$$\int_{0}^{\infty}\cos(2x)\prod_{k=1}^{\infty}\cos(x/k)\ dx=\frac{\pi}{8}+\epsilon,$$
where
$0<\epsilon<10^{-41}.$
... |
I need to write a coupon code system but I do not want to save each coupon code in the database. (For performance and design reasons.) Rather I would like to generate codes subsequent that are watermarked with another code.
They should like kind of fancy and random. Currently they look like this:
1: AKFCU2, 2: AK... |
Winding yarn into a ball suggests some mathematical questions:
<br />![Yarn][1]<br />
<ol>
<li>
Under some natural model, what paths should the yarn follow to
achieve the most dense ball?
One model is that used by
Henryk Gerlach and Heiko von der Mosel in their paper
"On sphere-filling ropes"
[arXiv:1005.4609... |
Winding yarn into a ball suggests some mathematical questions:
![Yarn][1]
Under some natural model, what paths should the yarn follow to achieve the most dense ball?
One model is that used by Henryk Gerlach and Heiko von der Mosel in their paper
"On sphere-filling ropes" [arXiv:1005.4609v1 (math.GT)][2].
(... |
The unit doublet is a symbolic object whose convolution with a differentiable function is supposed to give the derivative:
$$(x * u_1)(t) = \frac{dx(t)}{dt}$$
See also: http://en.wikipedia.org/wiki/Unit_doublet
Can this equation be made rigorous, in a way that it would be valid for all $x\in C^1(\mathbb R)$? ... |
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... |
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