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The coordinate transformation (due to Beukers, Calabi and Kolk)
$$x=\frac{\sin u}{\cos v}$$
$$y=\frac{\sin v}{\cos u}$$
transforms the square domain $0\lt x\lt 1$ and $0\lt y\lt 1$ into the triangle domain $u,v>0,u+v<\pi /2$ (in *[Proofs from the BOOK][1]* by M. Aigner and G. Ziegler).
Since the invert tran... |
I know that there was a guy that could get 100 decimal digits of $\pi$ before computers were able to get thousands.
How did the guy do that? |
Let $L,K$ be number fields and $L|K$ a galois extension.
Let $(0)\neq Q$ a prime ideal in $\mathcal O_L$ (=ring of integers in $L$) and $P=Q \cap \mathcal O_K$.
$Z_Q $ denotes the decomposition field of $Q$ and $T_Q$ denotes the inertia field of $Q$.
Now put $Q' :=Q\cap Z_Q$ and $Q'' :=Q\cap T_Q$.
How does o... |
A student came to me showing a question from his exam in basic group theory, in which they are asked to prove that $\mathbb{C}^*$ modulo the subgroup of roots of unity is isomorphic to $\mathbb{R}^+$ (in both cases we mean the multiplicative groups).
Now this seems to me to be a simple error in the question. I belie... |
Is $\mathbb{C}^*$ modulo the roots of unity isomorphic to $\mathbb{R}^+$? |
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... |
Algebraic structure cheat sheet anyone ? |
suppose, $f(x,y)$ is a bounded continuous function on $\mathbb R ^2$. Inspect
$\lim _{ y \rightarrow y' } sup_ {x \in \mathbb R } f(x,y)$.
In how far can you switch supremum and limes, or which possible term comes closest to switching these two? Ideally, there would be something like
$sup_ {x \in \mathbb R } \... |
I have an idea for solving the knapsack problem, but it looks too good to be true. I would like someone to explain potential problems with this approach. I'll give an example: I want to find a subset of {2,7,11} which has elements that sum to 13. Here is the algorithm for solving this:
In binary notation 2=0010, 7=0... |
suppose, $f(x,y)$ is a bounded continuous function on $\mathbb{R}^2$. Consider
$$\lim_{ y \rightarrow y' }\> \sup_{x \in \mathbb{R}} f(x,y).$$
In how far can you switch suprema and limits, or which possible term comes closest to switching these two? Ideally, there would be something like
$$\sup_{x \in \mathbb{R}}\... |
Your sum has both real and imaginary parts. Because
$$e^{ix}= \cos(x) + i \sin(x) \; .$$
Is $f(a,b)$ in your formula a real number? Then the transform has real and imaginary components in general. (For certain choices of $f(a,b)$, you can have purely real numbers.)
If you mean $e^{-i}$, then use the formula I ... |
Update: Steve points out in comments that many direct products of residually cyclic groups will be counterexamples. Indeed, I should have spotted this: every finitely generated abelian group is residually cyclic! I should have asked about **fully** residually cyclic groups, ie groups G such that for any finite subs... |
If G is fully residually cyclic, does G have at most one subgroup of each finite index? |
Just a quick question about the geometry of Hilbert spaces from an intuitive standpoint. Maybe just assuming we're working with $L^2$ would simplify the situation. Basically, in something like $\mathbb{R}^2$ we have the situation that $\cos(\theta)=\frac{\langle a,b\rangle}{\vert a\vert\cdot\vert b\vert}$, and the ide... |
One need only consult the history of algebra to find many informal
discussions on the existence and consistence of complex numbers.
Any *informal* attempt to justify the existence of $\mathbb C$
will face the same obstacles that existed in earlier times. Namely, the lack of any rigorous (set-theoretic) foundation... |
When ever I find definition of the quantum plane it says $A_q^2 = C<x,y>/I$, where $I = C<xy-qyx>$. What I want to know is whether they mean the unital free algebra or just the free algebra. In brief, is the quantum plane unital?
Moreover, when people write $A_q^N$, they mean the free (unital) algebra with $N$ gener... |
Prove or disprove:
Let $\rho : \mathbb{N} \rightarrow \mathbb{N}$ injective. Let $(a_{n})_{n \in \mathbb{N}}$ be a sequence.
(i) If $\displaystyle\sum\limits_{n=1}^\infty a_{n}$ absolutely converges then $\displaystyle\sum\limits_{n=1}^\infty a_{\rho(n)}$ also converges absolutely.
(ii) If $\displaystyle\sum\l... |
Let $G$ be a group and $G'$ its commutator subgroup. Let $\pi: G\to G/G'$ be the natural projection.
Statement 1: $G/G'$ is the largest Abelian quotient of $G$ in the sense that if $H\unlhd G$ and $G/H$ is Abelian, then $G'\le H$. Conversely, if $G'\le H$, then $H\unlhd G$ and $G/H$ is Abelian.
Statement 2: If $... |
These two group theory statements are "the same"? |
I have an idea for solving the knapsack problem, but it looks too good to be true. I would like someone to explain potential problems with this approach. I'll give an example: I want to find a subset of {2,7,11} which has elements that sum to 13. Here is the algorithm for solving this:
In binary notation 2=0010, 7=0... |
By the Homomorphism Theorem, any homomorphism $f\colon G\to K$ factors through $G/\mathrm{ker}f$, meaning that there is a map $\hat{f}\colon G/\mathrm{ker}f \to K$ such that $ f = \hat{f}\pi$. The map is indeed $\hat{f}(g\,\mathrm{ker}f) = f(g)$. This applies to the specific case given in Statement 2.
Why is 2 equiv... |
By the Homomorphism Theorem, any homomorphism $f\colon G\to K$ factors through $G/\mathrm{ker}f$, meaning that there is a map $\hat{f}\colon G/\mathrm{ker}f \to K$ such that $ f = \hat{f}\pi$. The map is indeed $\hat{f}(g\,\mathrm{ker}f) = f(g)$. This applies to the specific case given in Statement 2.
Why is 2 equiv... |
By the Homomorphism Theorem, any homomorphism $f\colon G\to K$ factors through $G/\mathrm{ker}f$, meaning that there is a map $\hat{f}\colon G/\mathrm{ker}f \to K$ such that $ f = \hat{f}\pi$. The map is indeed $\hat{f}(g\,\mathrm{ker}f) = f(g)$. This applies to the specific case given in Statement 2.
Why is 2 equiv... |
Say I have a numeric sequence A and a set of sequences B that vary with time.
I suspect that there is a relationship between one or more of the B sequences and sequence A, that changes in Bn are largely or wholly caused by changes in sequence A. However there is an unknown time delay between changes in A and their... |
**The context:**
I'm working on geocoded documents. This is, documents have latitude and longitude attributes, as well as some other geo attributes such as an address. Right now, I'm performing text search operations on the indexed docs and showing their geolocation on a map.
**The problem:**
Most of search crit... |
What exactly is "Approximation Theory"? If I read the wikipedia-article I doesn't get much clearer. Why are "pure" mathematicians interested in it? I see a lot of people that do harmonic analyse also do approximation theory. |
What exactly is "Approximation Theory"? If I read the wikipedia-article I doesn't get much clearer. Why are "pure" mathematicians interested in it? I see a lot of people that do harmonic analysis also do approximation theory. |
Is everything in mathematics built on add, subtract, multiply and divide? |
We can use your second estimate
$$\sum\limits_{p \leq x} \frac{1}{p} = \log{\log{x}} + A + R(x)$$
where $\displaystyle R(x) = \mathcal{O}\Bigl(\frac{1}{\log{x}}\Bigr)$
to show the asymptotic estimate:
$$\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} = \mathcal{\theta}\left(\frac{\sqrt{x}}{\log x}\right)$$
For... |
$S$ is a collection of disjoint sets. $(S,\cdot)$ is a commutative monoid and $(S,*)$ is a commutative semigroup. The identity element $e$ of $(S,\cdot)$ is the zero element of $(S,*)$.
The monoid $(S,\cdot)$ does not have a zero element. The binary operator $'*'$ is distributive over the binary operator $'\cdot'$. Wh... |
We can use your second estimate
$$\sum\limits_{p \leq x} \frac{1}{p} = \log{\log{x}} + A + R(x)$$
where $\displaystyle R(x) = \mathcal{O}\Bigl(\frac{1}{\log{x}}\Bigr)$
to show the asymptotic estimate:
$$\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} = \mathcal{\theta}\left(\frac{\sqrt{x}}{\log x}\right)$$
In ... |
We can use your second estimate
$$\sum\limits_{p \leq x} \frac{1}{p} = \log{\log{x}} + A + R(x)$$
where $\displaystyle R(x) = \mathcal{O}\Bigl(\frac{1}{\log{x}}\Bigr)$
to show the asymptotic estimate:
$$\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} = \mathcal{\theta}\left(\frac{\sqrt{x}}{\log x}\right)$$
In ... |
Approximation theory includes many subject areas of analysis, but the common idea is how well a target in a topological space (often a metric space) can be approximated by the points of a narrower subspace.
Some examples will illustrate the breadth of this topic. Given a real number x of a certain kind (e.g. algebr... |
I'm currently looking at the equation for a multiclass classifier of the form
$$\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/>
$N_I$ is the number of instances, <br/>
$N_K$ is the number of classes, <br/>
$N_... |
To construct a multivariate b-spline, we simply take the kronecker tensor product between the univariate basis functions constructed for each individual dimension.
What i'd like to know is how do you construct a b-spline derivative for a multivariate function? Is it the same method of using the tensor product, but w... |
To construct a multivariate B-spline, we simply take the Kronecker tensor product between the univariate basis functions constructed for each individual dimension.
What I'd like to know is how do you construct a B-spline derivative for a multivariate function? Is it the same method of using the tensor product, but w... |
What might be an even better question is How can I remember all of the different common algebraic structures? How do I keep track of all of the "different" definitions?
I use quotes because most of the definitions are very similar.
It is important that we understand homomorphisms : whenever someone says homomorphis... |
What might be an even better question is How can I remember all of the different common algebraic structures? How do I keep track of all of the "different" definitions?
I use quotes because most of the definitions are very similar.
It is important that we understand homomorphisms : whenever someone says homomorphis... |
I have an idea for solving the knapsack problem, but it looks too good to be true. I would like someone to explain potential problems with this approach. I'll give an example: I want to find a subset of {2,7,11} which has elements that sum to 13. Here is the algorithm for solving this:
In binary notation 2=0010, 7=0... |
This is an interesting rhetorical question. On the one hand, many theoretical mathematical concepts transcend arithmetic. However, computer algebra systems (such as Mathematica) are able to express answers to those sorts of questions using a CPU that performs a limited range of simple computations. So the answer is a m... |
I understand Lie groups are defined by the structure constants associated with the lie brackets, which are treated as commutators in quantum mechanics, but i dont know of a math theory related to group theory to define or use an anti commutator. If Lie groups theory uses the commutator, what theory uses the anti commut... |
What is the formal mathematical theory behind the concept of the anti commutator used to quantize fermions? |
This is in many places. I think that the whole point of model categories is that people saw you could get a lot of mileage out of the adjunction between Top and sSet (geometric realization and singular simplices). Just how good is the combinatorial model for topological spaces? The answer is that it is a Quillen equiva... |
I'm not sure how one sees that finding solutions to the linear system over {0,1} can be accomplished in polynomial time. Reducing the matrix to row-echelon form is not sufficient for finding such solutions. Suppose that the reduced system has a parametrized solution set, as happens in your example. Is there any way,... |
I want to why $\mathbb{Z}[x]/(1-x,p)$ is isomorphic to $\mathbb{Z_{p}}$ where p is a prime integer?
Here's what I have so far, but I am unsure if I am correct. Every $f\in \mathbb{Z}[x]$ can be written as $(1-x)q+ r$ where $q\in \mathbb{Z}[x]$ and $r$ is in $\mathbb{Z}$. Does It follows that there are $p$ cosets... |
Assuming I have linear equatiotion system
The solution is x1=x2=x3...=xn
What should be matrix properties or how do I prove that the system have such solution? |
What does $\mathbb{Z}[\omega]$ usually mean when $\omega$ is a primitive root of unity? |
Just to add something to the nice answers you already have. In fact this is a special case of a more general proposition. It can be shown that all the maximal ideals $\mathfrak{m}$ of the polynomial ring $\mathbb{Z}[x]$ have the form $\mathfrak{m} = (g(x), p)$ where $p \in \mathbb{Z}$ is a prime number and $g(x) \in \m... |
The idea of a read-only right-moving Turing Machine already exists [1].
Is there a relaxed version of this where an input-reading head only moves
right to read inputs? There may be other heads doing the computation
on work tapes; the movement of these heads is unrestricted.
I'm assuming this machine is Turing c... |
Have you tried an inductive proof?
1. Show 0+1=1+0
2. Suppose k+1=1+k. Use this to show k+1+1=1+(k+1). Hint: Apply ∀x ∀y [x + (y+1)= (x+y)+1] twice.
|
Assume I have linear equation system in $n$ variables and the solution is $x_1=x_2=\cdots = x_n$.
What properties must the matrix of the system have, or how do I prove that the system has such a solution? |
Suppose I have a function $f \in L^2(R^n)$ with compact support, and a mollifier $\phi(x) \in C^\infty(R^n)$ with compact support s.t. $\int \phi(x) dx = \int \phi_\epsilon(x) = 1$ and $lim_{\epsilon \to 0} \phi_\epsilon(x) = \delta(x)$ where $\phi_{\epsilon}(x) = \epsilon^{-n} \phi(x/\epsilon)$ and $\delta(x)$ denotes... |
Earlier I had posted question about Borel cantelli Lemma. Thanks to Nate and Shai for your kind response. Please see my profile for the earlier questions. As you can see I do not understand the lemma well. Here are some confusions I have. I hope someone will be kind enough to help me understand the following:
Is it ... |
The idea of a read-only right-moving Turing Machine already exists [1].
Is there a relaxed version of this where an input-reading head only moves
right to read inputs as needed? There may be other heads doing the computation
on work tapes; the movement of these heads is unrestricted.
Of course the right-moving h... |
I like to understand how things work. It's the only way I'll remember the rule. If anyone has a way of breaking this down into logical steps to get from one to the other... it would be greatly appreciated.
why is
$X^n\cdot X^m = X^{n+m}$ ?
|
![alt text][1]
[1]: https://i.stack.imgur.com/xLuHb.png
For part b, I'm getting a little stuck. So I'm trying to show that if $(x,y,r)R(a,b,s) and (a,b,s)R(x,y,r), then (x,y,r)=(a,b,s)$
And so $(x,y,r)R(a,b,s) implies \sqrt[]{(x-a)^2+(y-b)^2} \leq s-r$
and $(a,b,s)R(x,y,r) implies \sqrt[]{(a-x)^2+(b-y)^2}... |
![alt text][1]
[1]: https://i.stack.imgur.com/xLuHb.png
For part b, I'm getting a little stuck. So I'm trying to show that if $(x,y,r)R(a,b,s)$ and $(a,b,s)R(x,y,r)$, then $(x,y,r)=(a,b,s)$
And so $(x,y,r)R(a,b,s)$ implies $\sqrt{(x-a)^2+(y-b)^2} \leq s-r$
and $(a,b,s)R(x,y,r)$ implies $\sqrt{(a-x)^2+(b-y... |
Part (a) would give you the intuition. But the first thing to notice here (even before you notice that the square roots have the same value) is that since the square root is always nonnegative, the first condition gives you that $s-r\geq 0$ and the second gives you that $r-s\geq 0$. Those two together tell you *what* a... |
Hello all this is my first math question, I hope this is the right place to post.
Anyway, this is basic stuff but I'm a bit confused here, here's the problem's legend:
"Find the equation of the plane that passes through the intersection line ($l$) of the planes $x-y-z=2, x+y-3z+4=0$ and has a distance of 2 from the o... |
Given a number of 3in squares and 2in squares, how many of each are needed to get a total area of 35 in^2?
Through quick trial and error (the method they wanted I believe) you find that you need 3 3in squares and 2 2in squares, but I got to thinking on how to solve this exactly.
You have 2 unknowns and the follow... |
Part (a) would give you the intuition. But the first thing to notice here (even before you notice that the square roots have the same value) is that since the square root is always nonnegative, the first condition gives you that $s-r\geq 0$ and the second gives you that $r-s\geq 0$. Those two together tell you *what* a... |
I have read [this paper][1] and the wikipedia summary [here][2] and even [this homework problem][3], but I do not understand how 2SAT is in P.
Here is my concern: the idea is that we start by setting a random variable. Then we set any implied variables. Repeat, and if we eventually reach a contradiction, then the p... |
Hello all this is my first math question, I hope this is the right place to post.
Anyway, this is basic stuff but I'm a bit confused here, here's the problem's legend:
"Find the equation of the plane that passes through the intersection line ($l$) of the planes $2x-y-z=2, x+y-3z+4=0$ and has a distance of 2 from the ... |
Hello all this is my first math question, I hope this is the right place to post.
Anyway, this is basic stuff but I'm a bit confused here, here's the problem's legend:
"Find the equation of the plane that passes through the intersection line ($l$) of the planes $2x-y-z=2, x+y-3z+4=0$ and has a distance of 2 from the ... |
How to find the least positive root of the equation $\cos 3x + sin 5x = 0$?
My approach so far is to represent $\sin 5x$ as $\cos \biggl(\frac{\pi}{2} - 5x\biggr)$ then the whole equation reduces to $$2\cos \biggl(\frac{\pi}{4} - x\biggr)\cdot \cos \biggl(\frac{\pi}{4} - 4x\biggr) = 0$$
But am not much sure how t... |
How to find the least positive root of the equation $\cos 3x + \sin 5x = 0$?
My approach so far is to represent $\sin 5x$ as $\cos \biggl(\frac{\pi}{2} - 5x\biggr)$ then the whole equation reduces to $$2\cos \biggl(\frac{\pi}{4} - x\biggr)\cdot \cos \biggl(\frac{\pi}{4} - 4x\biggr) = 0$$
From here we can write:
... |
Prove or disprove:
Let $\rho : \mathbb{N} \rightarrow \mathbb{N}$ injective. Let $(a_{n})_{n \in \mathbb{N}}$ be a sequence.
(i) If $\displaystyle\sum\limits_{n=1}^\infty a_{n}$ absolutely converges then $\displaystyle\sum\limits_{n=1}^\infty a_{\rho(n)}$ also converges absolutely.
(ii) If $\displaystyle\sum\l... |
I'm reading a paper on fluid dynamics and it references a unit vector between two particles i and j. I'm not clear what it means by a unit vector in this instance. How do I calculate the unit vector between the two particles? |
How do you calculate the unit vector between two points? |
Hi I have a problem with quadratic splines, I am supposed to find $
S_1 $ and $S_2$ that interpolates the following points $S(-1)=0$, $S(0)=1$, $S(1)=2$, and at the same time we want to find $S$ such that $\int_{-1}^1 \! (S(x))^2 \, \mathrm{d}x$ is minimal. The answer is on the form
$S_1(x)=a_1 x^2 +b_1x +c_1$ on... |
If someone gives you a random large number, the last thing you want to do is perform a deterministic primality test -- it's very likely to be composite. Firstly, you should perform:
- trial division and
- [Fermat's test][1].
These can be performed using PFGW (from http://tech.groups.yahoo.com/group/primeform/... |
We can use your second estimate
$$\sum\limits_{p \leq x} \frac{1}{p} = \log{\log{x}} + A + R(x)$$
where $\displaystyle R(x) = \mathcal{O}\Bigl(\frac{1}{\log{x}}\Bigr)$
to show the asymptotic estimate:
$$\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} = \mathcal{\theta}\left(\frac{\sqrt{x}}{\log x}\right)$$
In ... |
if $\displaystyle \int_{1}^{\infty}f(x)dx$ converges, must $\displaystyle \int_{1}^{\infty}f(x)\sin xdx$ converge? |
Can we alter [Hilbert's axioms][1] to have $\mathbb{Q}^3$ as a unique model?
The critical axioms seem to be the [congruence axioms][2] IV.1 and IV.4, and presumably the [line completeness axiom][3] V.2.
But how are they to be modified?
IV.1 might be replaced by requiring that there are counter-examples (irrat... |
Can we alter Hilbert's axioms to have $\mathbb{Q}^3$ as a unique model? |
It can be proved that
$$\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}}.$$
This formula for $r_{n}$ provides a method for finding the estimate
$$\frac{1}{12\le... |
If your predictions are repetitive I think you can do this. For example, if you always predict 50% chance of one choice, 30% for a second and 20% for the third (even if they are permuted on a particular play) you can do a standard chi-square test. Did you in fact get 50% right on the one you claim?
If your predict... |
I have read [this paper][1] and the wikipedia summary [here][2] and even [this homework problem][3], but I do not understand how 2SAT is in P.
Here is my concern: the idea is that we start by setting a random variable. Then we set any implied variables. Repeat, and if we eventually reach a contradiction, then the p... |
Just to add something to the nice answers you already have. In fact this is a special case of a more general proposition. It can be shown that all the maximal ideals $\mathfrak{m}$ of the polynomial ring $\mathbb{Z}[x]$ have the form $\mathfrak{m} = (g(x), p)$ where $p \in \mathbb{Z}$ is a prime number and $g(x) \in \m... |
It can be proved 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 for finding the estimate
$$\frac{1}{12\le... |
if $\int_1^{\infty}f(x)\mathrm dx$ converges, must $\int_1^{\infty}f(x)\sin x\mathrm dx$ converge? |
Here's a paper [Stirling's Series Made Easy][stirling] by Chris Impens that states in Corollary 3
$$ \frac{1}{12n} - \frac{1}{360n^3} < r_n < \frac{1}{12n}$$
and (the interesting part) that the constant $1/360$ cannot be improved. I confess to not having
read the paper closely.
[Stirling]: http://biblio.ugent... |
Is there any way to decipher, manually of course , whether a number (may be large enough) is a prime? |
How do you determine if a number is prime or composite? |
Say I have a numeric sequence A and a set of sequences B that vary with time.
I suspect that there is a relationship between one or more of the B sequences and sequence A, that changes in Bn are largely or wholly caused by changes in sequence A. However there is an unknown time delay between changes in A and their... |
Set $S$ is a collection of disjoint sets each having cardinality of that of $\mathbb{R}$.The cardinality of set $S$ is also that of $\mathbb{R}$. $F$ is the set which is union of all the elements of $S$. Is the cardinality of set $F$ is equal to that of $\mathbb{R}$ ? |
It is proved in *Advanced Calculus* by Angus Taylor, § 20.8, that
$$\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}}.$$
This formula for $r_{n}$ provides a method ... |
It is proved in *Advanced Calculus* by Angus Taylor, § 20.8, that
$$\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}}.$$
This formula for $r_{n}$ provides a method ... |
We can use your second estimate
$$\sum\limits_{p \leq x} \frac{1}{p} = \log{\log{x}} + A + R(x)$$
where $\displaystyle R(x) = \mathcal{O}\Bigl(\frac{1}{\log{x}}\Bigr)$
to show the asymptotic estimate:
$$\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} = \mathcal{\theta}\left(\frac{\sqrt{x}}{\log x}\right)$$
In ... |
Your answer is mathematically correct, however i figured out another way to solve this equation, which leads to a simpler result.
I applied the technique for 2D projective geometry, which is described [here][1] to the 1D case and it works out fine.
[1]: http://books.google.de/books?id=si3R3Pfa98QC&lpg=PP1&pg=... |
Your answer is mathematically correct, however I figured out another way to solve this equation, which leads to a simpler result.
I applied the technique for 2D projective geometry, which is described [here][1] to the 1D case and it works out fine.
[1]: http://books.google.de/books?id=si3R3Pfa98QC&pg=PA35 |
We can use your second estimate
$$\sum\limits_{p \leq x} \frac{1}{p} = \log{\log{x}} + A + R(x)$$
where $\displaystyle R(x) = \mathcal{O}\Bigl(\frac{1}{\log{x}}\Bigr)$
to show the asymptotic estimate:
$$\sum\limits_{p \leq x} \frac{1}{\sqrt{p}} = \mathcal{\theta}\left(\frac{\sqrt{x}}{\log x}\right)$$
In ... |
$S$ is a collection of disjoint sets. $(S,\cdot)$ is a commutative monoid and $(S,*)$ is a commutative semigroup. The identity element $e$ of $(S,\cdot)$ is the zero element of $(S,*)$.
The monoid $(S,\cdot)$ does not have a zero element. The binary operator $'*'$ is distributive over the binary operator $'\cdot'$. Wh... |
In the following picture, the author of the *Field and Wave Electromagnetics* demonstrates that geometrical meaning of the direction of the gradient. Only follow the direction of the normal vector to the curve at that pointer could the rate of change be the maximum.
![alt text][1]
But what about the geometrical... |
What's the geometrical interpretation of the magnitude of gradient generally? |
In the following picture, the author of the *[Field and Wave Electromagnetics][1]* shows the geometrical meaning of the direction of the gradient. Only by following the direction of the normal vector to the curve at that pointer could the rate of change be the maximum.
![alt text][2]
But what about the geometri... |
In the following picture, the author of the *[Field and Wave Electromagnetics][1]* shows the geometrical meaning of the direction of the gradient. That is, only by following the direction of the normal vector to the curve at that pointer could the rate of change be the maximum.
![alt text][2]
But what about the... |
I have another question about correctness of my proof. We know, that $N$ and $G/N$ are p-groups, than $G$ is p-group.
Proof: $|N|=p^m$ and thus $|G|=p^m q$. Suppose that $(p^m,q)=1$, than in $G$ exists element of order $k$, such that $k | q$ and $(k,p^m)=1$ - suppose it's $b \in G$. Element $bN \in G/N$ is also of o... |
In the following picture, the author of the *[Field and Wave Electromagnetics][1]* shows the geometrical meaning of the direction of the gradient. That is, only by following the direction of the normal vector to the curve at that pointer could the rate of change be the maximum.
![alt text][2]
But what about the... |
My question is similar to [this][1], but not identical. I believe the following to be true, but I'd like a reference.
Given (quasicoherent?) sheaves of $\mathcal O_X$ modules $E$ and $F$ on a projective variety $X$,
$$
H^n(X, E\otimes F) \cong \bigoplus_{p+q=n} H^p(X, E) \otimes H^q(X, F).
$$
> Is this true,... |
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