earth - Asteroids in langrangian Points 4 & 5

There are Asteroids "trapped" in Jupiters Langrange points 4 and 5 called trojans and greeks. Are there any asteroids in the earths L4 and L5? Have we seen asteroids in Lagrange points of the earth and moon system? Why are only L4 and L5 stable and L1, L2 and L3 require corrections to remain at those positions?

nt.number theory - Algorithms for Diophantine Systems

No. Given any set of diophantine equations $f_1(z_1, ldots, z_n) = ldots = f_m(z_1, ldots, z_n)=0$, we can rewrite in terms of linear equations and quadratics. Create a new variable $w_{k_1 cdots k_n}$ for each monomial $z_1^{k_1} cdots z_n^{k_n}$ which occurs in the $f$'s, or which divides any monomial which occurs in the $f$'s. Turn each $f$ into a linear equation: For example, $x^3 y^2 + 7 x^2 y=5$ becomes $w_{32} + 7 w_{21} = 5$. Then create quadratic equations $z_i w_{k_1 cdots k_i cdots k_n} = w_{k_1 cdots (k_i +1) cdots k_n}$. For example, $x w_{22} = w_{32}.$ This shows that the solvability of Diophantine equations is equivalent to that of Diophantine equations of degree $leq 2$.

I'll also mention a very concrete case. The intersection of two quadrics in $mathbb{P}^3$ is a genus $1$ curve. To my knowledge, no algorithm is known to test for the existence of rational points even in this case. (But my knowledge is not very large.)

ag.algebraic geometry - "Algebraic" topologies like the Zariski topology?

Interesting questions.
Actually, this is indeed related to work on defining a natural topology on categories, which is part of noncommutative algebraic geometry.

A. Rosenberg defined the left spectrum for a noncommutative ring in 1981 (see The left spectrum, the Levitzki radical, and noncommutative schemes), and further generalized this spectrum to any abelian category (see reconstruction of schemes), and proved the so called Gabriel-Rosenberg reconstruction theorem which led to the correct definition of noncommutative scheme. I might have time to talk about this later. But for now, I shall just point out some papers, such as Spectra of noncommutative spaces.

In this paper, Rosenberg takes an abelian category as a "noncommutative space" and defines various spectra for different goals. (ONE remarkable destination is for representation theory of Lie algebras and quantum groups.)

One can not only define spectrum for abelian categories; this notion also makes sense in a non-abelian category and a triangulated category. In the paper Spectra related with localizations, Rosenberg defined the spectrum directly related to localization of categories. Roughly speaking, the spectrum of a category is a family of topologizing subcategories (which by definition, are closed under direct sum, sub- and quotient; in particular, thick or Serre subcategories) satifying some additional conditions.

There is also another paper, Underlying spaces of noncommutative schemes, trying to investigate the underlying space of a noncommutative scheme or other noncommutative "space" in noncommutative algebraic geometry. If we want to save flat descent in general, we might lose the base change property. In this work, Rosenberg deals with the "quasi-topology" (which means dropping the base change property) and defines the associative spectrum of a category.
Moreover: for the goals of representation theory, he built a framework relating representation theory with the spectrum of abelian category (in particular, categories of modules). Actually, in this language, irreducible representations are in one-to-one correspondence with the closed points in the spectrum; generic points in the spectrum also produce representations (not necessarily irreducible).

The most important part in this work is that it provided a completely categorical (algebro-geometric) way to do induction in an abelian category instead of the derived category. (I will explain this later if I have time). This semester, Rosenberg gave us a lecture course, using this framework to compute all the irreducible representations for the Weyl algebra, the enveloping algebra, quantized enveloping algebras, algebras of differential operators, $SL_2({mathbb R})$ and other algebraic groups, or related associative algebras. It works very efficiently. For example, computing irreducible representations of $U(sl_3)$ is believed to be very complicated, but using this spectrum framework, it becomes much simpler.

The general framework for these is contained in the paper Spectra, associated points and representation theory. If you want to see some concrete examples using this machine, you should look at Rosenberg's old book Noncommutative Algebraic Geometry And Representations Of Quantized Algebras.
There is another paper Spectra of `spaces' represented by abelian categories, providing the general theory for this machinery.

Furthermore, we can define the spectrum for an exact category; even more generally, for any Grothendieck site, and so for any category (because any category has a canonical Grothendieck pretopology). Rosenberg has recent work defining the spectrum for such categories -- Geometry of right exact `spaces' -- the main motivation for this work is to provide a background for higher universal algebraic K-theory for a right exact category (a category with a family of strict epimorphisms can be taken as a one-sided exact category). More important motivation is to study algebraic cycles for noncommutative schemes. (Warning: this paper is very abstract and hard to read. We will go through this paper in the lecture course this semester.)

All of these things will appear soon in his new book with Konstevich (but I am not sure of the exact time). If I have enough time to post, I will explain in more detail, how the theory of the spectrum for abelian categories comes into representation theory, and how this picture is related to the derived picture of Beilinson-Bernstein and Deligne.
In fact, today we have just learned Beck's theorem for Karoubian triangulated categories and will do the DG-version of Beck's theorem later. And then he will introduce the spectrum for triangulated categories, and explain the noncommutative algebraic geometry facts behind the BBD machine and the connection with his abelian machine.

co.combinatorics - String of integers puzzle

I apologize for not have the math background to put this question in a more formal way.
I'm looking to create a string of 796 letters (or integers) with certain properties.

Basically, the string is a variation on a De Bruijn sequence B(12,4), except order and repetition within each n-length subsequence are disregarded.
i.e. ABBB BABA BBBA are each equivalent to {AB}.
In other words, the main property of the string involves looking at consecutive groups of 4 letters within the larger string
(i.e. the 1st through 4th letters, the 2nd through 5th letters, the 3rd through 6th letters, etc)
And then producing the set of letters that comprise each group (repetitions and order disregarded)

For example, in the string of 9 letters:

A B B A C E B C D

the first 4-letter groups is: ABBA, which is comprised of the set {AB}
the second group is: BBAC, which is comprised of the set {ABC}
the third group is: BACE, which is comprised of the set {ABCE}
etc.

The goal is for every combination of 1-4 letters from a set of N letters to be represented by the 1-4-letter resultant sets of the 4-element groups once and only once in the original string.

For example, if there is a set of 5 letters {A, B, C, D, E} being used
Then the possible 1-4 letter combinations are:
A, B, C, D, E,
AB, AC, AD, AE, BC, BD, BE, CD, CE, DE,
ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, BDE, CDE,
ABCD, ABCE, ABDE, ACDE, BCDE

Here is a working example that uses a set of 5 letters {A, B, C, D, E}.

D D D D E C B B B B A E C C C C D A E E E E B D A A A A C B D D B

The 1st through 4th elements form the set: D
The 2nd through 5th elements form the set: DE
The 3rd through 6th elements form the set: CDE
The 4th through 7th elements form the set: BCDE
The 5th through 8th elements form the set: BCE
The 6th through 9th elements form the set: BC
The 7th through 10th elements form the set: B
etc.

* I am hoping to find a working example of a string that uses 12 different letters (a total of 793 4-letter groups within a 796-letter string) starting (and if possible ending) with 4 of the same letter. *

Here is a working solution for 7 letters:

AAAABCDBEAAACDECFAAADBFBACEAGAADEFBAGACDFBGCCCCDGEAFAGCBEEECGFFBFEGGGGFDEEEEFCBBBBGDCFFFFDAGBEGDDDDBE

zeta functions - Fractional powers of Dirichlet series?

For your first question: let $R$ be any commutative ring, and let $D(R)$ be the ring of formal Dirichlet series over $R$, i.e., the set of all functions $f: mathbb{Z}^+ rightarrow R$ under pointwise addition and convolution product.

Then the unit group of $R$ is precisely the set of formal Dirichlet series $f$ such that
$f(1)$ is a unit in $R$.

As for your second question, it is indeed equivalent to asking whether $U(D(R))$ is $n$-divisible. Here, if we take $R = mathbb{Z}$ as you asked, the answer is that for all $n geq 2$, $U(D(mathbb{Z}))$ is not $n$-divisible and that even the Dirichlet series $zeta(s)$ is not an $n$th power in $D(mathbb{Z})$.

[Now, for some reason, I switch back to the classical notation, i.e., I replace the arithmetical function $f$ by its "Dirichlet generating series" $sum_{n=1}^{infty} frac{f(n)}{n^s}$. It would have been simpler not to do this, but too late.]

Let
$f(s) = sum_{n=1}^{infty} frac{a_n}{n^s}$ be any formal Dirichlet series, and suppose
that $g(s) = sum_{n=1}^{infty} frac{b_n}{n^s}$ be a formal Dirichlet series such that
$g^2 = f$. Thus

$a_1 + frac{a_2}{2^s} + ldots = (b_1 + frac{b_2}{2^s} + ... )(b_1 + frac{b_2}{2^s} + ldots)$

$= b_1^2 + frac{2 b_1 b_2}{2^s} + frac{2 b_1 b_3}{3^s} + frac{2b_1 b_4 + b_2^2}{4^s} +
ldots$

(This multiplication is formal, i.e., it is true by definition.)

Thus $b_1 = pm sqrt{a_1}$. Suppose we take the plus sign, for simplicity. Then for all primes $p$,

$a_p = 2 b_1 b_p$, so

$b_p = frac{a_p}{2 sqrt{a_1} }$,

so we need $2 sqrt{a_1}$ to divide $a_p$, so at least we need $a_p$ to be even for all primes $p$. Further conditions will come from the composite terms.

These same considerations show that if we replaced the coefficient ring $mathbb{Z}$ by
$mathbb{Q}$ (or any coefficient field of characteristic $0$), then any formal Dirichlet series with $a_1 = 1$ is $n$-divisible for all positive integers $n$. In particular, you can write $zeta(s)^{frac{1}{n}}$ as a Dirichlet series with $mathbb{Q}$-coefficients just by applying the above procedure and successively solving for the coefficients. Whether there is a nice formula for these coefficients is a question for a better combinatorialist than I to answer.

EDIT: Based on your comments below, I now understand that you are looking for a characterization of $U(D(mathbb{Z}))$ as an astract abelian group. I believe it is isomorphic to
${ pm 1 } times prod_{i=1}^{infty} mathbb{Z}$. (Or, more transparently, to
the product of ${ pm 1}$ with the product of infinitely many copies of $prod_{i=1}^{infty} mathbb{Z}$, one for each prime number. But as abstract groups it amounts to the same thing.)

ag.algebraic geometry - what notions are "geometric" (characterized by geometric fibers)?

Sorry The title might not be suggestive enough.

The question is about things like the following: A reductive group scheme is defined to be a (really nice) group scheme whose geometric fibers are reductive groups. So in some sense, "reductiveness" is some kind of "geometric" notion.

So whatelse properties of schemes can be checked only on geometric fibers? What I know, for example, given a scheme over a field $k$, it is projective iff it is projective over $bar{k}$. But can this be extended to any base scheme?

In particular, is there any reference that collects such results? And "WHY" should this work? for example, WHY geometrically-reductive group schemes turn out to be the right generalization of reductive algebraic groups?

Sorry this question might be a little to vague, and thank you in advance.

matrices - Properties of Graphs with an eigenvalue of -1 (adjacency matrix)?

There are a lot of graphs with this property. I just introduce two classes that are very famous:

The Friendship graph $F_n$ that is $K_1nabla nK_2$, where $nabla$ means the join of two graphs. These graphs has $n-1$ eigenvalue $-1$.

The second class is graph $K_n$ that is removed $1,2,3$ or $4$ edges from it. So, this class contain also five different classes!!!