analytic number theory - If $2^x $and $3^x$ are integers, must $x$ be as well?

It sure feels like something that should be related to Schanuel's conjecture. Note that this is equivalent to finding integers $m$ and $q$ such that $$x = frac{ln m + 2pi i q}{ln 3}$$ is not an integer but $2^x$ is. [$3^x$ simplifies to $m$ by construction].

Continuing this, let us actually compute $2^x$. If we split it into real and imaginary parts, a large messy expression ensues. But it naturally splits into 2 reasonable cases, depending on whether $mgt 0$ or $m lt 0$. Let's deal with the positive case first. We get
$$2^x = m^{log_3 2}left( cos(2qlog_3(2) pi) + i sin(2qlog_3(2) pi)right) $$
For that to be an integer, it has to at least be real, but unless $2log_3(2)q$ is an integer, the $sin$ term will not be $0$. For $q=0$, this is $m^{log_3 2}$. We can rewrite that as $2^{log_3 m}$. But we assumed that $m$ was a power of $3$, so $log_3 m$ (and thus $x$) is an integer.

For the $m lt 0$, we get the slightly more complicated
$$2^x = (-m)^{log_3 2}left( cos(2(2q+1)log_3(2) pi ) + i sin(2(2q+1)log_3(2) pi )right) $$
Since $mlt 0$, the first term is real, so we need $2 (2q+1)log_3(2)$ to be an integer for the $sin$ term to disappear, which cannot happen.

Can someone find a flaw in my reasoning? I somehow expect so, as this did not seem difficult, and I would expect it to be if it's an open problem!

peano arithmetic - "Interesting" properties of sets of natural numbers

I'm glad you decided to follow up on my suggestion to look at Descriptive Set Theory! As I mentioned in my earlier answer, there are two flavors of Descriptive Set Theory: the "boldface theory" and the "lightface theory." As you observed, the boldface theory of sets of natural numbers is completely trivial since every subset of $mathbb{N}$ is clopen, but the lightface theory is not trivial at all.

The lightface theory is completely analogous to the boldface theory except that all operations are required to be computable, or effective in the descriptive lingo. Let's go through the first few levels of the boldface and lightface Borel hierarchies in parallel. Let $X$ be a Polish space and let $(U_n)_{n=0}^infty$ be a nice enumeration of a basis for the topology on $X$.

• A boldface $boldsymbol{Sigma}^0_1$ set is simply an open subset of $X$, or a countable union of basic open sets. Similarly, a lightface $Sigma^0_1$ set is an effective union of basic open sets, i.e. a union of the form $bigcup_{n=0}^infty U_{f(n)}$ where $f$ is a computable function. Well, not quite, since this informal definition technically excludes the empty set. A better definition is that the lightface $Sigma^0_1$ sets are the sets $G_e = bigcup_{n in W_e} U_n$ where $W_e$ is the $e$-th computably enumerable set. This also gives a better listing of the lightface $Sigma^0_1$ sets.




• A boldface $boldsymbol{Pi}^0_1$ set is simply a closed subset of $X$, or the complement of a boldface $boldsymbol{Sigma}^0_1$ set. Similarly, a lightface $Pi^0_1$ set is a the complement of a lightface $Sigma^0_1$ set. Note that we have a nice enumeration of all lightface $Pi^0_1$ sets, namely $F_e = X setminus G_e$ where $G_e$ was defined above.




• A boldface $boldsymbol{Sigma}^0_2$ set is a $F_sigma$ subset of $X$, or a countable union of boldface $boldsymbol{Pi}^0_1$ sets. Similarly, a lightface $Sigma^0_2$ set is an effective union of $Pi^0_1$ sets. As above, the lightface $Sigma^0_2$ sets are enumerated by $G'_e = bigcup_{n in W_e} F_n$ where $W_e$ is the $e$-th computably enumerable set and $F_e$ was defined above.




• A boldface $boldsymbol{Pi}^0_2$ set is a $G_delta$ subset of $X$, or the complement of a boldface $boldsymbol{Sigma}^0_2$ set. Similarly, a lightface $Pi^0_2$ set is a the complement of a lightface $Sigma^0_2$ set. Note that we again have a nice enumeration of all lightface $Pi^0_2$ sets, namely $F'_e = X setminus G'_e$ where $G'_e$ was defined above.

And so on, lightface $Sigma^0_{n+1}$ sets are effective countable unions of lightface $Pi^0_n$ sets, which in turn are complements of lightface $Sigma^0_n$ sets and at each step we have a nice enumeration of all of these sets.

Just as for the boldface Borel hierarchy, there is no reason to stop at $omega$; this process can be continued well into the transfinite, but there are a few roadblocks along the way. Because we want the hierarchy to be effective, we can only go up to the Church-Kleene ordinal $omega_1^{CK}$, which is the smallest non-computable ordinal. (Also, the hierarchy is defined using ordinal notations instead of actual ordinals.) In the end, we get the family lightface Borel sets, which is a countable subfamily of the usual Borel sets.

The analytic sets, or boldface $boldsymbol{Sigma}^1_1$ sets, also have a lightface analogue. The boldface $boldsymbol{Sigma}^1_1$ sets are precisely the projections onto the $X$-axis of the boldface $boldsymbol{Pi}^0_1$ subsets of the Polish space $X times mathbb{N}^mathbb{N}$, where $mathbb{N}^mathbb{N}$ denotes Baire space. Similarly, the lightface $Sigma^1_1$ subsets of $X$ are the projections onto the $X$-axis of the lightface $Pi^0_1$ subsets of $X times mathbb{N}^mathbb{N}$. Note that since we have a nice enumeration of the lightface $Pi^0_1$ sets, we also have a nice enumeration of the lightface $Sigma^1_1$ sets. There are similar lightface analogues of the entire projective hierarchy. A lot of the results of the boldface theory transfer to the lightface theory. For example, the Suslin Theorem that the boldface $boldsymbol{Delta}^1_1$ sets are precisely the Borel sets. A theorem of Kleene says that the same holds for the lightface $Delta^1_1$ sets, namely these are precisely the lightface Borel sets described above.

In the case of the discrete (and Polish) space $mathbb{N}$, the natural choice of basis are the singletons $U_n = {n}$. Working through the definitions, we see that the lightface $Sigma^0_1$ subsets of $mathbb{N}$ are precisely the computably enumerable subsets of $mathbb{N}$ and the lightface $Pi^0_1$ sets are the complements thereof. Hence, the $Delta^0_1$ subsets of $mathbb{N}$ are precisely the computable sets. In fact, the lightface $Sigma^0_n$, $Pi^0_n$, $Delta^0_n$, perfectly match the arithmetic hierarchy. The lightface Borel subsets of $mathbb{N}$ are commonly called hyperarithmetic sets.

Obviously, there are lots and lots of interesting properties of subsets of sets of natural numbers!

ag.algebraic geometry - Categorical Künneth theorem for module categories

What is the relationship between the category of modules over a product $Xtimes Y$ and the pair of module categories over $X$ and $Y$ separately? Ideally here $X$ and $Y$ are smooth projective varieties and we are considering the DG category of complexes of $mathcal{O}_X$-modules (with possibly some coherence condition if needed).

A more specific question which I am interested in is: say you have two sheaves of DG commutative algebras X, Y, both modules over the sheaf of DG commutative algebras Z (all sheaves over the same space), then how is the category of DG modules of the pushout of X and Y along Z related to the module categories for X,Y.

It seems that in Section 17 of this paper by Frenkel and Gaitsgory, there is a result to the effect that $DGQCoh(Xtimes Y)$ is a categorical tensor product of the categories $DGQCoh(X)$ and $DGCoh(Y)$, but I do not understand it, so if it does indeed answer my question, I would appreciate some remarks which may clarify exactly what is proved there.

gr.group theory - Results about the order of a group forcing a particular property.

The numbers n such that every group of order n is cyclic, abelian, nilpotent, supersolvable, or solvable are known. Most are described in an easy to read survey:

Pakianathan, Jonathan; Shankar, Krishnan. "Nilpotent numbers."
Amer. Math. Monthly 107 (2000), no. 7, 631-634.
MR 1786236
DOI: 10.2307/2589118

If you want to go beyond results like this, you may have better luck looking at a slightly more refined version of the order: the isomorphism type of the Sylow subgroups. Sometimes a p-group P has the property that every group G containing it as a Sylow p-subgroup has a normal subgroup Q of order coprime to p such that G is the semi-direct product of P and Q. An easy version of this that does appear in many group theory texts is that if n=4k+2, then in each group G of order n there is a normal subgroup Q of order 2k+1 so that G is the semi-direct product of any of its Sylow 2-subgroups and Q.

Groups all of whose Sylow p-subgroups are cyclic have very nice properties, subsuming those of groups of square-free order. Groups all of whose Sylows are abelian have more flexibility, but are still basically under control.

fa.functional analysis - Is this a $C^{infty}$ function ?

Here's an attempt at the second part. I'm much more familiar with the convenient calculus of Frölicher, Kriegl, and Michor than with Fréchet derivatives, but for Hilbert spaces, there shouldn't be too much in it.

Let $rho colon mathbb{R} to mathbb{R}$ be a smooth bump function at $0$ with support in $[-1/2,1/2]$. Since $|e_n - e_m| = sqrt{2}$ (for $n ne m$), if we define $f_n colon H to mathbb{R}$ by $f_n(x) = rho(|e_n - x|^2)$ then the $f_n$s have disjoint support. Then $f := sum n f_n colon H to mathbb{R}$ is locally smooth and hence smooth. In addition, $f(e_n) = n$ so $f$ maps the unit ball onto an unbounded set.

(How does that look? I'm well aware that I may have overlooked something really obvious!)

---

Edit 2012-11-12: I did overlook something that I shouldn't have done. The argument above is not quite correct: that the $f_n$s have disjoint support is not enough to know that the sum $sum n f_n$ is smooth. For that, I need to know that the $f_n$s have locally finite support. Fortunately, this is true. If $f_n(x) ne 0$ then $|e_n - x| < 1/4$. Thus for $y in H$ consider the ball of radius $1/4$ about $y$. If we have $x_1$ and $x_2$ in this ball and $n,m$ such that $f_n(x_1) ne 0$ and $f_m(x_2) ne 0$ then $|e_n - e_m| le |e_n - x_1| + |x_1 - y| + |y - x_2| + |x_2 - e_m| < 1$ whence $n = m$. Hence at most one $f_n$ has support intersecting this ball about $y$ and so the supports of the $f_n$ are locally finite.

st.statistics - Why do statistical randomness tests seem so ad hoc?

It's not clear that Marsaglia's tests are really good enough. See this Stack Overflow discussion.

Kolmogorov complexity is not the right criterion for statistical randomness tests, since any pseudorandom sequence has low Kolmogorov complexity. What you really want in a random number generator is for the sequence to be computationally pseudorandom; that is, without knowledge of the seed, no polynomial-time test can distinguish the sequence from a truly random sequence.

In fact, there are a number of random number generators which are believed to be computationally pseudorandom. Nobody uses these, however, partly because they are computationally too expensive, and partly because of inertia and partly because the existing pseudorandom generators we have are good enough most of the time. A while ago, for one of the most common methods of pseudorandom number generation (linear congruential), I ran into cases where they unexpectedly produced the wrong answer.

Marsaglia's tests were developed over a number of years, and I believe each was designed to detect certain flaws that many pseudorandom number generators contained at the time. Once good-enough pseudorandom number generators were available, nobody bothered creating a more stringent series of tests.

higher category theory - 2-groups are to crossed modules as 2-categories are to...?

Given a 2-group $mathcal{G}$, you can construct a crossed module $(G,H,t,alpha)$ and vice versa.

Is there something similar you can say for strict 2-categories?

In a personal attempt to understand strict 2-categories, I ended up constructing a speculative conceptual tool (whose validity remains to be seen) that I call the boundary of a 2-morphism. I've written up some raw notes here:

The basic idea is that given morphisms $f,g:xto y$ and a 2-morphism $alpha:fRightarrow g$, we define its boundary as an endomorphism

$$partialalpha:yto y$$

satisfying

$$partialalphacirc f = g.$$

When the source of the 2-morphism is an identity morphism, then we have

$$partialalpha = t(alpha),$$

which seems to relate things well to cross modules when all morphisms are invertible.

I'm curious if there is anything like a crossed module, but where we're not dealing with groups and morphisms are not invertible. What I'm trying to cook up seems like it might be related to such a thing if it exists.

Any thoughts and/or any comments on my notes would be greatly appreciated.

PS: Apologies in advance if my writing is not very clear. I'm not a mathematician, but am trying to teach myself some basic higher category theory.