Consider the following statement (which follows easily from various results found in the literature):

**(†)** There exists a primitive recursive (“p.r.”) relation $T$ on the ordinals such that, if $(f_e)_{e<\omega}$ is a standard numbering of the p.r. functions of one variable on the ordinals, we have $f_e(x)=y$ iff $\exists z.T(e,x,y,z)$; moreover, there then exists such a $z$ which is less than the smallest p.r.-closed ordinal $\geq x$.

One straightforward consequence of (†) is a Kleene normal form theorem for $\alpha$-recursion: there exists a p.r. relation $T'$ on the ordinals such that, if $(\varphi_e)_{e<\omega}$ is a standard numbering of the partial recursive functions of one variable on the admissible ordinal $\alpha$, we have $\varphi_e(x)\simeq y$ iff $\exists z<\alpha.T'(e,x,y,z)$. (One straightforward consequence of *that* is that there exists a “universal” partial recursive $g$ on $\alpha$ with the property that $g(e,x) \simeq \varphi_e(x)$.)

Now every proof of (†) that I was able to find boils down to something like this: if $f_e(x)=y$ then for $\beta$ large enough (larger than all the intermediate computation values) we have $L_\beta \models f_e(x)=y$, and “$L_\beta \models f_e(x)=y$” is a p.r. relation of the ordinals $e,x,y,\beta$ because it is a p.r. relation of the *sets* $e,x,y,\beta$ and p.r. relations on ordinals and on sets that happen to be ordinals coincide.

This proof is explicit, in the sense that it actually gives $T$, but the $T$ in question seems rather insanely complicated (as a p.r. relation on *ordinals*) because of the process of converting a p.r. relation on sets to one on ordinals and because of the route through formulas and the $L_\beta$.

**Question:** Can the statement (†) above be proved entirely at the level of p.r. functions and relations on the ordinals? **Variant:** Can we give an explicit $T$ that is reasonably short?

In the case of ordinary recursion, it is not too difficult. What I would like to understand is whether there is some reason why the transfinite ordinal case should be harder or whether it is just an artefact of the way things are written in the literature.

**Motivation:** Partially from trying to get a better understanding of this question I asked recently; partially from thinking about how, if $\alpha$-recursion is viewed as a transfinite programming language, one would proceed to write an “interpreter” for the language in the language itself (=universal function).

So, I just learned for the first time today that strict $\omega$-groupoids have strict inverses. I had always thought that they were just strict $\omega$-categories where all cells had weak inverses (that is, all cells are reversible in the sense of Lafont, Metayer, and Worytkiewicz (click me).

If we consider what I previously thought to be strict $\omega$-groupoids, call them weakly-groupoidal strict $\omega$-categories, do they model strictly more homotopy types than the true strict groupoids? If yes, can you think of an example?

Let V be a two dimensional vector space over k. Consider the set of isomorphism classes of all rank 1 $k[x]/x^2$-submodules of $V\otimes k[x]/x^2$. Does this set has a variety structure? Is it tangent bundle of $\mathbb{P}^1$?

I'm trying to either find a source or create a proof that solutions of the homogeneous Helmholtz equation with constant coefficients are infinitely differentiable. I don't know that the result is true, of course. If I did, I wouldn't have to ask.

I have one idea about how to go about it which is presented roughly below, but it's the result I'm interested in, not the method used to derive it.

Since the Helmholtz equation is a an eigenvalue problem for Laplace's equation $\Delta u=0$, this result can be shown in a straightforward way if it can be shown that in a very nice domain $\Omega$ (let's say a ball) and for $s\ge 0$, solutions to the weak version of the Poisson equation

$\Delta u=f$ on $\Omega$

with Dirichlet condition

$u|_{\partial \Omega}=g$

with $f\in H^{s}(\Omega)$, $g\in H^{s+\frac{1}{2}}(\partial\Omega)$ have

$u\in H^{s+1}(\Omega)$.

The result above is very well known to be true for $s=0$. I don't know if the result above is true or not for $s>0$, but it seems very plausible. We would only need to prove it for integer $s$, strictly speaking.

This would make it easy to see that homogeneous solutions of the Helmholtz equation are smooth because you could take a hypothetical homogeneous solution $w$ of the Helmholtz equation on $\Omega$ and write

$\Delta w=-k^2w$

$w|_{\partial\Omega}=w|_{\partial\Omega}$

which would reveal via a simple proof by induction–using the hypothetical regularity result for the Poisson equation–that if $w$ is at least in $H^{1}(\Omega)$ (definitely) or $L^2(\Omega)$ (probably can show this as well, but we need to be careful about the trace operator), $w\in H^s(\Omega)$ for all $s>0$.

This question is a duplicate of an existing MO question, but that other MO question has an accepted answer that does not actually answer the question, and I'm not sure how to fix that other than by re-asking the question.

On page 76 of Serre's book *A Course in Arithmetic*, he writes:

[T]here exist sets having an analytic density but no natural density. It is the case, for example, of the set $P^1$ of prime numbers whose first digit (in the decimal system, say) is equal to 1. One sees easily, using the prime number theorem, that $P^1$ does not have a natural density and on the other hand Bombieri has shown me a proof that the analytic density of $P^1$ exists (it is equal to $\log_{10}2 = 0.301029995\ldots$).

There is a slight misstatement here because literally speaking, $P^1$ has natural density zero, but clearly the intent is to speak of the *relative* density of $P^1$ inside the set $P$ of all primes. In other words, Bombieri's result is a kind of "Benford's law for primes":
$$\lim_{s\to1} {\sum_{m\in P_1} m^{-s} \over \sum_{p\in P} p^{-s}} = \log_{10}2.$$

My question is, how does one prove that the above limit (which goes by various names—relative analytic density, relative Dirichlet density, relative zeta density) exists? Serre does not say anything about this.

The accepted answer to the duplicate MO question cites two papers, one by Cohen and Katz, and one by Raimi. But the paper by Cohen and Katz simply restates what Serre says without giving a proof. The paper by Raimi cites a paper by R. E. Whitney (Initial digits for the sequence of primes, *Amer. Math. Monthly* **79** (1972), 150–152) but Whitney's paper considers *logarithmic density* rather than Dirichlet density:
$$\lim_{N\to\infty} {\sum_{m\in P_1, m\le N} 1/m \over \sum_{p\in P, p\le N} 1/p}.$$
It's not clear to me that this implies Bombieri's result.

In a comment to a now-deleted MO question, KConrad suggested looking in the book *Prime Numbers* by Ellison and Ellison, but I did not find the answer there either.

In the univariate case ($\chi^2$ distribution), I know we can expand the pdf into power series of the variance $\sigma^2$ with Laguerre polynomials. Does anyone know if we can do the same for the general multivariate case?

If this is possible, does the expansion still works for the singular case? (i.e., the sample size n is smaller than the dimension p)

Working in ZF, it's well-known that for any $n \ge 2,$ the claim that there is a $\Sigma_n$ well-ordering of the universe is equivalent to the axiom $V=HOD.$ It seems natural to believe there should be a similar theorem for $n=1,$ perhaps that there being a $\Sigma_1$ well-ordering of the universe is equivalent to $V=L.$ I can't find any counterexamples to this proposition, having checked various generic extensions of $L$ and canonical inner models like $L[U],$ so I'm guessing this is true.

It would suffice to check that if there a $\Sigma_1$ well-ordering of $V,$ then all sets of ordinals are in $L.$ This is true for subsets of $\alpha$ for any $\alpha$ countable in $V,$ by applying Mansfield's theorem that if there is a $\Sigma_2^1$ well-ordering of $\mathbb{R},$ then $\mathbb{R} \subset L.$ Then the first non-trivial case would be to show this holds for subsets of $\omega_1,$ but I haven't been able to do this (the proof of Mansfield's theorem doesn't seem to extend to this more general case).

So, are there any known results similar to what I'm asking?

I have $A_{n \times n}$ integer matrix. I want to obtain a matrix $B_{m \times m}$ such that the rows of $A$ are the diagonals of $B$ and the columns of $A$ are the anti-diagonals of $B$ ($m \ge n$). Equivalently, $B$ is the result of rotating $A$ clockwise $45°$ . Here is an example: $$\left( \begin{array}{cc} 1& 2 & 3\\ 4& 5 & 6 \\ 7& 8& 9 \end{array} \right)$$

is transferred to

$$\left( \begin{array}{cc} 0& 0& 1 & 0 &0 \\ 0& 4& 0& 2 &0 \\ 7& 0& 5& 0& 3 \\ 0 & 8& 0& 6 & 0 \\ 0& 0& 9& 0& 0 \end{array} \right)$$

Has anyone studied the properties of such transformation? Which algebraic operation does achieve such transformation? Is there any interesting group-theoretic uses of such transformation?

Serre's finiteness theorem says if $n$ is an odd integer, then $\pi_{2n}(S^{n + 1})$ is the direct sum of $\mathbb{Z}$ and a finite group. By looking at the table of homotopy groups, say on Wikipedia, one empirically observes that if $n \equiv 3 \pmod 4$, then we in fact have $$ \pi_{2n}(S^{n+1}) \cong \mathbb{Z} \oplus \pi_{2n-1}(S^n). $$ This holds for all the cases up to $n = 19$. On the other hand, for $n \equiv 1 \pmod 4$ (and $n \neq 1$), the order of the finite part drops by a factor of $2$ when passing from $\pi_{2n - 1}(S^n)$ to $\pi_{2n}(S^{n+1})$.

From the EHP sequence, we know that these two are the only possible scenarios. Indeed, we have a long exact sequence $$ \pi_{2n+2}(S^{2n+1}) \cong \mathbb{Z}/2\overset{P}{\to} \pi_{2n}(S^n) \overset{E}{\to} \pi_{2n+1}(S^{n+1}) \overset{H}{\to}\pi_{2n+1}(S^{2n+1}) \cong \mathbb{Z}. $$ Since $H$ kills of all torsion, one sees that the map $E$ necessarily surjects onto the finite part of $\pi_{2n+1}(S^{n+1})$. So the two cases boil down to whether or not $P$ is the zero map. What we observed was that it is zero iff $n \equiv 3 \pmod 4$, up to $n = 19$.

Since $\pi_{2n+2}(S^{2n+1}) \cong \mathbb{Z}/2\mathbb{Z}$ has only one non-zero element, which is the suspension of the Hopf map, it seems like perhaps one might be able to check what happens to this element directly. However, without a good grasp of what the map $P$ (or the preceeding $H\colon\pi_{2n+2}(S^{n+1}) \to \pi_{2n+2}(S^{2n+1})$) does, I'm unable to proceed.

Curiously, I can't seem to find any mention of this phenomenon anywhere. The closest I can find is this MO question, but this phenomenon is not really about early stabilization, since for $n = 3, 7$, the group $\pi_{2n-1}(S^n)$ is not the stable homotopy group, but something smaller. I'd imagine either this pattern no longer holds for larger $n$, or there is some straightforward proof I'm missing.

**Note:** Suggestions for a more descriptive title are welcome.

J. Bismut proved the asymptotic formula for the heat kernel of the Laplace-Beltrami operator $\Delta$ on a manifold $M$ in one of his well-known books. Later, in his paper on the index theorem, Bismut said that the result can be extended immediately to the heat equation semi-group for Hodge-Kodaira Laplacian or the Dirac Laplacian, since these (operators) are subordinated to the heat equation semi-group for $\Delta$. My question is: what is subordination or the subordination process in probability? If there is such subordination, how can one see that the result can be extended immediately? The proof of the asymptotic formula by Bismut in his book on Large Deviation is not easy to follow. Also Bismut mentioned that Malliavin has already used the subordination in his paper on vanishing theorem.

For a finite group $G$ one defines the *spectrum* $\omega(G)$ as the set of element orders of $G$. The set $\omega(G)$ is uniquely determined by the subset $\mu(G)$ consisting of those elements of $\omega(G)$ that are maximal with respect to the divisibility relation.

I would like to know whether for a finite non-abelian simple group $G$ one has $|\mu(G)| \geq 3$.

If this is the case, is there a proof that works for all finite simple groups of Lie type (without treating each family differently)?

Let $B$ an operator on a two-dimensional Hilbert space such that $B$ is not a scalar operator and let $\mathfrak{C}$ be the commutant of $B$ (i.e. $\mathfrak{C}$ of $B$ is the set of all Operators that commute with $B$.) If $T$ is in $\mathfrak{C}$, why there exist some complex numbers $a$ and $b$ such that $T = aB + b$ ?? Thank you

Let $X$ be a random variable with the distribution $F$.

What are the extreme points of the sets of the form: \begin{align} P_1&=\left\{ F: \int |x|^k dF\le c \right\},\\ P_2&=\left\{ F: |X| \le d \right\},\\ P_3&=\left\{ F: \int |x|^k dF\le c, \, |X| \le d \right\}.\\ \end{align} In this question it was show for the set $P_1$, the set of extrem points are all two mass disributions. What about $P_2$ and $P_3$?

It would also be nice if some one can provide a good reference where the subject of finding extreme points of a set of distributions can be found. I am familiar with this reference. However, was thinking maybe there is a more modern work or survey on this.

Let $I$ be an ideal in a regular ring $R$. Suppose $I$ can be generated by $n$ elements. Let $P$ be an associated primes of $I$. Is it true that the height of $P$ is bounded above by $n$?

Remark: (1) The question has an affirmative answer when $P$ is a minimal associated prime of $I$ by Krull's principal theorem.

(2) The question is false for general Noetherian rings Indeed, suppose $I = 0$ and $R$ has an embedded associated prime ideal.

I am reading a circle of papers which use arguments based on Fredholm determinants of nuclear operators to compute numerical quantities associated to real-analytic and holomorphic dynamical systems. In order to understand these papers I am learning about Grothendieck's theorems on traces and determinants of nuclear operators. (My reference for these results is the book *Traces and Determinants of Linear Operators* by Goh'berg, Goldberg and Krupnik, which I am finding easier to read than the original papers of Grothendieck.) The operators are typically defined on classical Banach spaces of holomorphic functions, sometimes of several variables.

In order for Grothendieck's results on the existence of traces and Fredholm determinants to be applied it seems that the Banach space $\mathfrak{X}$ on which the operators are to be studied must satisfy the *approximation property*: for every compact subset $K$ of $\mathfrak{X}$ and every $\varepsilon>0$ we must be able to find a finite-rank operator $F$ such that $\|x-Fx\|<\varepsilon$ for all $x \in K$. I was not previously familiar with the approximation property (my background is mainly in ergodic theory and dynamical systems) and am trying to better understand the scope of this particular hypothesis.

Unfortunately the approximation property does not seem to be very explicitly treated in the papers I am reading and I have not yet discovered any useful references. I wonder if anyone can point me to a reference for the following:

Let $D \subset \mathbb{C}^d$ be nonempty, open and bounded. Is it known whether or not the Banach space of bounded holomorphic functions $D \to \mathbb{C}$ with continuous extensions to $\overline{D}$, equipped with the uniform norm, has the approximation property?

Some qualifications:

- I would be happy with a reference which states that the problem is open, as well as with a definite answer either way.
- I'm interested both in cases where $D$ is connected, and in cases where it is not connected. Any information about either would be welcome.
- I understand that the approximation property for $H^\infty(D)$ is unknown when $D\subset \mathbb{C}$ is the unit disc. Can anyone give me a reference stating that this problem is open?

The Riemann hypothesis is equivalent to

$\forall n\geqslant2,\: \sigma(n)<H_n+e^{H_n}\log{H_n}$,

where $\sigma(n)$ is the divisor sum of $n$ and $H_n$ is the nth harmonic number.

For large $n$, $H_n$ is small. The only $n\geqslant 2$ I have found for which $\sigma(n)>e^{H_n}\log{H_n}$ are 2, 3, 4, 6, 12, 24, and 60. Are any other such $n$ known?

The famous and remarkable Voronin's universality theorem states:

**Theorem** (Voronin 75): let $0<r<1/4$ and suppose that $g(s)$ is a nonvanishing continuous function on the disk $ \vert s \vert \leq r$ that is analytic in the interior. Then for any $\epsilon>0$, there exists a positive real number $\tau$ such that: $$max_{\vert s \vert \leq r} \vert \zeta(s+3/4+i \tau)-g(s) \vert <\epsilon. $$

Which practically means that $g(s)$ could be approximated by $\zeta(s)$ for some "high enough" values of $\tau$ - on the right hand side of the critical strip.

The thing is that there is a certain "mystery" with respect to this theorem - when it comes to the question "how high is high enough"? Of course - one can give some analytic study trying to give an effective result. But nowadays we can also try to directly evaluate and verify it with a computer - the thing is that it turns out to be really hard to do that.

For instance - let us take the constant function $g(z)=e^{ \pm 3}$.

**Question:** Is there any estimate on $\tau$ for which Voronin's approximation works for $g(s)=e^{\pm 3}$?

It is important point out and compare, for instance, the following graph of $\log \vert \zeta(0.75+e^{0.0001 \tau} i) \vert$ for $\tau = 0,...,250000$ in this case:

As you can see - for quite big values - the function still doesn't seem to cross the bound $\pm 3$. So when does Voronin's theorem start to kick in?

*(Note also direct implication to zeros of zeta - RH, for instance.)*

- Do you think the letter $\wp$ has a name? It may depend on community - the language, region, speciality, etc, so if you don't mind, please be specific about yours. (Mainly I'd like to know the English names, if any, but other information is welcome.) If yes, when and how did you come to know it? When, how, and how often do you mention it? (See below.)
- What's the origin of the letter?

In computing, Unicode.org gave two names to the letter, but let us ignore that point here.

Background: (Sorry for being a bit chatty.)

Originally I raised a related question at Wikipedia. The user Momotaro answered that in math community it's called "Weierstrass-p". Momotaro also gave a nice reference to the book The Brauer-Hasse-Noether Theorem in Historical Perspective by Peter Roquette. The author's claim supports Momotaro. (The episode in the book about the use of $\wp$ by Hasse and Emmy Noether is very interesting - history amuses - but it's off topic. Read the above link to Wikipedia. :)

However I'm not completely sure yet, because the occasions on which the letter's name becomes a topic must be quite limited. For example perhaps in the classroom a professor draws $\wp$, and students giggle by witnessing such a weird symbol and mastery of handwriting it; then the professor solemnly announces "this letter is called Weierstrass-p", like that? And "Weierstrass-p" is never an alias of the p-function?

After reading Momotaro's comment, I think I've read somewhere that the letter was invented by Weierstrass himself, but my memory about it is quite vague. Does anyone know something about it? Is it a mere folklore, or any reference?

I don't think mathoverflow is a place for votes, but if it were, I'd like one: "Have you ever heard of the name of the letter $\wp$?

Slightly off-topic, about the p-function's name in Japanese; In Japanese, the names of the Latin alphabets are mostly of English origin, エー, ビー, シー... (eh, bee, cee, etc.) But $\wp$-function is called ペー (peh), indicating its German origin. See e.g. 岩波 数学公式 III, p34, footnote 2 I don't know the *name* of the letter in Japanese. (In fact, most non-English European languages read "p" as "peh"...)

I have a complicated 3rd-order ODE of the form $P(y, y', y'', y''') = 0$, where $P$ is a complicated polynomial (5th-order with 24 terms) and coefficients that are (unknown) functions of a parameter $\lambda$, say $$ P(z_0, z_1, z_2, z_3) = \sum c_{j_0 j_1 j_2 j_3}(\lambda) z_0^{j_0} z_1^{j_1} z_2^{j_2} z_3^{j_3}. $$ What I want to know is: Suppose that there exists a non-constant solution $y(x)$ that is independent of $\lambda$. What conditions does this force the coefficient functions $c_{j_0 j_1 j_2 j_3}(\lambda)$ to satisfy?

(An easy analog would be something like: a 1st-order ODE of the form $$y' + c_0(\lambda) y = 0 $$ has nonconstant parameter-independent solutions if and only if $c_0(\lambda)$ is a constant function.)

I'm sure that the specific conditions I'm looking for depend on the precise form of the ODE, which is pretty daunting in this case. I'm just wondering if there's a reasonable algorithm I could apply to find them.

Let $\mathbb{R}^\infty$ be the product of countably many real lines.

Assume that a finitely generated group $\Gamma$ acts on $\mathbb{R}^\infty$ (linearly and continuously) and there is a nonempty convex compact $\Gamma$-invariant subset $K$.

Is it true that $\Gamma$ has a fixed point in $K$?

(In my case, the group $\Gamma$ is NOT commutative and it is NOT amenable.)

*Please help.*