Recent MathOverflow Questions

Schwartz distributions, Colombeau algebra and applications

Math Overflow Recent Questions - Tue, 06/05/2018 - 16:36

I have studied "enough" the theory of distributions , I would like to deepen some topic with applications. With some research I arrived at this book:

"Geometric Theory of Generalized Functions with Applications to General Relativity (Mathematics and Its Applications) (Volume 537)"

You can find the index (and some pages) on google books or amazon. Does anyone know this theory? which prerequisites are more fundamental to study from this book? thanks for every answer.

Connectedness of the 1-skeleton of a locally polyhedral convex set

Math Overflow Recent Questions - Tue, 06/05/2018 - 15:56

A closed convex set $K\subset \mathbb{R}^n$ is the intersection of its supporting (closed) half-spaces. We call $K$ locally polyhedral if any intersection of $K$ with a convex polytope $P$ is a convex polytope. If $K$ is locally polyhedral, its extreme points are isolated. Moreoever, the intersection of $K$ with any polytope contained in a small enough neighborhood of an extreme point $x$ is the intersection of that polytope with a unique cone (the intersection of the supporting half-spaces that have the extreme point on the boundary). Let us call the intersections of the extreme rays of these cones with $K$ edges. Edges are either bounded and connect two extreme points, or unbounded. It seems fairly intuitive that any two extreme points of $K$ should be connected by a finite path of bounded edges. However, all the ways I've been able to think of to prove this seem overly complicated for such a simple result. I would appreciate it if someone were to able to point me to the correct reference or provide a simple argument I am failing to spot.

The Geometry of Jacobi Forms and their Asymptotic Expansions

Math Overflow Recent Questions - Tue, 06/05/2018 - 15:53

A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying

$$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}\bigg) = (c \tau + d)^{k} e^{\frac{2 \pi i mc}{c \tau +d} z^{2}}\varphi(\tau, z)$$

$$\varphi(\tau, z + \lambda \tau + \mu) = e^{-2 \pi i m(\lambda^{2}\tau + 2 \lambda z)} \varphi(\tau, z).$$

Let $\pi : \mathcal{C} \to \overline{\mathcal{M}}_{1,1}$ be the universal elliptic curve. We can identify the surface $\mathcal{C} \cong \overline{\mathcal{M}}_{1,2}$, and we can realize $\overline{\mathcal{M}}_{1,1}$ as a divisor of $\mathcal{C}$. The general fiber $F$ gives another independent divisor of $\mathcal{C}$.

The way I understand it (please correct me if I'm wrong) a Jacobi form $\varphi$ is a section of a line bundle $\mathcal{E} \to \mathcal{C}$ such that the weight and index are encoded as

$$k = \text{deg}\big( \mathcal{E}|_{\overline{\mathcal{M}}_{1,1}}\big), \,\,\,\,\,\,\,\,\,\,\,\,\, m = \text{deg}\big(\mathcal{E} |_{F}\big)$$

I believe one should think of $\tau$ as a coordinate on $\overline{\mathcal{M}}_{1,1}$ and $z$ as a coordinate in the fiber direction. If $z=0$, indeed we just get a section of a line bundle over $\overline{\mathcal{M}}_{1,1}$, consistent with $\varphi(\tau, 0)$ being an ordinary modular form of weight $k$.

The Jacobi form I'm interested in is $\varphi_{-2,1}$ which is the unique weak Jacobi form of weight -2 and index 1. Its inverse is a meromorphic Jacobi form of weight 2 and index -1. It is well known that we have an asymptotic expansion around $z=0$,

$$\frac{1}{\varphi_{-2,1}}(\tau, z) = \sum_{g=0}^{\infty} z^{2g-2} \mathcal{P}_{g}(\tau),$$

where $\mathcal{P}_{g}(\tau)$ is a quasi-modular form of weight $g$. Basically, I want to understand the geometry of an expansion of this form.

First off, applying the elliptic transformation law of Jacobi forms (the second of the two transformation equations above) to this expansion gives nonsense, which makes sense because it holds only for small $z$. In addition, we lose all information about the index, since that is associated to the fiber directions (i.e. general $z$). However, I would expect it to encode the weight perfectly. And it almost does! We can try to apply a modular transformation

$$\frac{1}{\varphi_{-2,1}}\big(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}\big)= \sum_{g=0}^{\infty} \big( \frac{z}{c \tau + d}\big)^{2g-2} \mathcal{P}_{g}\big(\frac{a \tau + b}{c \tau + d}\big),$$

and notice that if $\mathcal{P}_{g}$ were modular, the cancellation of the automorphy factors $c \tau + d$ would work out perfectly to give us the actual weight. So my first question is: why does the quasi-modularity arise to spoil this determination of the weight? It comes so close to working, which would make geometric sense.

Secondly, putting aside issues of quasi-modularity, $\mathcal{P}_{g}$ should be a section of a line bundle $\mathcal{E}_{g} \to \overline{\mathcal{M}}_{1,1}$. The equation just above seems to indicate that we can interpret $z$ as a coordinate on, or section of the dual bundle $\mathcal{E}_{g}^{\vee}$, at least in the expansion. This would explain the cancellation of the automorphy factors geometrically. Is this or something like it true?

Finally, doing an expansion around $z=0$, I would expect the normal bundle of $\overline{\mathcal{M}}_{1,1}$ in $\mathcal{C}$ to play a role. What is this normal bundle, and does it play a role in this story?

Difficul integral- Solow model

Math Overflow Recent Questions - Tue, 06/05/2018 - 15:49

My friends are preparing project about a Solow model. The asked me to calculate such integral: $ s(1-a)\int e^{(1-a)(b+c)t} \cdot (d-ge^{ft})^{1-a}dt$ where: $b,c,s,d,g>0$ and $a∈(0,1)$. $[a,b,c,d,g,f$-constans] They recived a hint to use hypergeometric series. Unfortunately, I do not know anything about hypergeometric series.

About the nature of the process which originates prime numbers [on hold]

Math Overflow Recent Questions - Tue, 06/05/2018 - 15:36

"Given the sporadic, random-like quality of the primes, it is quite surprising how much can be proved about them. Interestingly, theorems about the primes are usually proved by exploiting this seeming randomness...

Much research on prime numbers has this sort of flavour. Your first devise a probabilistic model for the primes - that is, you pretend to yourself that they have been selected according to some random procedure. Next, you work out what would be true if the primes really were generated randomly. That allows you to guess the answers to many questions. Finally, you try to show that the model is realistic enough for you guesses to be approximately correct...

It is interesting that the probabilistic model is a model not of a physical phenomenon, but of another piece of mathematics. Although the prime numbers are rigidly determined, they somehow feel like experimental data. Once we regard them that way, it becomes tempting to devise simplified models that allow us to predict what the answers to certain probabilistic questions are likely to be. And such models have indeed sometimes led people to proofs valid for the primes themselves."

T. Gowers, Mathematics: A Very Short Introduction (Oxford Univ. Press, 2002) p.120-121

In the spirit of this quote, are we allowed to assume that prime numbers are generated by an unique and invariable process?

Or could the prime number sequence be considered as the interplay of many distinct "random procedures" operating at once or subsequentially?

How to stablish a proof either way?

With this I mean that it isn't well stablished that subsequences of the primes, such as the twin prime pairs or the Mersenne primes, are not generated by a different process than ordinary primes.

This also means that it isn't well stablished that there is no prime number p such that primes greater than p aren't generated by a different process.

It seems obvious to consider it a single invariant process from the fact that there´s an algorithm, the sieve of Erastosthenes, that allows us to find all prime numbers greater than p from a list of primes from 2 to p.

If the process was not unique or invariable, one would need extra information than what is already present at that previous sequence of primes, right?

Also, in other context, can the small gap between consecutive twin primes be considered as evidence of a dynamical system which repeatedly approaches it´s initial condition?

Is the set $\{\zeta: \rho(A(\zeta))< 1\}$ connected for matrices under parameterization of first $m$ rows?

Math Overflow Recent Questions - Tue, 06/05/2018 - 15:19

Let $\zeta \in \mathcal{M}(m \times n; \mathbb R)$ and we fix a $\zeta_0 \in \mathcal M( (n-m) \times n; \mathbb R)$. Let us define a parameterization by \begin{align*} F : \zeta \mapsto \begin{pmatrix} \zeta \\ \zeta_0 \end{pmatrix} = A(\zeta). \end{align*} Let $S = \{ \zeta: \rho( A(\zeta) )<1\}$ and assume $S \neq \emptyset$. I am interested to know whether $S$ is connected?

Need a formula for a function which is not integrable on [0,1], but its square is [on hold]

Math Overflow Recent Questions - Tue, 06/05/2018 - 14:02

Are there any functions f : R → R which aren't integrable on [0, 1], but h(x) = f(x).f(x) is integrable on [0, 1].

Real polynomial bounded at inverse-integer points

Math Overflow Recent Questions - Tue, 06/05/2018 - 14:02

Let $p$ be a real polynomial and $N$ be a positive integer. Suppose I tell you that $|p(\frac{1}{k})| \le 1$ for all $k\in\{1,\ldots,N\}$, and also that $p(\frac{1}{N})\le -\frac{1}{2}$ while $p(\frac{2}{N})\ge \frac{1}{2}$. What bounds can you give me on the minimal possible degree $\deg(p)$?

An upper bound of $O(\sqrt{N})$ follows by considering the Chebyshev polynomials (which indeed are bounded everywhere in $[0,1]$, not just at the inverse-integer points).

On the other hand, the best lower bound I could show was $\deg(p)=\Omega(N^{1/4})$. This follows by restricting our attention to the interval $[\frac{1}{N},\frac{1}{\sqrt{N}}]$, which has no point more than about $1/N$ away from an inverse-integer point, and where (by assumption) $p$ also attains a large first derivative somewhere. We then apply standard results from approximation theory about polynomials bounded at discrete points, due to, e.g., Ehlich, Zeller, Coppersmith, Rivlin, and Cheney. Unfortunately the original papers seem to be paywalled, but the idea here is just to say that either $|p(x)|=O(1)$ in the entire interval $[\frac{1}{N},\frac{1}{\sqrt{N}}]$, in which case we can directly use Markov's inequality to lower-bound its degree, or else $p$ goes on some crazy excursion in between two of the discrete points at which it's bounded (say $\frac{1}{k}$ and $\frac{1}{k-1}$), in which case it attains a proportionately larger derivative there, so Markov's inequality can again be applied.

My question is whether there are any fancier tools from approximation theory that yield a better lower bound on the degree, like $\Omega(N^{1/3})$ or conceivably even $\Omega(\sqrt{N})$.

In case it helps: I already tried ransacking the approximation theory literature, but while I found many papers about polynomials bounded at evenly-spaced points, I found next to nothing about unevenly-spaced points (maybe I didn't know the right search terms). I also tried using Bernstein's inequality, which often yields better lower bounds on degree than Markov's inequality. But the trouble is that Bernstein's inequality is only useful if our polynomial attains a large first derivative far away from the endpoints of the interval where we're studying it (i.e., towards the center of the interval). And it seems that that can't be guaranteed here, basically because the interval $[0,1]$ has precious few inverse-integer points that are anywhere close to its endpoint of $1$.

The Logic of Buddha: A Formal Approach

Math Overflow Recent Questions - Tue, 06/05/2018 - 13:39

Buddhist logic is a branch of Indian logic (see also Nyaya), one of the three original traditions of logic, alongside the Greek and the Chinese logic. It seems Buddha himself used some of the features of such a non-standard logic in his philosophical reasoning which makes it important from a philosophical perspective. (see also Trairūpya and Hetucakra) However, in this post, I am going to tackle its mathematical aspects rather than philosophical ones.

I came across the term "Buddhist logic" in a personal discussion with an Indian fellow of analytic philosophy background who asked me whether Buddhist/Indian logic could have any applications in the modern (predominantly Greek-logic based) mathematics/physics. Eventually, we came up with the idea that in the absence of a robust formalism and a fully clarified set of axioms for any such "non-standard" logic, there is not much to say about its mathematical properties and (dis)advantages in comparison with other logics, let alone finding any actual applications in daily mathematics, computer science or theoretical physics.

My initial search revealed some intriguing pieces of information about Buddhist and Indian logic/mathematics which motivated me to go through a more thorough investigation of the topic:

  1. In the following papers, Graham Priest discusses the connection between some features of Buddhist logic such as Catuṣkoṭi and paraconsistent logic. He also provides some formalism for Jania logic a variant of Indian logic corresponding to Jainism.

    • Priest, Graham, None of the above: the Catuṣkoṭi in Indian Buddhist logic. New directions in paraconsistent logic, 517–527, Springer Proc. Math. Stat., 152, Springer, New Delhi, 2015. (MR3476967)

    • Priest, Graham, Jaina logic: a contemporary perspective. Hist. Philos. Logic 29 (2008), no. 3, 263–278. (MR2445859)

  2. Douglas Daye has published a series of papers investigating some issues with formalizing Buddhist logic. (See also his follow-up papers: MR0536102 and MR0547735)

    • Daye, Douglas Dunsmore, Metalogical incompatibilities in the formal description of Buddhist logic (Nyāya). Notre Dame J. Formal Logic 18 (1977), no. 2, 221–231. (MR0457129)
  3. Prior to Daye, there has been some effort by Staal in the direction of formalizing Buddhist logic:

    • Staal, J. F. Formal structures in Indian logic. Synthese 12 1960 279–286. (MR0131338)
  4. Also, some connections between Indian logic and more applied areas of research such as computer science has been investigated:

    • Sarma, V. V. S., A survey of Indian logic from the point of view of computer science. Sādhanā 19 (1994), no. 6, 971–983. (MR1362512)
  5. There are several references to logic in the context of Vedic mathematics including in Vyākaraṇa where a scholar named Pāṇini has developed a grammar which "makes early use of Boolean logic, of the null operator, and of context-free grammars, and includes a precursor of the Backus–Naur form (used in the description programming languages)" (cf. Wikipedia)

  6. Last but not least, it is worth mentioning that various set-theoretic concepts including the notion of infinity (see the Sanskrit terms Ananta and Purna) has a strong presence in the Buddhist/Indian logic literature where they make a significant distinction between various types (and sizes?) of infinities (i.e. Nitya, Anitya, Anadi, and Anant) in a terminology fairly similar to modern treatment of ordinals. For instance, the book Tattvacintāmaṇi "dealt with all the important aspects of Indian philosophy, logic, set theory, and especially epistemology, ..." (cf. Navya-Nyāya). There is also a very detailed treatment of transfinite and infinitesimal numbers in Jain mathematics.

I am not quite sure if this short list is comprehensive and satisfactory enough to grasp the whole mathematical theory behind Buddhist logic, particularly because I am not of some Buddhist background to be fully aware of terminology (which is a crucial factor in the searching process). Also, most of the mentioned articles deal with the topic only marginally or from a slightly philosophical perspective. So I am looking for the possible help of some native Indian mathematicians or expert logicians who have worked along these lines.

Question. What are some other examples of papers on Buddhist/Indian logic in which the axioms and properties of these logics are formally investigated?

I am particularly interested in rigorous mathematical papers (rather than historical and philosophical ones) in which some theorems about their logical properties such as expression power, syntax, semantics, number of values, compactness, interpolation, etc., have been discussed (in a possibly comparative way). Any reference to possible implications of these formal systems in the foundation of mathematics especially in connection with the concept of infinity is welcome.

On finding the critical points of $f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x$

Math Overflow Recent Questions - Tue, 06/05/2018 - 12:52

Given some constant $a\in\mathbb{R}-\{0\}$, find $x_0$ such that $f'(x_0)=0$ where $$f(x)=\left(x-a+\frac1{ax}\right)^a-\left(\frac1x-\frac1a+ax\right)^x.$$

I have managed to write $f'(x_0)=0$ as $$\begin{align}\small\dfrac{a(ax_0^2-1)}{x_0(ax_0^2-a^2x_0+1)}\left(\dfrac{ax_0^2-a^2x_0+1}{ax_0}\right)^a&=\small\left(\ln\left(\dfrac{a^2x_0^2-x_0+a}{ax_0}\right)+\dfrac{a(ax_0^2-1)}{a^2x_0^2-x_0+a}\right)\left(\dfrac{a^2x_0^2-x_0+a}{ax_0}\right)^x\end{align}$$

Note that for the simplest case when $a=1$, we have that $x_0=1$ and it is a point of inflexion.

I suppose there isn't a closed form for $x_0$ based on elementary functions, but I expect Lambert's $W$ function to be of use in the manipulation of $x_0$ as the subject of the equation above.

P.S. I asked this question on MSE a few months ago and have offered multiple bounties but I was not able to gather very useful observations.

Single quantum particle entropy

Math Overflow Recent Questions - Tue, 06/05/2018 - 12:08

Consider a wave function of a single particle in free space, whose evolution is described by the (non-dimensional) linear Schrodinger equation $$i\psi _t (t,\underline{x}) + \Delta \psi=V(\underline{x}) \, ,$$

where $V$ is a potential, and $\underline{x}\in \mathbb {R}^d$ with $d=1,2,3$.

My question: Does there exist a notion of entropy for the solution $\psi$? This might be vague, but I'm looking for an integral functional of $\psi$ which increases as time increases. Specifically, I'm interested in the free-space case, i.e., $V\equiv 0$.

What I know: First, for an entropy to be defined we may need some sort of a probabilistic structure which I did not specify, mostly because I don't know how.

Second, I know that in quantum field theory there is a sense of light-entropy, but that's a totally different model equation than that of the linear Schrodinger equation. See e.g., Loudon's Quantum Theory of Light.

(cross posted from physics.se)

Reference request: Newton above Hodge

Math Overflow Recent Questions - Tue, 06/05/2018 - 11:56

Let $K$ be a p-adic field, and let $\mathcal{O}$ be the ring of integers inside $K$ with residue field $k$. Let $\mathcal{X}$ be a smooth proper formal scheme over $\mathcal{O}$ (with topology given by $p$). In this situation, there is a beautiful/famous result goes by the saying "Newton above Hodge". The statement is that the Newton polygon of $H^i_{crys}(\mathcal{X}_0/W(k))[\frac{1}{p}]$ lies above or on the Hodge polygon of $H^i_{dR}(\mathcal{X}_{K})$.

The proof I'm aware of goes by proving Fontaine's Crystalline conjecture (due to Faltings, and many others, then most recently Colmez--Nizioł/Bhatt--Morrow--Scholze), which implies that the Crystalline cohomology $H^i_{crys}(\mathcal{X}_0/W(k))[\frac{1}{p}]$ as a filtered $\phi$ module is weakly admissible. (Obviously tons of details need to be said with this approach.)

This sounds like a somewhat complicated "proof". Does anyone know an alternative proof?

I have seen people stating the above result and refer to either Mazur's paper [Frobenius and the Hodge filtration (estimates)] (which is a special case) or the book by Berthelot--Ogus [Notes on Crystalline Cohomology, 8.36] (which shows the Newton polygon lies on or above the Hodge polygon $\textit{of the special fiber}$). I have also come across the paper by Ogus [Frobenius and the Hodge spectral sequence] (which generalizes the Berthelot--Ogus result above). These are certainly beautiful and related results, but it seems that one cannot quite deduce the statement in the first paragraph out of them.

Explicit permutation representation of the Schur double cover of the symmetric group

Math Overflow Recent Questions - Tue, 06/05/2018 - 10:10

Main question: How can we describe the double covers $(2\cdot\mathfrak{S}_n)^+$ and $(2\cdot\mathfrak{S}_n)^-$ of the symmetric group $\mathfrak{S}_n$ as permutation groups? (I.e., what sort of set with combinatorial structure do they act faithfully upon?)

I am not necessarily asking for the minimal permutation degree (I think this is an open problem; also see PS at bottom), but for some explicit combinatorial construction, reasonably uniform in $n$, that is hopefully reasonably small (perhaps merely exponential in $n$).

For a long time, I wrongly believed that one could form such a permutation representation of degree $2^{\lfloor n/2\rfloor}$ (or perhaps twice or four times that, or thereabouts) by taking an appropriate basis in the basic spin representation (and their negatives, and perhaps also times the imaginary unit), but I now realize that this does not work (at least, not as I thought it would). So I wonder if something else can be done.

The following question may is closely related to the above (trying to lift some $\mathfrak{S}_n$-set $X$ to a $(2\cdot \mathfrak{S}_n)$-set $\tilde X$ consisting of "signed" elements of $X$):

Alternate question: Given a set $X$ on which $\mathfrak{S}_n$ acts faithfully and transitively, with stabilizer $H$, is there some way to detect (merely by looking at $X$) whether $H$ lifts in $2\cdot \mathfrak{S}_n$ to a direct product $\{\pm 1\}\times H$, or equivalently (see below), whether the wreath product $1 \to \{\pm 1\}^X \to \{\pm 1\}\wr_X\mathfrak{S}_n \to \mathfrak{S}_n\to 1$ contains a non-split extension $1 \to \{\pm 1\} \to 2\cdot \mathfrak{S}_n \to \mathfrak{S}_n\to 1$?

Indeed, if $2\cdot \mathfrak{S}_n$ acts transitively on a set $\tilde X$, with stabilizer $H$, and the action does not factor through $\mathfrak{S}_n$, then $H$ intersects $\{\pm 1\}$ (center of $2\cdot\mathfrak{S}_n$) trivially, so maps isomorphically to a subgroup of $\mathfrak{S}_n$, which we can also call $H$, and the latter lifts to $2\cdot\mathfrak{S}_n$ as a direct product $\{\pm 1\}\times H$ (and we can see $\tilde X = (2\cdot\mathfrak{S}_n)/H$ as a double covering of $X = \mathfrak{S}_n/H$). But then, it follows from Derek Holt's paper "Embeddings of Group Extensions into Wreath Products" (Quarter. J. Math. Oxford 29 (1978) 463–468), theorem 2, that the wreath product contains our $2\cdot \mathfrak{S}_n$. Conversely, when this is the case, $2\cdot \mathfrak{S}_n$ acts faithfully and transitively (through the wreath product) on $\tilde X := \{\pm 1\}\times X$.

For example, if $H$ is generated by an $n$-cycle, and either $n$ is odd or we $n\equiv 2\pmod{4}$ if we are considering $(2\cdot\mathfrak{S}_n)^+$, we get a permutation action of degree $2(n-1)!$ on the set $\mathfrak{S}_n/H$ of cyclic orders with a sign added. But this isn't a great improvement over the regular action ($2n!$) and it's not very explicit, so I'm hoping we can do better.

PS: If my computations with Gap are correct, the minimal permutation representation for (Gap's choice) $(2\cdot\mathfrak{S}_n)^-$ is $16,48,80,240,240,480$ for $n=4,5,6,7,8,9$. This is not in the OEIS, I wonder if it's worth adding. I don't know how to construct $(2\cdot\mathfrak{S}_n)^+$ under Gap (I don't know how to perform central products, for example), and I don't see any reason why the degrees should be the same for the latter.

Singular values of Hadamard Product

Math Overflow Recent Questions - Tue, 06/05/2018 - 07:59

For two Matrices $A,B \in \mathbb{R}^{m \times n}$ the Hadamard Product is defined as $(A \circ B)_{i,j} = A_{i,j}B_{i,j}$.

For a proof of convergene I require an upper (and ideally a lower) bound on \begin{align} \sum_{i>k}^{\min(m,n)}{\sigma_i^2(A \circ B)} \end{align} for Matrices with $|A_{i,j}| < 1$ and $|B_{i,j}| < 1$ for all $i,j$ and an arbitrary $1 \leq k < \min(m,n)$

From tests with randomly generated matrices I suspect that the following strong pointwise property, that could be used for a good upper bound, holds:

\begin{align} \sigma_i(A \circ B) < \text{max}(\sigma_i(A),\sigma_i(B)) \end{align}

Generally a decrease in the singular values is expected due to the well known fact that $\lVert A \rVert_F = \sqrt{\sum_i{\sigma_i^2(A)}}$ and by definition $\lVert A \circ B \rVert_F < \min(\lVert A \rVert_F,\lVert B \rVert_F)$, but this does not imply either of the inequalities stated above.

Is there some known upper bound I could use for this problem? Does someone see a proof or counterargument to the strong bound I provided?

Combinatoric paths between the refined f- and h-partition polynomials of the associahedra

Math Overflow Recent Questions - Tue, 06/05/2018 - 01:26

The f-polynomials, $F_n(x)$ (cf. OEIS A126216, A033282, and A086810), and the h-polynomials, $H_n(x)$ (cf. A001263, the Narayana polynomials), of the family of simple convex polytopes the associahedra for the Coxeter group $A_n$, or Stasheff polytopes, are related by the simple analytic equation

$$F_n(x-1) = H_n(x).$$

McCammond in "Noncrossing partitions in surprising locations" presents a geometric/combinatoric path between the f- and h-polynomials of associahedra.

The f-polynomial $F_n(x)$ enumerates (is the o.g.f. of) the number of k-dimensional faces of the n-dimensional associahedron. Refined, signed versions of the f-polynomials for the associahedra, enumerating and labelling to a finer degree the faces of the associahedra, are given in A133437 as

$$FP_n(a_1,a_2,...) = (1/n!) [g(x)D_x]^n x|_{x=0}$$

with

$$g(x) = 1 / D_x f(x)= 1/ [a_1 + 2a_2 x + \cdots].$$

These partition polynomials give the coefficients of the power series that is the formal compositional inverse of the formal power series, or ordinary generating functions (o.g.f.s),

$$f(x) = a_1 x + a_2 x^2 + \cdots $$

and also correspond to enumerating dissections by polygons of polygons (see "Polygonal dissections and reversions" by Schuetz and Whieldon) as well as aspects of other combinatorial constructs, such as the Dyck lattice paths of A126216. For some more details see A145271, MO-Q, and MO-A.

The corresponding h-polynomials for the associahedra enumerate noncrossing partitions, and the corresponding refinement A134264 gives the partition polynomials $HP_n(b_0,b_1,...)$ for the compositional inverse of $f(x)$ in terms of the coefficients of the power series for the reciprocal

$$h(x) = x / f(x) = b_0 + b_1x + b_2 x^2 z + \cdots .$$

These partition polynomials are generated by

$$HP_n = (1/n!)[g(x)D_x]^n x|_{x=0}$$

with

$$g(x) = 1 / D_x [f(x)] = 1/ D_x[x/b(x)]$$

$$ = 1 /D_x [x/[b_0 + b_1 x + \cdots]]$$

and enumerate and label to a finer degree the noncrossing partitions as well as the Dyck lattice paths A125181.

Analytically, the refined f- and h-partition polynomials are related by the partition transformation A263633, a refined binomial convolution, connecting the indeterminates $a_n$ to $b_n$.

Question: What are some purely geometric/combinatoric paths/transformations, as opposed to the analytic path/transformation just skeched, between the refined f-an h-partition polynomials?

Mathematics without the principle of unique choice

Math Overflow Recent Questions - Mon, 06/04/2018 - 22:34

The principle of unique choice (PUC), also called the principle of function comprehension, says that if $R$ is a relation between two sets $A,B$, and for every $x\in A$ there exists a unique $y\in B$ such that $R(x,y)$, then there exists a function $f:A\to B$ such that $R(x,f(x))$ for every $x\in A$.

In ZF set theory (without the axiom of choice) this principle is provable; indeed it is basically the definition of what it means to be a "function". The same is true in weaker, e.g. intuitionistic set theories, and also for h-sets and h-relations in homotopy type theory / univalent foundations. However, PUC is not provable in (classical or intuitionistic) higher-order logic (HOL) — unless we assert it, or something that implies it, as an additional axiom.

Moreover, there are naturally-arising models of HOL in which PUC fails. For instance, we can take the types to be topological spaces, and the predicates on a space $A$ to be the subspaces of $A$ (subsets with the subspace topology); then a relation $R$ as in PUC corresponds to a span $A \leftarrow R \to B$ in which $R\to A$ is a continuous bijection, but need not have a continuous inverse enabling us to define a continuous map $A\to B$. (To be more precise, it would be better to replace topological spaces by something slightly better-behaved, such as a quasitopos.) We can also take the types to be partial equivalence relations on a partial combinatory algebra, or partial equivalence relations in a tripos.

Constructive mathematicians following Brouwer and Bishop have explored in depth what mathematics looks like in the absence of the full axiom of choice and the law of excluded middle. Usually it's not much different; one just has to take extra care in various places and state various theorems in idiosyncratic ways. Moreover, classical mathematics can be embedded in constructive mathematics, e.g. by judicious insertion of double-negations; thus constructive mathematics is simply "more informative", i.e. it "draws distinctions that classical mathematics ignores".

However, all constructive mathematicians that I know of accept the principle of unique choice. Has anyone seriously explored what mathematics would look like in the absence of PUC? I don't mean simply using the internal language of a quasitopos to prove a few things; I mean a serious development of large swaths of mathematics akin to Bishop's Constructive analysis.

Note that, as is the case with classical and constructive mathematics, ordinary mathematics with PUC should embed into mathematics without PUC by simply interpreting the word "function" to mean a total functional relation as appears in the hypothesis of PUC (which I call an anafunction after Makkai's "anafunctors"). So it seems that mathematics without PUC should again be simply "drawing previously-ignored distinctions", keeping track of which anafunctions are actually functions.

Extend a bundle "trivially"

Math Overflow Recent Questions - Mon, 06/04/2018 - 15:10

Suppose I have a fibre bundle $E\to B$ with compact fibre. Furthermore, $B$ is open in a larger, compact space, e. g. $B\subseteq B'$. I want to get a map $E'\to B'$ (not a bundle any more!) with

  1. $E'|_B = E$
  2. For $x\in B'\setminus B$ each fibre is only one point.
  3. $E'$ is a compact space.
  4. Each section $\omega:B\to E$ can be extended to $\omega':B'\to E'$.

The idea is to »collapse« the whole fibre when reaching the boundary of $B$.

For example, if $E:=[0,1]\times (0,1)$, $B:=(0,1)\subseteq [0,1]:=B'$ and the bundle $E\to B$ is the projection, this should be extended on $B':=[0,1]$ to $E'=\Sigma [0,1]$.

Is there a precise way to give this topological construction explicitely?

$R$ a DVR with fraction field $K.$ What are the $R$-submodules of $K^n?$

Math Overflow Recent Questions - Mon, 06/04/2018 - 13:50

It is known that if $R$ is a DVR with fraction field $K,$ then the $R$-submodules of $K$ are $0,K,x^nR,$ with $n$ any integer and $x$ a generator of the maximal ideal of $R.$ I was wondering if there is a simple structure theorem for the $R$-submodules of $K^n,$ the $n$-dimensional vector space over $K$ (similar to that for finitely generated torsion-free modules over a Dedekind domain). I know that the subroups of $\mathbb{Q}$ are somewhat complicated to describe, but was wondering, in particular, if the situation gets any nicer with, say, $\mathbb{Z}_P$-submodules of $\mathbb{Q}^n,$ where $P$ is a nonzero prime ideal of $\mathbb{Z}.$

Cops, Robbers and Cardinals: The Infinite Manhunt

Math Overflow Recent Questions - Sun, 06/03/2018 - 11:13

Cops & Robbers is a certain pursuit-evasion game between two players, Alice and Bob. Alice is in charge of the Justice Bureau, which controls one or more law enforcement officers, the cops. Bob controls a single robber, a guy whose main concern is to evade the cops at any costs. Both deploy their guys on the vertices of a city, a simple graph, and start a manhunt under the following natural rules:

  • On her turn, Alice chooses a (possibly empty) subset of the cops and moves them to some adjacent vertices.

  • On his turn, Bob may either move his robber to an adjacent vertex or stay still.

  • The game ends with a win for Alice whenever any of her cops manages to occupy the same vertex as the robber. Bob wins if this never happens in the game.

Cops & Robbers and its variants have been studied on finite graphs so extensively, while little is known in the case of the infinite graphs, particularly those of uncountable cardinality which are the subject of this question. One may take a look at the following references for a fairly comprehensive account of the topic (including some results concerning the infinite version) or watch this and this episodes of PBS Infinite Series show by Kelsey Houston-Edwards:

The cop number of a graph is the minimum number of the cops that Alice needs to have a winning strategy in the game. A graph with cop number $1$ is called cop-win. Otherwise, it is called robber-win. Before getting into the main question, let's observe that the cop number of a large infinite graph may or may not be large. For instance, $K_{\kappa}$ and $S_{\kappa}$, the complete and star graph of size $\kappa$, are clearly cop-win and therefore have the cop number $1$. In contrary, the totally disconnected graph of size $\kappa$ has the cop number $\kappa$. Some other examples are the infinite path $P_{\infty}$, and Rado graph, $\mathcal{R}$, which have the cop number $\aleph_0$.

Definition. We call connected infinite graphs with minimum and maximum possible cop-numbers (i.e. $1$ and $|G|$), law and crime neighborhoods respectively. A Mega City is an infinite connected graph, $M$, with the "dystopian" property that $M$ contains either a law or a crime neighborhood as big as $|M|$ as its sub-graph. An uncountable cardinal $\kappa$ is a Dredd cardinal (named after Judge Dredd) if every connected graph of size $\kappa$ is a Mega City.

The first question is about the possible large cardinal strength of the existence of such a Ramsey-like cardinal.

Question 1. Is there any Dredd cardinal? What is their large cardinal strength in comparison with other large cardinals of some Ramsey characterization such as weakly compact, Erdős, and Ramsey cardinals?

Remark 1. An approach towards question 1 might be through considering $\kappa$-Aronszajn trees and checking the tree property which provides a characterization of weakly compact cardinals. Maybe one can relate the cofinal branches in a $\kappa$-tree to the law or crime neighborhoods in a Mega City of size $\kappa$, given the fact that trees and paths are rather simple cities to analyze for the game of cops and robbers. Also, as Will mentioned in his comments here and here, in the definition of a Mega-City the connectedness assumption is essential for the whole graph and crime neighborhoods in order to avoid easy constructions.

The next question is about the possible characterization of infinite connected graphs with arbitrary cop-numbers.

Question 2. Let $\kappa$ be an infinite cardinal and $1\leq \lambda\leq \kappa$. Is there a connected graph of size $\kappa$ and cop number $\lambda$? How many graphs of this type are there up to isomorphism (for any given $\kappa$, $\lambda$)?

Remark 2. Concerning the question 2 in the special cases such as $\kappa=2^{\aleph_0}$, an approach could be thinking about using cardinal characteristics of the continuum for constructing examples of connected (directed) graphs of size continuum with cop numbers strictly less than $2^{\aleph_0}$. In order to observe this, note that cop number of any graph is less than or equal to its domination number because one may catch the robber in the first move simply by deploying one cop on each vertex in a minimal dominating set of the graph. The idea is that the set-theoretic dominating number, $\mathfrak{d}$, serves as the domination number of a graph, $G$, of size $2^{\aleph_{0}}$ (don't confuse the similar terms in set theory and graph theory with each other). At the request of Monroe, let me clarify the example a little bit more:

$G$ is defined by the sequence domination relation between reals (i.e. functions $f:\omega\rightarrow \omega$). In this graph, the vertex labeled by $g$ is adjacent to $f$ (i.e. $g\rightarrow f$) if $g$ dominates $f$ as a sequence (denoted by $f\leq^* g$) that is $f(n)\leq g(n)$ for all but finitely many $n\in \omega$. Now a dominating family, $\mathcal{F}$, of sequences for the entire real numbers is a set of functions from $\omega $ to $\omega$ such that every such function is either in $\mathcal{F}$ or is $\leq^*$-dominated by a member of $\mathcal{F}$ (i.e. $\mathcal{F}$ is a domination set of $G$ in the graph-theoretic sense) and $\mathfrak{d}$ is defined to be the minimum size of such a family (i.e. the domination number of the graph $G$). Consequently we have $c(G)\leq \mathfrak{d}\leq 2^{\aleph_0}$ (which $c(G)$ stands for the cop number of $G$). Finally using forcing one may make $\mathfrak{d}<2^{\aleph_0}$ and so the cop number of the graph $G$ in the generic extension will be strictly smaller than $2^{\aleph_{0}}$.

However, one should note that generally, the domination number is a very loose upper bound for the cop number of a graph. (Certainly, the Justice Bureau has much more clever strategies to control the crime rate in the city rather than placing a cop right in the front of every single home!) For instance, the domination number of the dodecahedron graph (of size $20$) is $7$ while its cop-number is only $3$. Here, the following question arises:

Question 3. What is the cop-number of $G$, the sequence domination graph of size continuum described in the previous remark? Can it be countable? If not, what is the precise value of this new cardinal characteristic of continuum and how does it relate to the other entities in the Cichoń's diagram?

Which utility functions are linearly transformed by normal perturbations?

Math Overflow Recent Questions - Sat, 06/02/2018 - 17:12

I'll ask this question as pure economics, as pure math, and showing the translation.

Economics (micro): Which utilities have the property that whenever $EU(X)>EU(Y)$, the same is true after perturbing both $X$ and $Y$ by $N(0,1)$? Is it just linear and exponential utilities?

Math (integro-differential equations): \begin{align} \text{For }EU(x+N(0,1))&=\int_{-\infty}^\infty \frac{e^{-(t-x)^2/2}}{\sqrt{2\pi}_\phantom{2_2}}\, U(t)\, dt,\\ \text{does }\ \frac{dEU(x+N(0,1))}{dx}&=c\ \frac{dU(x)}{dx} \ \text{ imply }\ U' U''' = U'' U'' \ ? \\ \end{align}

Translation:

Suppose my utility is a function of my wealth $x$, and my wealth may receive a normal shock.

Currently my utility is $U(x)$. My expected utility after the shock is $EU(x+N(0,1)$, as above.

  • For a linear utility function, the shock does not change my expected utility.

  • For a nonlinear polynomial utility function, the shock transforms it nonlinearly.

  • For an exponential utility function, of the form $$U(x)=a-be^{cx}$$ the expected utility becomes $$EU(x+N(0,1))\ =\ a-be^{c^2/2}e^{cx}\ =\ a-e^{c^2/2}(a-EU(x))$$ which is a linear transformation. Then integrating over many possibilities gives $$EU(X+N(0,1))>EU(Y+N(0,1))\ \leftrightarrow\ EU(X)>EU(Y),$$ and conversely we know from Von Neumann that this property holds iff $EU(X+N(0,1))$ and $EU(X)$ are linear transforms of each other.

The question is whether this property characterizes the exponential and linear utility functions, which are also characterized by the differential equation $U' U''' = U'' U''$.

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