Math Overflow Recent Questions


Is it always possible to calculate the limit of an elementary function?

Sat, 03/17/2018 - 04:06

I already asked this question on but as I received no answer; maybe it is not as obvious as I originally thought.

The precise formulation of my question is: define an "strong elementary function" by only admitting rational for the "constant function" in the usual definition of "elementary" function (see for example: Let $a$ be an "elementary real" if the constant function $f(x)=a$ is a strong elementary function. With this definition some non rational reals are elementary (for example $\pi=4⋅arctan(1)$); but there are reals that are not elementary. Now let $f(x)$ be a strong elementary function defined in an open interval of an elementary real $a$ with the possible exception of $a$. Suppose that $lim_{x\rightarrow a} f(x)$ exists. Is this limit necessarily an elementary real?

The idea behind this question is this: it seems that "limits" can always be calculated with some simple tricks (Hospital rule, etc...) in elementary calculus; but is there a general argument that shows that it is always possible? (the precise formulation of the question does not ask for an algorithm, I expect a positive answer to be constructive, but that is not entirely clear).

Update: clarification of the notion of elementary functions.

Definition By function $\mathbb{R}\rightarrow\mathbb{R}$; I mean a partial function. The class of (strong) elementary functions is the smallest class of functions such that:

  1. the constant function $f(x)=1$ is elementary.
  2. if $f$ and $g$ are elementary; so is $f+g$; $f-g$; $f\cdot g$; $f/g$. (the domain of $f/g$ is ${\rm dom}(f)\cap {\rm dom}(g)\cap \{x \ | \ g(x)\neq 0\}$.
  3. if $n$ is a natural number; $f(x)=x^n$ and $f(x)=\sqrt[n]{x}$ are elementary; the domain of the later is $\mathbb{R}^+$.
  4. $\sin$, $\cos$, $\tan$ are elementary
  5. $\arcsin$, $\arccos$ and $\arctan$ are elementary.
  6. $\exp$ is elementary.
  7. $\ln$ is elementary.
  8. if $f$ and $g$ are elementary; so is $f\circ g$; the domain of the latter is $\{x\ | \ x\in{\rm dom}(g) \wedge g(x)\in{\rm dom}(f)\}$.

I have not tried to avoid redundancy but note that the point 3 is not redundant because of the domain of the functions considered; for example $f(x)=x^2$ is defined on $\mathbb{R}$ but $f(x)=\exp(2\cdot\ln(x))$ is defined on $\mathbb{R}_0^+$. Also $\sqrt[n]{x}$ is defined for $x=0$ but not $\exp(\frac{\ln(x)}{n})$.

I think this is the class of functions we consider in the Risch algorithm ( except that I do not take all constant functions as elementary; that would obviously make no sense for my question.

I hope I have not missed something obvious. I do not think a small modification of my definition will make any difference; if it is it would be interesting to discus.

Help with understanding a proof on angle preservation

Sat, 03/17/2018 - 03:28

Let $R \in \mathbb{R}^{n,d}$ be a random Gaussian matrix comprised of independent $\operatorname{N}(0,\frac{1}{n})$ entries, let $\textbf{w}$ and $\textbf{x}$ be vectors in $\mathbb{R}^d$, and let $\epsilon \in (0,1)$. In Shi et al. 2012, it is claimed in Theorem 5 that if $\langle \textbf{w},\textbf{x} \rangle > 0$, then

$$\frac{1+\epsilon}{1-\epsilon}\frac{\langle \textbf{w},\textbf{x} \rangle}{\| \textbf{w} \| \| \textbf{x} \|} - \frac{2\epsilon}{1-\epsilon} \leq \frac{\langle R\textbf{w}, R\textbf{x}\rangle}{\| \textbf{w} \| \| \textbf{x} \|} \leq 1 - \frac{\sqrt{1-\epsilon^2}}{1+\epsilon} + \frac{\epsilon}{1+\epsilon} + \frac{1-\epsilon}{1+\epsilon}\frac{\langle \textbf{w}, \textbf{x} \rangle}{\| \textbf{w} \| \| \textbf{x} \|}$$ with probability at least $1 - 6\exp(-\frac{n}{2}(\frac{\epsilon^2}{2}-\frac{\epsilon^3}{3}))$.

An auxiliary lemma (which they call Lemma 10) that they use to help prove the theorem states that for any $\textbf{x} \in \mathbb{R^d}$, any random matrix $R \in \mathbb{R}^{n,d}$ comprised of independent $\operatorname{N}(0,\frac{1}{n})$ entries, and any $\epsilon \in (0,1)$: $$\operatorname{Pr}(1-\epsilon \leq \frac{\|R\textbf{x}\|^2}{\|\textbf{x}\|^2} \leq 1 + \epsilon) \geq 1 - 2\exp(-\frac{n}{2}(\frac{\epsilon^2}{2}-\frac{\epsilon^3}{3}))$$

The authors use the union bound in conjunction with Lemma 10 to establish Theorem 5, but their explanation of how to do so isn't clear to me. In particular, I'd like to know why they apply the union bound to the vectors $\textbf{w}$, $\textbf{x}$, and $\frac{\textbf{x}}{\|\textbf{x}\|} - \frac{\textbf{w}}{\|\textbf{w}\|}$ in order to obtain the probability lower bound of $1 - 6\exp(-\frac{n}{2}(\frac{\epsilon^2}{2}-\frac{\epsilon^3}{3}))$ from Theorem 5.

"Inner product" between prime factorizations

Fri, 03/16/2018 - 23:34

Let $x \in \mathbb{Q}^+$. Then $x$ can be expressed uniquely as a product over primes:

$$x = \prod_{p \text{ prime}} p^{\nu_p(x)}$$

where $\nu_p(x)$ is the p-adic valuation of $x$. Let $\nu(x) = \langle \nu_2(x), \nu_3(x), \ldots \rangle \in \mathbb{Z}^{\oplus \mathbb{N}}$. Then $\nu(x) + \nu(y) = \nu(xy)$ and $a \nu(x) = \nu(x^a)$. Thus the positive rationals can be viewed as a module where addition and multiplication of two module elements correspond to multiplication and exponentiation of their corresponding rationals. Suppose we define the following "inner product":

$$\nu(x) \cdot \nu(y) = \sum_{p \text{ prime}} \nu_p(x) \nu_p(y)$$

This gives rise to a notion of orthogonality, angles, lengths, and volumes between positive rationals. My question is this: Do this "inner product" and the aforementioned notions have any significance, from the perspective of number theory? Have they been studied before?

For example: if $x,y \in \mathbb{Z}^+$ then $\nu(x) \cdot \nu(y) =0$ iff $x$ and $y$ are coprime. Furthermore, if $x \in \mathbb{Z}^+$ then $\lVert \nu(x) \rVert_1 = \Omega(x)$ and $\lVert \nu(x) \rVert_0 = \omega(x)$.

Cardinality of the image of a polynomial modulo $p^n$

Fri, 03/16/2018 - 07:08

Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial and let $p$ be a prime number. I'm looking for results about $$N_f(p^k) := \#\{(f(n) \bmod p^k) : n \in \mathbb{Z}\},$$ as $k \to +\infty$, where $(a \bmod b)$ is the remainder of $a$ divided by $b$.

If $N_f(p) = p$ and $f^\prime(x) \not\equiv 0\bmod p$, for all $x \in \mathbb{Z}$, then using Hensel's lemma it can be proved that $N_f(p^k) = p^k$ for all $k \geq 1$. However, I did not find more general result.

Thank you for any suggestion.

Submanifolds of nilmanifolds coming from Lie subgroups

Fri, 03/16/2018 - 06:07

Let $G$ be a connected simply connected nilpotent real Lie group and $\Gamma$ a lattice in $G$, such that $M=\Gamma \backslash G$ is a compact nilmanifold. Let $p:G \to M$ be the projection. If $S$ is a Lie subgroup of $G$ (coming from a Lie subalgebra of $Lie(G)$), is $p(S) \subset M$ is a (embedded, immersed) submanifold?

My only intuition comes from the abelian case, where I think the answer yes. But this is very particular since in this case the nilmanifold itself is also a Lie group and everything works easier.

Parabolic subgroups of relatively hyperbolic and CAT(0) groups

Fri, 03/16/2018 - 05:01

Let $G$ be a finitely generated group. We say that $G$ is CAT(0) if it acts properly and co-compactly by isometries on a CAT(0) space. We say it is hyperbolic relative to a collection $\Omega$ of subgroups if it acts properly by isometries on a Gromov-hyperbolic space $X$ such that limit points are either conical or bounded parabolic and the stabilizers of the parabolic points are precisely the elements of $\Omega$. Those subgroups are called (maximal) parabolic subgroups or peripheral subgroups.

Question: Assume that $G$ is both CAT(0) and relatively hyperbolic. Are the peripheral subgroups CAT(0) themselves ?

A more general question would be: Is a quasi-convex subgroup of a CAT(0) is CAT(0) itself ? It somehow seems too much to ask since CAT(0) spaces do not behave so well with quasi-geodesics or quasi-isometries. However, we have more structure assuming the relatively hyperbolicity.

Connections in terms of tangent ($\infty$-)categories?

Fri, 03/16/2018 - 03:56

Given a commutative ring $k$ and a commutative $k$-algebra $A$, we know that the Kähler differential $\Omega_{A/k}^1$ could be described through machineries of tangent categories (see, for example, the nLab page), which could be adapted to $\infty$-categorical settings: there is a rich theory of the cotangent complex formalism in Lurie's Higher Algebra, section 7.3.

My question is that, whether we have an analogue for connections. Given a commutative ring $k$, a commutative $k$-algebra $A$ and an $A$-module $M$, we define a connection $\nabla$ to be a $k$-linear morphism $M\to\Omega_{A/k}^1\otimes_AM$ satisfying Leibniz rules. I wonder whether we can also describe this through category theory and imagine a link between this and the deformation theory, so that we can generalize the connection into $\infty$-categorical setting?

Access to a classic reference of Dold-Puppe

Fri, 03/16/2018 - 03:39

There is an old reference that I am unable to easily find. It is Dold-Puppe´s communication at a conference. More concretely, it is cited as:

A. Dold, D. Puppe: Duality, trace and transfer. Proceedings of the International Conference on Geometric Topology (Warsaw, 1978), PWN, Warsaw, 1980, pp. 81–102.

I don't know if it is due to difficulties associated to the iron curtain but I find no online access to this (apart from the russian version).

My library could ask for it, but I don't like this. Not only it is a tedious and costly process (request, scan at other library, send, print/collect), but it will not solve the problem for people willing to access it after me (let alone they must belong to an institution). Since it seems to be the classic reference for their notion of duality I would like to ask:

Is there a way to have online access to this reference?

If not

Would it be possible for someone with privileges for such things to upload it to lib genesis or some other place?

Transfer modules and Weyl algebra

Fri, 03/16/2018 - 02:52

Let $V$ be a $\mathbb{C}$-vectorial space of dimension $n$ and $V^*$ the complex dual space.

I would like to understand the following isomorphism $$D_{V^* \leftarrow V \times V^*} \overset{L}\otimes_{\mathcal{O}_{V \times V^*}} D_{V \times V^* \rightarrow V} \simeq D_{V \times V^*} \otimes det(V^*).$$ Here we use the standard notations for $D$-modules. In this case we work with Weyl algebras and $det(V^*)$ is the $n-th$ exterior product of $V^*$. The maps involved in the transfer modules are of course the projections.

Thank you for any help.

PS : This question has been asked on mathstackexchange without any answer but perhaps mathoverflow is more well suited for this.

Excel with 3 condition [on hold]

Fri, 03/16/2018 - 02:23

I want my excel to show the level of the agent (level 1, level 2, level 3) which will depend if the said agent was able to meet parameters.


Level 1 - <100,000; Level 2 - <1,000,000; Level 3 - >1,000,000 AND at least 5 cases with at least 20% growth

I know there is a way to do this in excel, just need the right person to help me fogure it out,

Thanks in advance

The upper bound of the number of points of a convex hull formed by external co-tangents of circles

Fri, 03/16/2018 - 01:48

Consider the following problem: Given a rope to surround some circles, and minimize the length of the rope.
In order to solve the problem, we shall calculate all external co-tangents of these circles, and use these points to get a convex hull. Then enumerate each edge of the convext hull to check whether it is part of the rope. The rope will be composed of such edges and some arcs.
So the crucial problem is to estimate the number of points of the convex hull. If the number of points can reach $\Theta(n^2)$ where $n$ is the number of circles, the algorithm will be slow. But I guess the number of points is $\Theta(n)$ from experiments.
Can we prove the upper bound is $\Theta(n)$? Or just construct an example to overthrow the claim.

Sum-regular $\{0,1\}$-matrices

Fri, 03/16/2018 - 00:56

Let $n\in\mathbb{N}$ be a positive integer. We say that an $n\times n$-matrix $A$ with all entries in $\{0,1\}$ is $k$-regular for some $k\in \{0,\ldots,n\}$ if the sum of every row and the sum of every column of $A$ equals $k$. Let $M(n, k)$ be the number of $k$-regular $n\times n$-matrices with all entries in $\{0,1\}$.

It's easy to see that $M(n,1)=n!$. Moreover, a symmetry argument shows that $M(n,k) = M(n, n-k)$ for all $k\in \{0,\ldots,n\}$.

Question. Given $n>1$, is it true that for all $k\in\{0,\ldots, n\}$ we have $M(n,k)\leq M(n, \lfloor\frac{n}{2}\rfloor)$?

Mellin-like Transform of Whittaker function

Thu, 03/15/2018 - 21:02

In computing the $L^2$ inner product of a non-holomorphic Eisenstein series with a Poincare series, the following integral appears:

$$\int_0^\infty e^{-2\pi my}y^{s+\frac{1}{2}}K_{s-\frac{1}{2}}(2\pi |m|y)\frac{dy}{y}$$ where $K_{s-\frac{1}{2}}(2\pi |m|y)$ denotes the Whittaker function, defined by $$K_{s-\frac{1}{2}}(2\pi |m|y)=\int_0^\infty t^{s-\frac{1}{2}}e^{-(t+\frac{1}{t})\pi|m|y}\frac{dt}{t}$$ I am looking either for resources or for tips on how to proceed in computing this integral. Any help is greatly appreciated.

Translation of Hilbert's paper on sums of squares

Thu, 03/15/2018 - 19:54

Does anyone know if there is a French or English translation of Hilbert's paper on sums of squares: Ueber die Darstellung definiter Formen als Summe von Formenquadraten?

Similarities between isomorphism classes of homeomorphic directed graphs

Thu, 03/15/2018 - 18:21

To clarify, I'm speaking of homeomorphisms in the graph theoretic context, obtained by subdivisions of the arcs of a directed graph. A subdivision of an arc $(x,z)$ in a directed graph is obtained by removing $(x,z)$ from the arc-set and adding both $(x,y)$ and $(y,z)$. This is the definition in [BJG2009; page 10]; intuitively, we add a vertex $y$ different from all existing vertices in the "middle" of our arrow, and then give it an arrow-"head" that is pointed in the same direction as the other "head":


Now, while studying the automorphism groups of digraphs. I.e., if $G=(V,E)$ is our digraph,

$\text{Aut}(G)=\{\sigma\in \text{Sym}(V):\ \forall x,y\in V \qquad (x,y)\in E\iff (\sigma(x),\sigma(y))\in E\ \}$.

I've noticed a number of relationships between homeomorphisms and automorphisms. For example, if $G$ is a finite(*) directed acyclic graph, then all orbits of the action of automorphism group on the vertex-set are anti-chains of the poset $(V,E^{+})$, where ${}^+$ denotes transitive closure. Also, if we call a vertex $v\in V$ "thin" iff $\text{deg}^{+}(v)=\text{deg}^{-}(v)=1$ while also referring to any path $P\subseteq G$ as a "thin path" iff every vertex of $P$ is a thin vertex in $G$, then every automorphism of $G$ permutes the maximal thin paths of like length in $G$. I.e., for each $\ell\in\omega$, there is a permutation action of $\mathrm{Aut}(G)$ on the set $\mathsf{Thin}_\ell(G)$ of all thin directed length-$\ell$-paths in $G$.

Now if we subdivide a single arrow in each of the maximal thin paths of some fixed length in $G$ it seems the order of the newly formed automorphism group is the same order as $\text{Aut}(G)$, in fact I'm pretty sure not only is this true but so are a number of other possible identities similar to this, for example here are two other propositions I'm confident to state (though I haven't formally proved them):

  1. Any finite(**) digraph is homeomorphic to another digraph which has a trivial automorphism group.
  2. Two directed acyclic graphs are isomorphic iff their barycentric subdivisions are isomorphic. Or more generally iff $\forall n\in \mathbb{N}$ their $n^{\text{th}}$ barycentric subdivisions are isomorphic.

With that in mind, are there other propositions of this kind that relate the automorphism groups of homeomorphic digraphs with one another, or equivalently, relate their isomorphism classes?

If anyone could point me to some writings that make such connections, that would be helpful as well.


(*)The following statement is obviously false for infinite digraphs.

(**)Without assuming finiteness, this is false: the two-way infinite directed path (i.e., the digraph $(\mathbb{Z},<)$) has the property that any subdivision still has the same, and hence nontrivial, automorphism group.

[BJG2009] Jørgen Bang-Jensen, Gregory Gutin, Digraphs: Theory, Algorithms and Applications, Springer Monographs in Mathematics, Second Edition, 2009, ISBN-13: 978-0857290410

Integral of fractional Laplacian is zero

Thu, 03/15/2018 - 17:34

Is it true that $$\int_{\mathbb{R}^N}(-\Delta)^su(x) dx = 0,$$ where $(-\Delta)^s$ is the fractional Laplacian?

Finding a solution to a 2nd order differential equation [on hold]

Thu, 03/15/2018 - 11:21

1)What is the solution to this equation- r"(t)=c*[r(t)]^(-2), $$r''(t)=c r(t)^{-2}$$ i.e. 2nd derivative of r(t)=c*[r(t)]^(-2) ; where r is a function of t and c is a constant.Also, the derivative is with respect to t.

I was thinking of finding a solution containing t... Any idea of a solution is welcome...

*-Monomorphic Images of C*-algebras [on hold]

Wed, 03/14/2018 - 20:40

Let A and B be C^* algebras. Let f be an injective $*$-homomorphism from A into B. Is f(A) necessarily an isometry?

Unitary dual of the complex motion group $\mathbb C^2 \rtimes SU(2)$?

Wed, 03/14/2018 - 09:17

The real motion group of $\mathbb R^2$, $M(2)$ is the semi-direct product of $\mathbb R^2$ with the special orthogonal group $K = SO(2)$. A well known fact is that the unitary dual $\hat{G}$, of $G$ is precisely the collection M. Sugiura: Unitary Representations and Harmonic Analysis. Kodansha Scientific books, Tokyo (1975), p: 165 $$\{\pi_\lambda,\, \lambda>0 \}\cup\{\chi_n: \, n\in \mathbb Z \},$$ where $[\pi_{\lambda}(z,e^{i\alpha})u](s) = e^{i\lambda \left<z,e^{is}\right>} u(s-\alpha)$ for $u\in L^2([0,2\pi])$ and where $\chi_n(z,e^{i\alpha})=e^{in\alpha}$.

Now, any references or someone can help me to write the analog of that in the cases of the complex motion group $G=\mathbb C^2 \rtimes SU(2)$ ? Thank you in advance $$................$$ Following the remark in Dual topology of the motion groups, p: 398, I tried to construct this representation!!

For each linear form $l$ on $\mathbb C^2$ and any irreducible unitary representation $\rho$ of the stabilizer $S_l$ of l in $SU(2)$, we have that $\, \sigma_{(\rho,l)}=\rho\otimes\chi_l \, $ is an irreducible unitary representation of $H_l = S_l\rtimes \mathbb C^2$ whose restriction to $\mathbb C^2$ is a multiple of the character $\chi_l$ of $\mathbb C^2$ given by $\chi_l(z) = e^{−i\left<l,z\right>}$ ($z\in \mathbb C^2$), and the induced representation $\pi_(\rho,l) := ind^{G}_{H_l} \sigma(\rho,l)$ is an irreducible representation of $G$.

If $r > 0$ is the radius of the sphere, we denote by $\chi_r$ the character associated with the linear form $l_r$ which is identified with the vector $(0, . . . , 0, r)^t$ .The stabilizer $S_{l_{r}}$ of $l_{r}$ is the subgroup $SU (1)$. Let us write $\rho_\mu$ instead of $\rho$ for the representation of $SU(1)$ with highest weight $\mu$ and $\pi(\mu,r)$ instead of $\pi(\rho_\mu,l_r)$. The representation $\pi(\mu,r)$ is realized on $L^2(SU(2))$ as follows; for all $(A, z)\in G$ and all $B \in SU(2)$ $$\pi(\mu,r)(A,z)F(B)=e^{i\left<B l_r, z\right>} F(A^{-1}B), \quad F\in L^2(SU(2)).$$ Finally, the unitary dual $\hat{G}$, of $G$ is precisely the collection: $$\{r>0, \rho\in\hat{SU(1)} \}\cup\{\pi \in\hat{SU(2)} \}.$$

Thank you in advance for any comments or suggestions!!

The closure of span of a linearly independent and convergent sequence in $\ell^2$ [on hold]

Tue, 03/13/2018 - 13:49

Let $\{v_n\}_{n \in \mathbb{N}} \subset \ell^2$ be a sequence in $\ell^2$ over $\mathbb{C}$ such that $\{v_n\}_{n \in \mathbb{N}}$ is linearly independent and $v_n \to v_0$.

I would like to know if exist a subsequence $\{v_{n_k}\}_{k \in \mathbb{N}}$ such that for each fixed $p \in \mathbb{N}$: $$ v_{n_p} \notin \overline{span} \{v_{n_k}\}_{k > p} $$