Recent MathOverflow Questions

When is "metric dimension" well defined?

Math Overflow Recent Questions - Wed, 08/02/2017 - 17:59

A subset $B$ of a metric space $(M,d)$ is called a metric generating set if and only if $$[\forall b \in B, d(x,b)=d(y,b)] \implies x = y \,. $$ A metric generating set $B$ is called a metric basis if it has minimal cardinality.

The metric dimension of $(M,d)$ is the cardinality of any metric basis.

Question: For which metric spaces is metric dimension well-defined? When can we be sure that any metric basis for a metric space has the same cardinality?

Sufficient criteria will suffice for answers, as will necessary criteria, although of course the holy grail of answers would be a non-trivial necessary and sufficient criterion.

Note: This is a follow-up to my previous question. There, the accepted answer pointed out that the notion of metric dimension does not make sense in arbitrary metric spaces.

In a matroid, any basis has the same cardinality, but there are metric spaces with metric generating sets of minimal yet non-equal cardinalities.

Nevertheless, it does seem possible that metric dimension may make sense for certain classes of metric spaces, e.g. Euclidean spaces (Murphy, A Metric Basis Characterization of Euclidean Space, 1975) or graphs (Ramirez-Cruz, Oellermann, Rodriguez-Velazquez, The Simultaneous Metric Dimension of Graph Families, 2015). It is unclear to me what property common to these two types of metric spaces allows the definition to be well-formed/well-defined for them.

In the case of Euclidean spaces, it seems intuitively clear that this notion should be related to that of affine independence, but coordinate-free definitions of affine independence (solely in terms of the metric) are rare (e.g. section 2.6 here), so I am still working on the algebra to show the connection.

Schubert calculus expressed in terms of the cotangent space of the Grassmannians

Math Overflow Recent Questions - Wed, 08/02/2017 - 13:23

Let $T^*_{\mathbb{C}}(Gr_{n,r})$ denote the cotangent space of the Grassmannian of $r$-planes in $\mathbb{C}^n$. Moreover, let $\Lambda^\bullet$ denote the exterior algebra of $T^*_{\mathbb{C}}(Gr_{n,r})$. Condsidering $Gr_{n,r}$ as the homogeneous space $U_n/(U_r \times U_{n-r})$, we have a unique representation of $U_n/(U_r \times U_{n-r})$ on $\Lambda^{\bullet}$ for which the associated homogeneous vector bundle is the direct sum $\bigoplus_{k \in \mathbb{N}} \Omega^k$.

(i) Just as for any homogeneous space, every de Rham cohomology class of $Gr_{n,r}$ has a $G$-invariant representative. Moreover, every $G$-invariant element must be harmonic, and so, gives by Hodge decomposition a cohomology class. Is it correct to conclude from this that the cohomology group $H^\bullet$ is isomorphic as a vector space to the space of $U(r) \times U(n-r)$-invariant elements in $\Lambda^\bullet$?

(ii) With respect to a standard weight basis of $T^*(Gr_{n,r})$, what do the $U(r) \times U(n-r)$-invariant elements look like, and how does this presentation of Schubert calculus relate to the partition presentation given in this question?

When are two pregeometries equivalent?

Math Overflow Recent Questions - Tue, 08/01/2017 - 15:04

Some model theorists / combinatorial geometers like to think about pregeometries (matroids with a weak finiteness condition) associated to first-order theories. But the usual way of constructing a pregeometry is not invariant under biinterpretability in any obvious sense. Is there a suitable notion of equivalence of pregeometries such that biinterpretable theories have equivalent pregeometries?

More precisely, if $T$ is a theory, then by "the pregeometry associated to $T$" I mean the following. Let $U$ be a monster model of $T$, and consider the algebraic closure operator on (the home sort of) $U$. This forms a pregeometry, the one I have in mind. So the question is

Question: Is there a notion of equivalence of pregeometries such that the pregeometry associated to a theory in the above sense is invariant under biinterpretability?

But trying to compare the pregeometries of biinterpretable theories is immediately problematic: biinterpretable theories could consider different sorts to be the "home sort". So it's not clear how to even produce a map of geometries (in the obvious sense) from an interpretation. Even if a map is produced, it seems unlikely to be an isomorphism.

Introductions to this topic seem to provide a candidate notion: a "geometry" is a pregeometry such that the closure of a singleton set is itself. Every pregeometry can be quotiented to yield a canonical geometry. But biinterpretable theories need not have isomorphic geometries -- in fact, the quotient seems even more problematic since biinterpretable theories need not agree on what the singletons are. So this doesn't seem to get us anywhere.

Bounds on the distance between probability distributions in terms characteristic functions

Math Overflow Recent Questions - Tue, 08/01/2017 - 13:34

I am looking for the bounds on the distance between probability distributions in terms characteristic functions.

For example, I am aware of the following bound \begin{align} d(P,Q) \le \frac{1}{T} \int_{-T}^T \frac{|\phi_P(t)-\phi_Q(t)|}{t} dt+ \frac{q_{max}}{ \pi T} \end{align} where the $d(P,Q)$ is the Kolmogorov-Smirnov distance and $q_{max}$ the maximum value of the pdf of $Q$.

Specifically, am looking for the bonds that would depend on the following difference

\begin{align} | t \phi_P^{\prime}(t)+\phi_P(t)-(t \phi_Q^{\prime}(t)+\phi_Q(t)) | \end{align}

Can anyone give me website to learn mathematics online? [on hold]

Math Overflow Recent Questions - Tue, 08/01/2017 - 13:12

I want to learn mathematics online. Can anyone give me website to learn mathematics from beginner to graduated level online?

Semi-embeddings and weak compactness

Math Overflow Recent Questions - Tue, 08/01/2017 - 13:11

Let $F$ and $H$ be normed spaces and let $E$ be a locally convex space. Let $T:F\to H$ and $S:H\to E$ be linear operators, such that $\|T\|= 1$, $S$ is an injective semi-embedding (i.e. $S\overline{B}_{H}$ is closed in $E$), and $ST$ is weakly compact.

Does it follow that $T$ is weakly compact?

The intuition is that if we consider $F^{**}\to ^{T^{**}} H^{**}\to^{S^{**}} E^{**}$, then $S^{**}T^{**}\overline{B}_{F^{**}}$ is equal to the closure of $ST\overline{B}_{F}$ in $E$, due to weak compactness of $ST$. Since $\|T\|=1$, it follows that $ST\overline{B}_{F}\subset S\overline{B}_{H}$, which is closed in $E$, and so $S^{**}T^{**}\overline{B}_{F^{**}}\subset S\overline{B}_{H}$. Hence, perhaps it is possible to show that $T^{**}\overline{B}_{F^{**}}\subset \overline{B}_{H}$, which is equivalent to weak compactness of $T$.

How to solve this operator equation numerically?

Math Overflow Recent Questions - Tue, 08/01/2017 - 13:00

I would like to know how one solves Sturm-Liouville problems on $\mathbb{R}$ numerically for the eigenvalues that are of the form

$$-f''(x)+\frac{1}{\sinh(x)^2}f(x)=\lambda f(x).$$

So even if there is a closed form solution to this problem, I would like to know how to treat singularities like this one.

Norm of an ideal

Math Overflow Recent Questions - Tue, 08/01/2017 - 12:38

Consider an irreducible polynomial $f(x) = x^3+3x+1$ over $\mathbb{Z}[x]$. Now, the algebraic factor base consists of all first degree prime ideals represented by a pair $(p,r)$ where $f(r) \cong 0\ ({\rm modulo}\ p)$. In this case, we consider all primes $\leq 10$ and their corresponding root modulo $p$.

So, ${\cal S}= \{ (3,2),(5,1),(5,2),(7,4) \}$.

  1. How to identify whether a prime is ramified or not and how to detect a $(p,r)$ in set ${\cal S}$ is a first degree prime ideal or not?
  2. Let $\alpha$ be a complex root of $f(x)$.

We define Norm of first degree ideal as follows:

$N(a-(b(\alpha))) = b^d * f(a/b)$ where $d$ is the degree of the polynomial. Here, ${\cal S}_1= \{ (0,1),(-4,13),(-1,5),(1,1),(-10,1),(9,4) \} $ consists of pairs smooth over Algebraic factor base.

Now, I am getting $\prod\limits_{(a_i,b_i)\in {\cal S}} N(a_i - b_i \alpha )=(180075)^2$ instead of $\prod\limits_{(a_i,b_i)\in {\cal S}} (a_i - b_i \alpha )$, a square, so how to get down from Norm level to product of $(a_i-b_i (\alpha))$ since the objective is latter product should be a square in $\mathbb{Z}[\alpha]$?

Spectral sequences in $K$-theory

Math Overflow Recent Questions - Tue, 08/01/2017 - 12:33

There is an algebraic analogue of the Atiyah-Hirzebruch spectral sequence from singular cohomology to topological $K$-theory of a topological space.

For a field $k$, let $X$ be smooth variety $X$ over $k$.

The following spectral sequence will be referred to in the sequel as the motivic spectral sequence: $$E_2^{i,j} := H^{i-j}(X, \mathbf{Z}(-j)) \Rightarrow K_{-i-j}(X).$$

See:

  1. the Bloch-Lichtenbaum motivic spectral sequence in [BL], and the generalizations by Levine [L] and Friedlander-Suslin [FS] to smooth varieties over $k$.
  2. the Voevodsky motivic spectral sequence [V].
  3. the Grayson motivic spectral sequence [G].

For $k$ and $X$ as in the foregoing, we may form the étale hypercohomology of the Bloch complex $z^{j}(X,\bullet)$ ([B]) on $X_{\rm\acute{e}t}$, denoted $H^{\bullet}_{L}(X, \mathbf{Z}(j))$ and usually called Lichtenbaum cohomology.

Questions:

  1. Is an "étale analogue" of the motivic spectral sequence from the foregoing, i.e.: $$E_2^{i,j} := H_L^{i-j}(X, \mathbf{Z}(j))\Rightarrow K_{-i-j}^{\rm\acute{e}t}(X)$$ available?
  2. If the answer to $(1)$ is "yes", what is the currently known generality?
  3. If the answer to $(1)$ is "yes", references?

References.

[BL] S. Bloch, S. Lichtenbaum, A spectral sequence for motivic cohomology, K-theory, 1995.

[L] M. Levine, Techniques of localization in the theory of algebraic cycles, 2001.

[FS] E. M. Friedlander, A. Suslin, The spectral sequence relating algebraic K-theory to motivic cohomology, 2002.

[V] V. Voevodsky, A possible new approach to the motivic spectral sequence for algebraic K-theory, 2002.

[G] A. Suslin, On the Grayson spectral sequence, 2003.

[B] S. Bloch, Algebraic cycles and Higher $K$-theory, 1986.

Reproducing Kernel Hilbert Spaces in Discrepancy Theory

Math Overflow Recent Questions - Tue, 08/01/2017 - 12:17

Let $(\Omega, \Sigma, \mu)$ be a probability space and ${\mathcal B}:=(B(\omega))_{\omega\in \Omega}$ be parameterized measurable sets, i.e. $B(\omega) \in \Sigma$ for all $\omega \in \Omega$. Now one can show that $$K_{\mathcal B}:\Omega \times \Omega \to [0,1],\ (x,y) \mapsto \int_\Omega 1_{B(\omega)}(x) 1_{B(\omega)}(y) d\mu(\omega)$$ defines a reproducing kernel on a Hilbert space $H(K_{\mathcal B})$ (constructed by Moore's theorem).

So we can observe that a system $\mathcal B$ of measurable sets induces a associated reproducing kernel. Now there arises a natural question:

If $K$ is a reproducing kernel can I always find a system $\mathcal B$ of measurable sets such that $K = K_{\mathcal B}$? Obviously that doesn't work, since it is necessary that $$ K(x,y) \in [0,1] \qquad \text{and} \qquad \vert K(x,y) \vert \leq \min\{K(x,x), K(y,y)\}.$$ So my question is if there are already any results that characterize such reproducing kernels? If so I would appreciate some literature on the topic. The problem might be very difficult in general though.

Characterisation of pronormality by transitive permutation representation of a group

Math Overflow Recent Questions - Tue, 08/01/2017 - 11:46

Definition: A subgroup $H$ of a group $G$ is said to be pronormal if every $g\in G$, there exists $x\in \langle H, H^g \rangle$ such that $H^x = H^g$ (note: $H^g:= gHg^{-1}$)

Let $G$ be a group and $H \leq G$. Then $H$ prn $G$ if and only if, in every transitive representation of $G$, $N_G(H)$ permutes the symbols left invariant by $H$.

Proof: Suppose that $G$ is represented transitively by permutations of a set $X$. Let $Y$ be the set of all $x\in X$ which are invariant under $H$. If $y\in Y$ and $a\in N_G(H)$, we have $Hy=y$, and so $Hay = aHy = ay$. Hence $ay \in Y$. Thus $N_G(H)$ leaves $Y$ invariant. Let $G_y$ be the stabiliser of $y$ in $G$. If $y$ and $z$ are in $Y$, we have that $gy=z$ for some $g\in G$, since $G$ permutes $X$ transitively. If $h\in H$, then $hy =y$, since $y\in Y$ is invariant under the action of $H$ by hypothesis. This implies that $h \in G_y$, and so $H \leq G_y$. Similarly, $H \leq G_z$. Thus $H \leq G_y \cap G_z$. Consequently, $gHg^{-1} = gG_yg^{-1} = G_z$. If $H$ prn $G$, then $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ and hence in $G_z$ as $\langle H, H^g \rangle \leq G_z$. Therefore there exists $b\in G_z$ such that $H^b = H^g$. This implies that $b^{-1}g \in N_G(H)$. Now $(b^{-1}g)y = b^{-1}(gy) = b^{-1}z = z$, as $b^{-1}\in G_z$ ($b\in G_z$ and $G_z \leq G$). Thus $N_G(H)$ permutes $Y$ transitively whenever $H$ prn $G$.

Conversely, suppose that in any transitive representation of $G$, $N_G(H)$ permutes the symbols left invariant by $H$. Let $g\in G$ and denote $J = \langle H, H^g \rangle$. Then $HJ = J$ and $H^gJ = J$, or equivalently, $Hg^{-1}J = g^{-1}J$. Therefore in the transitive permutation representation of $G$ on the left cosets of $J$ in $G$, the cosets $J$ and $g^{-1}J$ are invariant under the action of $H$. By hypothesis, $J = n(g^{-1}J)$ for some $n\in N_G(H)$. Then $ng^{-1} \in J$, and $(ng^{-1})gHg^{-1}(gn^{-1}) = nHn^{-1} = H$. This shows that $H^g$ and $H$ are conjugate in $J$, and since this is true for all $g\in G$, we deduce that $H$ prn $G$.

Question: Are there any questions one can derive from this result?

Math career - is it a good idea to use ADHD medication to improve research concentration? [on hold]

Math Overflow Recent Questions - Tue, 08/01/2017 - 11:30

Sorry if this might be off-topic or "sketchy".

I really love reading and studying mathematics in my undergraduate and attained top grades in my year in a top university in Canada. But I didn't display any super talent. For example, I did badly on the Putnam and was never able to reach the Putnam fellow state.

I then went on to graduate school, and really struggled through the PhD because I constantly got stuck on my proof and didn't know what to do. I felt it was partly related to my inability to concentrate on a proof for long time.

In my regular coursework, I never have problem concentrating during class or doing homeworks. But in research, I cannot come into the same office every day and concentrate for many hours being stuck on the same problem. I saw an interview of Len Adleman where he mentions that he can concentrate on the same problem for 10+ hours everyday for months at a time.

I feel that maybe if I self-medicated using mild ADHD drugs, then I can improve my concentration and make more breakthroughs in my math career. I don't drink coffee. Maybe that will help. Does anyone have any advice? Sorry if this is long and very personal.

Reducible binary quadratic form

Math Overflow Recent Questions - Tue, 08/01/2017 - 11:16

Let $f(x,y)=(ex+fy)(gx+hy); \ x,y,e,f,g,h \in \mathbb{Z}$ be a reducible integral binary quadratic form. Is there a criterion to determine if a number is represented by this form? In particular, does such a criterion exist for if an integral binary quadratic form has square discriminant?

Gysin map and blow up

Math Overflow Recent Questions - Tue, 08/01/2017 - 09:35

Let $X$ be a smooth projective variety and $W \subset X$ a smooth, projective subvariety. Let $\pi:\tilde{X} \to X$ be the blow-up of $X$ along $W$. Let $E$ be the exceptional divisor of $\pi$ and $i:E \hookrightarrow \tilde{X}$ the natural closed immersion. Let $\alpha \in H^k(\tilde{X})$ such that $i^*\alpha=0$ in $H^k(E)$. Does this imply that $\pi^*\pi_*\alpha=\alpha$?

Characterize constant objects in the internal language of a topos?

Math Overflow Recent Questions - Tue, 08/01/2017 - 09:05

A Grothendieck topos $\mathcal{E}$ is equivalent to the category of sheaves on some site $S$. We say a sheaf $X\colon S^{\text{op}}\to\mathsf{Set}$ is constant if it is the sheafification of a constant presheaf, i.e. one that factors through the terminal map $S^{\text{op}}\to \{*\}$.

But what if we forget the site $S$ and consider $X$ as an object in the topos? Can we characterize the property of $X\in\mathcal{E}$ being constant? More specifically, two questions:

  1. Is the property of $X$ being constant dependent on the choice of site $S$?
  2. Is there a way to characterize constant objects in the internal language of $\mathcal{E}$?

How to find a positive solution to an under-determined linear system (if such a solution exists)?

Math Overflow Recent Questions - Tue, 08/01/2017 - 04:28

Like the title says, if an under-determined system of linear equations does have at least one positive solution, how to find it efficiently?

Suppose we have an under-determined system:

$$Ax = b$$

where $A$ is an $m \times n$ matrix and $m < n$. How can we get a solution such that $x>0$?

I have tried to solve the system using linear programming: for the $Ax=b$ constraint, it is hard to find any solution. So I make it:

$$Ax \leq 1.01 b$$ $$Ax \geq 0.99 b$$

such that we have an approximated solution. But still had no success. To make a proper objective function is very tricky.

I wonder if there is some effective method to solve this problem?

btw: some papers might be relevant are listed:

Conditions for a unique non-negative solution to an underdetermined system. http://ieeexplore.ieee.org/document/5394815/

A Unique "Nonnegative" Solution to an Underdetermined System: from Vectors to Matrices https://arxiv.org/abs/1003.4778

The Farkas-Minkowski Theorem: www.math.udel.edu/~angell/Opt/farkas.pdf

Update 1:

By using linear least-squares to minimize $\|Ax-b\|_2$ and $x \geq 0$, it seems that we can get non-negative solution $x$, nevertheless, the zeros in $x$ and the error of $A x$ to $b$ are still not acceptable.

Update 2:

Problem solved! Now I can get a positive solution perfectly meets the equations (1 - 10), even N is big, e.g. N = 1000 unknowns.

Geometric or conceptual way to understand supersymmetry algebra

Math Overflow Recent Questions - Tue, 08/01/2017 - 02:07

Is there any geometric or more direct conceptual way to understand a supersymmetry algebra, rather than starting from a Lagrangian including boson and fermion fields, deriving all the expressions ensuring the supersymmetry invariance and then writing down the supersymmetry algebra? For a geometric or algebraic way, I mean to (at least partially) derive the supersymmetry algebra from pure geometry (spinors, spin group, etc.).

Finite order Hecke characters

Math Overflow Recent Questions - Tue, 08/01/2017 - 01:35

Let $\mathbb{A}_{\mathbb{Q}}^{\times}$ be the ring of ideles over the rational $\mathbb{Q}$. Let $\{ \chi : \mathbb{Q}^\times \backslash \mathbb{A}_{\mathbb{Q}}^{\times} : \chi_\infty = 1, \chi^n =1 \} $ be the set of Hecke characters of order $n$ with trivial infinity part.

I want to whether they are finite in number corresponding bijectively to the Dirichlect characters $(\mathbb{Z}/n\mathbb{Z})^\times \to \mathbb{C}^\times$.

A new $\ell_p$-metric on the hyperspace of finite sets?

Math Overflow Recent Questions - Sun, 07/30/2017 - 14:03

Let $(X,d)$ be a metric space and $Fin(X)$ be the family of all non-empty finite subsets of $X$. For every $n\in\mathbb N$ the elements of the power $X^n$ are thought as functions $f:n\to X$ where $n:=\{0,\dots,n-1\}$. For such function $f$ its image $f(n)\subset X$ will be denoted by $rg(f)$.

For every $p\in[1,\infty]$ the space $X^n$ is endowed with the $\ell_p$-metric $d_p$ defined by $$d_p(f,g)=\begin{cases}\sqrt[p]{\sum_{i\in n}d(f(i),g(i))^p}& \mbox{if $p<\infty$;}\\ \;\;\max_{i\in n}d(f(i),g(i))&\mbox{if $p=\infty$.} \end{cases} $$

I am interested in the metric $d_p$ on the hyperspace $Fin(X)$, defined by the formula$$d_p(A,B):=\inf\sum_{i=0}^kd_p(f_i,g_i)$$ where the infimum is taken over all chains $(f_0,g_0),\dots,(f_k,g_k)\in\bigoplus_{n\in\mathbb N}X^n\times X^n$ such that $A=rg(f_0)$, $B=rg(g_k)$ and $rg(g_i)=rg(f_{i+1})$ for all $i<k$.

The metric $d_p$ can be equivalently defined as the largest metric on $Fin(X)$ such that for every $n\in\mathbb N$ the map $X^n\to Fin(X)$, $(x_1,\dots,x_n)\mapsto\{x_1,\dots,x_n\}$, is non-expanding with respect to the $\ell_p$-metric on $X^n$.

It can be shown that the distance $d_\infty$ on $Fin(X)$ coincides with the well-known Hausdorff metric. I am interested in the metrics $d_p$ on $Fin(X)$ for $p<\infty$, especially in the metric $d_1$ on $Fin(X)$.

The distance $d_1$ can have applications in economics as it represent the cost of transportation of goods whose mass is negligible comparing to the mass of the transporting car.

For the metric $d_1$ many natural problems appear. In particular:

Problem. Find an (efficient) algorithm for calculation of the distance $d_1(A,B)$ between two finite subsets $A,B$ of $\mathbb R^n$ (at least for $n=2$).

Have such problems been studied in mathematical or economical literature?

Remark 1. In the simplest case of the 2-element set $\{a,b\}$ and a 1-element set $\{c\}$ in the plane the distance $d_1(\{a,b\},\{c\})=\|a-t\|+\|b-t\|+\|c-t\|$ where $t$ is the Fermat-Toricelli point of the triangle $\{a,b,c\}$, and $\|\cdot\|$ is the standard Euclidean norm on the plane.

Remark 2. It seems that for arbitrary finite sets $A,B$ in the plane the distance $d_1(A,B)$ is equal to the smallest total length of edges of a graph $\Gamma\subset\mathbb R^2$ whose any connected component intersects both sets $A$ and $B$.
The problem of finding such graph $\Gamma$ is related to the classical Steiner tree problem which is known to be difficult (at least, NP-hard), so the problem of calculating the distance $d_1$ on $Fin(\mathbb R^2)$ is also hard. But maybe for the Urysohn universal metric space $\mathbb U$ this problem is computationally more simple? In this case the distance $d_1$ between two finite subsets of $A,B$ of $\mathbb U$ depends only on the restrcition of the metric to $A\cup B$ and hence depends on finite number of real parameters; so, in principle, is computable.

Relative divisors

Math Overflow Recent Questions - Sat, 07/29/2017 - 15:42

Let $X\rightarrow T$ be a fibre bundles with smooth projective fibre $F$ and $X$ and $T$ are also smooth. Let $D$ is relative effective Weil divisor. Suppose $W_1 $ and $W_2$ are relative subvarieties which are isomorphic(by a relative map say $\phi$). Let $L:=\mathcal{O}(D)$. Let $L_t|_{W_{1,t}}\cong L_t|_{W_{2,t}}$ (via $\phi_t$). Is it true that $L|_{W_{1}}\cong L|_{W_{2}}$ (via $\phi$)?

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