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.

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?

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.

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}

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

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 $T$.

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.

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

- 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?
- 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 $\mathbb{Z}[\alpha]$?

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:

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

- 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?
- If the answer to $(1)$ is "yes", what is the currently known generality?
- 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.

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.

**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?

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.

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?

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 $\pi^*\pi_*\alpha=\alpha$?

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:

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

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.

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.).

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 $(\mathbb{Z}/n\mathbb{Z})^\times \to \mathbb{C}^\times$.

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.

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 $\phi$)?