Are metrizable subspaces of separable spaces separable?

Certainly subspaces of separable metrizable spaces are separable but subspaces of separable spaces need not be separable in general.

I am trying to solve an unconstrained optimization problem with the following properties: I am given a graph $G=(V,E)$ with functions $f_{ij}:\mathbb{R}^2\to \mathbb{R}$ for each edge $(i,j)$. I am trying to choose scalar variables $x_1,\dots,x_n$ for each vertex to minimize $$\sum_{(i,j)\in E } f_{ij} (x_i, x_j) $$. This is almost a separable problem except that a variable can appear in multiple functions. Are there any papers describing ways to exploit this special structure? The functions are continuous but not necessarily convex or concave.

Let $A=\{1,2\}$. For any $d \in A$ and any sequence $a=(a_1,a_2,\dots)\in A^{\mathbb N}$ the associated rooted tree $T(d,a)$ is recursively defined in the following way. The degree of the root of $T(d,a)$ is $d$, and the result of the removal of its root (together with the $d$ edges issued from it) is the tree $T(a_1,Sa)$ if $d=1$ and two trees $T(1,Sa), T(2,Sa)$ if $d=2$ (here $Sa=(a_2,a_3,\dots)$ denotes the shift of the sequence $a$).

I am interested in any references to the appearances of the family of trees $T(d,a)$ (or anything similar).

Consider a matrix $X \in \mathcal{R}^{n\times n}$ with rank $r$. Assume the matrix is sampled, using some distribution, and the resulting matrix $Y$ is defined as $Y_{i,j} = X_{i,j}$ if $(i,j)$ is sampled and $0$ otherwise. The interest is to determine how the rank of $X$ changes. For instance, if $X$ is invertible, what conditions, say on the sampling, ensure that $Y$ is also invertible? In particular, the interest is to do analysis on the change of eigenvalues under random sampling. I am looking for any references or works that deal with this question. Initial attempts using the Weyl's perturbation theorem was not fruitful.

Let $C$ be the Cartan matrix of a finite dimensional algebra $A$ with finite global dimension, then the Coxeter matrix is defined as $M=-C^{-1}C^T$. $A$ is called periodic in case $M^k=id$ for some $k \geq 1$. The smallest such $k$ is called the period of the algebra. See for example https://www.sciencedirect.com/science/article/pii/S0024379505001709 , https://www.sciencedirect.com/science/article/pii/S0024379502004056and related articles on that.

We can associate to every Dyck path from $(0,0)$ to $(2n-2,0)$ the Nakayama algebra with linear quiver and $n$ simple modules which has this Dyck path as the top boundary of its Auslander-Reiten quiver. This gives a natural bijection.

Now we call a Dyck path periodic in case its corresponding algebra is periodic and the periodic of the Dyck path is defined as the periodic of the corresponding algebra.

I am not experienced with this topic but I wondered:

What are the periodic Dyck paths and is there a formula for the period?

It seems to be very complicated, but I noted the following for $n \leq 9$ which I state here as a conjecture:

Mainconjecture: -The number of Dyck paths from $(0,0)$ to $(2n-2,0)$ with period $n+1$ is equal to $2^{n-2}$.

Other conjectures (that would follow from the main conjecture):

-Those with additionally having global dimension (=global dimension of the corresponding algebra) at most 2 are counted by the Fibonacci sequence (starting with 1,2,... for $n \geq 2$.)

-Those with additionally having global dimension at most 3 are counted by the Tribonacci sequence: https://oeis.org/A000073. (starting with 1,2,... for $n \geq 2$.)

-Those with additionally having global dimension at most 4 are counted by the Tetranacci sequence: https://oeis.org/A000078. (starting with 1,2,... for $n \geq 2$.)

-Probably one can guess how that might continue(Yes, global dimension 5 lead to the pentanacci numbers https://oeis.org/A001591). I found suprisingly much more nice sequences when restricting to certain nice homological conditions, but global dimension seems to be the best suited here.

The conjecture is checked for $n \leq 9$ with the computer. In case it is true, can those Dyck paths be nicely described? Here the Dyck paths for $n \leq 7$ with their period (period 0 means it is now periodic):

http://www.findstat.org/StatisticsDatabase/St001218

edit: Thanks to the comment of Michael Albert, I can see now what should be going on with period $n+1$, and it would be enough to prove the main conjecture to obtain the other conjectures as a corollary. I leave the other conjectures here for completeness.

Let $V$ be a complete discrete valuation ring whose residue field is a finite field $k=\mathbf{F}_q$. Let $\pi\in V$ be a uniformizer.

For any integer $d,n\ge 0$, define:

$${\pi^d \choose n} := \frac{\pi^d\cdot(\pi^d -1)\cdot\ldots\cdot(\pi^d-n+1)}{n!}.$$

- Is ${\pi^d\choose n}$ an element of $V$?
- For exactly what integers $n\ge 0$ is ${\pi^d\choose n}$ a unit?

**Example.** If $V = \mathbf{Z}_p$, $\pi = p$, then ${p^d\choose n}$ is zero unless $0\le n\le p^d$, and its $p$-adic valuation is $d-v_p(n)\ge 0$ for $1\le n\le p^d$. In all cases, the answer to the first question is yes, and the answer to the second question is: for $n = 0, c\cdot p^d$, with $(c,p) = 1$.

This question has come out while reading J. Moser "*New Aspects in the Theory of Stability
of Hamiltonian Systems*". I'm particularly interested to the Appendix, where one investigates the stability of elliptic fixed points of Hamiltonian dynamical systems, in the time independent case. I start presenting the framework.

Let us consider the Hamiltonian dynamical system $$ \dot{x}_\nu = H_{y_\nu}(x,y), \qquad \dot{y}_\nu=-H_{x_\nu}(x,y) \qquad \qquad (1) $$

where $\nu=1,2,\dots,n$. Hamiltonian $H$ does not depend on time $t$ and it is assumed to be a real analytic function of $x_\nu,\,y_\nu$, with $\nu=1,2,\dots,n$ in the neighboorhood of $x=y=0$, the expansion of which starts with quadratic terms. Then $x=y=0$ is an equilibrium solution.

One can construct a fundamental system of solutions of exponential form $$ w^{(\nu)}=e^{\gamma_\nu t}p^{(\nu)} $$ where $p^{(\nu)}$ are constant vectors or, in the case of multiple eigenvalues $\gamma_{\nu}$, possibly polinomials in $t$. The numbers $\gamma_\nu$ are obtained as the eigenvalues of the matrix determined by the linear terms of the right-hand side of (1). Suppose that all eigenvalues are distinct and purely imaginary, i.e. of the type $\gamma_\nu=i\beta_\nu$, with $\beta_\nu$ real. So the spectrum has the form $$ \pm i \beta_1, \quad \pm i \beta_2, \,\dots,\, \pm i \beta_n. $$ So one obtains a collection of distinct numbers $\beta_\nu$, $-\beta_\nu$ with $\nu=1,2,\dots,n$

So far, so good.

For later purposes, one needs to define the sign of $\beta_\nu$. The Author says that the sign of $\beta_\nu$ is taken in such a way that $$ \mathcal{Im}\left[w^{(\nu)},\overline{w^{(\nu)}}\right]<0. $$ Square brackets are defined as Lagrange Brackets (an outdated therminology, nowadays called symplectic form [see comments]). More precisely, given any two $2n$-dimensional vectors, $x$ and $\tilde{x}$, with components $x_\nu$ and $\tilde{x}_\nu$, their Lagrange Bracket (read: symplectic form) is defined as $$ [x,\tilde{x}]=\sum_{\nu=1}^n (x_\nu\tilde{x}_{\nu+n}-x_{\nu+n}\tilde{x}_\nu) $$ In passing, one has to remember that an Hamiltonian sysytem is marked by the fact that, for any two solutions $x$ and $\tilde{x}$, the Lagrange bracket (read: symplectic form) $[x,\tilde{x}]$ is $t$-independent.

**Questions:** can someone please show

1) how to explicitly compute:

$$ [w^{(\nu)},w^{(\mu)}] = ? $$

$$ [w^{(\nu)},\overline{w^{(\mu)}}] = ? $$

2) how to prove that $$ [w^{(\nu)},\overline{w^{(\nu)}}] = [p^{(\nu)},\overline{p^{(\nu)}}] $$

$$ [p^{(\nu)},\overline{p^{(\nu)}}] \text{ is purely imaginary} $$ 3) How to define the sign $\beta_{\nu}$ in such a way that $$ \mathcal{Im}\left[w^{(\nu)},\overline{w^{(\nu)}}\right]<0. $$ I am from the Physics community, so I kindly ask to display all important passages.

Let $X$ be a scheme and $U$ be an open subscheme. The proof of the Thomason-Trobaugh Theorem implies that under some mild assumptions, for any perfect complex $F$ on $U$, we have that $F\oplus F[1]$ can be extended to a perfect complex on $X$. I'm just wondering whether there exists examples where $F$ is a perfect complex on $U$ but $F$ itself cannot be extended to $X$? I've found an example when $X$ is the cone $xy-z^{2}=0$ and $U$ is the complement of the origin. Is there an example for smooth $X$?

Let $X$ be a topological space. Set
$K(X) := \{ A\subseteq X\mid A$ is quasi-compact and open $\}.$ A topological space $X$ is called **spectral**,
if it satisfies all of the following conditions:

1) $X$ is quasi-compact and $T_0$. 2) $K(X)$ is a basis of open subsets of $X$. 3) $K(X)$ is closed under finite intersections. 4) $X$ is sober, i.e. every nonempty irreducible closed subset of $X$ has a (necessarily unique) generic point.

Let $C$ be a closed subset of a spectral topological space $X$. I am looking for equivalent conditions on $C$ under which if $A$ is a clopen(=Closed+Open) subset of $C$, then there exists a clopen subset $B$ of $X$ such that $A=C\cap B$?

Let $G$ be a finite group of order $144$. Suppose there is an irreducible $\mathbb{C}$-character $\theta$ such that $\theta(1)=9$.

QUESTION: Prove $G\cong A_4\times A_4$.

By using Magma, we know there is only one group of order $144$ with an irreducible $\mathbb{C}$-character $\theta$ of degree $9$. Now I want to prove this result without using Magma.

A friend of mine, obtained a lower bound for the trace norm of matrices described in this question (for the special case $a_{ij} = \pm 1$). That lower bound is $ \frac{f(n)}{2\pi}$ where $$ f(n) := \int_0^\infty \log\left( \frac{(1+t)^n +(1-t)^n}{2} +n(n-1) t(1+t)^{n-2}\right)t^{- 3/2} \ \mathrm{d}t $$ Numerical computaions suggest that $$ f(n) = 4 \pi n + o(n) $$ How to justify it? Moreover, is it possible to obtain a good rate of convergence?

Today I was explaining to some one the notion of $\mathcal{G}$ spaces, covering spaces over orbifolds from Orbifolds as Groupoids: an Introduction.

Definition : Let $X$ be a locally compact Hausdorff space. An orbifold structure on $X$ is given by an orbifold groupoid $\mathcal{G}$ and a homeomorphism $f:|\mathcal{G}|\rightarrow X$. If $\mathcal{H}\rightarrow\mathcal{G}$ is a equivalnece, then $|\phi|:|\mathcal{H}|\rightarrow |\mathcal{G}|$ is a homeomorphism and the composition $f\circ|\phi|:|\mathcal{H}|\rightarrow |\mathcal{G}|\rightarrow X$ is viewed as defining an equivalent orbifold structure on $X$. An orbifold $\underline{X}$ is a space $X$ equipped with an equivalnece class of orbifold structures.

I have given definition of a $\mathcal{G}$-space as a smooth manifold $E$ with smooth maps $\pi:E\rightarrow \mathcal{G}_0$ and $\mu:E\times_{\mathcal{G}_0}\mathcal{G}_1\rightarrow E$ behaving like a group action map.

Then, I said, given a morphism of Lie groupoids $\phi:\mathcal{H}\rightarrow \mathcal{G}$ there is a functor from category of $\mathcal{G}$ spaces to the category of $\mathcal{H}$ spaces $\phi^*:(\mathcal{G}-\text{spaces})\rightarrow (\mathcal{H}-\text{spaces})$.

Definition : A covering space over a groupoid $\mathcal{G}$ is a $\mathcal{G}$ such that the map $\pi:E\rightarrow \mathcal{G}_0$ is a covering projection.

Then, I said the equivalnece of categories $\phi^*:(\mathcal{G}-\text{spaces})\rightarrow (\mathcal{H}-\text{spaces})$ restircted to covering spaces is still an equivalence. So, I can talk about covering space over an orbifold.

Definition : A covering space over an orbifold $\underline{X}$ is a covering space over Lie groupoid $\mathcal{G}$ representing $\underline{X}$ (in the sense defined above).

This notion is well defined : Suppose there is another Lie groupoid representing $\underline{X}$ in the equivalnece class, say $\mathcal{H}$ then, this $\mathcal{H}$ has to be morita equivalent with $\mathcal{G}$. So, category of covering spaces over $\mathcal{G}$ would be same(equivalent) as that of category of covering spaces over $\mathcal{H}$. So, the notion makes sense.

Then, he asked

"why do you do this much just to define the notion of covering space over a space $X$?"...

I did not think about this before and said at that moment that, $X$ is not merely a topological space in which case you can define covering map to be just local homeomorphism plus something, the usual definition.

One standard example in the classical notion of orbifold is quotient space of a manifold by a Lie group $M/G$.

Suppose you want to define the notion of covering space over $M/G$, you can not treat $M/G$ as simply a topological space, in which case you can define covering space to be a map $\pi:E\rightarrow X=M/G$ such that given $x\in X$ there is an open set $U$ containg $x$ and $\pi^{-1}(U)$ is disjoint union $\bigsqcup V_\alpha$ where $V_\alpha$ is mapped **homeomorphically** onto $U$ under $\pi$. You can not even treat $M/G$ as a smooth manifold where you would define smooth covering map just by replacing **homeomorphically** in above definition by **diffeomorphically**. In general $M/G$ is not a smooth manifold. It is more than a topological space and less than a smooth manifold. It is not easy/obvious to define what is a covering space over $M/G$ would be.. So, to make sense of covering spaces, we have to go to orbifold groupoid setup where notion of covering space would be just simple as above.

To summarize I said the following :

Orbifolds are neither just topological space where you can define covering maps to be local homeomorphisms plus something nor smooth manifolds where you can define covering maps to be local diffeomorphisms plus something. Orbifolds are something more than just a topological space and less than a smooth manifold. So, you need a separate approach to define covering spaces and groupoid approach is useful/straight-forward.

I just want to know if what I have said is actually a reason or did I misunderstood the idea.

Any comments are welcome.

Edit: This is just to bump this question up so that it gets some attention and in turn some comments.

The concept of curvature is defined for any linear connection on any vector bundle $E \to M$, but the concept of torsion is only defined for connection on the tangent bundle $TM$ of a manifold $M^n$, or for a connection obtained as the pullback of a connection on a vector bundle $E \to M$ *isomorphic to $TM$* via an isomorphism $\theta \colon TM \to E$ equivalent to a solder form.

Why is that so ? If torsion can be interpreted as the twist of a moving frame along a curve, the same phenomena should occur for a connection on any vector bundle.

Is there a way to define a notion of torsion for any vector bundle ?

If $T$ is a countable complete first-order theory with infinite models, the number of countable models it has, $I(T,\omega)$, must be an element of $N=\{1,3,4,5,6,7,\dots,\omega,\omega_1,2^\omega\}$ (although we don't know if $\omega_1$ can happen). For which pairs $n,m\in N$ does there exist a countable complete theory $T$ with $n$ countable models but $m$ countable models after adding finitely many constants to the theory? Countably many new constants? In particular can we have $m<n$? EDIT: By 'adding constants,' I mean adding constants whose type is completely specified, i.e. expanding by constants and then passing to a complete theory in the expanded language.

Let $n\rightarrow m$ denote the statement "There exists a complete countable theory $T$ and a finite tuple of constants $\overline{a}$ such that $I(T,\omega)=n$ and $I(T_\overline{a},\omega)=m$." And let $n\rightarrow_\omega m$ denote the statement "There exists a complete countable theory $T$ and a countable set of distinct constants $A$ such that $I(T,\omega)=n$ and $I(T_A,\omega)=m$." Some easy results and relevant observations:

- If $n\rightarrow m$ (resp. $n\rightarrow_\omega m$) and $k \rightarrow \ell$ (resp. $k \rightarrow_\omega \ell$), then $nk \rightarrow m\ell$ (resp. $nk \rightarrow_\omega m\ell$). (Take the disjoint union of the relevant theories.)
- $n \rightarrow n$ for every $n\in N-\{\omega_1\}$. (This is obvious for $n=1$. There are easy examples for $n=\omega,2^\omega$ and the standard examples for $n=3,4,5\dots$ all have constants which do not increase the number of countable models.)
- $n^2+n\rightarrow (n+1)^2$ for any $1<n<\omega$. (DLO with $n-1$ colors and a countable set of constants of order type $\omega + \omega^\ast$. By itself this theory has $n^2 + 1$ countable models. Adding a constant in between $\omega$ and $\omega^\ast$ makes the theory have $(n+1)^2$ models.)
- $1\not\rightarrow n$ and $n\not\rightarrow 1$ for any $n\in N - \{1\}$.
- $1\rightarrow_\omega 2^\omega$ (For example: DLO.)
- $1\rightarrow_\omega \omega$ (For example: A structureless set.)
- $n\not\rightarrow_\omega 1$ for any $n\in N$.
- $1 \rightarrow_\omega n$ for every $2<n<\omega$. (The standard examples of Ehrenfeucht theories are $\omega$-categorical theories with countably many constants added.)
- If a theory is not small, then it will have $2^\omega$ countable models after adding any countable set of constants.

Let $I$ and $J$ be finite sets of open intervals $(a,b)\subset\mathbb R$. For a finite set of points $P\subset \mathbb R$ we denote those subsets of intervals from $I$ and $J$ containing some point from $P$ by $I_P,J_P$. Now suppose that \begin{align}\tag{*}\label{IP JP ineq}\lvert I_P\rvert\le \lvert J_P\rvert+1\end{align} for all finite subsets $P$, and that the inequality \eqref{IP JP ineq} is optimal in the sense that there exists at least one finite $P$ for which equality is achieved. Here $\lvert\cdot\rvert$ is simply the counting measure.

I have the strong suspicion (supported by numerical experimentation with random sets) that there has to exist some specific interval $(a,b)\in I$ such that $(a,b)$ contains at least one point from each set $P$ for which equality is achieved in \eqref{IP JP ineq}.

It is clear that the statement cannot be true for more general subsets than intervals but I couldn't come up with any argument yet. It is also clear that the claim cannot hold true if the $1$ in \eqref{IP JP ineq} is replaced by $0$.

What would be some examples in the mathematical sciences of what Feynman once colorfully described as Cargo Cult Science (CCS) that are certifiably bogus according to reliable sources? Feynman's original essay can be found here. The formation of the cargo cults in the New Guinea Mountains in the 1930s is analyzed in Tomasz Witkowski's *Psychology Led Astray: Cargo Cult in Science*, on pages 17-20.

In the natural sciences there are some classical examples like phlogiston, ether (these two may have been historically justifiable and not in the same category as the other examples), and Lysenkoism. A more recent example is the Bogdanov affair attested to by an internal report of the CNRS to the effect that the theory had "no scientific value".

But it seems that in mathematics the examples are more rare. There is a number of published articles about $\tau (=2\pi)$ but this is not so much a CCS as a triviality. I remember reading somewhere that around 1900 there was some work on evaluating improper integrals that was genuinely bogus (as opposed to the techniques of summation of divergent series which are of course a legitimate field, with applications to physics, as nicely described in Varadarajan's 2007 article on Euler). Another example is Sergeyev's "grossone" CCS (now officially attested to by the unanimous statement of the EMS Surveys editors; meanwhile, Zentralblatt recycles Sergeyev's claims on how the grossone outperforms $\infty,\aleph_0,\omega$), involving numerous publications in refereed journals, books, collaborators, "international" conferences featuring keynote dinners in "exclusive" restaurants, "international" prizes, etc.

Are there other examples? Here it seems reasonable to impose a criterion of at least one publication in a refereed venue so as to filter out any number of oddballs posting at arxiv or vixra.

The examples developed so far (that meet the criteria for inclusion as above) are the following, in alphabetical order:

(1) Bogdanov, Grichka (a thesis in mathematics; a CNRS opinion of "no value");

(2) Santilli, Ruggero (publications in refereed venues; detailed rebuttal in MathSciNet review by Magill;

(3) Sergeyev, Yaroslav (publications in refereed venues, including books; unanimous negative opinion by editorial board as above. Note that this refers to his work in the *grossone* whereas his work on optimisation is another matter).

(4) Smarandache, Florentin (some of the publications seem to be legitimate, particularly in elementary geometry).

There is a couple of examples mentioned in the *comments* that I haven't had a chance to examine in detail yet.

While working on a problem, I constructed something which looked like an induced representation, but with a tensor product instead of a direct sum.

Here is a special case. Let $G$ be a group, with $H$ a subgroup of index $2$. Choose $s \in G$ which is not in $H$. For $(\pi,V)$ a representation of $H$, define a representation $\sigma$ of $G$ with underlying space $V \otimes V$ as follows: if $h \in H$, define $\sigma(h) = \pi(h) \otimes \pi(shs^{-1})$. If $g \in G, \not\in H$, define $\sigma(g)$ on generators by

$$\sigma(g) v \otimes w = \pi(gs^{-1})w \otimes \pi(sg)v$$

If we had used $V \oplus V$ instead of $V \otimes V$, then following the above construction would have produced the representation $\operatorname{Ind}_H^G(\pi)$. So this is like a tensor product version of induced representation.

I found some papers which talk about "tensor induction," and I believe the above construction is a very simple case of that. What is tensor induction, generally speaking? Is it adjoint to some functor, or satisfy a universal property?

I was reading Chernikov's notes about stable theories, and he mentions the following fact:

If $T$ is stable and $A$ is some set of parameters large enough then there is some indiscernible sequence $I \subseteq A$ such that $|I|=|A|$.

I have been trying to find a reference of the previous fact but I have not been lucky. Any comment, hint or reference is highly appreciated!

The Fejer-Jackson-Gronwall inequality involving the sine function is as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$

Here I ask the following related question.

QUESTION: Do we have $$\sum_{k=1}^n(-1)^k\left(\frac{\sin kx}k\right)^m<0<\sum_{k=1}^n\left(\frac{\sin kx}k\right)^m$$ for all $m,n=1,2,3,\ldots$ and $0<x<\pi$ ?

Actually I formulated this question in 2013. My numerical computation suggests that the answer should be positive. How to prove this?

Let $C$ be a locally presentable, locally cartesian closed $\infty$-category. Then I think it's not hard to show that the class of effective epimorphisms in $C$ is closed under colimits and cobase-change, and is accessible, so that it forms the left class of a factorization system. If $C$ is an $\infty$-topos, then the corresponding right class consists of the monomorphisms in $C$. How about in general?

I guess I'm particularly interested in the case where $C$ is the unstable motivic category over a scheme $S$.