I have one scenario In which there are two ranges for height and weight lets say **H1-H2** and **W1-W2** respectively. value of weight is depend on height.
more the heigh the more is the weight.
I need some formula to calculate weight from other category by giving height as a input.

If height given is lowest in range then weight is also must be lowest in range lowest. and same for other values also.

Is there a Riemannian metric on $\mathbb{R}^2$ (or a $2$ dimensional manifold) such that the intersection of every two open discs is an open disc, again?

I'm looking for the paper

*Extensions of general algebras*, S. Eilenberg, Ann. Soc. Polon. Math. 21, (1948). 125–134 (MR0026647 in Mathscinet)

It is supposedly hosted at the Digital Repository of Scientific Institutes (RCIN) here, but I cannot access it, although I have registered in the site.

Can someone locate another online copy, or send one to me?

Let $G$ be a complex reductive group acting linearly on a complex affine variety $X\subseteq\mathbb{C}^n$. Then, there is a stratification by orbit type of the GIT quotient
$$X//G=\operatorname{Spec}{\mathbb{C}[X]^G}.$$
Namely, if $\pi:X\to X//G$ is the quotient map and $H\subseteq G$ a subgroup, then $p\in X//G$ has *orbit type* $(H)$ if for a point $x\in\pi^{-1}(p)$ such that $G\cdot x$ is closed, the stablizer $G_x$ is conjugate to $H$ in $G$. This gives a stratification
$$X//G=\bigcup_{(H)}(X//G)_{(H)}.$$
Is this a Whitney stratification?

Let us say that a commutative ring with unity $R$ has the Krull Intersection Property (KIP in short) if the following holds : For every finitely generated module $M$ over $R$ and every proper ideal $I$ of $R$, $I(\cap_{n=1}^\infty I^n M)=\cap_{n=1}^\infty I^n M$ .

Now let $R$ be an integral domain having KIP , and let $\bar R$ be the integral-closure of $R$ in its fraction field (i.e. Normalization) ; then does $\bar R$ have KIP too ?

Let $(P,\leq)$ be a countable poset with the property that whenever $a<b\in P$ then $P\cong [a,b]$.

**Question.** If $P$ does contain elements $a,b$ with $a<b$, does this imply that $P \cong [0,1]\cap\mathbb{Q}$, or that $P$ is isomorphic to the nonzero countable atomless Boolean algebra?

**Note.** Thanks to user @YCor for suggestions to make this a better question.

Here we use $\omega_1^{CK}$ to denote the least nonrecursive ordinal. The following theorem is well known.

**Theorem** $\omega_1^{CK}$ is an admissible ordinal.

But its proof seems weird. The usual proof uses a nonstandard technique. I wonder whether there exists a pure standard proof. Or, to negate it, whether there is an $\omega$-model $M$ of $KP$ so that $$M\models \omega_1^{CK}\mbox{ exists but is not admissible ?}$$

If there exists such a model, then it means that $KP$ is not enough to prove the theorem. So one has to use some assumptions beyond $KP$ and probably requires the existence of Kleene's $O$. So nonstandard techniques can be applied.

This question is inspired by another recent one here, Characterization of the hypergeometric function. The latter is about the classical result of Riemann characterizing the hypergeometric functions as sections of any two-dimensional subsheaf of the sheaf of functions analytic away from $\{0,1\}$ and having local expansions of the form $z^{\alpha_0}f_0(z)$ near zero and $(1-z)^{\alpha_1}f_1(z)$ near 1 with $f_i$ holomorphic at $i$. The additional condition is closure under analytic continuation which I don't really understand how to formulate in purely sheaf-theoretic terms (probably some pushforwards must be involved but I do not see any details). I also suspect one should say something about behavior at $\infty$ but again, I am not sure. In any case, the question cites a paper by Deligne-Mostow, as well as the original paper of Riemann, while in a comment Alexandre Eremenko refers to the very last entry in the "Complex Analysis" by Ahlfors for a detailed treatment.

My question is whether a similar characterization of the basic hypergeometric functions is known. I mean their characterizations in terms of some sheaf of functions with prescribed singularities at given points, or finite amount of similar kind of local data, without mentioning any $q$-difference equations or any explicit series expansions.

I tried to look in Gasper & Rahman, Fine, Bailey, Slater, as well as Kac-Cheung and Etingof-Frenkel-Kirillov. I might well overlook it but the fact is I could not find anything.

Suppose $c,t$ are such that, $0< c< 1$ constant and $cn\leq t \leq n$. I want to have an estimation of

$\sum _{i=0}^{cn} {cn\choose {i}}{(1-c)n \choose t-i} 2^{t-i}$

when n goes to infinity.

Can I bound it by $2^{c'n}$ for some $0<c'<1$?

I have no idea to do that.Is there any hint?

Thanks!

I am trying to understand how to obtain the Picard group for general toric varieties. So far, I have been using information found in https://arxiv.org/pdf/1003.5217.pdf .

Here, a toric variety has homogeneous coordinates $H:=\{x_i : i=1,\ldots, I\}$ equipped with a number $R$ of equivalence relations $$ (x_1,\dots,x_I)\sim (\lambda_r^{Q^{(r)}_1} x_1, \ldots, \lambda_r^{Q^{(r)}_I} x_I), \\ $$ for $r=1,\dots, R$ with the weights $Q^{(r)}_i\in \mathbb Z$ and $\lambda_r\in \mathbb{C}^* = \mathbb{C}-\{0\}$.

They go on to say (above equation (7)) that for each divisor $D_i$ of a toric variety, there is a line bundle $$\tag{7} L_i={\cal O}_X \bigl(Q^{(1)}_i, \ldots, Q^{(R)}_i \bigr)\; . $$

It is not clear to me why the weights $Q^{(r)}_i\in \mathbb Z$ govern the classification of line bundles, but this seems to imply that for a particular toric variety, $X$, the Picard group is
$$
Pic(X)=\mathbb{Z}^R.
$$
However, I have also read here -Reference for Weighted Projective Stacks - that for weighted projective spaces, the Picard group is cyclic. (Edit: It seems that this link talks about weighted projective *stacks*, which are not toric varieties.)

What is the rationale for describing line bundles in terms of weights as in equation (7), and how does one find the Picard group of a toric variety in general?

For the purpose of this post, I will say that *the* Gale-Stewart game is the infinite two-player game of perfect information where players I and II alternate playing natural numbers, with I going first. A round of the game looks like: I plays $n_0$, II plays $n_1$, I plays $n_2$, II plays $n_3$, etc. The outcome of this round is the sequence $(n_i)_{i\in\omega}\in\omega^\omega$.

Given a set $A\subseteq\omega^\omega$, I will speak of a player *having a strategy for playing into $A$*, rather than using the usual winning/losing terminology. If $\sigma$ is a strategy for one of the players, I denote by $[\sigma]\subseteq\omega^\omega$ the set of all outcomes of the game when that player follows $\sigma$.

**Question:** Suppose we are given a set $C\subseteq\omega^\omega\times\omega^\omega$. Under which circumstances do there exists strategies $\sigma$ and $\tau$ for players (either one) in the Gale-Stewart game such that one of $[\sigma]\times[\tau]\subseteq C$ or $([\sigma]\times[\tau])\cap C=\emptyset$ holds?

Is it sufficient that $C$ is determined? To put that another way, is there a game which encodes this property? Or is there a (reasonably definable) counterexample?

The first thing that comes to mind is the game where the two players alternate, with I going first, playing pairs of natural numbers $(n_i,m_i)$, and whose outcome is the pair $((n_i)_{i\in\omega},(m_i)_{i\in\omega})\in\omega^\omega\times\omega^\omega$. If $C$ is determined, then either I has a strategy for playing into $C$ or II has a strategy for playing into its complement in this game. Suppose player I has a strategy for playing into $C$. From this, it is easy to construct two strategies $\sigma$ and $\sigma'$ in the Gale-Stewart game for player I with $[\sigma]\subseteq\pi_0(C)$ and $[\sigma']\subseteq\pi_1(C)$, where $\pi_0$ and $\pi_1$ are the first and second coordinate projections, respectively: read off the first (or second) coordinate of I's play in the game with pairs while II plays their move in the Gale-Stewart in the first (or second) coordinate, and an arbitrary number in the other coordinate. However, I have no reason to suspect that $[\sigma]\times[\sigma']\subseteq C$. One issue seems to be a lack of independence in the coordinates played according to a strategy; each of played coordinates can depend on either of the previously played coordinates.

The question is maybe a bit technical, but I find the related construction very beautiful.

In the very famous work - "$C^1$-isometric imbeddings" by J.Nash (1954) the author presented the fundamental theorem (which was especially recognized some years after) about isometric embeddings of Riemannian manifolds.

$\textbf{Theorem of J. Nash:}$ Any Riemannian $n$-manifold has $C^{1}$ an isometric imbedding in $E^{2n+1}$ (Euclidean $2n+1$-dimensional space).

There are also some similair other theorems of J.Nash and N.Kuiper which nowadays are formulated in a bit different form, especially in a view of famous Gromov's H-principle.

I have a question regarding the considerations of J.Nash in the aforementioned paper. I don't detail all the proof of this theorem, but I recall the main ingredient of the construction of such immersion (imbedding) - "short" immersion or imbedding.

$\textbf{Definition}:$ Immersion (imbedding) $z : (M,g) \rightarrow (E^k, h)$ is short if $$ z^*h \leq g \text{ in the sense of quadratic forms}, $$ where $g$ is a metric on $M$, $h$ is euclidean metric on $E_k$.

Having initial short immersion or imbedding J.Nash presents some sequence of immersions (imbeddings) $\{z_n\}_{n=1}^{\infty}$, where all $z_n$ are all short and monotonically increase induced metric $z_n^*h$. This sequence of $z_n$ converges in $C^1$ sense and also gives the desired metric tensor $g$ for induced metric. So we naturally come up to the following question:

$\textbf{Important question:}$ How to construct an initial short immersion (imbedding)?

1) For compact manifolds the answer is simple: use Whitney theorem and multiply the map by a small $\varepsilon > 0$ to decrease the induced metric.

2) For open (non-compact) manifolds it is a bit trickier. The construction is as follows:

Let $\{U_i\}_{i=1}^{\infty}$ locally finite open cover of $M$, $\{\phi_i\}_{i=1}^{\infty}$ is the partition of unity subordinated to this cover. Using Dawker's theorem (or Borsuk theorem) we choose $\{U_i\}$ such that no more than $n$ charts intersect other chart. So we say that the multiplicity of the cover $\{U_i\}$ is $s$.

We divide $\{U_i\}$ into $s$ separate classes: it is easy, we take the charts in order and give them any class that is not yet given to any other neighbor. Next we construct an imbedding in $s(n+2)$ dimensional space by defining the maps: \begin{align} &u_\sigma(x) = \begin{cases} \varepsilon_i \phi_i(x), \text{ if $x$ belongs to vicinity of class } \sigma,\\ 0, \text{ otherwise}, \end{cases}\\ &v_\sigma(x) = \begin{cases} \varepsilon^2_i \phi_i(x), \text{ if $x$ belongs to vicinity of class } \sigma,\\ 0, \text{ otherwise}, \end{cases}, \\ &w_{\sigma j}(x) = \begin{cases} \varepsilon_i \phi_i(x) x_{ij}, \text{ if $x$ belongs to vicinity of class } \sigma,\\ 0, \text{ otherwise}, \end{cases}, \end{align} where $x_{ij}$ stands for the $j$-th coordinate in chart $U_i$. The numbers $\varepsilon_i$ is the sequence of positive constants monotonously decreasing to zero.

One can see that: if two points $x$ and $y$ lie in different charts $U_i, U_j$, then they are separated by maps $u_\sigma, \, v_\sigma$. If they belong to the same chart, then they are separated by $w_{\sigma i}$. All maps are $C^{\infty}$-smooth and, by the latter considerations, are injective. From the form of $w_{\sigma i}$ one can see that the differential is injective, so it is an immersion and as it is injective it is an imbedding. Well, injectivity of differential and global injectivity do not yet imply an embedding, but one can analyze the afforementioned maps and see that it is a homeomorphism onto its image.

Controllong the $\varepsilon_i$ we can make it short. So we've constructed short imbedding in $E^{s(n+2)}$.

Then J.Nash says that since the image of the manifold in $E^{s(n+2)}$ is $n$-dimensional set, "the classical process of generic linear projection can be applied and one can reduce the dimension of surrounding Euclidean space to $2n+1$ without introducing any singularities and self intersections".

$\textbf{Question:}$ What is this "classical process of generic linear projection?"

p.s. I found the latter construction for the case of open manifolds very beautiful. It is sad to miss the last point of such beauty!

The Golomb space $\mathbb G$ is the set of positive integers endowed with the topology generated by the base consisting of thearithmetic progressions $a+b\mathbb N_0$ with relatively prime $a,b$ and $\mathbb N_0=\{0\}\cup\mathbb N$. It is know that the space $\mathbb G$ is connected and Hausdorff.

It is also easy to check that the multiplication map $\cdot:\mathbb G\times \mathbb G\to\mathbb G$, $\cdot:(x,y)\mapsto xy$, is continuous, so $\mathbb G$ is a commutative topolological semigroup.

**Problem.** Is the Golomb space $\mathbb G$ topologically homogeneous? Or maybe rigid?

We recall that a topological space $X$ is *rigid* if its homeomorphism group is trivial.

This problem was motivated by this question, which discusses the relation of the Golomb space to another countable connected Hausdorff space, called the rational projective space $\mathbb QP^\infty$. This space is easily seen to be topologically homogeneous.

Let $\mu$ be a **compactly supported** probability measure on a finite-dimensional euclidean space (for simplicity) $\mathbb E$, and suppose $\mu$ **has density**. For a random point $x \sim \mu$,
consider the Boltzmann distribution at temperature $T$, on $N$ states, viz

$$p_j(x; T) = \frac{\exp(-\frac{1}{T}E_j(x))}{\sum_{l=1}^N\exp(-\frac{1}{T}E_l(x))},$$

where energies $E_j(x) = x^Ty_j + c_j$ is the energy at state $j$, and $y_1,\ldots,y_N \in \mathbb E$, $c_1,\ldots, c_N \in \mathbb R$ are fixed.

QuestionWhat can be said about the expectation integral $$e_j^\mu(T) := \int p_j(x; T) d\mu(x) $$

Is it a known quantity / transform ?

Is there an efficient way to compute $e_j^\mu(T)$ which is "cheaper" (in the sense faster convergence / lower variance of

estimates) than just sampling $x_1,\ldots, x_M \sim \mu$ and computing the empirical mean $\frac{1}{M}\sum_{i=1}^M p_j(x_i;T)$ ?

For the last two questions, for convenience one may assume $T \rightarrow 0^+$ (low-temperature limit). In the high-temperature limit, it's not hard to show that $e_j^\mu(\infty) = N^{-1},\; \forall j=1,\ldots,N$.

Let $(E, \langle\cdot\;, \;\cdot\rangle)$ be a complex Hilbert space, with the norm $\|\cdot\|$ and let $\mathcal{L}(E)$ be the algebra of all bounded linear operators from $E$ to $E$. Let $M\in \mathcal{L}(E)^+$ (i.e. $M^*=M$ and $\langle Mx\;,x\;\rangle\geq 0$ for all $x\in E$.

Assume that $T(ker(M))\nsubseteq ker(M)$. We define the following subset:

\begin{eqnarray*} S_M(T) &=&\{\lambda\in \mathbb{C}\,;\;\; \exists\,(\alpha_n,\beta_n)\in ker(M)\times \overline{Im(M)}\,;\;\;\|M^{1/2}\beta_n\|=1, \displaystyle\lim_{n\rightarrow+\infty}\langle MT \alpha_n\; |\;\beta_n\rangle+\langle MT \beta_n\; |\;\beta_n\rangle=\lambda,\\ &&\phantom{+++++}\;\hbox{and}\;\displaystyle\lim_{n\rightarrow+\infty}\|M^{1/2} T(\alpha_n+\beta_n)\|<\infty\;\}. \end{eqnarray*} What do you think about the convexity of $S_M(T)$?? I try with an example of $M$ and $T$ such that $T(ker(M))\nsubseteq ker(M)$, I get $S_M(T)=\mathbb{C}$.

I claim that $S_M(T)=\mathbb{C}$. Do you think that my claim is true?

Thank you for your help!!

Is it something known about $L^2$-Betti numbers for Golod-Shafarevich groups?

The following conjecture is known as ``Table problem on $\Bbb S^2$ "

**Conjecture (Table problem on $\Bbb S^2$):** Suppose $x_1, x_2,x_3,x_4 \in\Bbb S^2 \subseteq \Bbb R^3$ are the vertices of a
square that is inscribed in the standard $2$-sphere, and let $h : \Bbb S^2\to \Bbb R$ be a smooth function.
Then there exist a rotation $\rho\in \rm{SO}(3)$ such that $h(\rho(x_1)) =\cdots= h(\rho(x_4))$.

**Question 1:** What is the means of ``standard $2$-sphere"? topological $2$-sphere or geometrical?

**Question 2:** Is this conjecture open still?

I want to calculate this equation, and want to understand this flow. But, calculation not easy. Please help me.

[Recurrence and topology]

I have seen the following statement being used in different papers but never saw a proof:

If $f:X\rightarrow Y$ is a flat morphism between normal varieties and $\mathcal F$ is a reflexive sheaf on $Y$ (i.e. $\mathcal F^{\vee\vee}\cong\mathcal F$). Then the pull-back $f^*\mathcal F$ is a reflexive sheaf on $X$.

Does someone know an easy way to prove this or a paper or book where it is proven?

Let $k$ be a field and $A$ and $B$ be two graded $k$-algebras satisfying $A\subset B$. The $\operatorname{Proj}$ construction is not functorial but is there nothing to say about $\operatorname{Proj}(A)$ and $\operatorname{Proj}(B)$ ? (even under some additional assumptions you are free to make for exemples)

Many thanks for your enlightenments!