I am a little stuck with my project. In the calculations of my project, I need to calculate the spread of some random variables. Up to now, there was no special difficulty to analytically calculate the standard deviation of correlated and uncorrelated random variables.

The point is that I am dealing now with variances and covariances of ratios between 3 different random variables X, W and Y. How to calculate the following? $$ cov(X/Y,W/Y)=? $$ Since the ratios share the same denominator, is it correct to write : $$ cov(X/Y,W/Y)=\frac{1}{\left[E(Y)\right]^2}\,cov(X,W) $$ Thanks for your kind help.

Define $E_{i,j} \in \mathbb{R}^{n \times n}$ to be the canonical basis (that is all elements set to zero except the entry $i,j$ ) let the bloc matrix $M \in \mathbb{R}^{n^2 \times n^2}$ defined by :

$M = \begin{pmatrix}E_{\sigma(i), \sigma(j)} \end{pmatrix}_{ 1\leq i,j \leq n}$ for some permutation $\sigma$ of ${1,2,..,n}$

can you find the convex hull of this set of points ? Or at least is the number of equations that define this polytope is exponential?

How can be large the chromatic number of a $K_{r}$-free graph, a graph with clique number at most $r-1$, comparing with the number of its vertices? Especially, I am interested in any good upper bound for the case $r=4$.

$\omega^{CK}_1$ is the supremum of all the recursive ordinals, where an ordinal $\alpha$ is recursive if there is a computable ordering of a subset of the naturals with order type $\alpha$.

For a Turing degree $D$, we will say that an ordinal $\alpha$ is $D$-recursive if there is a $D$-computable ordering of a subset of the naturals with order type $\alpha$. We will also say that the supremum of the $D$-recursive ordinals is $\omega^{CK}_D$.

This has some interesting properties that connects Turing degrees and countable ordinals. For example, for any countable ordinal $\alpha$ there is a Turing degree $D$ such that $\alpha$ is $D$-recursive (simply choose a ordering of the natural numbers with order type $\alpha$, and construct an oracle that computes that ordering). This in particular implies that supremum of the $\omega^{CK}_D$ over all Turing degrees $D$ is $\omega_1$. Additionally, the order type of the $\omega^{CK}_D$ over all Turing degrees $D$ is also $\omega_1$. Also, for each Turing degree $D$, we can construct an ordinal notation for the ordinals $< \omega^{CK}_D$, similar to Kleene's O.

My question is, has this relationship between Turing degrees and countable ordinals been explored before?

I am looking for a book or paper that discusses basic notions of calculus of variations on metric/topological measure spaces.

I have a problem: Let $(X,\mu)$ be a measure space with a positive measure $\mu$ and let $u:X\rightarrow \mathbb{R}$ be continuous (here is where I need a toplogy on X if not a metric). Suppose $F:\mathbb{R}\rightarrow \mathbb{R}$ is smooth and satisfies $\int_{X}|F(u)| d\mu <\infty$.

Are the Euler-Lagrange equations $F^{\prime}(u)=0$ necessary for the existence of an extremizer for the functional $u\rightarrow \int_{X} F(u) d\mu$ over a given subset of $X$?

This is a simple consequence of the Euler-Lagrange equations when $\mu$ is a Lebesgue measure. What about more general topological measure spaces ?

A reference would be appreciated.

In this question - On a Hirzebruch surface. , the Hirzebruch surface is shown to be isomorphic to a hypersurface in $\mathbb{P}^1\times \mathbb{P}^2$.

My question is, does such an isomorphism exist for all toric varieties (or at least simplicial ones)? To be precise, given a toric variety $$ (\mathbb{C}^N \backslash U)/(\mathbb{C}^*)^m, $$ can we show that it is isomorphic to a hypersurface of a product of $m$ projective spaces?

At least for complete intersection Calabi-Yaus, this seems to be true, based on https://arxiv.org/abs/0805.2875.

I have a big problem to solve this system $\Delta f-hf^2=0$

$|\nabla f|^2+hf^3=0$

where $h$ is a constant, $f$ is a 2-dimensional smooth function, $\Delta f$ is Laplacian of $f$ (i.e. $\Delta f=f_{xx}+f_{yy}$) and $\nabla f$ is the gradient of $f$.

ADD In first case $f$ is defined on $R^2$ and in second case $f$ is defined on surface $S$ ($f:S \rightarrow (0, \infty)$). is there a solution? Thank you for help

MODIFICATION after Igor Khavkine answer: and if the system is

$\Delta f-hf^2+cf=0$

$|\nabla f|^2+hf^3=0$

(c is another constant)

I am currently trying to read Colmez' "Série principale unitaire pour $Gl_2(\mathbb{Q}_p)$ et représentations triangulines de dimension 2", that you can find here https://webusers.imj-prg.fr/~pierre.colmez/triangulines . In the proof of Lemma 4.1. at the very end I can not follow anymore.

The statement is the following: For $b\in \mathcal{E}^{\dagger}$ (i.e. an element of the Robba ring that is bounded at 0) apparently can find an element $c$ of $\mathcal{B}^\dagger$ with $\varphi(c)=bc$, where $\varphi$ is the operator with $T\mapsto (1+T)^p-1$. Furthermore there is stated that $\mathcal{B}^\dagger$ is absolutely non-ramified over $\mathcal{E}^\dagger$. In the same paper I can not find a definition of $\mathcal{B}^\dagger$, but in another paper of Colmez, "Représentations cristallines et représentation de hauteur finie", that you can find here https://webusers.imj-prg.fr/~pierre.colmez/hauteurfinie.pdf it is defined. I am still quite new to the theory of $(\varphi,\Gamma)$-modules and did not work with Witt-vectors yet. So I have truble understanding what exactly $\mathcal{B}^\dagger$ is and how those statements would follow.

I am looking for a reference where these two statements are proven. It feels like they follow fast from the definition, so I would also be happy about some reference where $\mathcal{B}^\dagger$ is regarded in greater detail.

The following question came up while trying to determine whether the extension problems in a spectral sequence are trivial.

Given a noetherian ring $R$ and a finitely generated $R$-module $M$ with a filtration $M=F_0 \supset F_1 \supset \ldots \supset F_n \supset F_{n+1}=0$ such that $M \cong \bigoplus_{i=0}^n F_i/F_{i+1}$ (abstractly, but I do not want to assume that this isomorphism is induced by the maps in the filtration), is the filtration split in the sense that all the short exact sequences $0 \to F_{i+1} \to F_i \to F_i/F_{i+1} \to 0$ are split?

I can prove this for $R=\mathbb{Z}$, but my proof does not seem to generalize well. In that case, one can work $p$-locally and then show that a short exact sequence $0 \to A \to B \to C \to 0$ of $p$-groups,

$|\{x \in B|ord(x)\text{ divides }p^k\}|\geq |\{x \in A|ord(x)\text{ divides }p^k\}|\cdot|\{x \in C|ord(x)\text{ divides }p^k\}|$

, with equality iff the sequence is split, and then use that by the assumed isomorphisms, the number of these elements can never strictly increase.

Steven Landsburgs answer in Do all exact sequences $0 \rightarrow A \rightarrow A \oplus B \rightarrow B \rightarrow 0$ split for finitely generated abelian groups? got me hoping that something like this might hold for finitely generated modules over noetherian rings, but a simple induction argument does not seem to work.

Does anyone know if this generalisation is true, and knows a proof?

I have asked this question exactly here. The question is as follows:

I am interested deeply in the following problem:

Let $f$ be a (fixed) positive definite quadratic form; and let $n$ be any arbitrary natural number; then find a closed formula for number of solutions to the equation $f=n$.

For special case $f_1(x,y)=x^2+y^2$, here gives a closed formula for number of solutions.

Also you can find another formulas

for the special cases $f_5(x,y)=x^2+5y^2$ and $f_7(x,y)=x^2+7y^2$ there.You can finde a close formula here for $f_2(x,y)=x^2+2y^2$, here.

You can finde a close formula here for $g(x,y)=x^2+xy+y^2$, here. Also you can find the answer for (only finitely many) other forms there, may be this helps too.

You can finde a close formula here for $f(x,y,z,w)=x^2+y^2+z^2+w^2$, here.

As an example of the case which contains more than one class per genus; see here.

By a more Intelligently search through the web; you can find similar formulas for only finite limited number of positive definite quadratic forms.

[I think there exists such an explicit formula

at most for $10000$ quadratic forms.**Am I right?**]

As I have mentioned (I am not sure of it!) only for finite number of quadratic forms we have such a explicit, closed, nice formula; and this way goes in dead-end for arbitrary quadratic forms.

So *Dirichlet* tries to find the (weighted) sum of such representations by binary quadratic forms of the same discriminat.

That formula works very nice for our purpose if the genera contains exactly one form. In the dirichlet formula each binary quadratic forms apears by weight one in the (weighted) sum.

More precisely let $f_1, f_2, ..., f_h=f_{h(D)}$ be a complete set of representatives for reduced binary quadratic forms of discriminant $D < 0$;
then for every $n \in \mathbb{N}$, with $\gcd(n,D)=1$ we have:

$$ \sum_{i=1}^{h(D)} N(f_i,n) = \omega (D) \sum_{d \mid n} \left( \dfrac{D}{d}\right) ; $$

where $\omega (-3) =6$ and $\omega (-4) =4$ and for every other (possible) value of $D<0$ we have $\omega (D) =2$. Also by $N(f,n)$; we meant number of integral representations of $n$ by $f$; i.e. :

$$ N(f,n) := N\big(f(x,y),n\big) = \# \{(x,y) \in \mathbb{Z}^2 : f(x,y)=n \} . $$

I have hered that there is a generalization of dirichlet's theorem for ;quadratic forms, **in more variables**; due to Siegel.

I have searched through the web;
but I have found only this link : Smith–Minkowski–Siegel mass formula ;
also I confess that I can't understand whole of this wiki-article.

Could anyone introduce me a simple reference in english; for Siegel mass formula ?

Also you can find better informations here, and may be here & here.

Denote $\mathcal R_2[n]=\mathcal R[n] + \mathcal R[n]$ to be sumset of integers in $\mathcal R[n]$ where $\mathcal R[n]$ to be set of permanents possible with permanents of $n\times n$ matrices with $0/1$ entries.

- Is $\mathcal R_2[n]\subseteq\mathcal R[n+1]$?

Example:

$$\mathcal R_2[1]=\mathcal R[1]+\mathcal R[1]=\{0,1\}+\{0,1\}=\{0,1,2\}\subseteq\mathcal R[2]$$

$$\mathcal R_2[2]=\mathcal R[2]+\mathcal R[2]=\{0,1,2\}+\{0,1,2\}=\{0,1,2,3,4\}\subseteq\mathcal R[3]$$

$$\mathcal R_2[3]=\mathcal R[3]+\mathcal R[3]=\{0,1,2,3,4,5,6,7,8,9,10,12\}\subseteq\mathcal R[4]$$

It suffices (but may not be necessary) to show $\{0,1,2,\dots,2(n!)\}\subseteq R[n+1]$.

Denote $\mathcal T[n,i]$ to be set of $n\times n$ matrices with $0/1$ entries with permanent $i$.

- Is there a $c>0$ such that $$\big|\big\{i:\mathcal T[n+1,i] > \sum_{\substack{1=a<i-1\\a<b<i-1\\a+b=i}}\big(\mathcal T[n,a]+\mathcal T[n,b]\big)\big\}\big|>\frac{n!}{n^c}$$ when $1\leq i\leq 2(n!)$?

Note I avoided $a=0$ or $b=0$.

Let $\mathbb G$ be an abelian vatiety over an $\mathbb E_\infty$-ring $A$. That is to say, it consists of an abelian group object in the $\infty$-category of relative schemes $\mathbb G\to \operatorname{Spét} A$ which are flat (+ maybe more conditions). Denote its underlying classical abelian variety over $\pi_0A$ by $\mathbb G_0$.

Recall from Lurie's "Survey of Elliptic Cohomology":

**Definition:** A **preorientation** on $\mathbb G$ is a map of abelian topological groups
$
\mathbf{CP}^\infty\to \mathbb G(A)
$
or equivalently a map of formal spectral group schemes
$
\operatorname{Spf}A^{\mathbf {CP}^\infty}\to\mathbb G,
$
or equivalently yet an element of $\pi_2\mathbb G(A)$.

Viewed as a map $S^2\to\mathbb G(A)$, a preorientation induces through some adjunctions and restriction to $\pi_0$ a map $\beta :\omega\to \pi_2$, where $\omega$ denotes the invariant differentials of $\mathbb G_0$ over $\pi_0A$.

**Question:** *In what way are the following two conditions related?*

A) The preorientation map exhibits an equivalence $\operatorname{Spf}A^{\mathbf {CP}^\infty}\simeq \widehat{\mathbb G}$, where the RHS is the formal completion of $\mathbb G$ at the identity.

B) The preorientation is an *orientation*, in the sense that

- The map of underlying ordinary schemes $\mathbb G_0\to \operatorname{Spec} \pi_0A$ is smooth of relative dimnension $1$.
- For all $n,$ the composition $$\pi_nA\otimes_{\pi_0A}\omega\xrightarrow{\operatorname{id}\otimes\beta}\pi_nA\otimes_{\pi_0A}\pi_2A\to\pi_{n+2}A,$$ where the unlabeled arrow is the multiplication in the graded ring $\pi_*A$, is an isomorphism.

Lurie seems to assert in "Survey" that they are equivalent, or at the very least that B) implies A). I would be very happy if somebody could explain why that is.

**Remark:**
A surely relevant fact is that for the classical formal group $\mathbb G_0 = \operatorname{Spf}\pi_0\left(A^{\mathbf {CP}^\infty}\right)=\operatorname{Spf}A^0(\mathbf {CP}^\infty)$, the $n$-th tensor power of its module of invariant differentials $\omega^n$ is isomorphic to $\pi_{2n}A$, as proved e.g. in Rezk's notes. But I don't see how this shows that B) $\Rightarrow$ A).

Namely, I feel like this should be saying something about the $\operatorname{Spf}A^{\mathbf{CP}^\infty}$ and its module (spectrum) of invariant differentials, but all I see is a statement about (tensor powers of) the module of invariant differentials of its underlying classical counterpart.

Any help will be warmly appreciated!

I have read a bit about the torsion of an acyclic complex. One of my concrete hopes was that I could understand why $L(7,1)$ and $L(7,2)$ are not homeomorphic - I am under the impression that classifying lens spaces was I problem that motivated Reidemeister to introduce torsion.

All of the definitions of torsion that I have seen are totally opaque to me. How do people think of the torsion of a chain complex and how in trying to classify lens spaces could I have been led to defining/computing torsions?

In these days I am studying some properties of Kan extensions. In order to do so I am looking at some general observations on Kan extensions that I would like to share with the community. A feedback would be very important to me.

Give a look at the following diagram to set the notation.

$\require{AMScd}$ \begin{CD} A @>f>> C\\ @VgVV \\ B \end{CD}

I will suppose that $g$ is fully faithful, that $C$ is cocomplete and that the kan extension is pointwise.

My interest is to understand how properties of $f$ interact with properties of Lan$_g f.$

To do so, I would start by giving a factorization of the Kan extension in the following way:

$$B \stackrel{\text{Elts}^g}{\to} \text{Fib}(A) \stackrel{f_*}{\to} \text{Fib}(C) \stackrel{\text{colim}}{\to} C.$$

Let's give a description of these functors.

Elts$^g$: $B \to \text{Fib}(A)$

By e $\text{Fib}(A)$ we mean the category of fibrations over $A$. This functor takes an object $b \in B$ into the category Elts($B(g\_, b)$).

The functor $f^*$ is just composition. So we map the fibration $E \to A$ to $E \to A \stackrel{f}{\to} C.$

Colim is just the functor that takes the colimit of the diagram induced by the fibration.

Under these notations it looks to me (please confirm, I am following the construction presented by Borceux (3.7.4, HoCA vol II)) that one can writhe down the equation: $$\text{Lan}_gf = \text{colim} \circ f_* \circ \text{Elts}^g. $$

Here come my questions:

What do we know about the functors **Elts**$^g$ and **colim**?
More precisely,

**Q1** Are they faithful or conservative?

**Q2** What (co)limits do they preserve?

As a final remark, I would point out that any reference about Kan extensions which goes deeply in their properties and their behave is absolutely welcome.

**Motivations:**

My motivating questions are the following ones:

In the setting of the question, under what assumptions can I hope that $f$ conservative implies that Lan$_g f$ is conservative?

What about faithfulness?

Again, any reference for an answer to these questions is absolutely welcome.

These questions are not so desperate ad they may appear. If $B = \text{Set}^{A^{\text{op}}}$ and $g$ is the yoneda embedding, then the functor Elts$^y$ should be faithful and conservative because $$\text{PseudoPres}(A) \cong \text{Fib}(A) $$ precisely under that functor.

Let $X$ be an algebraic variety (separated quasi-compact scheme of finite type) over a field $k$.

One of the possible definitions of an ample line bundle goes as follows:

**Def 1:** A line bundle $\mathcal{L}$ on $X$ is said to be **ample** iff some tensor power of it $\mathcal{L}^{\otimes k}$ admits $n+1$-generating sections (for some $n$) s.t. the associated morphism $X \to \mathbb{P}^n$ is a closed embedding (then $\mathcal{L}^{\otimes k}$ is said to be **very ample**).

I always found this definition rather subtle and mysterious. The classical story goes through showing that this definition is equivalent to the following one (which is manifestly much more useful in practice and much less easy to check):

**Def 2:** A line bundle $\mathcal{L}$ on $X$ is said to be **ample** if for every coherent sheaf $\mathcal{F}$ there exists some $n>0$ (depending on $\mathcal{F}$) such that for all $m>n$ the sheaf $\mathcal{L}^{\otimes m} \otimes_{\mathcal{O}_X} \mathcal{F}$ is generated by global sections (i.e. is a quotient of a trivial vector bundle).

The proof I know of this equivalence is subtle and goes through a reduction argument to the projective case and using serre vanishing (notice that may be why the relation between $k$ in the first definition and $n$('s) in the second is highly indirect).

Let $QCoh(X)$ denote the derived (stable $\infty$-)category of sheaves of quasi-coherent $\mathcal{O}_X$-modules with symmetric monoidal structure given by $\otimes_{\mathcal{O}_X}$.

Given this structure we can easily to detect (shifted) line bundles inside $QCoh(X)$ as those are given by the $\otimes$-invertible objects. Here's the question:

**Questions:** Given a (shifted-)line bundle $\mathcal{L}$ in $QCoh(X)$ can we... (increasing level of difficulty).

Detect whether $\mathcal{L}$ is

**ample**(in the classical sense above)**without "leaving" the derived category**$QCoh(X)$? (using $\otimes$-structure).Detect whether $\mathcal{L}$ is

**ample**by considering $QCoh(X)$ without the $\otimes$-structure, but**remembering the action of $Pic(X)$**(the $\infty$-picard groupoid of line bundles) on it.Detect whether $\mathcal{L}$ is

**very ample**without using the $\otimes$-structure at all?. ֿ

Please help with my past year paper question, answer not given to us. The question tell us that it is aritmetic, however it says the three numbers are in geo progession. I do not understand the question. Kindly help to solve the quesiton. Thank you

The sum of three number in an arithmetic progession is 24. If the first term is decreased by 1 and the second term is decreased by 2, the three numbers are in a geometric progression.

a) Let a be the first term and d be the common difference of the arithmetic progession. Use this information to write down the second and third terms in terms of a and d.

b) Solve for the values of a and d, by first formulating two equations.

c) List down the values of the three number.

Let $F$ be the set of mappings $f: (\mathbb{R}^n,d) \rightarrow (\mathbb{R}^n,d)$ of the real metric space to itself.

Considering the two connected components $SO(n)$ and $O_{\_}(n)$ of the orthogonal group $O(n)$, what possible mappings $T(f): F \rightarrow \mathbb{R}$ are there, such that the restrictions $T|_{SO(n)}(f) = 1$ and $T|_{O_{\_}(n)}(f) = -1$ ?

It is clear that on $O(n)$, uniquely $T|_{O(n)} (f) = \det f$.

What other possibilities are there for $T$ in the general case? Is $T$ is unique on $F$?

Let us say a group $(G,\cdot)$ has *character $0$* if for all $g\in G\setminus\{e\}$ and for all positive integers $n$ we have $g^n \neq e$ (where $e$ is the neutral element of the group.

Is there a collection ${\cal G}$ of countably infinite groups, such that

- each $G\in {\cal G}$ has character $0$,
- if $G_1\neq G_2\in{\cal G}$ then $G_1\not\cong G_2$, and
- $|{\cal G}| = 2^{\aleph_0}$?

?

The space $A$ is called homotopy dominated by the space $X$ if there are maps $f:A\longrightarrow X$ and $g:X\longrightarrow A$ so that $g\circ f\simeq id_A$.

Question: If $A$ is homotopy dominated by $X_1\vee X_2$, then is $A$ of the form $A_1 \vee A_2$ (up to homotopy equivalent) where $A_i$ is homotopy dominated by $X_i$ for $i=1,2$?

Let $\Gamma^{op}$ be the category of finite pointed sets. A special $\Gamma$-category is a functor $Y:\Gamma^{op}\to Cat$ such that the canonical maps $Y[n]\to Y[1]^n$ are equivalences of categories, for all $n\geq 0$.

There is a canonical way of associating to an unbiased symmetric monoidal category a special $\Gamma$-category (called a “homotopy monoidal category” by Tom Leinster in Higher Operads, Higher Categories).

I think that Moritz Groth, in his Example 3.4 of his course in infinity categories, is saying that this association is an equivalence.

However, Tom Leinster, pages 120-121 of the aforementioned book, says that “there is reasonable hope” that this is true (via unbiased symmetric monoidal categories).

I would like to know whether this result is true or not, and where do the delicate points lie. If it is true, I what are some references where the result is proven in detail?