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!

Are there algorithms for finding $\omega_1\dots,\omega_n$, so that

$\omega_i+\omega_j\le \|e_{ij}\|\ \forall i,j\quad\wedge\quad\sum{\omega_i}=max$

that are *not* based on linear programing, e.g. graph theoretic algorithms?

$a_n = \sum\limits_{k=0}^n \frac{1}{k + 1} {{n + k} \choose {n - k}} {{2k} \choose {k}} $

Does anyone know how to find the generating function of this sequence

I am trying to find the most efficient way to select points on a 2d plane from a set that maximizes the area of the of the shape they define when joined together.

The points are all paired (sharing the same A->B vector), with these pairs also appearing mirrored about the origin. Here is an example:

Sometimes a single point is used from the pair, and sometimes both.

For any given problem I can easily draw the solution by hand, but I am struggling to formulate an efficient solution to program. I can loop through and for each point, check adjacent points, using the solution for which no other point in the set is further from the origin than the joining line, but it seems messy. I feel like there should be a more elegant solution.

Does anyone have any ideas?

Suppose I have two random variables $X$ and $Y$, sampled from two independent probability distributions. From the simulations, I observe $$ \mathbb{E}[X] \rightarrow \mu_{X} \text{ and } \mathbb{E}[Y] \rightarrow \mu_{Y} $$ with $\vert \mu_{X} - \mu_{Y}\vert = \mathcal{O}(1)$. But both $\mathbb{E}[X^{2}]$ and $\mathbb{E}[Y^{2}]$ are divergent in the sense that $$ a_{X} < \mathbb{E}[X^{2}] < b_{X} \\ a_{Y} < \mathbb{E}[Y^{2}] < b_{Y} $$ fluctuating between some ranges, i.e. neither is converging to some value, like the expectations do. ($a$'s and $b$'s are simply constants and are not equal) All the above are obtained from the law of large number sense, i.e., $$ \frac{\sum_{i=1}^{n}X_i}{n} \rightarrow \mu \text{ in prob} $$ More specifically, I run the problem for $2000$ times and get the mean. Similar procedure applies to the second moment (and thus the variance)

From the knowledge of convergence in distribution, I know that if we want such convergence, it is equivalent to prove the the expectations are equal. So I am wondering if I can draw the conclusion that $$ F_{X} \overset{\mathcal{D}}{\rightarrow} F_{Y} $$

The sections of the holomorphic line bundle $\mathcal{O}(n)$ are acted on by the covariant derivative $$ d+nA, $$ where $A$ is the connection on the $U(1)$ bundle to which $\mathcal{O}(n)$ is associated.

My question is, are there holomorphic line bundles associated to $U(1)^M$ bundles, where $M$ is some integer? The sections of such a holomorphic line bundle would be acted on by the covariant derivative

$$ d+\sum_b^M n_b A_b, $$ where $n_b$ denotes the representation w.r.t. each $U(1)$ factor. It would be natural to denote such a line bundle as $\mathcal{O}(n_1,n_2,\ldots,n_b)$. Do such holomorphic line bundles exist in the mathematical literature? References would be highly appreciated.

**Edit:** To be more precise, I am interested in the holomorphic line bundles described in the previous paragraph defined over toric manifolds $X=(\mathbb{C}^N-\mathcal{P})/({\mathbb{C}^{\times}})^k$, especially the case where $k>1$.

(Here $\mathcal{P}$ denotes a subset of $\mathbb{C}^N$ fixed under a continuous subgroup of $({\mathbb{C}^{\times}})^k$.)

In particular, I would like to understand the generalization of the line bundles over $\mathbb{P}^1 \times \mathbb{P}^1$ which are denoted as $\mathcal{O}(m,n)$, mentioned in Line bundles and vector bundles on $\mathbb P^1 \times \mathbb P^1$. There, it is explained in the comments that $\mathcal{O}(m,n)= p_1^*\mathcal{O}(m)\otimes p_2^*\mathcal{O}(n)$, where $p_1$ and $p_2$ denote the projections onto the two factors.

Recall that a space $X$ is called homotopy dominated by a space $Y$ if there are maps $f:X\longrightarrow Y$ and $g:Y\longrightarrow X$ so that $g\circ f\simeq id_X$.

If $X$ is homotopy dominated by $Y$, then is $X$ homotopy equivalent to $f(X)$?

Let $(p_1, p_2)$ be a twin prime pair, where we include $(2, 3)$. If $p_1 \equiv 3$ mod $4$ then we let $t_{(p_1, p_2)} := p_2 ^ 2 / p_1 ^ 2$ otherwise, we let $t_{(p_1, p_2)} := p_1 ^ 2 / p_2 ^ 2$. Also define $t_{(2,3)}:=3 ^ 2/2 ^ 2$.

I conjecture that the product $$ \prod_{(p_1, p_2): \text{twin primes}}t_{(p_1, p_2)} =\tfrac{3 ^ 2}{2 ^ 2} \cdot \tfrac{5 ^ 2}{3 ^ 2}\cdot \tfrac{5 ^ 2}{7 ^ 2}\cdot\tfrac{13 ^ 2}{11 ^ 2} \cdot\tfrac{17 ^ 2}{19 ^ 2} \cdot\tfrac{29 ^ 2}{31 ^ 2} \cdot\tfrac{41 ^ 2}{43 ^ 2} \cdot \tfrac{61 ^ 2 }{59 ^ 2} \cdot \tfrac{73 ^ 2}{ 71 ^ 2}\cdot \tfrac{101 ^ 2 }{ 103 ^ 2}\cdots $$

is equal to $\pi$. (If this is true then twin prime numbers are infinity many.)

Some numerical values of partial products:

3.1887755102040816321 to $10^1$,

3.2055606708805624550 to $10^2$,

3.1290622219773513145 to $10^3$,

3.1364540609918890779 to $10^4$,

3.1384537326021492746 to $10^5$,

3.1417076006640026373 to $10^6$,

3.1417823471756806475 to $10^7$,

3.1415377533170544536 to $10^8$,

3.1415215264211035597 to $10^9$,

3.1415248453830039795 to $10^{10}$,

3.1415126339547108140 to $10^{11}$,

3.1415144504088659201 to $10^{12}$,

3.1415142045284687040 to $10^{13}$,

3.1415144719058962626 to $10^{14}$,

3.1415384423175311229 to $10^{15}$

Can we find a few more decimal places using the extrapolation method?

I am reading this paper about "Numerical approximation of the logarithmic capacity of domains", and there (on the third page) I found simple formulas for logarithmic capacity of simple figures like squares and equilateral triangles.

For a better understanding, I tried to recalculate those by myself several times, but I could not succeed. Is there a way that I can find those calculations or something similar that I can use as a guide?

Is there any irrational number that is known the probability distribution of digits?

Something like 0 appears 10% of time, 1 appears 10% of time, etc.

Probably irrational numbers that are defined by a construction on digits like:

1234567891011121314....

You can prove digits distribution but I am interested in numbers that is not defined this way like pi, e, or square root of prime.

Is there any advance on this topic?

Assume we have a set of equations in $x \in \mathbb{R}^n$

$$|a_i\cdot x|=b_i$$

where $a_i \in \mathbb{R}^n$ and $b_i>0$ are given. Could such a system be solved efficiently?

- In a theoretical machine storing reals with perfect accuracy.
- An approximate solution taking rounding errors into account.

This is equivalent to having $2$ possible values for each linear combination, which is a valid question over any field. Does this equivalent problem over a fixed finite field have an efficient solution? Or is it perhaps known to be NP-complete?

So far, I concluded that squaring both sides of the equation we get linear equations in $y_{ij} = x_i x_j$. However, this helps only if the system is quadratically overdetermined.

**Question:** How to find **the smallest** value $x$ satisfying the equation: $x^2 = a \pmod c$ (known is $a$ and $c$, $c$ is not the prime)?

Using the Tonelli-Shanks algorithm and the Chinese remainder theorem does not always give me the smallest $x$ satisfying condition.

Is there any solution for calculating the smallest $x$?

Does anyone have an idea?

**---- Edit:**

I will describe more accurately my problem:

Let's assume that we are looking for a solution: $x^2 \equiv 1024 \pmod{1302}$.

We need to know the distribution of the factors $1302$, so $1302 = 2 \cdot 3 \cdot 7 \cdot 31$.

Now, using the Tonelli-Shanks algorithm we calculate for all divisors:

**For 2:**

$1024 \equiv k_1 \pmod 2$

$k_1 = 0$

$x_1 ^ 2 \equiv k_1 \pmod{2}$

$x_1^2 \equiv 0 \pmod{2}$

$x_1 = 0$

**For 3:**

$1024 \equiv k_2 \pmod 3$

$k_2 = 1$

$x_2^2 \equiv k_1 \pmod{3}$

$x_2^2 \equiv 1 \pmod{3}$

$x_2 = 1$

**For 7:**

$1024 \equiv k_3 \pmod 7$

$k_3 = 2$

$x_3 ^ 2 \equiv k_3 \pmod{7}$

$x_3^2 \equiv 2 \pmod {7}$

$x_3 = 4$

**For 31:**

$1024 \equiv k_4 \pmod 31$

$k_4 = 1$

$x_4 ^ 2 \equiv k_4 \pmod{31}$

$x_4 ^ 2 \equiv 1 \pmod{31}$

$x_4 = 1$

Then we solve the system of equations from the Chinese remainder theorem. We know the factors and also the values of $x$ from the formula: $x ^ 2 \equiv c \pmod {p}$ where $p$ and $c$ are known.

We solve the system of equations.

$x \equiv 0 \pmod{2}$

$x \equiv 1 \pmod{3}$

$x \equiv 4 \pmod{7}$

$x \equiv 1 \pmod{31}$

The Chinese remainder theorem comes out $x = 32$, and this is the good, smallest solution: $32 ^ 2 \equiv 1024 \pmod {1302}$ - quite trivial case.

The problem, however, is that it does not always agree. And so I write why.

In the above case, for example, for the first factor $31$ I assumed that I found a result equal to $1$. I do not necessarily have to find exactly $1$ as well:

$(31-1)^2 \equiv k_4 \pmod{31}$

$(31-1)^2 \equiv 1 \pmod{31}$

$30^2 \equiv 1 \pmod{31}$

The above is that for $30$ will also be $1$.

For such a system of equations (new value at $31$):

$x \equiv 0 \pmod{2}$

$x \equiv 1 \pmod{3}$

$x \equiv 4 \pmod{7}$

$x \equiv 30 \pmod{31}$

From the Chinese remainder theorem we get $x = 2944$. This also agrees, because $2944 ^ 2 \equiv 1024 \pmod {1302}$ but this is no longer the samllest possible value (smallest possible is $x = 32$).

Knowing the first value ($1$ for factor = $31$), the second one ($30$ for factor = $31$) that fits is easy to calculate as I did here.

However, since all combinations of values will be $2 ^ k$ (where $k$ is the number of prime factors (in different example). I have not found a way to do some search for these combinations in a better way than brutal-force.

Any ideas for that so I'm looking for.

Can someone suggest something?

Let $G=(V,E)$ be a simple, undirected graph such that every vertex has degree at least $2$. Given $n\in\mathbb{N}$, a map $c:E \to \{1,\ldots, n\}$ is said to be a *weak coloring* if for every $v\in V$ the edges adjacent to $v$ do not all have the same color. (More formally, we want the restriction $c|_{E(v)}$ to be non-constant, where $E(v) = \{e\in E: v\in e\}$.)

These two nice posts by Mikail Tikhomirov and Brendan McKay respectively show that for every finite graph there is a weak edge coloring with $3$ colors. I tried to carry through their arguments with transfinite induction to infinite graphs - without success.

**Question.** If $G=(V,E)$ is an infinite simple undirected graph, is there a weak edge coloring $c:E \to \{1,2,3\}$?

I'm not exactly a differential geometer, so I hope this isn't too elementary a question.

From a naive point of view, it seems as if there are two natural group actions on the space of connections on the tangent bundle of a manifold $X$: the group of diffeomorphisms, which acts by pull-backs, and the group of gauge transformations, which acts by, well, gauge transformations.

Now, there isn't any obvious relationship between the two groups and, as far as I can tell, the gauge group is a much more natural and much more convenient object to study.

However, on the other hand, it is certainly possible that for given diffeomorphism $\phi: X \to X$ and a connection $A$, we have a gauge equivalence between $\phi^{*}(A)$ and $A$. I'm interested in understand when this occurs in general. In particular, for which diffeomorphisms is it true that $\phi^*(A)$ is gauge equivalent to $A$, for *every* connection $A$?

I tried attacking this, at least for diffeomorphisms isotopic to the identity, by viewing the isotopy as coming from a flow on $X$ and writing down a differential equation satisfied by a one-parameter family of gauge transformations inducing the same map on some fixed connection, but it didn't seem to lead anywhere.

Thanks.

Let $G$ be a connected Lie group and $\Sigma$ a closed oriented surface. We know that principal $G$-bundles $P$ can be topologically classified by a characteristic class $c(P)\in H^2(\Sigma,\pi_1G)\cong\pi_1G$.

The following is my question:

Let $G$ be a semisimple connected (or even compact) Lie group, $\beta\colon\pi_1(\Sigma)\to G$ a group homomorphism, and $\Sigma$ a closed oriented surface. Consider the universal cover $\tilde{\Sigma}\to\Sigma$ which is a principal $\pi_1(\Sigma)$-bundle. Form the associated bundle $\tilde{\Sigma}\times_\beta G$ which is necessarily a principal $G$-bundle over $\Sigma$. Hence, it should have a characteristic class $c(\tilde{\Sigma}\times_\beta G)\in H^2(\Sigma,\pi_1G)\cong\pi_1G$. Is there a way to compute this characteristic class in terms of $\beta$?

The difficulty here is that $\pi_1(\Sigma)$ is not connected, and hence I cannot use the functoriality of the characteristic class. Another one is that I am not sure how to compute the induced map $B\beta\colon B\pi_1(\Sigma)\to BG$ concretely.

I've asked this question on MSE and haven't got any answers yet.

Let $\mathcal{R}$ be the Robba ring and $\mathcal{E}^{\dagger}$ the elements of $\mathcal{R}$ that are bounded at 0 (so the coefficients of the powerseries are bounded. Is it true that $x, y\in \mathcal{R}$, $z\in (\mathcal{E}^\dagger)^*$ with $x\cdot y= z$ implies $x,y\in (\mathcal{E}^\dagger)^*$?

If not, is it then true that $x\in \mathcal{R}$, $ y, z\in (\mathcal{E}^\dagger)^*$ with $x\cdot y= z$ implies $x\in (\mathcal{E}^\dagger)^*$?

This kind of seems to be implied in some of the papers of Colmez about $(\varphi, \Gamma)$-modules, for example here https://webusers.imj-prg.fr/~pierre.colmez/triangulines in Proposition 4.2, but I might be mistaken and some other argument is implicitly used.

Let $\mathcal S'=\mathcal S'(\mathbb R^n)$ be the Schwartz distribution space. Suppose $A\colon\mathcal S'\to\mathcal S'$ is linear, continuous and microlocal. By being microlocal I mean that the wave front sets satisfy $WF(Af)\subset WF(f)$ for all $f$. (For another version, one could consider the singular supports instead of wave fronts, but I assume the answer wouldn't be different.) Does it follow that $A$ is a pseudo-differential operator?

This is a variation on this question about continuous endomorphisms on the Schwartz space. There one only assumed linearity and continuity, and the answer was negative. The other question was about $\mathcal S$ instead of $\mathcal S'$, but I need distributions to allow singularities.

I realize that there are many classes of pseudo-differential operators. The question is whether such a microlocal operator is always a ΨDO of some kind. Any details on what kind would be very interesting, of course.

**Edit:** According to the interesting comments of Michael Albanese and Nick L we revise the question as follows:

By manifold compactification of a manifold $M$ we mean a compact manifold $\tilde{M}$ which contains $M$ as an open dense subset.

Assume that $M$ is an open connected manifold which admits a manifold compactification. Does $M$ necessarily admit a manifold compactification with zero Euler characteristic?

Write $\mathbb{Z}^{a\times b}$ for the $a\times b$ integer matrices. Let $Q\in\mathbb{Z}^{n\times n}$. Let $G=O(Q)$ be the orthogonal group of $Q$. For $X_0\in \mathbb{Z}^{n\times 2}$, set $$ R'(T,Q,X_0) = \{ X\in X_0G(\mathbb{Z}) : |X_{ij}|\leq T\}. $$

**Dubious conjecture:** If $Q$ is indefinite then
$$
\#R'(T,Q,X_0) \ll_{Q,\epsilon}
T^{\max\{0,2n-6\}+\epsilon}
\tag{$\star$}
$$
for all $\epsilon>0$, with an implicit constant depending only on $Q$ and $\epsilon$.

**Question:** Is this true? How about simple (?) examples like $Q=I_3$ or $Q=I_4$?

Let $B\in\mathbb{Z}^{2\times 2}$ be nonsingular and symmetric. I am interested in the number of solutions $$ R(T,Q,B) = \{ X\in \mathbb{Z}^{n\times 2} : XQX^T =B, \, |X_{ij}|\leq T\} $$ with height up to $T$, when $Q$ is indefinite. A naive guess based on the circle method would be that $$ \#R(T,Q,B) \ll_Q 1+T^{2n-6} \tag{$\ast$} $$ with an implicit constant depending at most on $Q$. The set $R(T,Q,B)$ is the union of finitely many $G(\mathbb{Z})$-orbits $R'(T,Q,X_0)$, so the conjecture above is a weak version of this.

Note that ($\ast$) is true for large $n$. For instance when $Q$ is indefinite and $n\geq 12$ this (and much more) follows from the asymptotic result of Brandes. (The condition $n\geq 12$ follows from the comment at the end of section 1 in the subsequent paper.) On the other hand ($\ast$) is certainly false for small $n$, though I don't know of a good example in the indefinite case.

If $B$ is positive definite and $Q$ has signature $(p,q)$ then it is a special case of Theorem 1.4 of Eskin, Mozes and Shah that $$ \# R(T,Q,B) \sim C T^{\min\{2,q\}(n-1-\min\{2,q\})} $$ for some $C= C(Q,B)$, unless a certain volume is infinite. In the latter case, the comments after formula (1.4) of Duke, Rudnick and Sarnak suggest that the right-hand side of ($\ast$) just changes by a factor of $\log T$. Perhaps a proof of ($\star$) in this case is implicit in their work, but I have not been able to verify this.

can anybody here help me, please?I really fed up. I am a master student and I have a numerical analysis course and it's very difficult for me cause I have not had experience in this subject. whatever I read I just mix-up more. Thank you.