Let $H$ denote the irreducible component of $\text{Hilb}^{3t+1}\mathbb{P}^3$ whose general member corresponds to a non-singular twisted cubic. Let $C$ be a subscheme lying in the boundary of $H$ and assume it lies in a surface $S \subseteq \mathbb{P}^3$.

Then why is it possible that we can find families $C_R, S_R \subseteq \mathbb{P}^3_R$ over a DVR $R$ with fraction field $K$ such that

1) $C_R \subseteq S_R$

2) The generic fiber $C_K$ is a non-singular twisted cubic

3) $C \subseteq S$ are the closed fibers of the family.

The authors in "Hilbert Scheme Compactification of the Space of Twisted Cubics": https://www.uio.no/studier/emner/matnat/math/MAT4230/h10/undervisningsmateriale/Hilbertscheme.pdf make the claim on page 4 (pg 763), line 7 of the proof. Although they are studying embedded points, this claim seems to be something more general about flat limits?

More generally, is it true that if something on the boundary of my component in a Hilbert scheme lied in a hypersurface, then I could find a family over a DVR like above?

Pick a random pair $(a,b)\in\mathbb Z_n^2\backslash\{0,0\}$. Denote $N_r(a,b)$ to be minimum $\ell_r$ norm of vector $(x',y')$ as $(x',y')$ ranges over all non-zero integral solutions to $(x',y')\equiv t(a,b)\bmod n$ where $t\in\mathbb Z$.

Then Akshay Venkatesh's paper 'Spectral theory of automorphic forms, a very brief introduction' published in ' Equidistribution in Number Theory, An Introduction edited by Andrew Granville, Zeév Rudnick' says it is true as $n\rightarrow\infty$ the distribution of $N_2(a,b)/\sqrt{n}$ coincides with distribution of $1/\sqrt y$ where $x+iy$ is picked at random with respect to hyperbolic measure from $\{z:|z|\geq1,|\mathcal R(z)|\leq\frac12\}$ when $n$ is prime.

From Probability density from standard domain we know the density of $v=1/\sqrt y$ is \begin{equation} h(v):= \int_{-\infty}^\infty g(u,v)\,du= \begin{cases} 6v/\pi &\text{ if } 0<v<1, \\ 6v\left(1-2 \sqrt{1-v^{-4}}\right)/\pi &\text{ if } 1\leq v<\sqrt[4]{4/3}, \\ 0&\text{ if }v\ge \sqrt[4]{4/3}. \end{cases} \end{equation}

I am looking for the analogous result for $N_2(a_1,a_2,\dots,a_{m-1},a_m)$.

- What is the distribution of $N_2(a_1,a_2,\dots,a_{m-1},a_m)/n^{(m-1)/m}$?

From above computations $N_2(a,b)/\sqrt{n}\asymp n^{-\epsilon}$ at given sequence $(a,b)$ happens with probability that scales as $\frac6{\pi n^{\epsilon}}\asymp 1/O(n^{\epsilon})$.

*Following is the most important question I need an answer. This alone will get the bounty*.

- Does $N_2(a_1,a_2,\dots,a_{m-1},a_m)/n^{(m-1)/m}<n^{-\epsilon}$ hold with probability that scales at most as much as $1/O(n^{\epsilon})$ or at least at most as much as $1/O(n^{poly(m)\epsilon})$ when $n$ is semiprime?

I was working on some mathematics of Wasserstein GAN and found out a seemingly interesting research problem but *I am not quite sure whether it has already been studied in some recent literature of Optimal Transport Theory* (As far as I know, it hardly is)

The observation is as follows (which has been validated with some toy simulations).

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**Problem (not rigorously stated)**

Let $\mu, \nu$ probabilistic measures on *regular* manifolds $\mathcal{M}, \mathcal{N}$, $C^{\infty}(\mathcal{M}, \mathcal{N})$ the set of continuous mapping from $\mathcal{M}$ to $\mathcal{N}$, and $\Pi(\mu,\nu)$ the set of measures on $\mathcal{M}\times\mathcal{N}$ s.t. its marginal distributions are respectively $\mu, \nu$.

Consider the following optimization problem

$$ (*) = \min_{T\in{C^{\infty}(\mathcal{M}, \mathcal{N})}} \inf_{\gamma\in\Pi(\mu,\nu)} \int d^{2}(T(p), q) d\gamma(p,q) $$ where the cost function can be considered as the $L_2$ distance on $\mathcal{N}$'s total space as a real vector space.

My question is that whether $(*) \propto h(\chi(\mathcal{M}), \chi(\mathcal{N}))$ where $h$ is certain metric function and $\chi(\cdot)$ denotes *Euler characteristic*. **Generally, would it be possible that the minimum cost of the Wasserstein game is deeply related with the difference between some topological invariants of underlying manifolds'?**

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Look forward to any feedbacks and welcome discussions and potential references :D. I am willing to provide details of my toy experiments if one is interested in this problem.

It is well-known that any positive rational number can be written as the sum of finitely many distinct unit fractions. This is easy since $$\frac1n=\frac1{n+1}+\frac1{n(n+1)}\quad\text{for all}\ n=1,2,3,\ldots.$$ In 2015 I thought that this easy fact should have a further refinement which is somewhat sophisticated. Note that the series $\sum_p\frac1{p-1}$ and $\sum_p\frac1{p+1}$ (with $p$ prime) diverge just like the harmonic series $\sum_{n=1}^\infty\frac1n$. Also, for any positive integer m there are infinitely many primes $p$ congruent to $1$ (or $-1$) modulo $m$ (by Dirichlet's theorem). Motivated by this, I made the following conjecture in Sept. 2015.

**Conjecture**. For any rational number $r>0$, there are finite sets $P_r^-$ and $P_r^+$ of primes such that
$$r=\sum_{p\in P_r^-}\frac1{p-1}=\sum_{p\in P_r^+}\frac1{p+1}.$$

This appeared as Conjecture 4.1 of this published paper of mine. For example, $$2=\frac1{2-1}+\frac1{3-1}+\frac1{5-1}+\frac1{7-1}+\frac1{13-1}$$ with $2,3,5,7,13$ all prime, and $$1=\frac1{2+1}+\frac1{3+1}+\frac1{5+1}+\frac1{7+1}+\frac1{11+1}+\frac1{23+1}$$ with $2,3,5,7,11,23$ all prime. Also, \begin{align*}\frac{10}{11}=&\frac1{3-1}+\frac1{5-1}+\frac1{13-1}+\frac1{19-1}+\frac1{67-1}+\frac1{199-1} \\=&\frac1{2+1}+\frac1{3+1}+\frac1{5+1}+\frac1{7+1}+\frac1{43+1}+\frac1{131+1}+\frac1{263+1} \end{align*} with $2,3,5,7,13,19,43,67,131,199,263$ all prime. The reader may see more numerical data in my detailed introduction to this conjecture.

After learning this conjecture from me, Prof. Qing-Hu Hou and Guo-Niu Han checked my above conjecture seriously and their computational results support my conjecture. For example, in 2018 Prof. Han found 2065 distinct primes $p_1<\ldots<p_{2065}$ with $p_{2065}\approx 4.7\times10^{218}$ such that $$\frac1{p_1+1}+\ldots+\frac1{p_{2065}+1}=2.$$

My question is whether the above conjecture is true. I would like to offer 500 US dollars as the prize for the first correct solution.

**Remark**. Let $r$ be any positive rational number, and let $\varepsilon\in\{\pm1\}$. As the series $\sum_p\frac1{p+\varepsilon}$ (with $p$ prime) diverges, there is a unique prime $q$ such that $$\sum_{p<q}\frac{1}{p+\varepsilon}\le r<\sum_{p\le q}\frac1{p+\varepsilon}.$$ Thus
$$0\le r_0:=r-\sum_{p<q}\frac1{p+\varepsilon}<\frac1{q+\varepsilon}\le1.$$
If $r_0=\sum_{j=1}^k\frac1{p_j+\varepsilon}$ with $p_1,\ldots,p_k$ distinct primes, then $p_1,\ldots,p_k$ are all greater than $q$, and
$$r=\sum_{p<q}\frac1{p+\varepsilon}+\sum_{j=1}^k\frac1{p_j+\varepsilon}.$$ Therefore it suffices to consider the conjecture only for $r<1$.

Take classic problem of finding $P,Q$ in balanced semi-prime $N=PQ$. Is there evidence that no extension of Coppersmith technique will accomplish factoring $N=PQ$ in polynomial time?

Technically I am asking possibility of no possible way to reduce factoring to finding integer roots of polynomials. Is there contradiction in literature that supports possibility that if you can solve polynomial equations within bounds on the unknowns that are large enough to allow factorization, then you should also be able to solve polynomial equations with exponentially many solutions, which cannot be done in polynomial time?

This seems to imply no such evidence exists Reduction from factoring to solving Pell equation. However I do not know surely.

Is there any other such reduction known? Is there literature references?

Recently I was playing several rounds of the game of pairs with my children. I was surprised that almost every time, one matching pair was adjacent (either next to each other in a row, or vertically). This led to the following question.

Let $n$ be a positive integer. Consider the set $$C_n = \{1,\ldots, 2n^2\}\times\{0,1\}.$$ We say that $(k,0)$ and $(k,1)$ for $k\in \{1,\ldots,2n^2\}$ is a *matching pair* in $C_n$.

Let $G_n = \{1,\ldots, 2n\} \times \{1,\ldots,2n\}\subseteq \mathbb{R}^2$ be thought of the "gaming grid" where we place the numbers. Formally: shuffling and distributing the cards corresponds to a bijection $\varphi: C_n \to G_n$. So we can define the *minimum distance of a matching pair* by $$M_n = \min\big\{||\varphi(k,0), \varphi(k,1)||: k\in \{1,\ldots,2n^2\}\big\}.$$

By $||\cdot||$ we denote the Euclidean distance.

$M_n$ is a random variable, so we can calculate its expected value $E(M_n)$.

**Question.** In plain English: if we play the game of pairs on an ever growing quadratic play-field, can we expect to find some matching pair in a reasonably close distance? Or more formally: Do we have $\lim_{n\to\infty} E(M_n) < \infty$?

Now suppose that a power series $p(x)$ has same radius of convergence ($=\rho$) both in $ \mathbb{Q}_p$ and $ \mathbb{R}$ with respect to $p$-adic absolute value $\lvert\cdot\rvert_p$ and usual absolute value $\lvert\cdot\rvert$ on $ \mathbb{Q}$.

I.e., the power series $p(x)$ converges in $ |x|_p < \rho \ $ in $\mathbb{Q}_p$ and $|x|< \rho$ in $\mathbb{R}$.

What are the rationals satisfying $|x|< \rho$ and $ |x|_p< \rho$?

Let $G$ be the free pro-p group with $n$ generators; we can assume $n=2$ and generators are $x,y$ at first. Let $G_0=G$ and $G_{n+1}=[G,G_n]$ be the group generated by certain commutators. Then the quotient group $\Delta_n=G/G_n$ is a nilpotent group of class $n$.

Now I want to know more about the structure of the groups $\Delta_n$. For example, $\Delta_1=\mathbb{Z}_px\bigoplus\mathbb{Z}_py$ and $\Delta_2$ is isomorphic to the group of upper triangular matrix with diagonal equal to 1 and coefficients belonging to $\mathbb{Z}_p$.

Moreover, I guess that $\Delta_{n+1}/\Delta_n$ (it is abelian) can be viewed as a free $\mathbb{Z}_p$-module, am I right? (If so, then the structure of the associated graded Lie algebra is known, c.f. Serre "Lie Algebras And Lie Groups".)

Any ideas or references will be welcome. Thanks!

Consider function $f: (\mathbb{R}^{d})^{n} \rightarrow \mathbb{R}$ with spatial invariance property of the form : $f(x_1,x_2,...,x_n) = f(x_1 + \zeta, x_2 + \zeta,..., x_n + \zeta)$ for $\zeta \in \mathbb{R}^{d} $.

I want to show that $\int \limits_{(\mathbb{R}^{d})^{n}} \exp(\,f(x_1,...,x_n) \,) dx_1..dx_n \,=\, \infty $

I would like some help if possible.

My only idea is to fix ball $B(0,r) \subseteq \mathbb{R}^{d}$ and $\zeta \in \mathbb{R}^{d}$ and then compute $\int \limits_{{B(0,r)}^{n}} \exp(\,f(x_1,...,x_n) \,) dx_1..dx_n \, = \int \limits_{{B(\zeta,r)}^{n}} \exp(\,f(y_1 - \zeta,...,y_n - \zeta) \,) dy_1..dy_n = \int \limits_{{B(\zeta,r)}^{n}} \exp(\,f(y_1,...,y_n) \,) dy_1..dy_n \, $ where the first equality comes from setting $x_i = y_i - \zeta $ and the second from the spatial invariance property.

So we get that the integral of this strictly positive f over any ball is the same. But does it imply that the integral over $(\mathbb{R}^{d})^{n}$ is $\infty$ ?

I'm trying to solve below probability question ;

fuel station has $5$ filling points of which $3$ are use to fill petrol and remaining $2$ are use to fill diesel for different kind of vehicles.It was noticed that an average $3$ minutes to fill fuel to any vehicle (diesel/petrol).number of petrol vehicles to enter the fuel station within an hour estimated is $24$ and the diesel vehicle estimated 36 within an hour.

want to find below probabilities :

- all petrol filling points are idle
- all diesel filling points are busy
- all petrol filling points are busy
- all filling points are busy

Let $G$ be a torsion free group. Let $\alpha$ be an element in $\mathbb CG$, the group algebra of $G$, with $\|\alpha\|_1=1$ and assume that

- $\{1,\alpha,\alpha^2,\dotsc\}$ is linearly independent,
- $(\alpha^n)_{n\in\mathbb N}$ converges to 0 in strong operator topology, so in particular $\lim_n\|\alpha^n\|_2=0$.

Assume that $K$ is the closed linear span of $\{1,\alpha,\alpha^2,\dotsc\}$ in $\ell^2(G)$. Is $\{1,\alpha,\alpha^2,\dotsc\}$ a Schauder basis for $K$?

Let's suppose we have an objective function $\max_\limits{x} \sum_\limits{i} f_i(x_i)$ with the constraint that $\ x_i \geq 0, \sum_\limits{i} x_i = 1$.

Each function $f_i$ is continuous and differentiable within the boundary, with the following properties: $f_i(x_i) \geq 0$, $f_i(0) = 0$, $f'_i(x) > 0$, $f'_i(1) > 0$, $f_i''(x_i) < 0$.

By adding a Lagrange term and taking the partial derivative, we get

$\frac{\partial \sum_i f_i(x_i) + \lambda (1 - \sum_i x_i)}{\partial x_i} = f'_i(x_i) - \lambda$.

By setting the above expression to zero, moving $\lambda$ to the right side of the equation and multiplying an $x_i$ with each side, we would come up with a fixed-point update $x_i^\mbox{new} \propto x_i \cdot f'_i(x_i)$.

Would this iterative algorithm monotonically increase the concave objective and is it guaranteed to converge?

I asked this question at MSE now I repeat it at MO:

Let $A,B,C$ be $3$ unital $C^*$ algebras. Assume that we have the following short exact sequence of $C^*$-algebras:

$$0\to A\to C\to B\to 0$$

Assume that $A,B$ are generated by their projections. Is $C$ necessarily generated by its projections, too?

Assume that $A,B$ are von Neumann algebras, is $C$ necessarily a von Neumann algebra, too?

Does the last question has an obvious answer when $A,B$ (hence $C$) are commutative algebras?

Let $k$ be a field of characteristic $0$, and let $\mathcal{C}= \mathbf{Vect}_k^{\leq 0}$ be the $\infty$-category of vector spaces concentrated in degrees $\leq 0$. Consider the category $\mathbf{Pr}(\mathcal{C}):= \operatorname{Fun}(\mathbf{CAlg}(\mathcal{C}), \mathcal{S})$ of prestacks over $k$, where $\mathcal{S}$ is the $\infty$-category of spaces or $\infty$-groupoids.

Suppose we have a grouplike prestack $G \in \mathbf{Pr}(\mathcal{C})$. That is, a functor $G: \mathbf{CAlg}(\mathcal{C}) \to \mathbf{Sp}^{\text{cn}}$, where $\mathbf{Sp}^{\text{cn}}$ is the $\infty$-category of connective spectra, thought of as a functor to spaces by composing with the forgetful functor $\mathbf{Sp}^{\text{cn}} \to \mathcal{S}$. We can then form the iterated classifying spaces $B^nG$.

Suppose we have a nice enough stack $X \in \mathbf{Pr}(\mathcal{C})$ (e.g. a perfect stack). When will the category $\mathbf{QCoh}(\text{Map}(X,B^nG))$ of quasicoherent sheaves on the mapping stack be compactly generated? Is the assumption that $X$ be perfect enough? Do we have to make any assumptions on $G$?

Sage 50 Error solution Ireland 353-766-803-988. The main cause of this error 1628 is:-

Corrupt download file of sage. An invalid process of installation. Incomplete installation of sage in the system. Windows may be corrupt due to frequent installation. When the .NET framework is expired. Missing company files corrupt software. Some programs are delete Virus and malware corrupt the file of sage Let Us discuss how to Troubleshoot the Error as follows:-

SOLUTION 1:-

Start system [Windows, Mac, iPhone, iPad]. Insert the installation CD. Install the .NET file in the system. Install the setup file. SOLUTION 2:-

Run Microsoft easy fix utility. SOLUTION 3:-

Repair all windows Run system scan > remove the virus, malware, and threats. Delete junk and temporary files Check notification > run updates. Undo recent changes. Uninstall sage 50 software Re-install accounting software Run windows system Clean system installation SOLUTION 4:-

Rename the InstallShield folder > My computer > Click InstallShield folder > Click Rename > Press Enter SOLUTION 5:-

Install latest window installer > Install the .exe file > Click install or download.

There is one sentence I don't understand in some paper.

"A simply connected and conformally flat three mainifold can be conformally immersed into $S^3$" by the means of a developing map.

Is any reference about this short argument? Maybe it is a direct consequence from definition. Could anyone explain a little bit to me?

**Question:** Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe just a lot harder than known proofs?

I recently decided I would like to attempt to learn a proof of the theorem that for a hyperbolic group $G$ with $\partial G \cong S^1$ implies that the group is virtually Fuchsian. From what I understand the proof goes through showing that convergence group acting on $S^1$ conjugate to Fuchsian groups, and then show that hyperbolic group act on their boundary as convergence groups.

Something else I would like to learn more about is applications of bounded cohomology. It is my understanding that it is useful in determining things like conjugacy of representations into $\mathrm{Homeo}^+(S^1)$. So, I would guess (perhaps naively) that it would be useful or an alternative approach to the above question. If there was it would be a cool application...

I am having a hard time finding information on this (and can't find a copy of *Groups acting on the circle* by Ghys for some reason which is where I would think it would be discussed if it was a plausible approach) so I am guessing there isn't much connections, but maybe someone has thought about this before or is an expert who can see why bounded cohomology would not be useful.

I'm wondering if the following argument is correct:

Consider optimizing a complex functional S[x(t)]. Since S is complex, it only has an optimum with respect to the lexicographic order of the complex numbers. (This is different from Cauchy-Riemann stability, where δS = 0, but there is no notion of optimality).

To optimize with respect to the lexicographic order of the complex numbers, we first optimize Real(S)

0 = δ(Real S)

Next, we optimize imaginary(S) while keeping δ(Real S) = 0.

This is the part I'm unsure of: The Lagrange multiplier method is not suitable here, and in general, we cannot say that optimizing Imaginary S will also stabilize it. However, we can say that the difference between stabilizing Imaginary S and optimizing it is unpredictable noise, which is 0 on average.

δ(Imaginary S) = noise

Adding these together, gives δS = i noise.

Is my argument that δ(Imaginary S) = noise correct? Furthermore, what can we say about this noise?

If $a_n$ satisfies the linear recurrence relation $a_n = \sum_{i=1}^k c_i a_{n-i}$ for some constants $c_i$, then is there an easy way to find a linear recurrence relation for $b_n = a_n^2$ ?

For example, if $a_n = a_{n-1} + a_{n-3}$, then $b_n=a_n^2$ seems to satisfy $b_n=b_{n-1}+b_{n-2}+3b_{n-3}+b_{n-4}-b_{n-5}-b_{n-6}$.

I remember reading somewhere that the complement of a meagre set in a Baire space is also a Baire space and this is in fact easy to prove. Looking for this result in the standard collection of Topology books I could not find it. Can anyone help me locate a reference?