Let be $S$ a separable(non compact) metric space and $X=C_b(S)$ the set of all bounded continuous functions, then it's topological dual $X^{\star}=rba(S)$ is the set of all regular Borel additive measures endowed with the variation norm. Denote by $\mathscr{P}(S)$ the subset of $rba(S)$ of all additive probability measures, endowed with the weak$^{*}$ topology.

Fix $\mu$ a extremal point of $ \mathscr{P}(S)$ and a closed set $\mathcal{F}$ in $\mathscr{P}(S)$ such that the convex hull $co(\mathcal{F})$ do not contains $\mu,$ then it is possible to show that there is an affine functional $\ell_{\mu,\mathcal{F}}:\mathscr{P}(S)\to \mathbb{R}$ such that $\ell_{\mu,\mathcal{F}}(\mu)=0$ and $\ell_{\mu,\mathcal{F}}$ is strictly positive in $\mathcal{F}$

**Question 1:** Is there some way to use the above stated to get a global result, that is, I want to show that there is a affine functional $\ell_{\mu}:\mathscr{P}(S)\to \mathbb{R}$ such that $\ell_\mu(\mu)=0$ and $\ell_\mu(\nu)>0$ for all $\nu\in \mathscr{P}(S)\setminus \{\mu\}$ ?

**Question 2:** If not, is there some other approach to show that there is a affine functional $\ell_{\mu}:\mathscr{P}(S)\to \mathbb{R}$ such that $\ell_\mu(\mu)=0$ and $\ell_\mu(\nu)>0$ for all $\nu\in \mathscr{P}(S)\setminus \{\mu\}$ ?

**Edit:** Following the suggestions of Jochen Glueck I had made some edits in the second paragrapher

For any integer $m>2$, let $P_m$ be the set of primes less than $m$, and let $$ f(m) = \sum\limits_{p \in P_m} \frac{1}{m-p}. $$ For example, $f(3)=\frac{1}{3-2}=1$, $f(4)=\frac{1}{4-2}+\frac{1}{4-3}=\frac{3}{2}$, and so on.

The question is to estimate $I=\inf\limits_{m>2} f(m)$.

A simple Mathematica calculation shows that $f(m)\geq f(223)\approx 0.60178$ for all $m$ up to $10,000$. It is true that $I>0$? Is $I>0.5$? Is $I=f(223)$?

What is the finite description of Rayo's number? And how does it compare in theoretical size to transfinite numbers? How many operator levels are required to comprehend Rayo's number?

Let x be a random n-bit string, and let $I ={i_1,i_2,...,i_n}$ be the starting indexes of the longest 0-runs of x, sorted in decreasing order (so $i_1$ is the starting index of the longest (~$\log n$) 0-run, and ties are broken by lexicographic order). Note that the entropy of $I$ is at most $H(I) <= n$ bits (since $I$ is determined by $x$ and $H(x)=n$). What I'm trying to show is that "most" of this entropy is actually concentrated on a rather *small* number of indices with "large" entropy. More precisely, is it true that there is a subset $S \subseteq I$, $|S|=n/poly\log(n)$, s.t $H(S) > |S|*h$ , where $h >= \log^{\epsilon}n$ or even $h =\Omega(\log(\log(n)$)).

Note $H(i_1) = \log n$ by symmetry. The intuition is that $i_2,\ldots, ,i_{\Omega(\sqrt{\log n)}}$ are *determined* by $i_1$ since there's a gap of at least ~$\Omega(\sqrt{\log n)}$ between the first and second longest 0-runs. Then there's another discontinuous "random jump" to the next non-overlapping run, etc.

Would appreciate if anyone can point out relevant references.. Thanks!

Let $(R, \mathfrak m)$ be a valuation ring such that $\mathfrak m$ and every ideal of finite height is principal. Then is $R$ Noetherian , i.e. a discrete valuation ring ?

Let $d(n)$ denote the number of positive divisors of $n$. Is it known how to evaluate the sum

$$\displaystyle \sum_{1 \leq m < n \leq X} d(m) d(n) d(n-m)?$$

A slightly more difficult question is if we change the height condition in the summation, to obtain the sum

$$\displaystyle \sum_{\substack{1 \leq mn(n-m) \leq X \\ 1 \leq m < n}} d(m) d(n) d(n-m).$$

This is a generalization of the single variable case, where it is known how to evaluate sums of the form

$$\displaystyle \sum_{1 \leq n \leq X} d(an + b) d(cn+d)$$

for fixed positive integers $a,b,c,d$.

The *density* of a set
$X\subseteq\omega$ refers to:
$\limsup\limits_{n\rightarrow\infty}\dfrac{C\cap n}{n}$.

Given a set of positive integers $F= \{m_0<\cdots<m_{k-1}\}$, let $C\subseteq \omega$ be such that for every $x$ there exists $y\in (\{x\}\cup x+F)\cap C$.

Q1. Given $F$, what is the smallest possible density of $C$? Is it always $1/(1+k)$?

Q2. Given $F$, if one build $C$ stochastically as following, what is the density of $C$? For every $x$, if there is not any element in the current $(\{x\}\cup x+F )\cap C $, then select an element $y$ uniformly randomly from $ \{x\}\cup x+F$ and add $y$ into $C$.

Let $G$ be locally compact group and let $H$ be a open subgroup in $G$. Then the full group $C^*$-algebra of $H$, $C^*(H)$, is a subalgebra of $C^*(G)$ and there is a conditional expectation $$E\colon C^*(G)\to C^*(H),$$ which is induced by restriction $f\in L^1(G) \mapsto f_{|H}\in L^1(H)$ of functions which are integrable w.r.t. the left Haar measure on $G$, see Rieffel, induced representations of $C^*$-algebras, Proposition 1.2.

Note that $E$ is'n faithful in general, there is an example in this paper in the section above definition 3.: Consider $G$ a nonamenable discrete group and $H$ an open subgroup consisting of the identity element of $G$. Then there are nonzero elements $c$ in the kernel of the left regular representation of $G$ (since $G$ is not amenable) and they satisfy $E(c^*c)=0$.

My Question: Now, let $G$ be a locally compact amenable group and $H$ be an open compact (amenable) subgroup.

Is then $E$ faithful?

I think yes (I have considered some examples), but I am stuck with a proof.

If I additionally assume $G$ (and $H$) to be discrete I can prove it considering $E$ as a conditional expectation $C_r^*(G)\to C_r^*(H)$ and then it is $\tau_G=\tau_H\circ E$, where $\tau_G$ and $\tau_H$ are the canonical faithful tracial states on $C_r^*(G)$, $C_r^*(H)$ respectively. It follows that $E$ must be faithful.

For the more general case, I thought about trying a similar strategy using the fact that $C_r^*(G_1)$ of a locally compact group $G_1$ which contains a non-trivial amenable open subgroup $H_1$ has a tracial state $\tau^{G_1}$ satisfying $$\tau^{G_1}( \lambda_{G_1}(f))=\int_{H_1}f(s)d\mu(s),$$ see corollary 4.1 in 'embedding theorems in group $C^*$-algebra' by Lee for this fact. If one can check that the tracial state $\tau^{H_1}$ is faithful, then this together with $\tau^{G_1}=\tau^{H_1}\circ E$ implies that $E$ is faithful. But I am stuck with proving faithfulness of the trace. Probably I am on the wrong track..Other strategies regarding my question are welcome.

I am interested in the generating function of $SO(N)$ random matrix, that is, I want to compute $$ Z_N[J]=\int dM e^{{\rm Tr} (J^T M)}, $$ where $dM$ is the $SO(N)$ Haar measure, and $J$ is an arbitrary $N\times N$ matrix. From this generating function, I can generate all correlations $\langle M_{ij}M_{kl}\cdots\rangle$ by taking derivatives with respect to the elements of $J$.

Due to the invariance of the measure, one sees that $Z[J]=Z[U^TJV]$ with $U,V\in SO(N)$, and thus $Z$ only depends on the singular values of $J$. (Stated otherwise, $Z$ only depends on the $N$ invariants ${\rm Tr}((J^TJ)^n)$, $n=1,...,N$).

Finally, at least for $N=2$ and $N=3$, one can show that $Z$ is also invariant under permutations of the singular values of $J$ (maybe it can be generalized for all $N$ ?).

It is not too hard to compute explicitly $Z_2[J]$, which is given in terms of a Bessel function of the sum of the two singular values of $J$.

Is there a way (or has it been done in the literature) to compute $Z_N$ for any $N$ ? I would already be happy with $Z_3$, which I cannot manage to compute explicitly.

EDIT : Here is an attempt, which kind of works for $N=2$, but for which I am stuck for $N=3$. If we define a Laplacian $\Delta=\sum_{ij}\frac{\partial^2}{\partial J_{ij}^2}$ (with $J_{ij}$ the elements of $J$), one shows easily that $$ \Delta Z[J]=N Z[J]. \tag{1} $$ If we call $\lambda_i$ the singular values of $J$ (with $\lambda_1>\lambda_2>\ldots$), using the fact that $Z[J]=Z[\lambda_1,\lambda_2,\ldots]$, one shows (at least for $N=2$ and $N=3$, but it might be generalizable to $N\geq4$) that $$ \Delta Z=\frac{1}{D}\sum_{i}\frac{\partial}{\partial \lambda_i}\left(D\frac{\partial}{\partial \lambda_i}Z\right), $$ where $D=\prod_{i< j}(\lambda_i^2-\lambda_j^2)$ is related to the Jacobian to go from $J_{ij}$ to $\lambda_i$. This equation looks nice enough, so my hope is that a solution exists, I am not quite sure how to find it for $N=3$.

In the case $N=2$, we can compute $Z_2$ exactly via its definition, and it reads $Z_2[\lambda_1,\lambda_2]=I_0(\lambda_1+\lambda_2)$, with $I_\nu$ the modified Bessel function of the first kind. One checks that this is indeed a solution of Eq. (1).

Unfortunately, even in that case, it is not clear to me how to find this solution starting from Eq. (1) only. Given that $D=\lambda_1^2-\lambda_2^2$, it is tempting to defined $u=(\lambda_1+\lambda_2)/2$ and $v=(\lambda_1-\lambda_2)/2$. Then Eq. (1) is solved by separation of variables and we find a family of solution $Z_{2,\mu}$ (where I have already use the fact that $Z_N[0]=1$) : $$ Z_{2,\mu}[u,v]=I_0(\sqrt{\mu}u)I_0(\sqrt{4-\mu}v). $$ Clearly the solution to my problem corresponds to $\mu=4$, but it is not clear to me what is the rigorous argument to pick this value of $\mu$ (since $Z_N>0$ $\forall J$, we must have $\mu\geq 4$ as $I_0$ can be negative for imaginary variables; but how to select $\mu=4$ as the only viable solution ?).

One way to solve this issue of $\mu$ is to use the fact that $Z_2[J=\lambda Id_2]$ can be computed explicitly: $Z_2[J=\lambda Id_2]=I_0(2\lambda)$, which unambiguously selects $\mu=4$.

Since we can always compute the expansion of $Z_N[\lambda Id_N]$ explicitly at least for small $\lambda$, this kind of argument might be enough to fix the constant also for $N>2$. For $N=3$, a few special cases can be computed explicitly, which might help too.

Any insight for the solution of Eq. (1) would be greatly appreciated.

For $n=1,2,3,\ldots$ let $a_n$ denote the determinant $\det[(i^2+j^2)^n]_{0\le i,j\le n-1}$. Then $$a_1=0,\ a_2=-1,\ a_3=-17280,\ a_4= 1168415539200.$$

QUESTION: Is it true that $(2n)!\mid a_n$ for all $n=3,4,\ldots$?

I even conjecture that $$a_n'=\frac{(-1)^{n(n-1)/2}a_n}{2\prod_{k=1}^n(k!(2k-1)!)}$$ is a positive integer for every integer $n>2$. Note that \begin{gather*}a_3'=1,\ a_4'=559,\ a_5'=10767500,\ a_6'=9372614611500. \end{gather*}

The question is similar to my previous question http://mathoverflow.net/questions/302130. But it seems that darij grinberg's method there does not work for the present question.

Any comments are welcome!

Let $\phi:X\to Y$ be an etale morphism of noetherian scheme. Does $\phi$ have to be quasi-affine? In other wards, if $Y$ is affine does it mean that $X$ is quasi-affine?

It will follow from the fact that quasi-finit morphisms are quasi-affine, but I do not know whether this is true.

Calculate the Chern classes $$ c_n \in H^{2n}(S^{2n})$$ for the generator of the group $$ K(S^{2n})$$ where $S^{2n} $ - sphere of dimension $ 2n $, $ K(S^{2n})$ - group from K-theory.

I found the following fact about the K-group $$ K(S^0) = K(S^{2k})= \mathbb{Z} \times \mathbb{Z},\quad K(S^1) = K(S^{2k+1}) =\mathbb{Z} $$ How to calculate the Chern classes for sphere $S^{2n}$ if we know that $\eta$ is generator for $K(S^{2n})$?

For $n=1,2,3,\ldots$ let $a(n)$ denote the determinant $\det[(i+j)^n]_{0\le i,j\le n-1}$.

QUESTION: Is it true that $n^2\mid a(n)$ for all $n=3,4,\ldots$?

I even conjecture that $$b(n)=\frac{(-1)^{n(n-1)/2}a(n)}{(n-2)!n\prod_{k=1}^nk!}$$ is a positive integer for every integer $n>2$. Note that \begin{gather*}b(3)=4,\ b(4)=229,\ b(5)=89200,\ b(6)=336775500, \\ b(7)= 15858447494400,\ b(8)= 11358391301972951040. \end{gather*}

Any comments are welcome!

I have studied "enough" the theory of distributions , I would like to deepen some topic with applications. With some research I arrived at this book:

"Geometric Theory of Generalized Functions with Applications to General Relativity (Mathematics and Its Applications) (Volume 537)"

You can find the index (and some pages) on google books or amazon. Does anyone know this theory? which prerequisites are more fundamental to study from this book? thanks for every answer.

A closed convex set $K\subset \mathbb{R}^n$ is the intersection of its supporting (closed) half-spaces. We call $K$ *locally polyhedral* if any intersection of $K$ with a convex polytope $P$ is a convex polytope. If $K$ is locally polyhedral, its extreme points are isolated. Moreoever, the intersection of $K$ with any polytope contained in a small enough neighborhood of an extreme point $x$ is the intersection of that polytope with a unique cone (the intersection of the supporting half-spaces that have the extreme point on the boundary). Let us call the intersections of the extreme rays of these cones with $K$ *edges*. Edges are either bounded and connect two extreme points, or unbounded. It seems fairly intuitive that any two extreme points of $K$ should be connected by a finite path of bounded edges. However, all the ways I've been able to think of to prove this seem overly complicated for such a simple result. I would appreciate it if someone were to able to point me to the correct reference or provide a simple argument I am failing to spot.

A Jacobi form of weight $k$ and index $m$ is a meromorphic function $\varphi: \mathbb{H} \times \mathbb{C} \to \mathbb{C}$ satisfying

$$\varphi\bigg(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}\bigg) = (c \tau + d)^{k} e^{\frac{2 \pi i mc}{c \tau +d} z^{2}}\varphi(\tau, z)$$

$$\varphi(\tau, z + \lambda \tau + \mu) = e^{-2 \pi i m(\lambda^{2}\tau + 2 \lambda z)} \varphi(\tau, z).$$

Let $\pi : \mathcal{C} \to \overline{\mathcal{M}}_{1,1}$ be the universal elliptic curve. We can identify the surface $\mathcal{C} \cong \overline{\mathcal{M}}_{1,2}$, and we can realize $\overline{\mathcal{M}}_{1,1}$ as a divisor of $\mathcal{C}$. The general fiber $F$ gives another independent divisor of $\mathcal{C}$.

The way I understand it (please correct me if I'm wrong) a Jacobi form $\varphi$ is a section of a line bundle $\mathcal{E} \to \mathcal{C}$ such that the weight and index are encoded as

$$k = \text{deg}\big( \mathcal{E}|_{\overline{\mathcal{M}}_{1,1}}\big), \,\,\,\,\,\,\,\,\,\,\,\,\, m = \text{deg}\big(\mathcal{E} |_{F}\big)$$

I believe one should think of $\tau$ as a coordinate on $\overline{\mathcal{M}}_{1,1}$ and $z$ as a coordinate in the fiber direction. If $z=0$, indeed we just get a section of a line bundle over $\overline{\mathcal{M}}_{1,1}$, consistent with $\varphi(\tau, 0)$ being an ordinary modular form of weight $k$.

The Jacobi form I'm interested in is $\varphi_{-2,1}$ which is the unique weak Jacobi form of weight -2 and index 1. Its inverse is a meromorphic Jacobi form of weight 2 and index -1. It is well known that we have an asymptotic expansion around $z=0$,

$$\frac{1}{\varphi_{-2,1}}(\tau, z) = \sum_{g=0}^{\infty} z^{2g-2} \mathcal{P}_{g}(\tau),$$

where $\mathcal{P}_{g}(\tau)$ is a quasi-modular form of weight $g$. Basically, I want to understand the *geometry* of an expansion of this form.

First off, applying the elliptic transformation law of Jacobi forms (the second of the two transformation equations above) to this expansion gives nonsense, which makes sense because it holds only for small $z$. In addition, we lose all information about the index, since that is associated to the fiber directions (i.e. general $z$). However, I would expect it to encode the weight perfectly. And it almost does! We can try to apply a modular transformation

$$\frac{1}{\varphi_{-2,1}}\big(\frac{a \tau + b}{c \tau + d}, \frac{z}{c \tau + d}\big)= \sum_{g=0}^{\infty} \big( \frac{z}{c \tau + d}\big)^{2g-2} \mathcal{P}_{g}\big(\frac{a \tau + b}{c \tau + d}\big),$$

and notice that if $\mathcal{P}_{g}$ were modular, the cancellation of the automorphy factors $c \tau + d$ would work out perfectly to give us the actual weight. So my first question is: **why does the quasi-modularity arise to spoil this determination of the weight? It comes so close to working, which would make geometric sense.**

Secondly, putting aside issues of quasi-modularity, $\mathcal{P}_{g}$ should be a section of a line bundle $\mathcal{E}_{g} \to \overline{\mathcal{M}}_{1,1}$. The equation just above seems to indicate that we can interpret $z$ as a coordinate on, or section of the *dual* bundle $\mathcal{E}_{g}^{\vee}$, at least in the expansion. This would explain the cancellation of the automorphy factors geometrically. **Is this or something like it true?**

Finally, doing an expansion around $z=0$, I would expect the *normal bundle* of $\overline{\mathcal{M}}_{1,1}$ in $\mathcal{C}$ to play a role. **What is this normal bundle, and does it play a role in this story?**

My friends are preparing project about a Solow model. The asked me to calculate such integral: $ s(1-a)\int e^{(1-a)(b+c)t} \cdot (d-ge^{ft})^{1-a}dt$ where: $b,c,s,d,g>0$ and $a∈(0,1)$. $[a,b,c,d,g,f$-constans] They recived a hint to use hypergeometric series. Unfortunately, I do not know anything about hypergeometric series.

"Given the sporadic, random-like quality of the primes, it is quite surprising how much can be proved about them. Interestingly, theorems about the primes are usually proved by exploiting this seeming randomness...

Much research on prime numbers has this sort of flavour. Your first devise a probabilistic model for the primes - that is, you pretend to yourself that they have been selected according to some random procedure. Next, you work out what would be true if the primes really were generated randomly. That allows you to guess the answers to many questions. Finally, you try to show that the model is realistic enough for you guesses to be approximately correct...

It is interesting that the probabilistic model is a model not of a physical phenomenon, but of another piece of mathematics. Although the prime numbers are rigidly determined, they somehow feel like experimental data. Once we regard them that way, it becomes tempting to devise simplified models that allow us to predict what the answers to certain probabilistic questions are likely to be. And such models have indeed sometimes led people to proofs valid for the primes themselves."

T. Gowers, Mathematics: A Very Short Introduction (Oxford Univ. Press, 2002) p.120-121

In the spirit of this quote, are we allowed to assume that prime numbers are generated by an unique and invariable process?

Or could the prime number sequence be considered as the interplay of many distinct "random procedures" operating at once or subsequentially?

How to stablish a proof either way?

With this I mean that it isn't well stablished that subsequences of the primes, such as the twin prime pairs or the Mersenne primes, are not generated by a different process than ordinary primes.

This also means that it isn't well stablished that there is no prime number p such that primes greater than p aren't generated by a different process.

It seems obvious to consider it a single invariant process from the fact that there´s an algorithm, the sieve of Erastosthenes, that allows us to find all prime numbers greater than p from a list of primes from 2 to p.

If the process was not unique or invariable, one would need extra information than what is already present at that previous sequence of primes, right?

Also, in other context, can the small gap between consecutive twin primes be considered as evidence of a dynamical system which repeatedly approaches it´s initial condition?

Let $\zeta \in \mathcal{M}(m \times n; \mathbb R)$ and we fix a $\zeta_0 \in \mathcal M( (n-m) \times n; \mathbb R)$. Let us define a parameterization by \begin{align*} F : \zeta \mapsto \begin{pmatrix} \zeta \\ \zeta_0 \end{pmatrix} = A(\zeta). \end{align*} Let $S = \{ \zeta: \rho( A(\zeta) )<1\}$ and assume $S \neq \emptyset$. I am interested to know whether $S$ is connected?

Are there any functions f : R → R which aren't integrable on [0, 1], but h(x) = f(x).f(x) is integrable on [0, 1].