## reference request – D-modules as ind-coherent sheaves over positive characteristics?

There is an interpretation of D-modules over “sufficiently nice” prestacks $$X$$ (read: various finiteness conditions apply, perhaps even smoothness) by Gaitsgory and Rozenbylum (see chapter I.4 here and this paper), in which one views D-modules as ind-coherent sheaves (i.e. filtered colimits of coherent sheaves) over so-called de Rham spaces $$X_{dR}$$ attached to the previously mentioned “nice” prestacks $$X$$. As I understand it, this is essentially using the fact that each crystal in quasi-coherent sheaves comes canonically equipped with a flat connection, and thus can be seen as a D-module; the approach by Gaitsgory-Rozenblyum is therefore a version of infinitesimal cohomology (in the sense of Grothendieck-Ogus) wherein establishing the six functors is somewhat easier, as now the six functors for D-modules can be deduced from the general theory of ind-coherent sheaves. There is, however, a caveat: our prestack $$X$$ has to be an object in characteristic $$0$$, and preferably over a field of characteristic $$0$$, as smooth schemes over fields are automatically reduced.

Now I am aware of the fact that there is also a theory of “arithmetic” D-modules, developed by Berthelot, wherein one replaces infinitesimal sites and all the businesses involving de Rham spaces with crystalline sites, whose objects are pd-immersions and whose coverage is the usual Zariski coverage. Given the somewhat ad hoc definition of pd-structures, is it also possible (at least in principle) to reformulate the theory of arithmetic D-modules in the style of Gaitsgory and Rozenblyum ? Have there been any attempts at this, and if this is not possible, why so ?

## rootfinding – How many positive roots can \$sum_{i}frac{a_i}{x+b_i}\$ have where \$b_i\$’s are all positive?

What is the maximum number of positive roots $$sum_{i}^Nfrac{a_i}{x+b_i}$$ can have where $$b_i$$‘s are all positive? (everything here is a real number. To provide context, I encountered this problem while doing theoretical neuroscience research where I am modeling a biological neuronal network as an artificial neural network.)

$$x$$ is our variable, $$a_i$$ is a constant (can be either positive or negative), and $$b_i$$ is always a positive constant. $$a_i$$ and $$b_i$$ have unique values at each $$i$$.

In other words, how many $$x>0$$ can satisfy $$sum_{i}^Nfrac{a_i}{x+b_i}=0$$?

If $$a_i$$ happens to be all positive or negative, I see that there are no roots at positive $$x$$. For example, the following shows $$y=frac{1}{x+1}+frac{1}{x+2}+frac{1}{x+3}$$. You can see that there are poles at -1, -2, and -3 (which are $$-b_i$$‘s), and the roots exist between the poles. Since $$b_i$$‘s are all positive, the roots between the poles need to be all negative.

However, if $$a_i$$‘s are a mix of positives and negatives (and $$b_i$$‘s are still all positive), there can be root(s) outside the poles, making it possible to have a root when $$x>0$$. For example, $$y=frac{1}{x+1}+frac{1}{x+2}-frac{3}{x+3}$$ is $$0$$ at $$x>0$$ as shown below:

If I zoom in to the positive $$x$$ part, we see the following:

It overshoots below 0, and then asymptotically approaches 0.

So far, no matter how large my $$N$$ is, a randomly generated function $$sum_{i}^Nfrac{a_i}{x+b_i}$$ seemed to have only one root at positive $$x$$, if there was any, when I swept through $$x$$ on my computer. However, I still believe the number of positive roots should be dependent on $$N$$. Any thoughts?

## equation solving – Why isn’t this expression returning true to being positive when it is clearly positive?

Consider:

``````Expre10 = (B^2/C + 2 B + B^2/D + (B C)/D + (B D)/C) +
1 (C + D) - (A (Beta) (Sigma))/(B C D) (C + D);
Assuming({A > 0, B > 0, C > 0,
D > 0, (Beta) > 0, (Sigma) > 0, (A (Beta) (Sigma))/(B C D) <=
1}, Simplify(Expre10 > 0))
``````

Returns the inequality:

B (B + C) (B + D) > A (Beta) (Sigma)

But we clearly see if `(A (Beta) (Sigma))/(B C D) <= 1` holds then our inequality will always be true, so why am I getting the wrong output?

## If \$f(x,y) = cx \$ and \$0<x<y<1\$ prove that \$E[x] = 1/2\$ where X,Y are positive random variables

I worked as such: $$int_0^1int_0^y cxdxdy=1 rightarrow c=6$$. Also:
$$f_x(x) = int_x^1 6xdy = 6(1-x)x$$
So now I need: $$E(X) = int_0^yx(6(1-x))dx$$. This is y-dependent , the result is not. What am I missing?

## How to prove the Fourier transform of \$e^{-x^p}\$ is positive

I wonder how to prove that

$$int_0^inftyexp(-x^p)cos(tx)dtgeq 0, frac{1}{2}

This conclusion is used in the answer to another question here
Looking for sufficient conditions for positive Fourier transforms

## real analysis – Mid-point convex measurable subset of \$mathbb{R}\$ with positive Lebesgue measure is an interval

The question is right as the title:

Let $$E$$ be a measurable subset of $$mathbb{R}$$ w.r.t. Lebesgue measure, and has positive measure. For any $$x,yin E$$, $$frac{x+y}{2}in E$$. Prove that $$E$$ is an interval(like $$(a,b),(a,b),(a,b)$$ etc., possibly infinity endpoint).

The hint is to find some function on it, and I tried looking at its characteristic function, but I have no clue.

## factorial – If n is a positive integer that is four digits long and is relatively prime to 100!, why must n be prime?

Suppose there is some positive integer n that is four digits long and is relatively prime to 100! (meaning n and 100! have no common factors other than 1). n must be prime, but why?

100! is a composite number, but composite numbers can be relatively prime to other composite numbers, so that can’t be the reason n is prime. n being four digits long and 100! having factors that are all less than four digits must have something to do with it, but I can’t wrap my head around the exact reason.

So, why does n have to be prime in this situation?

## linear algebra – Inner product matrix have positive determinant

Suppose $$|v_1rangle, |v_2rangle ,cdots |v_krangle in S_N$$ and suppose a matrix $$G$$ form by the inner product of these vectors
$$G_{ij}=langle v_i|v_jrangle$$
I’m trying to prove that $$text{det} (G)geq 0$$ where equality hold for if vectors are linearly dependent.

Since
$$langle v_i|v_jrangle =langle v_j|v_irangle ^*rightarrow G_{ij}=G_{ji}^*rightarrow G=G^dagger$$
this means that eigenvalues will be positive. In the diagonal form,
$$G=text{diag}(g_1,g_2,cdots ,g_k)$$
The determinant can be written as
$$text{det}(G)=prod_ig_i$$
To prove this is positive, We can prove that all the eigenvalues are positive but that is much more restrictive. I’m not sure, How do I proceed? Please help me with this.

## algebraic number theory – Extension of morphism in local fields of positive characterisic

Consider $$theta:mathbb F_q(T)mapstomathbb F_q(T)$$ defined by $$theta(Q)=Q(T^q)$$. It is a morphism of fields. Let $$P$$ be an irreducible polynomial of $$mathbb F_q(T)$$. Then, $$theta$$ can be uniquely extended to $$mathbb F_q(T)_P$$ by continuity (denote it $$theta$$ again), where $$mathbb F_q(T)_P$$ is the completion of $$mathbb F_q(T)$$ for the topology induced by the $$P$$-valuation . It is a morphism of fields yet and for every $$xin K_P$$, one has $$theta(x)=x^q$$.
My question: can $$theta$$ be extended continuously to $$Omega_P$$, the topological closure of an algebraic closure $$overline{K_P}$$ of $$K_p$$ such that $$theta$$ is a morphism of fields in $$overline{K_P}$$ and $$theta(x)=x^q$$ for any element $$xinoverline{K_P}$$?

## Question on integral expression of a positive definite matrices

Is there any integral expression for $$log (X + Y) – log (X)$$ if $$X$$ and $$Y$$ are positive definite matrices?

Could anyone give some suggestion as to how to find such an integral expression if there is any?

Thanks a bunch.