The space of integral liftings of a variety

This question consists of two slightly similar questions.

a) Let $ F$ be a field of positive characteristic. Let $ X$ be a smoooth proper geometrically connected $ F$ -scheme.

Let $ C$ be the category of local integral domains together with an isomorphism from the residue field to $ F$ .

Consider the (covariant?) functor from $ C$ to sets, sending a ring $ R$ to the set of $ R$ -isomorphism classes of smooth proper $ R$ -schemes whose special fiber, which is naturally an $ F$ -scheme, is isomorphic to $ X$ (or take values in groupoids if it helps).

Is this (co-)representable by something geometric? What if restrict to Noetherian rings, or DVR’s?

b) Let $ R$ be a DVR with the residue field $ F$ of positive characteristic. Consider the functor sending smooth proper geometrically connected $ F$ -schemes to the set of liftings to $ R$ -schemes as above. Is this (co-)representable by something geometric?

Solving numerically a special singular integral equation

I am trying to code the following integral equation to find the solution numerically using Mathematica. In fact the exact solution is x^2 (1 - x).

First we define the following functions:

phi[x_]:=Piecewise[{{1, 0 <= x < 1}}, 0] 
f[x_] := 1/    1155 (112 (-1 + x)^(3/4) +       x (144 (-1 + x)^(3/4) +          x (1155 + 256 (-1 + x)^(3/4) -             1280 x^(3/4) - (1155 + 512 (-1 + x)^(3/4)) x +             1024 x^(7/4)))); exactsoln[x_] := x^2 (1 - x); 

I am trying to solve the following integral equation for u (x) (numerically). where

u[x] - Integrate[(x - t)^(-1/4)*u[t], {t, 0, x}] -     Integrate[(x - t)^(-1/4)*u[t], {t, 0, 1}] = f[x]; 

where f[x] is defined as above. Here is the numerical scheme. Our goal is to find the coefficients c[j, k]. We approximate the solution u by the approximated solution \approx[x] which can be written as

approxsoln[x_, n_] :=   Sum[c[j, k]*psijk[x, j, k], {j, -n, n}, {k, -2^n, 2^n - 1}] 

If you plug the function approxsoln[x,n] in the integral equation, we will end up by

Sum[c[j, k]*(psijk[x, j, k] -       Integrate[(x - t)^(-1/4)* psijk[t, j, k], {t, 0, x}] -       Integrate[(x - t)^(-1/4)* psijk[t, j, k], {t, 0, 1}]), {j, -n,     n}, {k, -2^n, 2^n - 1}]; 

Now every thing is known except the coefficients c[j,k]. We need to use a suitable subdivision, may be divide the interval [0,1] into (2n+1) 2^(n+1) points (to meet the size of the truncated sum in approxsoln[x,n]) to be used in the equation to construct a system of linear equation, to find these coefficients in order to find the approximated solution (approxsoln[x,n]) defined above. Is there any way to code this problem using Mathematica. I think it is worth to try for n=10 first.

Stoke’s Theorem: Integral limits

I am tasked with finding: $ $ \iint_{C}F(x,y,z) \cdot dr$ $

Where $ C:x=y$ formed by the curves $ x=z$ and $ x^{2}=z$ , traversed anti-clockwise with respect to the normal vector $ n=(1,-1,0)$ . The field is $ F=(7xy,-z,3xyz)$

Here is my attempt:

$ $ curl(F)=\begin{vmatrix}i & j & k \ \delta_x & \delta_y & \delta_z \ 7xy & -z & 3xyz\end{vmatrix}=(3xz-1, -3yz, -7x)$ $

It’s product with the normal resulted in:

$ (3xz-1, -3yz, -7x) \cdot (1, -1, 1) = 3xz+3yz+1$

Hence I arrive at integral (since $ x=y$ ): $ $ \iint_{R}6yz+1 \ dA$ $

However, I’m not sure how to get past this point. What would the limits be to finish the integral and why? Assuming that I’m correct thus far, do the limits have something to do with $ x=z$ and $ x^2=z \ $ ?

Failure of Tate acyclicity for integral structure sheaves

Let $ (A,A^+)$ be a sheafy Tate-Huber pair, and let $ X=\operatorname{Spa}(A,A^+)$ . It is well-known that $ H^i(X,\mathcal{O}_X)=0$ for $ i>0$ . I assume it is generally not true that $ H^i(X,\mathcal{O}_X^+)=0$ for $ i>0$ , but I don’t think that I have seen an explicit counterexample. Is there a simple example of a nonzero cohomology class in some $ H^i(X,\mathcal{O}_X^+)$ for $ i>0$ ?

Ratio of exponentially weighted Selberg integral

I’m interested in bounding the following ratio of integral: $ $ \frac{\int_{0<x_k<…<x_1<1}\prod_{i=1}^kx_i^{m-\frac{k+1}{2}}\prod_{i<j}(x_i-x_j)\exp(-\sum_{i=1}^kw_ix_i)}{\int_{0<x_k<…<x_1<1}\prod_{i=1}^kx_i^{n-\frac{k+1}{2}}\prod_{i<j}(x_i-x_j)\exp(-\sum_{i=1}^kw_ix_i)}$ $ where $ \frac{k+1}{2}<m<n$ and $ w_i>0, i=1,…,k$ are any positive numbers. This is related to Selberg integral. Does there exist an upper bound of the above ratio only depending on $ m,n,k$ and independent of $ w_i$ ?

Differential equation with Fresnel integral

We have $ \frac{y'(x)}{cos(x)}=C(x)$ and need to find y(x). Generally we should express $ y(x)$ through $ C(x)$ and elementary functions. I can only do it through $ C(x)$ and $ S(x)$ , or through $ \Phi(x)$ . This is my solution: $ y(x)=\int cos(x)C(x)dx=sin(x)C(x)-\int sin(x)cos(x^2)dx=sin(x)C(x)-\int (\frac{sin(x^2+x)}{2}-\frac{sin(x^2-x)}{2})dx=sin(x)C(x)-\int(\frac{sin((x+\frac{1}{2})^2-\frac{1}{4})}{2}-\frac{sin((x-\frac{1}{2})^2-\frac{1}{4})}{2})dx,$ $ x+\frac{1}{2}=a,x-\frac{1}{2}=b =>y(x)=sin(x)C(x)-\int \frac{sin(a^2-\frac{1}{4})}{2}da+\int \frac{sin(b^2-\frac{1}{4})}{2}db=sin(x)C(x)-\frac{cos(\frac{1}{4})}{2}(S(a)-S(b))+\frac{sin(\frac{1}{4})}{2}(C(a)-C(b))=sin(x)C(x)-\frac{cos(\frac{1}{4})}{2}(S(x+\frac{1}{2})-S(x-\frac{1}{2}))+\frac{sin(\frac{1}{4})}{2}(C(x+\frac{1}{2})-C(x-\frac{1}{2})).$ Is there any way to improve the result of this solution and express $ y(x)$ only through $ C(x)$ ?

Change of variables, why does integral disappear?

In a PDF about Gaussian-Stochastic models for spectral diffusion, there is a simplification made by change of variable between equations 7.21 and 7.22 that I just can’t seem to make.

Writing it here,

$ $ \tfrac{1}{2}\int^t_0 d\tau’ \int^t_0 d\tau” \langle d\omega(\tau’ – \tau”) d\omega(0) \rangle ~~~~~~~~~~~~7.21$ $

Substitute in $ \tau = \tau’ – \tau”$

$ $ \int^t_0 d\tau (t – \tau) \langle d\omega(\tau) d\omega(0) \rangle ~~~~~~~~~~~~7.22$ $

How did it happen? Please show me the intermediate steps.

Integral lifts of families of varieties over a finite field

Let $ X\rightarrow \mathrm{Spec}\:F_q[[t]]$ be a flat morphism with smooth proper geometrically connected fibers. Suppose the central fiber lifts to a scheme $ X’_0$ smooth proper over $ W(F_q)$ . Is our family the base change of a flat morphism with smooth proper fibers $ X’\rightarrow \mathrm{Spec}\:W(F_q)[[t]]$ whose central fiber is $ X’_0$ ? is our family the base change of any flat morphism with smooth proper fibers $ X’\rightarrow \mathrm{Spec}\:W(F_q)[[t]]$ ?

If the answer is “no”, what additional conditions are required? Is it enough to demand that the central and the generic fiber lift to smooth proper schemes in possibly unrelated ways?

An indefinite Integral Problem with algebric numerator and trigonometric denominator

$ $ \int \frac{x^2+(n(n-1))}{(xsinx +ncosx )^2 } dx$ $ I know this is an homework problem, but I really couldn’t think of any way to solve it. Like DI Method (No go) , What kind of substitution as denominator is trigonometric whereas Numerator is algebric. Thought of n(n-1) can come by double differentiating but.. like how would we have it here … etc confusing and weird thoughts. Please help me out