## Allow Sharepoint REST API app access to a singular list and nothing else

By default the permissions of a sharepoint app seem limited to read/write/full control over the entire site.

I’m looking for a way to grant a supplier access to a list but do not want them to be able to access the information stored on the site other than the list.

Preferred solution would be to limit the app using user permissions like a regular user.

## Ideal of strictly singular operators

Let $$X$$ be a Banach space. An operator $$T:X\to X$$ is called strictly singular iff for any infinite dimensional subspace $$Y\subseteq X,$$ $$T|_{Y}:Y\to T(Y)$$ is not an isomorphism.

It is known that for $$X=\ell_p,$$ $$1\leq p<\infty,$$ an operator is strictly singular iff it is compact. Also $$T:\ell_\infty\to\ell_\infty$$ is strictly singular iff $$T$$ is weakly compact. Can someone provide me proofs for these facts? I could not really locate proofs of the above mentioned facts in literature.

## Random matrix with given singular value distribution

Let $$X\in\mathbb{R}^{n\times n}$$ be a random matrix defined in the following way: $$X=U\Sigma V^T$$ where $$U,V$$ are iid uniformly distributed on $$O(n)$$, $$\Sigma=diag\{\sigma_1,…,\sigma_n\}$$ and $$(\sigma_1,…,\sigma_n)$$ have joint density $$f(\sigma_1,…,\sigma_n)$$ with respect to the Lebesgue measure on $$\mathbb{R}^n$$. Moreover, we assume $$f$$ is invariant under permutation of coordinates and change of sign of each coordinate.

Now the question is, what is the density of $$X$$ with respect to the Lebesgue measure on $$\mathbb{R}^{n\times n}$$

## Can the cardinality of the set of all intervening cardinals between sets and their power sets be always singular?

This is a question that I’ve posted to Mathematics Stack Exchange, that was un-answered.

To re-iterate it here:

Is the following known to be consistent relative to some large cardinal assumption?

$$\forall \kappa [\kappa >2 \to \kappa < \kappa^* < 2^\kappa \wedge singular(\kappa^*)]$$

where $$\kappa$$ is a cardinal and $$“<“$$ is strict cardinal smaller than, and $$\kappa^* = |\{\lambda|\kappa < \lambda < 2^\kappa\}|$$

## 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.

## Solve using singular perturbation method

Solve the following using singular perturbation method:

εy′′ – y = 0, y(0) = 1, y(1) = 0

ε is a small parameter.

## Bound for Expectation of Singular Value

In my case, $$X_{\boldsymbol{\delta}}\in\mathbb{R}^{d\times M}$$ is a function of Rademacher variables $$\boldsymbol{\delta}\in\{1,-1\}^M$$ with $$\delta_i$$ independent uniform random variables taking values in $$\{−1, +1\}$$. $$X_{\boldsymbol{\delta}}=[\sum_{i=1}^{I_1}\delta_{i}\mathbf{x}_{i},\sum_{i=I_1+1}^{I_2}\delta_{i}\mathbf{x}_{i},…,\sum_{i=I_{M-1}+1}^{I_M}\delta_i\mathbf{x}_{i}]$$ is a group-wise sum with known $$I_1,I_2,…,I_M$$ and non-singular $$X=(\mathbf{x}_1,\mathbf{x}_2,…,\mathbf{x}_N)\in\mathbb{R}^{d\times N}$$ where $$N>M\gg d$$.

Given that $$\sigma_i(X_{\boldsymbol{\delta}})$$ denotes $$i$$-th smallest singular value, how can I find the lower bound of the expectation $$\underset{\boldsymbol{\delta}}E\left[\sum_{i=1}^{k} \sigma_{i}^{2}\left(X_{\boldsymbol{\delta}}\right)\right]$$ assuming $$k?

Note: I can find an upper bound by Jensen’s inequality and concavity of sum of $$k$$ smallest eigenvalue, but I am curious about whether it is possible to get a lower bound.

I have also posted the question here.

## Failure of entropy condition for a singular limit

Consider a regularization of the conservation law $$\partial_t u + \partial_x f(u) + \epsilon \partial_{x}^3 u = 0$$

How does one prove that the limit function $$u$$ of $$\{u_\epsilon\}_\epsilon$$ as $$\epsilon \to 0$$ is not an entropy solution of $$\partial_t u + \partial_x f(u)=0 \ ?$$

This problem was studied in a series of papers by Lax and Levermore (1983). Here I’m looking for a direct proof that $$u$$ violates the entropy condition (or Olienik inequality).

## Do we know the consistency strength of the Singular Cardinal Hypothesis failing on an uncountable cofinality?

Suppose that $$\kappa$$ is a strong limit cardinal. The singular cardinal hypothesis states $$2^\kappa=\kappa^+$$. We know that the failure of SCH requires large cardinals, and in fact is equiconsistent with a measurable cardinal $$\kappa$$ satisfying $$o(\kappa)=\kappa^{++}$$.

But this failure is at $$\aleph_\omega$$. Suppose we wanted more.

Suppose that we wanted the failure to happen on a couple isolated points. Well, it’s not hard to redo the standard constructions and get just that. But what happens when we have limit points?

Even more, by Silver’s theorem if SCH fails at $$\kappa>\operatorname{cf}(\kappa)>\omega$$, then there is a stationary subset of $$\kappa$$ where SCH failed.

What would be the consistency strength when $$\kappa$$ is a singular limit of singular cardinals, and SCH fails cofinally below $$\kappa$$? What if we require $$\kappa$$ to be of uncountable cofinality?

As a side question, what if $$\kappa$$, with uncountable cofinality, does satisfy SCH, but an unbounded subset (which has to be non-stationary, of course) of it does not?

## Smallest singular value distribution

Let $$G_\mathbb{R}\in\mathbb{R}^{n\times n}$$ and $$G_\mathbb{C}\in\mathbb{C}^{n\times n}$$ denote the real and complex Ginibre random matrices, i.e. random matrices with independent real/complex Gaussian entries of zero mean and variance $$\mathbb{E}\lvert(G_K)_{ij}\rvert^2=n^{-1}$$ for $$K=\mathbb{R},\mathbb{C}$$. Denote the singular values of the matrix $$X$$ by $$\sigma_1(X)\ge\dots\ge \sigma_n(X)$$. It follows from the work of Edelman that $$$$\mathbb{P}\Big( \sigma_n(G_K)\le\frac{x}{n}\Big) = \begin{cases} 1-e^{-x^2/2-x}+o(1),&K=\mathbb{R}\ 1-e^{-x^2},&K=\mathbb{C}. \end{cases}\tag{*}\label{sing val}$$$$ In particular, the smallest singular value is of order $$n^{-1}$$.

Now I am wondering about additive perturbations to $$G_K$$, say $$G_K+\lambda I$$, where $$\lambda$$ is some real (or even complex) parameter and $$I$$ is the identity matrix. In general the singular values of $$G_K$$ give little information about the singular values of $$G_K+\lambda I$$. The most one can hope for is bounds like $$\sigma_n(G_K+\lambda I)\ge \lvert\lambda\rvert-\sigma_1(G_K).$$ It is known that $$\sigma_1(G_K)$$ is Tracy-Widom distributed around $$2$$, so in particular, $$\sigma_n(G_k+\lambda I)$$ is bounded away from $$0$$ as long as $$\lvert \lambda\rvert>2$$.

Question: Is the analogue of $$\eqref{sing val}$$, i.e. the distribution of $$\sigma_n(G_K+\lambda I)$$ known for $$G_K+\lambda I$$? If exact formulae are not available, I would be interested in the average scaling of $$\sigma_n(G_K+\lambda I)$$. I guess there should be some phase transition of the type $$\mathbb E \sigma_n(G_K+\lambda I)\sim\begin{cases}n^{-1},&\lvert\lambda\rvertc.\end{cases}$$ I think $$c=1$$, but am unsure about the critical exponent.