## Do negative indecomposable bundles on curves have sections?

Let $$X$$ be a smooth projective curve, and $$E$$ an indecomposable vector bundle on $$X$$ with $$\mathrm{deg} E<0$$. Is it true that $$H^0(X,E)=0$$?

This is true if $$E$$ is a line bundle, which means it is also true whenever $$X$$ is $$\mathbb{P}^1$$, since all vector bundles split here.

It is also true by results of Atiyah if $$X$$ is an elliptic curve. What about for curves of higher genus?

The assumption that $$E$$ is indecomposable is of course necessary.

## Matlab: Matrix with negative numbers

I have a matrix, A = [745 x 678], which consists of negative and positive values. I would like to make all the negative numbers zero but am not sure how to go about doing this.

I was thinking of something like:

n = 678 for i = 1:n A(A<0) = 0 end

I am not sure to how to apply this to every element of the matrix though. Any help would be appreciated.

## Why are the mollifiers negative in this sequence?

I’m reading a proof that starts the following way:

Assume $$E$$ is open and $$u \geq 0$$ a.e. Given $$K \subset E$$ compact, let $$\psi\in C_0^{\infty}(\mathbb{R}^n)$$ satisfy: $$\psi = 1 \quad \text { on } \{x:\text{dist}(x,K) \leq \frac{1}{2}\text{dist}(\mathbb{R}^N-E,K)$$ $$\psi = 0 \quad \text { on } \mathbb{R}^N-E$$ and $$0 \leq \psi \leq 1$$. Let $$\phi_\epsilon(x)$$ be a family of mollifers with support $$\phi_\epsilon \subset B_\epsilon(0)$$ and set $$w_n(x) = u * \phi_{\epsilon_n}(x) = \int_{B_{\epsilon_n}}u(y)\phi_{\epsilon_n}(x-y)\,dy$$ where $$\epsilon_n \rightarrow 0$$ monotonically and $$\epsilon_0 < \text{dist}(\mathbb{R}^N-E,K)$$. So $$w_n \leq 0$$ on $$K$$.

Why is $$w_n \leq 0$$ on $$K$$? Also the proof switches notation from $$\psi$$ to $$\phi$$. I’m assuming this was in error, but perhaps there’s something missing?

## Proof that shortest path with negative cycles is NP hard

I’m looking into the shortest path problem and am wondering how to prove that shortest path with neg. cycles is NP-hard. (Or is it NPC? Is there a way to validate in P time that the path really is shortest?)

How would I reduce the SAT problem into the shortest path problem in polynomial time?

## What are the negative and positive aspects of dark color scheme?

I am using Light-on-dark color scheme (dark background with light text) in my application.

My question is, what are the negative and positive aspects of this color scheme and why this color scheme is not widely used?

## Relation between two Fox-H function with positive and negative argument

Is there the relation between Fox-H function with positive argument and Fox-H function with negative argument? My question is attached in the image.

## Problem

The negative log-likelihood of Dirichlet multinominal distribution is given by $$f(\alpha)=-L(\alpha)=\sum_{i=1}^n\sum_{j=1}^d \log\frac{\Gamma(\alpha_j)}{\Gamma(\alpha_j+\mathbf{x}_{ij})}+\text{constant}$$ where $$\mathbf{x}_{ij}\in \{0, 1\}$$ and $$\alpha\in \mathbb{R}^d, \alpha_j >0$$

Then determined whether $$f(\alpha)$$ is convex or not.

## What I Have Done

It is natural to compute the Hessian $$\nabla_\alpha^2 f(\alpha)$$. However, when I checked Wikipedia, the derivative of Gamma function looks much too complicated.

Then I tried to find some counter examples for $$d=1$$ case. It seems that this function is indeed convex according to several simulations I carried out.

I am wondering if we could analytically determine the convexity of this function.

## How to format a negative number red within a mixed formula result

We use the following formula:

="Stu/W: " & TEXT((A\$  2/B2),"[Red]#,###,##0.00") &"€" 

and wish to have the resulting negative currency numbers colored red and positive blue.

Unfortunately neither this or other combinations we tried works. Neither did we find a solution using Conditional Formatting.
How can we have the resulting negative currency numbers colored red as well as the positive blue?

## Discrete spectrum of potential with negative integral

I have a schrödinger operator $$H=-\frac(d^{2}}{dx^{2}}+V$$. Let’s assume that $$V$$ is compactly supported and that $$\int_{-infty}^{+infty}V(x)dx<0$$. Does this guarantee that $$\sigma_{p}(H)\subset [-\infty,0]$$ (where $$\sigma_{p}(H)$$ is the discrete spectrum of $$H$$) and that it is nonempty. If this holds, it keeps being true for $$V$$ in some broader class of functions ($$L^p$$ etc..)