Ms. Podolskaya,

Your 2nd question is related to [matrix copositivity][1], I believe. Take a look at the 5th chapter of [Parrilo's doctoral dissertation][2].

A quadratic form in $\mathbb{R}[x_1,x_2,x_3]$ is of the form

$P (x_1,x_2,x_3) = \left[\begin{array}{c} x_1\\\ x_2\\\ x_3\end{array}\right]^T \left[\begin{array}{ccc} q_{11} & q_{12} & q_{13}\\\ q_{12} & q_{22} & q_{23}\\\ q_{13} & q_{23} & q_{33}\end{array}\right] \left[\begin{array}{c} x_1\\\ x_2\\\ x_3\end{array}\right]$

or, more compactly, $P (x) = x^T Q x$. You ask: when is $P$ positive on the positive octant? If $P > 0$ when $x > 0$, then

$(\forall x \in \mathbb{R^3}) (x > 0 \implies x^T Q x > 0)$
 
and, in theory, one could use [quantifier elimination][3] to obtain conditions on the $q_{ij}$ coefficients so that $P > 0$ on the positive octant. The following [REDLOG][4] script

    % positivity on the positive octant
    
    load_package redlog;
    rlset ofsf;
    
    % define quadratic form 
    P := 1 * q11 * x1 * x1 +
       + 1 * q22 * x2 * x2 +
       + 1 * q33 * x3 * x3 +
       + 2 * q12 * x1 * x2 +
       + 2 * q13 * x1 * x3 +
       + 2 * q23 * x2 * x3;
    
    % define universally quantified formula
    phi := all({x1,x2,x3}, (x1 > 0 and x2 > 0 and x3 > 0) impl P>0);
    
    % perform quantifier elimination
    rlqe phi;
    
    end;

produces results in a few seconds, but the conditions on the $q_{ij}$ coefficients are enormously long quantifier-free formulas. So enormous that [REDLOG][4] crashed!

If $P$ is nonnegative on the nonnegative octant, then

$(\forall x \in \mathbb{R^3}) (x \geq 0 \implies x^T Q x \geq 0)$

which is equivalent to saying that matrix $Q = Q^T$ is [copositive][5].


  [1]: http://en.wikipedia.org/wiki/Copositive_matrix
  [2]: http://resolver.caltech.edu/CaltechETD%3Aetd-05062004-055516
  [3]: http://en.wikipedia.org/wiki/Quantifier_elimination
  [4]: http://redlog.dolzmann.de/
  [5]: http://en.wikipedia.org/wiki/Copositive_matrix