Solving q-Virasoro constraints

We show how q-Virasoro constraints can be derived for a large class of (q, t)-deformed eigenvalue matrix models by an elementary trick of inserting certain q-difference operators under the integral, in complete analogy with full-derivative insertions for $$\beta $$ β -ensembles. From free field point of view, the models considered have zero momentum of the highest weight, which leads to an extra constraint $$T_{-1} \mathcal {Z} = 0$$ T - 1 Z = 0 . We then show how to solve these q-Virasoro constraints recursively and comment on the possible applications for gauge theories, for instance calculation of (supersymmetric) Wilson loop averages in gauge theories on $$D^2 \times S^1$$ D 2 × S 1 and $$S^3$$ S 3 .

Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone

The present investigation deals with global instability of a generaln-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaev's method, assuming that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.