Overgroups in GL(nr, F) of certain subgroups of SL(n, K), I

AbstractFor any pair of division rings k and F with K ⊃ F and dimFK = r we determine all the overgroups of SL(n, K) in GL(nr, F), as well as the overgroups of Sp(n, K) in GL(nr, F) (for commutative K and even n). The overgroups of SU(n, K, f) and Ω(n, K, Q) in GL(nr, F) will be determined in another paper, “Overgroups in GL(nr, F) of certain subgroups of SL(n, K), II”

Invariant measures and random attractors of stochastic delay differential equations in Hilbert space

This paper is devoted to a general stochastic delay differential equation with infinite-dimensional diffusions in a Hilbert space. We not only investigate the existence of invariant measures with either Wiener process or Lévy jump process, but also obtain the existence of a pullback attractor under Wiener process. In particular, we prove the existence of a non-trivial stationary solution which is exponentially stable and is generated by the composition of a random variable and the Wiener shift. At last, examples of reaction-diffusion equations with delay and noise are provided to illustrate our results

Invariant measures and random attractors of stochastic delay differential equations in Hilbert space

This paper is devoted to a general stochastic delay differential equation with infinite-dimensional diffusions in a Hilbert space. We not only investigate the existence of invariant measures with either Wiener process or Lévy jump process, but also obtain the existence of a pullback attractor under Wiener process. In particular, we prove the existence of a non-trivial stationary solution which is exponentially stable and is generated by the composition of a random variable and the Wiener shift. At last, examples of reaction-diffusion equations with delay and noise are provided to illustrate our results

Synchronous dynamics of a delayed two-coupled oscillator

This paper presents a detailed analysis on the dynamics of a delayed two-coupled oscillator. Linear stability of the model is investigated by analyzing the associated characteristic transcendental equation. By means of the equivariant Hopf bifurcation theorem, we not only investigate the effect of time delay on the spatio-temporal patterns of periodic solutions emanating from the trivial equilibrium, but also derive the formula to determine the direction and stability of Hopf bifurcation. Moreover, we illustrate our results by numerical simulations