Transversely holomorphic structures with many invariant hypersurfaces

In this paper we show that a transversely holomorphic foliation in a compact manifold $M$ having an infinite number of invariant hypersurfaces admits a basic transversely meromorphic first integral $f:M?\overline{\mathbb{C}}$.     

Ergodic properties of some piecewise-deterministic Markov process with application to gene expression modelling

We investigate a piecewise-deterministic Markov process with a Polish state space, whose deterministic behaviour between random jumps is governed by a finite number of semiflows. We provide tractable conditions ensuring a form of exponential ergodicity and the strong law of large numbers for the chain given by the post-jump locations. Further, we establish a one-to-one correspondence between invariant measures of the chain and those of the continuous-time process. These results enable us to derive the strong law of large numbers for the latter. The studied dynamical system is inspired by certain models of gene expression, which are also discussed here.     

Polycyclic codes as invariant subspaces

Polycyclic codes are a powerful generalization of cyclic and constacyclic codes. Their algebraic structure is studied here by the theory of invariant subspaces from linear algebra. As an application, a bound on the minimum distance of these codes is derived which outperforms, in some cases, the natural analogue of the BCH bound.     

Group properties of the extended Green–Naghdi equations

The group analysis method is applied to the extended Green–Naghdi equations. The equations are studied in the Eulerian and Lagrangian coordinates. The complete group classification of the equations is provided. The derived Lie symmetries are used to reduce the equations to ordinary differential equations. For solving the ordinary differential equations the Runge–Kutta methods were applied. Comparisons between solutions of the Green–Naghdi equations and the extended Green–Naghdi equations are given.     

Improved moment estimates for invariant measures of semilinear diffusions in Hilbert spaces and applications

We study regularity properties for invariant measures of semilinear diffusions in a separable Hilbert space. Based on a pathwise estimate for the underlying stochastic convolution, we prove a priori estimates on such invariant measures. As an application, we combine such estimates with a new technique to prove the L¹-uniqueness of the induced Kolmogorov operator, defined on a space of cylindrical functions. Finally, examples of stochastic Burgers equations and thin-film growth models are given to illustrate our abstract result.     

On the elementary divisors of the Sylvester and Lyapunov maps

The paper addresses the problem of computing the elementary divisors of the tensor product of linear transformations using the analysis of the tensor products of polynomial models, as developed in Fuhrmann and Helmke [5]. We use this to study the elementary divisors of the Lyapunov and complementary Lyapunov maps.     

New invariant sets to nonlinear diffusion equations with x-dependent convection and absorption

In this paper, we introduce a new invariant set

     

Elliptic equations for measures: Regularity and global bounds of densities

We consider elliptic equations of the form L^*μ=ν for measures on . The membership of solutions in the Sobolev classes is established under appropriate conditions on the coefficients of L. Bounds of the form (x)CΦ(x)⁻¹ for the corresponding densities are obtained.     

Coexistence of domains of attraction of nonautonomous neural networks with time-varying delays

This paper presents new dynamical behavior, i.e., the coexistence of 2^N domains of attraction of N-dimensional nonautonomous neural networks with time-varying delays. By imposing some new assumptions on activation functions and system parameters, we construct 2^N invariant basins for neural system and derive some criteria on the boundedness and exponential attractivity for each invariant basin. Particularly, when neural system degenerates into periodic case, we not only attain the coexistence of 2^N periodic orbits in bounded invariant basins but also give their domains of attraction. Moreover, our results are suitable for autonomous neural systems. Our new results improve and generalize former ones. Finally, computer simulation is performed to illustrate the feasibility of our results.