A fundamental result in linear algebra states that if a homogenous equation system has only the trivial solution, then there are at most as many variables equations. We prove following generalisation of this phenomenon. If possibly infinite with finitely each exists an injection from to equations maps variable which it appears.

The Kolmogorov-Sinai entropy is a far reaching dynamical generalization of Shannon entropy of information systems. This entropy works perfectly for probability measure preserving (p.m.p.) transformations. However, it is not useful when there is no finite invariant measure. There are certain successful extensions of the notion of entropy to infinite measure spaces, or transformations with ...

We introduce the class of contraction algebras in which the algebra of infinite and finite terms is free over the set of variables. We develop a general theory of systems of equations at the level of categories in close connection with the Banach Principle of Contraction. Two applications of this theory are given. The first is the case of regular systems of equations with arbitrary terms. The s...

In this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional Brownian motion in a Hilbert space. We establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz conditions with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory

We show that lattice dynamical systems naturally arise on infinite-dimensional invariant manifolds of reaction-diffusion equations with spatially periodic diffusive fluxes. The result connects wave pinning phenomena in lattice differential equations and in reaction-diffusion equations in inhomogeneous media. The proof is based on a careful singular perturbation analysis of the linear part, wher...