Max-min algebra is an algebraic structure in which classical addition and multiplication are replaced by ⊕ and \kr, where a⊕b=max{a,b},a\krb=min{a,b}. The notation \mbfA\kr\mbfx=\mbfb represents an interval system of linear equations, where \mbfA=[\pA,\nA], \mbfb=[\pb,\nb] are given interval matrix and interval vector, respectively, and a solution is from a given interval vector \mbfx=[\px,\nx]. We define six types of solvability of max-min interval systems with bounded solution and give necessary and sufficient conditions for them.
During the last ten some years, many research works were devoted to the chaotic behavior of the weighted shift operator on the Köthe sequence space. In this note, a sufficient condition ensuring that the weighted shift operator $B_{w}^{n}\colon \lambda _{p}(A)\to \lambda _{p}(A)$ defined on the Köthe sequence space $\lambda _{p}(A)$ exhibits distributional $\epsilon $-chaos for any $0< \epsilon < \mathop{\rm diam} \lambda _{p}(A)$ and any $n\in \mathbb {N}$ is obtained. Under this assumption, the principal measure of $B_{w}^{n}$ is equal to 1. In particular, every Devaney chaotic shift operator exhibits distributional $\epsilon $-chaos for any $0< \epsilon < \mathop{\rm diam} \lambda _{p}(A)$.
Several abstract model problems of elliptic and parabolic type with inhomogeneous initial and boundary data are discussed. By means of a variant of the Dore-Venni theorem, real and complex interpolation, and trace theorems, optimal Lp-regularity is shown. By means of this purely operator theoretic approach, classical results on Lp-regularity of the diffusion equation with inhomogeneous Dirichlet or Neumann or Robin condition are recovered. An application to a dynamic boundary value problem with surface diffusion for the diffusion equation is included.
We consider the dynamics of spatially periodic nematic liquid crystal flows in the whole space and prove existence and uniqueness of local-in-time strong solutions using maximal L^{p} regularity of the periodic Laplace and Stokes operators and a local-intime existence theorem for quasilinear parabolic equations à la Clément-Li (1993). Maximal regularity of the Laplace and the Stokes operator is obtained using an extrapolation theorem on the locally compact abelian group G: = \mathbb{R}^{n - 1} \times \mathbb{R}/L\mathbb{Z} to obtain an R-bound for the resolvent estimate. Then, Weis’ theorem connecting R-boundedness of the resolvent with maximal L^{p} regularity of a sectorial operator applies., Jonas Sauer., and Obsahuje seznam literatury
Local well-posedness of the curve shortening flow, that is, local existence, uniqueness and smooth dependence of solutions on initial data, is proved by applying the Local Inverse Function Theorem and $L^2$-maximal regularity results for linear parabolic equations. The application of the Local Inverse Function Theorem leads to a particularly short proof which gives in addition the space-time regularity of the solutions. The method may be applied to general nonlinear evolution equations, but is presented in the special situation only.
Max-min algebra and its various aspects have been intensively studied by many authors \cite{Baccelli,Green79} because of its applicability to various areas, such as fuzzy system, knowledge management and others. Binary operations of addition and multiplication of real numbers used in classical linear algebra are replaced in max-min algebra by operations of maximum and minimum. We consider two-sided systems of max-min linear equations
\begin{math}\emph{A}\otimes\emph{x}= B\otimes\emph{x}\end{math}, with given coefficient matrices \emph{A} and \emph{B}. We present a polynomial method for finding maximal solutions to such systems, and also when only solutions with prescribed lower and upper bounds are sought.
Let M be a given nonempty set of positive integers and S any set of nonnegative integers. Let δ(S) denote the upper asymptotic density of S. We consider the problem of finding µ(M) := sup S δ(S), where the supremum is taken over all sets S satisfying that for each a, b ∈ S, a − b ∉ M. In this paper we discuss the values and bounds of µ(M) where M = {a, b, a + nb} for all even integers and for all sufficiently large odd integers n with a < b and gcd(a, b) = 1.