Almost Automorphic Dynamics of Generalized Li\'enard Equation

In this study, we focus on the solutions of the Li\''enard equation being bounded in the future and characterize the almost automorphic, asymptotically almost automorphic, and weighted pseudo almost automorphic dynamics. An example is presented to illustrate the main findings.     

Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations

In this paper, applying the theory of semigroups of operators to evolution family and Banach fixed point theorem, we prove the existence and uniqueness of an (a) almost automorphic (weighted pseudo almost automorphic) mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) in Banach space under conditions.     

Quaternionic structures

Any oriented 4-dimensional real vector bundle is naturally a line bundle over a bundle of quaternion algebras. In this paper we give an account of modules over bundles of quaternion algebras, discussing Morita equivalence, characteristic classes and K-theory. The results have been used to describe obstructions for the existence of almost quaternionic structures on 8-dimensional Spin^c manifolds in ?adek et al. (2008) [5] and may be of some interest, also, in quaternionic and algebraic geometry.     

Almost Periodic Solutions of Nonautonomous Linear Difference Equations

Explicit almost periodic solutions are obtained for a class of almost periodic linear difference equations. The stability characteristics of the almost periodic solution are investigated. The results are applied to a nonautonomous hyperbolic difference equation modelling the dynamics of a single species population in temporally varying environments.     

Generic Initial Ideals,Graded Betti Numbers,and k-Lefschetz Properties

We introduce the k-strong Lefschetz property and the k-weak Lefschetz property for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results are:

1. Let I be an ideal of R = K[x ₁, x ₂,…, x _n ] whose quotient ring R/I has the n-SLP. Suppose that all kth differences of the Hilbert function of R/I are quasi-symmetric. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I.

2. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.     

Almost Split Sequences in Dimension One

Let R be a complete discrete valuation ring with quotient field K, and let Λ be an R-order in a semisimple K-algebra. For an indecomposable Λ-lattice E, a sublattice Bi E satisfying Rad E ? Bi E is defined, and it is shown that the middle term H of an almost split sequence τE ? H ? E can be obtained by an amalgamation of E/Bi E with E′/τE for a suitable overlattice E′ of τE. The method is bound to dim R = 1.     

A Non-Existence Theorem for Almost Split Sequences

Let k be a field, Q a quiver with countably many vertices and I an ideal of kQ such that kQ/I is a spectroid. In this note, we prove that there is no almost split sequence ending at an indecomposable not finitely presented representation of the bound quiver (Q, I). We then get that an indecomposable representation M of (Q, I) is the ending term of an almost split sequence if and only if it is finitely presented and not projective. The dual results are also true.     

Semigroups That Are Almost N 3

A construction is given for semigroups S such that S ³ has just two elements.     

Constructing Almost Symmetric Numerical Semigroups from Irreducible Numerical Semigroups

Every almost symmetric numerical semigroup can be constructed by removing some minimal generators from an irreducible numerical semigroup with its same Frobenius number.