On the class of almost L-weakly and almost M-weakly compact operators

In this paper, we introduce and study new concepts of almost L-weakly and almost M-weakly compact operators.     

Einstein almost cokähler manifolds

We study an odd‐dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non‐compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with nonnegative *‐scalar curvature are cokähler (indeed, transversely Calabi–Yau); more generally, we give a lower and upper bound for the *‐scalar curvature in the case that the structure is not cokähler. We prove similar bounds for almost Kähler Einstein manifolds that are not Kähler.     

Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations

The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs).     

Compact indefinite almost Kähler Einstein manifolds

In the present article, we extend the integral formula on a compact almost Kähler manifold with positive-definite metric to the one on a compact indefinite almost Kähler manifold and give its applications for some special indefinite almost Kähler Einstein manifolds taking the related problems to the indefinite analogy of Goldberg conjecture into consideration.     

Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces

This work aims to study the existence and uniqueness of pseudo compact almost automorphic solution for some dissipative ordinary and functional differential equations. We prove the existence and uniqueness of pseudo compact almost automorphic solution for dissipative differential equations in Banach spaces and then we apply this result to show the existence of pseudo compact almost automorphic solutions for some functional differential equations.     

Collectively compact sets of operators and almost periodic functions

A characterization of collectively compact sets of linear operators on the Banach algebra of almost periodic functions is obtained by using Gelfand-Naimark theory. Properties of collectively compact sets are then used to prove that the Fourier coefficients of an almost periodic function are approximately the eigenvalues of Fredholm type integral operators.     

Balanced Hermitian structures on almost abelian Lie algebras

We study balanced Hermitian structures on almost abelian Lie algebras, i.e. on Lie algebras with a codimension-one abelian ideal. In particular, we classify six-dimensional almost abelian Lie algebras which carry a balanced structure. It has been conjectured in [1] that a compact complex manifold admitting both a balanced metric and an SKT metric necessarily has a Kähler metric: we prove this conjecture for compact almost abelian solvmanifolds with left-invariant complex structures. Moreover, we investigate the behaviour of the flow of balanced metrics introduced in [2] and of the anomaly flow [3] on almost abelian Lie groups. In particular, we show that the anomaly flow preserves the balanced condition and that locally conformally Kähler metrics are fixed points.     

Actions of almost simple groups on compact eight-dimensional projective planes are classical,almost

We show that the existence of an almost simple group of automorphisms of dimension greater than 10 characterizes the Hughes planes (including the quarternion plane) among the 8-dimensional compact projective planes.Dedicated to Prof. Helmut R. Salzmann on his 65th birthday     

Characterizations of almost periodic homeomorphisms

In this paper we study recurrent and almost periodic homeomorphisms on the Euclidean space Rm; we give conditions under which recurrent implies periodic. On the other hand we give properties of elements of compact groups of diffeomorphisms on manifolds.     

Four-dimensional almost Kähler Einstein manifolds

Summary We prove that a four-dimensional compact almost Kähler manifold which is Einsteinian, and Einsteinian is a Kähler manifold.     

Measure-compact operators, almost compact operators, and linear functional equations in L p

Under study are the measure-compact operators and almost compact operators in L _p . We construct an example of a measure-compact operator that is not almost compact. Introducing two classes of closed linear operators in L _p , we prove that the resolvents of these operators are almost compact or measure-compact. We present methods for the reduction of linear functional equations of the second kind in L _p with almost compact or measure-compact operators to equivalent linear integral equations in L _p with quasidegenerate Carleman kernels.     

Hyperbolic embeddability of locally complete almost complex submanifolds

In this Note, we generalize to the almost complex setting, a theorem of Zaidenberg (1983) [13] and Thai (1991) [12] by giving a characterization on hyperbolic embeddability of a locally complete and relatively compact almost complex submanifold in terms of extension of pseudo-holomorphic disks from the punctured unit disk and of limit J-complex lines.     

Weak stability of almost regular contact foliations

We prove that on a compact manifold, a contact foliation obtained by a smallC ¹ perturbation of an almost regular contact flow has at least two closed characteristics. This solves the Weinstein conjecture for contact forms which areC ¹-close to almost regular contact forms.Supported in part by NSF Grant DMS 90-01861     

Dynamics of Bohr almost periodic motions of topological abelian semigroups

We study the topological and ergodic dynamics of Bohr almost periodic motions of a topological abelian semigroup acting continuously on a compact metric space.     

Weakly almost periodic functionals on the measure algebra

It is shown that the collection of weakly almost periodic functionals on the convolution algebra of a commutative Hopf von Neumann algebra is a C*-algebra. This implies that the weakly almost periodic functionals on M(G), the measure algebra of a locally compact group G, is a C*-subalgebra of M(G)* = C ₀(G)**. The proof builds upon a factorisation result, due to Young and Kaiser, for weakly compact module maps. The main technique is to adapt some of the theory of corepresentations to the setting of general reflexive Banach spaces.     

Note on the existence of almost omplex structures on compact manifolds

In the paper we define a multiplicative genus of a compact orientable manifold. We use this genus for the study of the existence of almost complex structures on manifolds. A few applications are given, namely, we prove the nonexistence of an almost complex structure on quaternionic flag manifolds and give a theorem on the existence of an almost complex structure on the product of manifolds.     

Existence of periodic solutions of Hamiltonian systems on almost every energy surface

We estab lish the existence of periodic solutions of Hamiltonian systems on almost every smooth, compact energy surface in the sense of Lebesgue measure.     

Compact almost automorphic solutions to semilinear Cauchy problems with non-dense domain

In this work we study the existence and uniqueness of compact almost automorphic solutions to a first-order differential equation with a linear part dominated by a Hille-Yosida type operator with non dense domain.     

Compact almost automorphic solutions for some nonlinear dissipative differential equations in Banach spaces

The aim of this work is to prove the existence and uniqueness of compact almost automorphic solutions for some dissipative differential equations in Banach spaces when the input function is only almost automorphic in the sense of Stepanov. Examples and a numerical simulation are provided to illustrate the theoretical findings.     

Measure compact,almost compact,and integral operators of the first,second, and third kind

Under study are the metric, algebraic, spectral, and characteristic properties of measure compact, almost compact, and integral operators of the first, second, and third kind in spaces of integrable functions.