On Kaufmanʼs theorem

Kaufman?s theorem (Kaufman, 1978 [9]) on representing closed linear operators as quotients of bounded operators is given a new constructive proof, and is extended to operators between two Hilbert spaces. Two different definitions of operator quotients existing in the literature are compared.     

Funayama’s theorem revisited

Funayama’s theorem states that there is an embedding e of a lattice L into a complete Boolean algebra B such that e preserves all existing joins and meets in L iff L satisfies the join infinite distributive law (JID) and the meet infinite distributive law (MID). More generally, there is a lattice embedding e: L → B preserving all existing joins in L iff L satisfies (JID), and there is a lattice embedding e: L → B preserving all existing meets in L iff L satisfies (MID). Funayama’s original proof is quite involved. There are two more accessible proofs in case L is complete. One was given by Grätzer by means of free Boolean extensions and MacNeille completions, and the other by Johnstone by means of nuclei and Booleanization. We show that Grätzer’s proof has an obvious generalization to the non-complete case, and that in the complete case the complete Boolean algebras produced by Grätzer and Johnstone are isomorphic. We prove that in the non-complete case, the class of lattices satisfying (JID) properly contains the class of Heyting algebras, and we characterize lattices satisfying (JID) and (MID) by means of their Priestley duals. Utilizing duality theory, we give alternative proofs of Funayama’s theorem and of the isomorphism between the complete Boolean algebras produced by Grätzer and Johnstone. We also show that unlike Grätzer’s proof, there is no obvious way to generalize Johnstone’s proof to the non-complete case.     

An application of ramsey’s theorem to groups with homogeneous theory

For any integers n >m ≥ 2, we say that a complete theory T is (m, n)-homogeneous if, for each model M of T, two n-tuples [abar],[bbar] in M have the same type if the corresponding m-tuples from [abar] and [bbar] have the same type. It was conjectured by H. Kikyo that, if M is an infinite group, with possibly additional structure, then the theory of M is not (m, n)-homogeneous. We prove a general result on structures with (m, n)-homogeneous theory which implies that, if M is a counterexample to this conjecture, then there exists an integer h such that each abelian subgroup of M has at most h elements. It follows that there exist an integer k such that M ^k = 1, and an integer l such that each finite subgroup of M has at most l elements.     

Favardʼs theorem of piecewise continuous almost periodic functions and its application

In this paper, by constructing Bochner–Fejér polynomials for piecewise continuous almost periodic functions (PCAP, for short), the authors establish Favard?s theorem of PCAP functions, which illustrates when the primitive function of PCAP function is a PCAP function. As its application, combining coincidence degree theory, we consider the existence of PCAP solution of impulsive single population model with hereditary effects. To our best knowledge, it is the first time when coincidence degree theory is used to study the existence of PCAP solution of impulsive differential equation.     

Yan’s oscillation theorem revisited

Yan’s contribution [J. Yan, Oscillation theorems for second order linear differential equations with damping, Proc. Amer. Math. Soc. 98 (1986) 276–282] was an important breakthrough in the development of the Theory of Oscillation. This frequently cited paper has stimulated extensive investigations in the field. During the last decade, an integral oscillation technique has been developed to such an extent as to allow us to revisit Yan’s fundamental oscillation theorem and remove one of the conditions, leaving the other assumptions and the conclusion intact, thus enhancing this keystone result.     

Grothendieck’s comparison theorem and multivariable Fredholm theory

We use a variant of Grothendieck’s comparison theorem to show that, for a Fredholm tuple T ∈ L(X)ⁿ on a complex Banach space, there are isomorphisms . We conclude that a Fredholm tuple T ∈ L(X)ⁿ satisfies Bishop’s property (β) at z = 0 if and only if the vanishing conditions hold for . We apply these observations and results from commutative algebra to show that a graded tuple on a Hilbert space is Fredholm if and only if it satisfies Bishop’s property (β) at z = 0 and that, in this case, its cohomology groups can grow at most like k^p. Received: 14 January 2009     

Harnackʼs inequality for solutions to the linearized Monge–Ampère operator with lower-order terms

Under minimal geometric assumptions, we prove a Harnack inequality for non-negative solutions to the non-homogeneous linearized Monge–Ampère equation with bounded lower-order terms compatible with the maximum principle.     

Bergeʼs maximum theorem for noncompact image sets

This note generalizes Berge?s maximum theorem to noncompact image sets. It also clarifies the results from Feinberg, Kasyanov and Zadoianchuk (2013) [7] on the extension to noncompact image sets of another Berge?s theorem, that states semi-continuity of value functions. Here we explain that the notion of a K$\mathbb{K}$-inf-compact function introduced there is applicable to metrizable topological spaces and to more general compactly generated topological spaces. For Hausdorff topological spaces we introduce the notion of a KN$\mathbb{K}\mathbb{N}$-inf-compact function (N$\mathbb{N}$ stands for “nets” in K$\mathbb{K}$-inf-compactness), which coincides with K$\mathbb{K}$-inf-compactness for compactly generated and, in particular, for metrizable topological spaces.     

Nonstandard second-order arithmetic and Riemannʼs mapping theorem

In this paper, we introduce systems of nonstandard second-order arithmetic which are conservative extensions of systems of second-order arithmetic. Within these systems, we do reverse mathematics for nonstandard analysis, and we can import techniques of nonstandard analysis into analysis in weak systems of second-order arithmetic. Then, we apply nonstandard techniques to a version of Riemannʼs mapping theorem, and show several different versions of Riemannʼs mapping theorem.     

Egoroff’s theorem in measurable operator spaces associated with a Von Neumann algebra

In this article, some kinds of convergence about the τ–measurable operators affiliated with a von Neumann algebra are defined. Moreover, the relationships among these kinds of convergence are considered.     

An application of Chen’s theorem to convergence of Fourier multiplier transforms

Let V be a star shaped region. In 2006, Colzani, Meaney and Prestini proved that if function f satisfies some condition, then the multiplier transform with the characteristic function of tV as the multiplier tends to f almost everywhere, when t goes to∞. In this paper we use a Theorem established by K.K.Chen to show that if we change their multiplier, then the condition on f can be weakened.