Computing maximal chains

In (Fund Math 60:175–186 1967), Wolk proved that every well partial order (wpo) has a maximal chain; that is a chain of maximal order type. (Note that all chains in a wpo are well-ordered.) We prove that such maximal chain cannot be found computably, not even hyperarithmetically: No hyperarithmetic set can compute maximal chains in all computable wpos. However, we prove that almost every set, in the sense of category, can compute maximal chains in all computable wpos. Wolk’s original result actually shows that every wpo has a strongly maximal chain, which we define below. We show that a set computes strongly maximal chains in all computable wpo if and only if it computes all hyperarithmetic sets.     

Singular sets and maximal topologies

Spaces which are maximal with respect to a semi-regular property are characterised. Furthermore, a method to construct such topologies is given. Consequently, new characterisations of maximal pseudocompact spaces and of maximal Q.H.C. spaces are presented. Known characterisations of maximal connected spaces and of maximal feebly compact spaces are given alternative proofs.

     

Dunkl-spherical maximal function

The purpose of this paper is to present some recent results for maximal functions and to study the \(L^p\)-boundedness of the spherical maximal function associated to the Dunkl operators.     

Almost maximal spaces

A topological space X is called almost maximal if it is without isolated points and for every x∈X, there are only finitely many ultrafilters on X converging to x. We associate with every countable regular homogeneous almost maximal space X a finite semigroup Ult(X) so that if X and Y are homeomorphic, Ult(X) and Ult(Y) are isomorphic. Semigroups Ult(X) are projectives in the category F of finite semigroups. These are bands decomposing into a certain chain of rectangular components. Under MA, for each projective S in F, there is a countable almost maximal topological group G with Ult(G) isomorphic to S. The existence of a countable almost maximal topological group cannot be established in ZFC. However, there are in ZFC countable regular homogeneous almost maximal spaces X with Ult(X) being a chain of idempotents.     

Cubic maximal nontraceable graphs

We determine a lower bound for the number of edges of a 2-connected maximal nontraceable graph, and present a construction of an infinite family of maximal nontraceable graphs that realize this bound.     

Normalizers of maximal tori

Normalizers and p-normalizers of maximal tori in p-compact groups can be characterized by the Euler characteristic of the associated homogeneous spaces. Applied to centralizers of elementary abelian p-groups these criteria show that the normalizer of a maximal torus of the centralizer is given by the centralizer of a preferred homomorphism to the normalizer of the maximal torus; i.e. that “normalizer” commutes with “centralizer”. Received April 1, 1995; in final form August 11, 1997     

The boundedness of maximal Bochner-Riesz operator and maximal commutator on Morrey type spaces

The boundedness of maximal Bochner-Riesz operator Bδ* and that of maximal commutator Bδb,* generated by this operator and Lipschitz function on the classical Morrey space and generalized Morrey space are established.     

Some maximal arcs in Hall planes

In 1974 J.A. Thas constructed a class of maximal arcs in certain translation planes of square order, including the Desarguesian ones, but not the Hall planes. We construct a family of maximal arcs in the Hall planes inherited from the Thas maximal arcs in the Desarguesian planes. In particular, maximal arcs are shown to exist in all Hall planes of even order.The author gratefully acknowledges the support of an Australian Postgraduate Research Award.     

On maximal triangle-free graphs

A triangle-free graph is maximal if adding any edge will create a triangle. The minimal number of edges of a maximal triangle-free graph on n vertices having maximal degree at most D is denoted by F(n, D). We determine the value of lim_n-∞ F(n, cn)/n for 2/5 < c < 1/2. This investigation continues work done by Z. Füredi and Á. Seress. Our result is contrary to a conjecture of theirs.     

A class of maximal plurisubharmonic functions

In this note, we introduce a class of maximal plurisubharmonic functions and use that class to prove some properties of maximal plurisubharmonics functions.     

Almost maximal volume entropy

We prove the existence of manifolds with almost maximal volume entropy which are not hyperbolic.     

On some maximal non-selfadjoint operator algebras

We show that the reflexive algebra given by the lattice generated by a maximal nest and a rank one projection is maximal with respect to its diagonal.     

On maximal curves with Frobenius dimension 3

Frobenius dimension is one of the most important birational invariants of maximal curves. In this paper, a characterization of maximal curves with Frobenius dimension equal to 3 is provided. Our main tool is the Natural Embedding Theorem for maximal curves. As an application, maximal curves with Frobenius dimension 3 defined over the fields with 16 and 25 elements are completely classified.     

A class of affine maximal hypersurfaces

We describe a class of affine maximal surfaces lying in the threedimensional affine space. Every bounded domain in the plane may covered by an unbounded domain which is the range of definition of an affine maximal surface.     

Remarks on maximal left ideals in semigroups

Some similar results to those for maximal (two-sided) ideals in a compact semigroupS are obtained for maximal left ideals inS, with one exception i.e. the intersection of all maximal left ideals inS may be empty. The maximal left ideals in the convolution semigroup of measures onS are also considered.     

Symmetric sign patterns with maximal inertias

The inertia of an n by n symmetric sign pattern is called maximal when it is not a proper subset of the inertia of another symmetric sign pattern of order n. In this note we classify all the maximal inertias for symmetric sign patterns of order n, and identify symmetric sign patterns with maximal inertias by using a rank-one perturbation.     

Efficient initials for computing maximal eigenpair

This paper introduces some efficient initials for a well-known algorithm (an inverse iteration) for computing the maximal eigenpair of a class of real matrices. The initials not only avoid the collapse of the algorithm but are also unexpectedly efficient. The initials presented here are based on our analytic estimates of the maximal eigenvalue and a mimic of its eigenvector for many years of accumulation in the study of stochastic stability speed. In parallel, the same problem for computing the next to the maximal eigenpair is also studied.     

Isomorphisms between sub-strongly reducible maximal triangular algebras

In this paper, we introduce the concept of sub-strongly maximal triangular algebras which form a large class of maximal triangular algebras, and prove that every algebraic isomorphism of sub-strongly maximal triangular algebras is spatially implemented, which generalizes the result by Ringrose in two respects.

     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.