Maximality Theorem Research Articles

Minimal Terracini Loci in a Plane and Their Generalizations

We study properties of the minimal Terracini loci, i.e., families of certain zero-dimensional schemes, in a projective plane. Among the new results here are: a maximality theorem and the existence of arbitrarily large gaps or non-gaps for the integers x for which the minimal Terracini locus in degree d is non-empty. We study similar theorems for the critical schemes of the minimal Terracini sets. This part is framed in a more general framework.

A Maximality Theorem for Continuous First Order Theories

In this paper we prove a Lindström like theorem for the logic consisting of arbitrary Boolean combinations of first order sentences. Specifically we show the logic obtained by taking arbitrary, possibly infinite, Boolean combinations of first order sentences in countable languages is the unique maximal abstract logic which is closed under finitary Boolean operations, has occurrence number ω1, has the downward Lüowenheim-Skolem property to ωand the upward Lüowenheim-Skolem property to uncountability, and contains all complete first order theories in countable languages as sentences of the abstract logic. We will also show a similar result holds in the continuous logic framework of [5], i.e. we prove a Lindström like theorem for the abstract continuous logic consisting of Boolean combinations of first order closed conditions. Specifically we show the abstract continuous logic consisting of arbitrary Boolean combinations of closed conditions is the unique maximal abstract continuous logic which is closed under approximate isomorphisms on countable structures, is closed under finitary Boolean operations, has occurrence number ω1, has the downward Lüowenheim-Skolem property toω, the upward Lüowenheim-Skolem property to uncountability and contains all first order theories in countable languages as sentences of the abstract logic.

A relational improvement of a true particular case of Fierro’s maximality theorem

In this paper, by using relational notations, we improve and supplement a true particular case of an inaccurate maximality theorem of Ra?l Fierro from 2017, which has to be proved in addition to Zorn?s lemma and a famous maximality principle of H. Br?zis and F. Browder.

A solution of the maximality problem for one-parameter dynamical systems

We prove a maximality theorem for one-parameter dynamical systems that include W*-, C*- and multiplier one-parameter dynamical systems. Our main result is new even for one-parameter actions on commutative multiplier algebras including the algebra Cb(R) of bounded continuous functions on R acted upon by translations. The methods we develop and use in our characterization of maximality include harmonic analysis, topological vector spaces and operator algebra techniques.

Presence or absence of analytic structure in maximal ideal spaces

We study extensions of Wermer's maximality theorem to several complex variables. We exhibit various smoothly embedded manifolds in complex Euclidean space whose hulls are non-trivial but contain no analytic disks. We answer a question posed by Lee Stout concerning the existence of analytic structure for a uniform algebra whose maximal ideal space is a manifold.

The arithmetic of curves defined by iteration

We show how the size of the Galois groups of iterates of a quadratic polynomial $f(x)$ can be parametrized by certain rational points on the curves $C_n:y^2=f^n(x)$ and their quadratic twists. To that end, we study the arithmetic of such curves over global and finite fields, translating key problems in the arithmetic of polynomial iteration into a geometric framework. This point of view has several dynamical applications. For instance, we establish a maximality theorem for the Galois groups of the fourth iterate of quadratic polynomials $x^2+c$, using techniques in the theory of rational points on curves. Moreover, we show that the Hall-Lang conjecture on integral points of elliptic curves implies a Serre-type finite index result for these dynamical Galois groups, and we use conjectural bounds for the Mordell curves to predict the index in the still unknown case when $f(x)=x^2+3$. Finally, we provide evidence that these curves defined by iteration have geometrical significance, as we construct a family of curves whose rational points we completely determine and whose geometrically simple Jacobians have complex multiplication and positive rank.

Omitting uncountable types and the strength of [formula omitted]-valued logics

We study a class of [0,1]-valued logics. The main result of the paper is a maximality theorem that characterizes these logics in terms of a model-theoretic property, namely, an extension of the omitting types theorem to uncountable languages.

Uniform algebras and approximation on manifolds

Let $\Omega \subset \mathbb{C}^n$ be a bounded domain and let $\mathcal{A} \subset \mathcal{C}(\bar{\Omega})$ be a uniform algebra generated by a set $F$ of holomorphic and pluriharmonic functions. Under natural assumptions on $\Omega$ and $F$ we show that the only obstruction to $\mathcal{A} = \mathcal{C}(\bar{\Omega})$ is that there is a holomorphic disk $D \subset \bar{\Omega}$ such that all functions in $F$ are holomorphic on $D$, i.e., the only obstruction is the obvious one. This generalizes work by A. Izzo. We also have a generalization of Wermer's maximality theorem to the (distinguished boundary of the) bidisk.

Regular embeddings of the stationary tower and Woodin's maximality theorem

AbstractWe present Woodin's proof that if there exists a measurable Woodin cardinal δ then there is a forcing extension satisfying all sentences ϕ such that CH + ϕ holds in a forcing extension of V by a partial order in Vδ. We also use some of the techniques from this proof to show that if there exists a stationary limit of stationary limits of Woodin cardinals, then in a homogeneous forcing extension there is an elementary embedding j: V → M with critical point such that M is countably closed in the forcing extension.

The Craig Interpolation Theorem in abstract model theory

The Craig Interpolation Theorem is intimately connected with the emergence of abstract logic and continues to be the driving force of the field. I will argue in this paper that the interpolation property is an important litmus test in abstract model theory for identifying “natural,” robust extensions of first order logic. My argument is supported by the observation that logics which satisfy the interpolation property usually also satisfy a Lindström type maximality theorem. Admittedly, the range of such logics is small.

Modules over subalgebras of the disk algebra

This paper deals with the problem of characterizing sub- modules of C(T) over certain subalgebras of the disk algebra A. We obtain results that are analogues of the classical characterizations of sub- spaces of C(T) that are invariant under multiplication by z, i.e., that are submodules over A. These characterizations yield generalizations of Wermer's maximality theorem applicable to these subalgebras. We also present an invariant subspace theorem that seems to be of independent interest and show the equivalence of the classical theorem of the brothers Riesz on analytic measures with the theorem of Fatou.

A Characterization of Generalized Příkrý Sequences

A generalization of Přikrý's forcing is analyzed which adjoins to a model of ZFC a set of order type at most ω below each member of a discrete set of measurable cardinals. A characterization of generalized Přikrý generic sequences reminiscent of Mathias' criterion for Přikrý genericity is provided, together with a maximality theorem which states that a generalized Přikrý sequence almost contains every other one lying in the same extension. This forcing can be used to falsify the covering lemma for a higher core model if there is an inner model with infinitely many measurable cardinals – changing neither cardinalities nor cofinalities. Another application is an alternative proof of a theorem of Mitchell stating that if the core model contains a regular limit θ of measurable cardinals, then there is a model in which every set of measurable cardinals of K bounded in θ has an indiscernible sequence but there is no such sequence for the entire set of measurables of K below θ.

A note on a theorem of Ahern and Sarason

A strong weak* maximality theorem due to Ahern and Sarason is simply deduced from a more easily obtained and weaker maximality theorem of Garnett.

A maximality theorem concerning extreme points

A maximality theorem concerning extreme points

On Cluster Sets which Satisfy a Maximality Theorem

On Cluster Sets which Satisfy a Maximality Theorem

A radial uniqueness theorem for meromorphic functions

A classical theorem of Lusin and Privalov states that a meromorphic function in the unit disc, which has radial limit zero on a set which is both of second category and metrically dense in some boundary arc, must vanish identically. We prove below a radial uniqueness theorem which includes the Lusin-Privalov theorem as a special case and which also generalises the Barth-Schneider-Tse asymptotic analogue of the F. and M. Riesz radial uniqueness theorem. The part of the proof relating to Baire category is disposed of by using the Collingwood maximality theorem.

A generalization of Hartog's theorem relating to the subalgebras of continuous functions on the n-dimensional torus

The polydisk algebra, A, generated by the n-coordinate functions, ZI ,***, n,, on D” (D denotes the unit disk) identifies with a closed subalgebra of C(7’“) (T denotes the unit circle) and it is natural to study those closed subalgebras of C(P) containing A. Unlike the case for n = 1 (here, A is maximal in C(T) by Wermers maximality theorem), there are many algebras lying properly in between A and C(7”) for n > 1. Although a reasonable characterization of these algebras appears to be exceedingly difficult, the characterization of those whose maximal ideal spaces imbed “nicely” inside D” ‘is accomplished by Theorems I and I’. Theorem I also generalizes Hartog’s classical extension theorem [3; Theorem 5, Chap. I, Sect. C, p. 201, giving rise to a large class of “extension” subsets of D”, of which its topological boundary is but one example. Other examples relating to a certain class of subalgebras of C(P) containing A occur in Section iii.

On Wermer's maximality theorem

On Wermer's maximality theorem

A Maximality Theorem in Fourier Analysis

A Maximality Theorem in Fourier Analysis

A maximality theorem in Fourier analysis

Wermer's well-known maximality theorem [4], [5] has been generalized by Hoffman and Singer [2] as follows. Let G be an ordered abelian group with nonnegative class G+, and dual group P. Then the subalgebra A of C(F) generated by G+ can be extended to a maximal subalgebra of C(F), provided there exists a homomorphism t:#90 of G into the additive group R with t < 0 on G+. The homomorphism t is unique to within multiplication by a positive scalar, and the maximal subalgebras so obtained are translates of each other, as set forth precisely in [2]. (See [1, pp. 193-194] for an interesting investiga tion of maximality. A is a Dirichlet algebra, [1].) Our theorem is a converse: suppose that B is a maximal proper closed subalgebra of C(F), or of li(G), and that for each element g of G, either gEB or g-'eB. Of course, l1(G) is construed as a (dense) subalgebra of C(F) via the Fourier transform.