Wesley Calvert
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View article: Homogeneous Linear Orderings: Index sets, Approximations and Categoricity
Homogeneous Linear Orderings: Index sets, Approximations and Categoricity Open
We study linear orderings expanded by functions for successor and predecessor. In particular, the sp-homogeneous and weakly sp-homogeneous linear orderings are those which are homogeneous or weakly homogeneous with this additional structur…
View article: Generically computable Abelian groups and isomorphisms
Generically computable Abelian groups and isomorphisms Open
Approximate computability, in the form of generically computable sets introduced by Jockusch and Schupp, was motivated by asymptotic density problems studied by Gromov in combinatorial group theory. More recently, we have defined notions o…
View article: Generically computable linear orderings
Generically computable linear orderings Open
We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the Σβ hierarchy. We focus on linear orderings. We show that at the Σ1 level, all linear orderings have both …
View article: In Memory of Martin Davis
In Memory of Martin Davis Open
This is a memorial tribute to the distinguished scholar Martin Davis, who gave outstanding contributions to the development of computability theory and of symbolic logic
View article: Generically Computable Linear Orderings
Generically Computable Linear Orderings Open
We study notions of generic and coarse computability in the context of computable structure theory. Our notions are stratified by the $Σ_β$ hierarchy. We focus on linear orderings. We show that at the $Σ_1$ level all linear orderings have …
View article: In Memory of Martin Davis
In Memory of Martin Davis Open
The present paper gives an account for the general mathematical reader of the life and work of Martin Davis. Since two rather comprehensive autobiographical accounts and two long biographical interviews already exist, the present work focu…
View article: Normality, Relativization, and Randomness
Normality, Relativization, and Randomness Open
Normal numbers were introduced by Borel and later proven to be a weak notion of algorithmic randomness. We introduce here a natural relativization of normality based on generalized number representation systems. We explore the concepts of …
View article: Computability in infinite Galois theory and algorithmically random algebraic fields
Computability in infinite Galois theory and algorithmically random algebraic fields Open
We introduce a notion of algorithmic randomness for algebraic fields. We prove the existence of a continuum of algebraic extensions of $\mathbb{Q}$ that are random according to our definition. We show that there are noncomputable algebraic…
View article: STRUCTURAL HIGHNESS NOTIONS
STRUCTURAL HIGHNESS NOTIONS Open
We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and s…
View article: INTERPRETING A FIELD IN ITS HEISENBERG GROUP
INTERPRETING A FIELD IN ITS HEISENBERG GROUP Open
We improve on and generalize a 1960 result of Maltsev. For a field F , we denote by $H(F)$ the Heisenberg group with entries in F . Maltsev showed that there is a copy of F defined in $H(F)$ , using existential formulas with an arbitrary n…
View article: Structural Highness Notions
Structural Highness Notions Open
We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and s…
View article: Strong jump inversion
Strong jump inversion Open
We say that a structure $\\mathcal{A}$ admits \\emph{strong jump inversion}\nprovided that for every oracle $X$, if $X'$ computes $D(\\mathcal{C})'$ for some\n$\\mathcal{C}\\cong\\mathcal{A}$, then $X$ computes $D(\\mathcal{B})$ for some\n…
View article: Generically Computable Equivalence Structures and Isomorphisms
Generically Computable Equivalence Structures and Isomorphisms Open
We define notions of generically and coarsely computable relations and structures and functions between structures. We investigate the existence and uniqueness of equivalence structures in the context of these definitions
View article: Genericity and UD-random reals
Genericity and UD-random reals Open
Avigad introduced the notion of UD-randomness based in Weyl's 1916 definition of uniform distribution modulo one. We prove that there exists a weakly 1-random real that is neither UD-random nor weakly 1-generic. We also show that no 2-gene…