Partial metric spaces with negative distances and fixed point theorems

In this paper we consider partial metric spaces in the sense of O'Neill. We introduce the notions of strong partial metric spaces and Cauchy functions. We prove a fixed point theorem for such spaces and functions that improves Matthews' contraction mapping theorem in two ways. First, the existence of fixed points now holds for a wider class of functions and spaces. Second, our theorem also allows for fixed points with nonzero self-distances. We also prove fixed point theorems for orbitally $r$-contractive and orbitally $\phi_r$-contractive maps. We then apply our results to give alternative proofs of some of the other known fixed point theorems in the context of partial metric spaces.Comment: 19 page

Multiplicative Valued Difference Fields

The theory of valued difference fields $(K, \sigma, v)$ depends on how the valuation $v$ interacts with the automorphism $\sigma$. Two special cases have already been worked out - the isometric case, where $v(\sigma(x)) = v(x)$ for all $x\in K$, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where $v(\sigma(x)) > n\cdot v(x)$ for all $n\in\mathbb{N}$ and $x\in K^\times$ with $v(x) > 0$, has been worked out by Salih Azgin. In this paper we deal with a more general version, called the multiplicative case, where $v(\sigma(x)) = \rho\cdot v(x)$, where $\rho (> 0)$ is interpreted as an element of a real-closed field. We give an axiomatization and prove a relative quantifier elimination theorem for such a theory.Comment: 37 page

Conformal integrals in various dimensions and Clifford groups

Conformal transformations of the Euclidean space of any dimension can be formulated as the M\"obius group in terms of Clifford algebras. Correlation functions of a conformal field theory are expressed in terms of conformal integrals, which furnish solutions to linear systems on the configuration space of points on the Euclidean space. Expressing the conformal integral in terms of a Clifford algebra the linear system is related to toric GKZ systems. Explicit series solutions for conformal integrals are obtained using the power of toric methods as GKZ hypergeometric functions. Partial permutation symmetry of the solutions dictated by the toric method is used to relate the solutions to Feynman trees, or, channels, in field theory parlance.Comment: Largely expanded with conformal integral written explicitly as A-hypergeometric series solutions for GKZ systems for N=4,5,6 points. Symmetry of solutions discussed and connection to channels established. LaTeX 1+20 pages. 2 figure

Basic Relevant Theories for Combinators at Levels I and II

The system B+ is the minimal positive relevant logic. B+ is trivially extended to B+T on adding a greatest truth (Church constant) T. If we leave ∨ out of the formation apparatus, we get the fragment B∧T. It is known that the set of ALL B∧T theories provides a good model for the combinators CL at Level-I, which is the theory level. Restoring ∨ to get back B+T was not previously fruitful at Level-I, because the set of all B+T theories is NOT a model of CL. It was to be expected from semantic completeness arguments for relevant logics that basic combinator laws would hold when restricted to PRIME B+T theories. Overcoming some previous difficulties, we show that this is the case, at Level I. But this does not form a model for CL. This paper also looks for corresponding results at Level-II, where we deal with sets of theories that we call propositions. We adapt work by Ghilezan to note that at Level-II also there is a model of CL in B∧T propositions. However, the corresponding result for B+T propositions extends smoothly to Level-II only in part. Specifically, only some of the basic combinator laws are proved here. We accordingly leave some work for the reader.&nbsp