Michael A. Bukatin

Partial Metrics and Co-continuous Valuations

The existence of deep connections between partial metrics and valuations is well known in domain theory. However, the treatment of non-algebraic continuous Scott domains has been not quite satisfactory so far.

Incorporation of reaction field effects into density functional calculations for molecules of arbitrary shape in solution

Abstract An attempt is made to combine continuum reaction field approaches with DFT ab initio calculations for quantitative evaluation of salvation effects in chemical processes. The formalism of the combined method is delineated along with its possibilities and limitations, and applied to several small model systems. It is found that DFT can provide dipole moments in vacuum and in solution (e.g., for water) with accuracies (0.1 D) that have not been reported with other methods. The results obtained suggest that agreement within

Towards Computing Distances Between Programs via Scott Domains

˜1 kcal/mole can be expected between calculated and experimental hydration enthalpies of polar unchanged solutes. The results for ions are not as consistent as for dipolar molecules, suggesting that accurate multipole representations of the electron density of solutes may be required especially for ionic solutes.

A view of thermodynamics of hydration emerging from continuum studies.

This paper introduces an approach to defining and computing distances between programs via continuous generalized distance functions ρ: A×A→D, where A and D are directed complete partial orders with the induced Scott topology, A is a semantic domain, and D is a domain representing distances (usually, some version of interval numbers). A continuous distance function ρ can define a To topology on a nontrivial domain A only if the axiom ∃0 e D.∀x e A.ρ(x,x)=0 does not hold. Hence, the notion of relaxed metric is introduced for domains — the axiom ρ(x,x)=0 is eliminated, but the axiom ρ(x,y)=ρ(y,x) and a version of the triangle inequality tailored for the domain D remain.

An intelligent theory of cost for partial metric spaces

Main physical-chemical features of hydration found in continuum studies and possible limitations of the method are analyzed. Particular attention is given to: the choice of thermodynamic observables to be compared to the calculations; representations of the solute polarizability; compensation between the loss of hydration enthalpy and gain in Coulomb interactions upon a complex formation; two minima in interaction potentials between polar groups in solution; similarities and dissimilarities between interaction potentials in solution from continuum and molecular theories; continuum calculations of entropies of hydration; and evaluation of a temperature dependence of thermodynamic characteristics of hydration with continuum methods.

Continuum based calculations of hydration entropies and the hydrophobic effect

Partial metric spaces generalise metric spaces, allowing non zero self distance. This is needed to model computable partial information, but falls short in an important respect. The present cost of computing information, such as processor time or memory used, is rarely expressible in domain theory, but contemporary theories of algorithms incorporate precise control over cost of computing resources. Complexity theory in Computer Science has dramatically advanced through an intelligent understanding of algorithms over discrete totally defined data structures such as directed graphs, without using partially defined information. So we have an unfortunate longstanding separation of partial metric spaces for modelling partially defined computable information from the complexity theory of algorithms for costing totally defined computable information. To bridge that separation we seek an intelligent theory of cost for partial metric spaces. As examples we consider the cost of computing a double negation ¬¬p in two-valued propositional logic, the cost of computing negation as failure in logic programming, and a cost model for the hiaton time delay.