We present a dual sequent calculus for the necessity fragment of the constructive modal logic S4 and show its adequacy for proof-search in the style of modern interactive theorem provers. The main feature of dual systems is the use of two contexts to capture the notions of true and valid formulas, without using any formal semantics. This attribute allows us to give simple rules for the □ operator that, most of the time, grant the substitution of a strict modal reasoning by a pure propositional one, thus simplifying the proof-search process. Moreover, we introduce a formal notion of backward proof corresponding to a bottom-up construction of a derivation tree by means of a left-to-right depth-first proof-search.

A domain-theoretic method for solving initial value problems (IVPs) is presented, together with proofs of soundness, completeness, and some results on the algebraic complexity of the method. While the common fixed-precision interval arithmetic methods are restricted by the precision of the underlying machine architecture, domain-theoretic methods may be complete, i.e., the result may be obtained to any degree of accuracy. Furthermore, unlike methods based on interval arithmetic which require access to the syntactic representation of the vector field, domain-theoretic methods only deal with the semantics of the field, in the sense that the field is assumed to be given via finitely-representable approximations, to within any required accuracy.In contrast to the domain-theoretic first-order Euler method, the second-order method uses the local Lipschitz properties of the field. This is achieved by using a domain for Lipschitz functions, whose elements are consistent pairs that provide approximations of the field and its local Lipschitz properties. In the special case where the field is differentiable, the local Lipschitz properties are exactly the local differential properties of the field. In solving IVPs, Lipschitz continuity of the field is a common assumption, as a sufficient condition for uniqueness of the solution. While the validated methods for solving IVPs commonly impose further restrictions on the vector field, the second-order Euler method requires no further condition. In this sense, the method may be seen as the most general of its kind.To avoid complicated notations and lengthy arguments, the results of the paper are stated for the second-order Euler method. Nonetheless, the framework, and the results, may be extended to any higher-order Euler method, in a straightforward way.

The bond-calculus is a language for modelling interactions between continuous populations of biomolecular agents. The calculus combines process-algebra descriptions of individual agent behaviour with affinity patterns, which can specify a wide variety of patterns of interactions between the sites of different agents. These affinity patterns extend binary molecular affinities to multiway reactions, general kinetic laws, and cooperative interactions. In this paper we explore bond-calculus modelling of gene regulation at both the molecular and network levels. At the molecular level, we show how affinity patterns can succinctly describe the λ-switch, a prototypical example of cooperative regulation. Moving to the network level, we develop a general model of gene regulatory networks using affinity patterns and an expanded Hill kinetic law. We illustrate the approach with a specific example: the complex plant circadian clock. We analyse these models via the bond-calculus's differential equation and stochastic semantics, and validate our results against existing models from the literature.

Medical imaging help medical doctors provide faster and more efficient diagnoses to their patients. Medical image quality directly influences diagnosis. However, when medical images are acquired, they often present degradations such as poor detail or low contrast. This work presents an algorithm that improves contrast and detail, preserving the natural brightness of medical images. The proposed method is based on multiscale top-hat transform by reconstruction. It extracts multiple features from the image that are then used to enhance the medical image. To quantify the performance of the proposed method, 100 medical images from a public database were used. Experiments show that the proposal improves contrast, introducing less distortion and preserving the average brightness of medical images.

Bisimulation is a concept that captures behavioural equivalence. It has been studied extensively on nonprobabilistic systems and on discrete-time Markov processes and on so-called continuous-time Markov chains. In the latter, time is continuous but the evolution still proceeds in jumps. We propose two definitions of bisimulation on continuous-time stochastic processes where the evolution is a flow through time. We show that they are equivalent and we show that when restricted to discrete-time, our concept of bisimulation encompasses the standard discrete-time concept. The concept we introduce is not a straightforward generalization of discrete-time concepts.