Extending relAPS to first order logic

RelAPS is an interactive system assisting in proving relation-algebraic theorems. The aim of the system is to provide an environment where a user can perform a relation-algebraic proof similar to doing it using pencil and paper. The previous version of RelAPS accepts only Horn-formulas. To extend the system to first order logic, we have defined and implemented a new language based on theory of allegories as well as a new calculus. The language has two different kinds of terms; object terms and relational terms, where object terms are built from object constant symbols and object variables, and relational terms from typed relational constant symbols, typed relational variables, typed operation symbols and the regular operations available in any allegory. The calculus is a mixture of natural deduction and the sequent calculus. It is formulated in a sequent style but with exactly one formula on the right-hand side. We have shown soundness and completeness of this new logic which verifies that the underlying proof system of RelAPS is working correctly.

An Interactive Theorem Prover for First-Order Dynamic Logic

Dynamic logic is an extension of modal logic originally intended for reasoning about computer programs. The method of proving correctness of properties of a computer program using the well-known Hoare Logic can be implemented by utilizing the robustness of dynamic logic. For a very broad range of languages and applications in program veri cation, a theorem prover named KIV (Karlsruhe Interactive Veri er) Theorem Prover has already been developed. But a high degree of automation and its complexity make it di cult to use it for educational purposes. My research work is motivated towards the design and implementation of a similar interactive theorem prover with educational use as its main design criteria. As the key purpose of this system is to serve as an educational tool, it is a self-explanatory system that explains every step of creating a derivation, i.e., proving a theorem. This deductive system is implemented in the platform-independent programming language Java. In addition, a very popular combination of a lexical analyzer generator, JFlex, and the parser generator BYacc/J for parsing formulas and programs has been used.

A System for Models of First Order Theories

If you want to know whether a property is true or not in a specific algebraic structure,you need to test that property on the given structure. This can be done by hand, which can be cumbersome and erroneous. In addition, the time consumed in testing depends on the size of the structure where the property is applied. We present an implementation of a system for finding counterexamples and testing properties of models of first-order theories. This system is supposed to provide a convenient and paperless environment for researchers and students investigating or studying such models and algebraic structures in particular. To implement a first-order theory in the system, a suitable first-order language.( and some axioms are required. The components of a language are given by a collection of variables, a set of predicate symbols, and a set of operation symbols. Variables and operation symbols are used to build terms. Terms, predicate symbols, and the usual logical connectives are used to build formulas. A first-order theory now consists of a language together with a set of closed formulas, i.e. formulas without free occurrences of variables. The set of formulas is also called the axioms of the theory. The system uses several different formats to allow the user to specify languages, to define axioms and theories and to create models. Besides the obvious operations and tests on these structures, we have introduced the notion of a functor between classes of models in order to generate more co~plex models from given ones automatically. As an example, we will use the system to create several lattices structures starting from a model of the theory of pre-orders.

Cure Kinetics and Sorption Characteristics of Neoprene-Based Rubber Ferrite Composites

Rubber ferrite composites were prepared by incorporating nickel ferrite in a neoprene rubber matrix. Kinetics of the cure reaction were determined from the rheometric torque values and found to follow first-order kinetics. Analysis of the swelling behavior of the rubber ferrite composites in toluene elucidates the mechanism of solvent penetration and sorption characteristics, and reveals the extent of the physical interaction of the ferrite particles with the neoprene rubber matrix. Mechanical properties of rubber ferrite composites were determined, which support the reinforcing nature of nickel ferrite to the neoprene rubber matrix. These results show that magnetic composites with the required processing safety can be prepared economically by incorporating higher amounts of nickel ferrite in the neoprene rubber matrix

First-order molecular descriptors for molecular steric similarity

En aquest treball s'analitza la contribució estèrica de les molècules a les seves propietats químiques i físiques, mitjançant l'avaluació del seu volum i de la seva mesura de semblança, a partir d'ara definits com a descriptors moleculars de primer ordre. La difeèsncia entre aquests dos conceptes ha estat aclarida: mentre que el volum és la magnitud de l'espai que ocupa la molècula com a entitat global, la mesura de semblança ens dóna una idea de com està distribuïda la densitat electrònica al llarg d'aquest volum, i reflecteix més les diferències locals existents. L'ús de diverses aproximacions per a l'obtenció d'ambdós valors ha estat analitzat sobre diferents classes d'isòmers

The kinetics of the 2π+2π photodimerisation reactions of single-crystalline derivatives of trans-cinnamic acid: a study by infrared microspectroscopy

The kinetics of the photodimerisation reactions of the 2- and 4-β-halogeno-derivatives of trans-cinnamic acid (where the halogen is fluorine, chlorine or bromine) have been investigated by infrared microspectroscopy. It is found that none of the reactions proceed to 100% yield. This is in line with a reaction mechanism developed by Wernick and his co-workers that postulates the formation of isolated monomers within the solid, which cannot react. β-4-Bromo and β-4-chloro-trans-cinnamic acids show approximately first order kinetics, although in both cases the reaction accelerates somewhat as it proceeds. First order kinetics is explained in terms of a reaction between one excited- and one ground-state monomer molecule, while the acceleration of the reaction implies that it is promoted as defects are formed within the crystal. By contrast β-2-chloro-trans-cinnamic acid shows a strongly accelerating reaction which models closely to the contracting cube equation. β-2-Fluoro- and β-4-fluoro-trans-cinnamic acids show a close match to first order kinetics. The 4-fluoro-derivative, however, shows a reaction that proceeds via a structural intermediate. The difference in behaviour between the 2-fluoro- and 4-fluoro-derivative may be due to different C–HF hydrogen bonds observed within these single-crystalline starting materials.

Determination of the degradation kinetics of anthocyanins in a model juice system using isothermal and non-isothermal methods

The effect of temperature on the degradation of blackcurrant anthocyanins in a model juice system was determined over a temperature range of 4–140 °C. The thermal degradation of anthocyanins followed pseudo first-order kinetics. From 4–100 °C an isothermal method was used to determine the kinetic parameters. In order to mimic the temperature profile in retort systems, a non-isothermal method was applied to determine the kinetic parameters in the model juice over the temperature range 110–140 °C. The results from both isothermal and non-isothermal methods fit well together, indicating that the non-isothermal procedure is a reliable mathematical method to determine the kinetics of anthocyanin degradation. The reaction rate constant (k) increased from 0.16 (±0.01) × 10−3 to 9.954 (±0.004) h−1 at 4 and 140 °C, respectively. The temperature dependence of the rate of anthocyanin degradation was modelled by an extension of the Arrhenius equation, which showed a linear increase in the activation energy with temperature.

The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory

We investigate Fréchet differentiability of the scattered field with respect to variation in the boundary in the case of time–harmonic acoustic scattering by an unbounded, sound–soft, one–dimensional rough surface. We rigorously prove the differentiability of the scattered field and derive a characterization of the Fréchet derivative as the solution to a Dirichlet boundary value problem. As an application of these results we give rigorous error estimates for first–order perturbation theory, justifying small perturbation methods that have a long history in the engineering literature. As an application of our rigorous estimates we show that a plane acoustic wave incident on a sound–soft rough surface can produce an unbounded scattered field.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.