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.

The Borel Complexity of Isomorphism for Some First Order Theories

In this work we consider several instances of the following problem: "how complicated can the isomorphism relation for countable models be?"' Using the Borel reducibility framework, we investigate this question with regard to the space of countable models of particular complete first-order theories. We also investigate to what extent this complexity is mirrored in the number of back-and-forth inequivalent models of the theory. We consider this question for two large and related classes of theories. First, we consider o-minimal theories, showing that if T is o-minimal, then the isomorphism relation is either Borel complete or Borel. Further, if it is Borel, we characterize exactly which values can occur, and when they occur. In all cases Borel completeness implies lambda-Borel completeness for all lambda. Second, we consider colored linear orders, which are (complete theories of) a linear order expanded by countably many unary predicates. We discover the same characterization as with o-minimal theories, taking the same values, with the exception that all finite values are possible except two. We characterize exactly when each possibility occurs, which is similar to the o-minimal case. Additionally, we extend Schirrman's theorem, showing that if the language is finite, then T is countably categorical or Borel complete. As before, in all cases Borel completeness implies lambda-Borel completeness for all lambda.

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.

An existence theorem for an analytic first order PDE in the framework of Colombeau's theory

We study the existence of a holomorphic generalized solution u of the PDE[GRAPHICS]where f is a given holomorphic generalized function and (alpha (1),...alpha (m)) is an element of C-m\{0}.

Quantum transition state theory and solving a first order master equation for complex reactions numerically

The field of chemical kinetics is an exciting and active field. The prevailing theories make a number of simplifying assumptions that do not always hold in actual cases. Another current problem concerns a development of efficient numerical algorithms for solving the master equations that arise in the description of complex reactions. The objective of the present work is to furnish a completely general and exact theory of reaction rates, in a form reminiscent of transition state theory, valid for all fluid phases and also to develop a computer program that can solve complex reactions by finding the concentrations of all participating substances as a function of time. To do so, the full quantum scattering theory is used for deriving the exact rate law, and then the resulting cumulative reaction probability is put into several equivalent forms that take into account all relativistic effects if applicable, including one that is strongly reminiscent of transition state theory, but includes corrections from scattering theory. Then two programs, one for solving complex reactions, the other for solving first order linear kinetic master equations to solve them, have been developed and tested for simple applications.

Foundational, First-Order, and Second-Order Classification Theory

Both basic and applied research on the construction, implementation, maintenance, and evaluation of classification schemes is called classification theory. If we employ Ritzer’s metatheoretical method of analysis on the over one-hundred year-old body of literature, we can see categories of theory emerge. This paper looks at one particular part of knowledge organization work, namely classification theory, and asks 1) what are the contours of this intellectual space, and, 2) what have we produced in the theoretical reflection on con- structing, implementing, and evaluating classification schemes? The preliminary findings from this work are that classification theory can be separated into three kinds: foundational classification theory, first-order classification theory, and second-order classification theory, each with its own concerns and objects of study.

A class of singular first order differential equations with applications in reaction-diffusion

Agências Financiadoras: FCT e MIUR

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.

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.

Electromagnetic field at finite temperature: A first order approach

In this work we study the electromagnetic field at finite temperature via the massless DKP formalism. The constraint analysis is performed and the partition function for the theory is constructed and computed. When it is specialized to the spin 1 sector we obtain the well-known result for the thermodynamic equilibrium of the electromagnetic field. (c) 2006 Elsevier B.V. All rights reserved.

Electromagnetic field in Lyra manifold: a first order approach

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

First order thermal phase transition with 126 GeV Higgs mass

We study the strength of the electroweak phase transition in models with two light Higgs doublets and a light SU(3)c triplet by means of lattice simulations in a dimensionally reduced effective theory. In the parameter region considered the transition on the lattice is significantly stronger than indicated by a 2-loop perturbative analysis. Within some ultraviolet uncertainties, the finding applies to MSSM with a Higgs mass mh ≈ 126 GeV and shows that the parameter region useful for electroweak baryogenesis is enlarged. In particular (even though only dedicated analyses can quantify the issue), the tension between LHC constraints after the 7 TeV and 8 TeV runs and frameworks where the electroweak phase transition is driven by light stops, seems to be relaxed.

First order mem-circuits: modeling, nonlinear oscillations and bifurcations

This paper presents a theoretical framework intended to accommodate circuit devices described by characteristics involving more than two fundamental variables. This framework is motivated by the recent appearance of a variety of so-called mem-devices in circuit theory, and makes it possible to model the coexistence of memory effects of different nature in a single device. With a compact formalism, this setting accounts for classical devices and also for circuit elements which do not admit a two-variable description. Fully nonlinear characteristics are allowed for all devices, driving the analysis beyond the framework of Chua and Di Ventra We classify these fully nonlinear circuit elements in terms of the variables involved in their constitutive relations and the notions of the differential- and the state-order of a device. We extend the notion of a topologically degenerate configuration to this broader context, and characterize the differential-algebraic index of nodal models of such circuits. Additionally, we explore certain dynamical features of mem-circuits involving manifolds of non-isolated equilibria. Related bifurcation phenomena are explored for a family of nonlinear oscillators based on mem-devices.