Conceptual Unification of Gravity and Quanta

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra \mbox{{\cal A}} on the groupoid \Gamma = E \times G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of \mbox{{\cal A}}, is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein's equation. The algebra \mbox{{\cal A}}, when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra \mathcal{M} of random operators representing the quantum sector of the model. The Tomita-Takesaki theorem allows us to define the dynamics of random operators which depends on the state \phi . The same state defines the noncommutative probability measure (in the sense of Voiculescu's free probability theory). Moreover, the state \phi satisfies the Kubo-Martin-Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra \mbox{{\cal A}}, one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not ``feel'' singularities.Comment: 28 LaTex pages. Substantially enlarged version. Improved definition of generalized Einstein's field equation

Limiting modular symbols and their fractal geometry

In this paper we use fractal geometry to investigate boundary aspects of the first homology group for finite coverings of the modular surface. We obtain a complete description of algebraically invisible parts of this homology group. More precisely, we first show that for any modular subgroup the geodesic forward dynamic on the associated surface admits a canonical symbolic representation by a finitely irreducible shift space. We then use this representation to derive an `almost complete' multifractal description of the higher--dimensional level sets arising from Manin--Marcolli's limiting modular symbols.Comment: 20 pages, 1 figur

Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra ${\cal A}$ on a transformation groupoid $\Gamma = E \times G$ where $E$ is the total space of a principal fibre bundle over spacetime, and $G$ a suitable group acting on $\Gamma$. We show that every $a \in {\cal A}$ defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita-Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra ${\cal A}$ which can be used to define a state dependent dynamics; i.e., the pair $({\cal A}, \phi)$, where $\phi$ is a state on ${\cal A}$, is a ``dynamic object''. Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on $\phi$ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair $({\cal A}, \phi)$ defines the so-called free probability calculus, as developed by Voiculescu and others, with the state $\phi$ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.Comment: 13 pages, LaTe

Biologically inspired distributed machine cognition: a new formal approach to hyperparallel computation

The irresistable march toward multiple-core chip technology presents currently intractable pdrogramming challenges. High level mental processes in many animals, and their analogs for social structures, appear similarly massively parallel, and recent mathematical models addressing them may be adaptable to the multi-core programming problem

Modelling of Modular Robot Configurations Using Graph Theory

Modular robots are systems that can change its geometry or configuration when connecting more modules or when rearranging them in a different manner to perform a variety of tasks. Graph theory can be used to describe modular robots configurations, hence the possibility to determine the flexibility of the robot to move from one point to another. When the robot’s configurations are represented in a mathematical way, forward kinematics can be obtained

A Unified Approach for Representing Structurally-Complex Models in SBML Level 3

The aim of this document is to explore a unified approach to handling several of the proposed extensions to the SBML Level 3 Core specification. The approach is illustrated with reference to Simile, a modelling environment which appears to have most of the capabilities of the various SBML Level 3 package proposals which deal with model structure. Simile (http://www.simulistics.com) is a visual modelling environment for continuous systems modelling which includes the ability to handle complex disaggregation of model structure, by allowing the modeller to specify classes of object and the relationships between them.

The note is organised around the 6 packages listed on the SBML Level 3 Proposals web page (http://sbml.org/Community/Wiki/SBML_Level_3_Proposals) which deal with model structure, namely comp, arrays, spatial, geom, dyn and multi. For each one, I consider how the requirements which motivated the package can be handled using Simile's unified approach. Although Simile has a declarative model-representation language (in both Prolog and XML syntax), I use Simile diagrams and equation syntax throughout, since this is more compact and readable than large chunks of XML.

The conclusion is that Simile can indeed meet most of the requirements of these various packages, using a generic set of constructs - basically, the multiple-instance submodel, the concept of a relationship (association) between submodels, and array variables. This suggests the possibility of having a single SBML Level 3 extension package similar to the Simile data model, rather than a series of separate packages. Such an approach has a number of potential advantages and disadvantages compared with having the current set of discrete packages: these are discussed in this paper

A validated computational framework to evaluate the stiffness of 3D printed ankle foot orthoses

The purpose of this study was to create and validate a standardized framework for the evaluation of the ankle stiffness of two designs of 3D printed ankle foot orthoses (AFOs). The creation of four finite element (FE) models allowed patient-specific quantification of the stiffness and stress distribution over their specific range of motion during the second rocker of the gait. Validation was performed by comparing the model outputs with the results obtained from a dedicated experimental setup, which showed an overall good agreement with a maximum relative error of 10.38% in plantarflexion and 10.66% in dorsiflexion. The combination of advanced computer modelling algorithms and 3D printing techniques clearly shows potential to further improve the manufacturing process of AFOs

Realising the open virtual commissioning of modular automation systems

To address the challenges in the automotive industry posed by the need to rapidly manufacture more product variants, and the resultant need for more adaptable production systems, radical changes are now required in the way in which such systems are developed and implemented. In this context, two enabling approaches for achieving more agile manufacturing, namely modular automation systems and virtual commissioning, are briefly reviewed in this contribution. Ongoing research conducted at Loughborough University which aims to provide a modular approach to automation systems design coupled with a virtual engineering toolset for the (re)configuration of such manufacturing automation systems is reported. The problems faced in the virtual commissioning of modular automation systems are outlined. AutomationML - an emerging neutral data format which has potential to address integration problems is discussed. The paper proposes and illustrates a collaborative framework in which AutomationML is adopted for the data exchange and data representation of related models to enable efficient open virtual prototype construction and virtual commissioning of modular automation systems. A case study is provided to show how to create the data model based on AutomationML for describing a modular automation system

Integrated analysis of engine structures

The need for light, durable, fuel efficient, cost effective aircraft requires the development of engine structures which are flexible, made from advaced materials (including composites), resist higher temperatures, maintain tighter clearances and have lower maintenance costs. The formal quantification of any or several of these requires integrated computer programs (multilevel and/or interdisciplinary analysis programs interconnected) for engine structural analysis/design. Several integrated analysis computer prorams are under development at Lewis Reseach Center. These programs include: (1) COBSTRAN-Composite Blade Structural Analysis, (2) CODSTRAN-Composite Durability Structural Analysis, (3) CISTRAN-Composite Impact Structural Analysis, (4) STAEBL-StruTailoring of Engine Blades, and (5) ESMOSS-Engine Structures Modeling Software System. Three other related programs, developed under Lewis sponsorship, are described