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This paper explores the observability and estimation capability of dynamical systems using predominantly relative measurements of the system's state-space variables, with minimal to no reliance on absolute measurements of these variables. We concentrate on linear time-invariant systems, in which the observation matrix serves as the algebraic representation of a graph object. This graph object encapsulates the availability of relative measurements. Utilizing algebraic graph theory and abstract linear algebra (geometric) tools, we establish a link between the structure of the graph of relative measurements and the system-theoretic observability subspace of linear systems. Special emphasis is given to multi-agent networked systems whose dynamics are governed by the linear consensus protocol. We demonstrate the importance of absolute information and its placement to the system's dynamics in achieving full-state estimation. Finally, the analysis shifts to the synthesis of a distributed observer with relative measurements for single integrator dynamics, exemplifying the relevance of the preceding analytical findings. We support our theoretical analysis with numerical simulations.

In the present paper, we outline and expound the fundamental and novel qualitative-cum-philosophical premises, principles, ideas, concepts, constructions and results that originate from our ongoing research project of applying the conceptual panoply and the technical machinery of Abstract Differential Geometry (ADG) to various persistently outstanding issues in Quantum Gravity (QG) research. At the end of the paper, we discuss the potential philosophical repercussions of two possible future research routes that the main stream of our applications of ADG to QG may bifurcate towards in view of three independent, but overlapping, research papers that are currently under development., Comment: Corrected, updated and slightly expanded Version II of the paper, to replace Version I

A new algebra, hitherto not encountered in the usual Lie algebraic varieties or supervarieties, is introduced. The paper explores the rich and novel structure of the algebra, and it compares it on the one hand with the Jordan-Lie Superalgebras studied by Okubo and Kamiya, and on the other, with the four usual Euclidean division rings of the reals (R), the complexes (C), the quaternions (H) and the octonions (O), key algebraic properties of which the algebra is seen to combine, alter, extend and generalise. A potential physical application of the algebra is briefly alluded to at the end., Comment: arXiv admin note: substantial text overlap with arXiv:math-ph/0110030

We continue ongoing research work on applying the homological algebraic conceptual and technical machinery of Abstract Differential Geometry towards formulating a finitary, causal and quantal version of vacuum Einstein Lorentzian gravity and free Yang-Mills theories., Comment: This is an invited paper contribution to a Special Volume titled `Physical Geometry: Unravelling the Weave of Quantum Geometry' in memory and honour of Professor Anastasios Mallios, edited by Dr Elias Zafiris. Extended latest version includes two Addenda and an Appendix at the back

This paper introduces a distributed leaderless swarm formation control framework to address the problem of collectively driving a swarm of robots to track a time-varying formation. The swarm's formation is captured by the trajectory of an abstract shape that circumscribes the convex hull of robots' positions and is independent of the number of robots and their ordering in the swarm. For each robot in the swarm, given global specifications in terms of the trajectory of the abstract shape parameters, the proposed framework synthesizes a control law that steers the swarm to track the desired formation using the information available at the robot's local neighbors. For this purpose, we generate a suitable local reference trajectory that the robot controller tracks by solving the input-output linearization problem. Here, we select the swarm output to be the parameters of the abstract shape. For this purpose, we design a dynamic average consensus estimator to estimate the abstract shape parameters. The abstract shape parameters are used as the swarm state feedback to generate a suitable robot trajectory. We demonstrate the effectiveness and robustness of the proposed control framework by providing the simulation of coordinated collective navigation of a group of car-like robots in the presence of robots and communication link failures.

Large-Scale Actuator Networks (LSAN) are a rapidly growing class of electromechanical systems. A prime application of LSANs in the industrial sector is distributed manipulation. LSAN's are typically implemented using: vibrating plates, air jets, and mobile multi-robot teams. This paper investigates a surface capable of morphing its shape using an array of linear actuators to impose two dimensional translational movement on a set of objects. The collective nature of the actuator network overcomes the limitations of the single Degree of Freedom (DOF) manipulators, and forms a complex topography to convey multiple objects to a reference location. A derivation of the kinematic constraints and limitations of an arbitrary multi-cell surface is provided. These limitations determine the allowable actuator alignments when configuring the surface. A fusion of simulation and practical results demonstrate the advantages of using this technology over static feeders.

This paper deals with the problem of designing a distributed fault detection and isolation algorithm for nonlinear large-scale systems that are subjected to multiple fault modes. To solve this problem, a network of communicating detection nodes is deployed to monitor the monolithic process. Each node consists of an estimator with partial observation of the system's state. The local estimator executes a distributed variation of the particle filtering algorithm using the partial sensor measurements and the fault progression model of the process. During the implementation of the algorithm, each node communicates with its neighbors by sharing pre-processed information. The communication topology is defined using graph theoretic tools. The information fusion between the neighboring nodes is performed by means of a distributed average consensus algorithm to ensure the agreement over the value of the local likelihood functions. The proposed method enables online hypothesis testing without the need of a bank of estimators. Numerical simulations demonstrate the efficiency of the proposed approach.

We summarize the twelve most important in our view novel concepts that have arisen, based on results that have been obtained, from various applications of Abstract Differential Geometry (ADG) to Quantum Gravity (QG). The present document may be used as a concise, yet informal, discursive and peripatetic conceptual guide-cum-terminological glossary to the voluminous technical research literature on the subject. In a bonus section at the end, we dwell on the significance of introducing new conceptual terminology in future QG research by means of `poetic language', Comment: 16 pages, preliminary version

Based on the algebraico-categorical (:sheaf-theoretic and sheaf cohomological) conceptual and technical machinery of Abstract Differential Geometry, a new, genuinely background spacetime manifold independent, field quantization scenario for vacuum Einstein gravity and free Yang-Mills theories is introduced. The scheme is coined `third quantization' and, although it formally appears to follow a canonical route, it is fully covariant, because it is an expressly functorial `procedure'. Various current and future Quantum Gravity research issues are discussed under the light of 3rd-quantization. A postscript gives a brief account of this author's personal encounters with Rafael Sorkin and his work., Comment: 43 pages; latest version contributed to a fest-volume celebrating Rafael Sorkin's 60th birthday (Erratum: in earlier versions I had wrongly written that the Editor for this volume is Daniele Oriti, with CUP as publisher. I apologize for the mistake.)

This is a two-part, `2-in-1' paper. In Part I, the introductory talk at `Glafka--2004: Iconoclastic Approaches to Quantum Gravity' international theoretical physics conference is presented in paper form (without references). In Part II, the more technical talk, originally titled ``Abstract Differential Geometric Excursion to Classical and Quantum Gravity'', is presented in paper form (with citations). The two parts are closely entwined, as Part I makes general motivating remarks for Part II., Comment: 34 pages, in paper form 2 talks given at ``Glafka--2004: Iconoclastic Approaches to Quantum Gravity'' international theoretical physics conference, Athens, Greece (summer 2004)

Previous work on applications of Abstract Differential Geometry (ADG) to discrete Lorentzian quantum gravity is brought to its categorical climax by organizing the curved finitary spacetime sheaves of quantum causal sets involved therein, on which a finitary (:locally finite), singularity-free, background manifold independent and geometrically prequantized version of the gravitational vacuum Einstein field equations were seen to hold, into a topos structure. This topos is seen to be a finitary instance of both an elementary and a Grothendieck topos, generalizing in a differential geometric setting, as befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies. The paper closes with a thorough discussion of four future routes we could take in order to further develop our topos-theoretic perspective on ADG-gravity along certain categorical trends in current quantum gravity research., Comment: 49 pages, latest updated version (errata corrected, references polished) Submitted to the International Journal of Theoretical Physics

The tomographic histories approach is presented. As an inverse problem, we recover in an operational way the effective topology of the extended configuration space of a system. This means that from a series of experiments we get a set of points corresponding to events. The difference between effective and actual topology is drawn. We deduce the topology of the extended configuration space of a non-relativistic system, using certain concepts from the consistent histories approach to Quantum Mechanics, such as the notion of a record. A few remarks about the case of a relativistic system, preparing the ground for a forthcoming paper sequel to this, are made in the end., Comment: 19 pages, slight chang in title and corrected typos in second version. To appear to a special proceedings issue (Glafka 2004) of the International Journal of Theoretical Physics

The smooth gravitational singularities of the differential spacetime manifold based General Relativity (GR) are viewed from the perspective of the background manifold independent and, in extenso, Calculus-free Abstract Differential Geometry (ADG). In particular, the inner Schwarzschild singularity is being `resolved' ADG-theoretically in two different ways. A plethora of important mathematical, physical and philosophical issues in current classical and quantum gravity research are addressed and tackled., Comment: Size: 481 pages. The document is pretty much in its final form. We are still groping with quotations' referencing in the text. We shall rectify things duly

A `resolution' of the interior singularity of the spherically symmetric Schwarzschild solution of the Einstein equations for the gravitational field of a point-particle is carried out entirely and solely by finitistic and algebraic means. To this end, the background differential spacetime manifold and Calculus-free purely algebraic (:sheaf-theoretic) conceptual and technical machinery of Abstract Differential Geometry (ADG) is employed via Sorkin's finitary (:locally finite) poset substitutes of continuous manifolds in their Gel'fand-dual picture in terms of discrete differential incidence algebras and the finitary spacetime sheaves thereof. It is shown that the Einstein equations hold not only at the finitary poset level of `discrete events', but also at a suitable `classical continuum limit' of the said finitary sheaves and the associated differential triads that they define ADG-theoretically. We infer that the law of gravity does not break down in any (differential geometric) sense in the vicinity of the locus of the point-mass as the usual manifold based analysis of spacetime singularities in General Relativity has hitherto maintained. Various possible implications that such a total evasion of smooth gravitational singularities, as well as some anticipations of the wider significance that the general ADG-framework, may have for certain current `hot' issues in both classical and quantum gravity research are briefly discussed at the end., Comment: 41 pages, to appear in the International Journal of Theoretical Physics

We claim that both multipartiteness and localization of subsystems of compound quantum systems are of an essentially relative nature crucially depending on the set of operationalistically available states. In a more general setting, to capture the relativity and variability of our structures with respect to the observation means, sheaves of algebras may need be introduced. We provide the general formalism based on algebras which exhibits the relativity of multipartiteness and localization., Comment: 10 pages, LaTeX, Procs. of the International Conference `Quantum Structures 2002'

We continue recent work and formulate the gravitational vacuum Einstein equations over a locally finite spacetime by using the basic axiomatics, techniques, ideas and working philosophy of Abstract Differential Geometry. The whole construction is `fully covariant', `inherently quantum' (both expressions are analytically explained in the paper) and genuinely smooth background spacetime independent., Comment: 133 pages, revised and slightly corrected final version

A recently proposed algebraic representation of the causal set model of the small-scale structure of space-time of Sorkin et al. is briefly reviewed and expanded. The algebraic model suggested, called quantum causal set, is physically interpreted as a locally finite, causal and quantal version of the kinematical structure of general relativity: the 4-dimensional Lorentzian space-time manifold and its continuous local orthochronous Lorentz symmetries. We discuss various possible dynamical scenarios for quantum causal sets mainly by using sheaf-theoretic ideas, and we entertain the possibility of constructing an inherently finite and genuinely smooth space-time background free quantum theory of gravity. At the end, based on the quantum causal set paradigm, we anticipate and roughly sketch out a potential future development of a noncommutative topology, sheaf and topos theory suitable for quantum space-time structure and its dynamics., Comment: 8 pages, ReV-TeX, substantially revised, expanded and updated version

An algebra A not encountered in either the usual algebraic varieties or supervarieties is introduced. A is a graded and deformed version of the quaternions, with structure similar to that of a Jordan-Lie superalgebra as defined by Okubo and Kamiya, but it is shown to be neither that of a purely associative Lie superalgebra, nor that of a purely antiassociative Jordan-Lie superalgebra. Rather, it exhibits a novel kind of associativity, here called `ordered graded associativity', that is somewhat `in between' pure associativity and pure antiassociativity. In addition to graded associativity, the generators of A obey graded commutation relations encountered in both the usual Lie superalgebras and in graded Jordan-Lie algebras. They also satisfy new graded Jacobi identities that combine characteristics of the Jacobis obeyed by the generators of ungraded Lie, graded Lie and graded Jordan-Lie algebras. Mainly due to these three features, A is called a `mixed' Jordan-Lie superalgebra. The present paper defines A and compares it with the Jordan-Lie superalgebra defined by Okubo and Kamiya., Comment: 17 pages

A brief synopsis of recent conceptions and results, the current status and future outlook of our research program of applying sheaf and topos-theoretic ideas to quantum gravity and quantum logic is presented., Comment: 12 pages; paper of a talk given at the 5th Biannual International Quantum Structures Association Conference in Cesena, Italy (March-April 2001)

Isham's topos-theoretic perspective on the logic of the consistent-histories theory is extended in two ways. First, the presheaves of consistent sets of history propositions in the topos proposed by Isham are endowed with a Vietoris-type of topology and subsequently they are sheafified with respect to it. The category resulting from this sheafification procedure is the topos of sheaves of sets varying continuously over the Vietoris-topologized base poset category of Boolean subalgebras of the universal orthoalgebra of quantum history propositions. The second extension of Isham's topos consists in endowing the stalks of the aforementioned sheaves, which were originally inhabited by structureless sets, with further algebraic structure, that also enjoys a quantum causal interpretation, so as to arrive at the topos of consistent-histories of quantum causal sets. The resulting quantum causal histories topos is compared with Markopoulou's quantum causal histories, with Mallios and this author's finitary spacetime sheaves of quantum causal sets, as well as with Butterfield and Isham's topos perspective on the Kochen-Specker theorem of quantum logic., Comment: 34 pages

The notion of finitary spacetime sheaves is introduced based on locally finite approximations of the continuous topology of a bounded region of a spacetime manifold. Finitary spacetime sheaves are seen to be sound mathematical models of approximations of continuous spacetime observables.

A mathematical formalism for treating spacetime topology as a quantum observable is provided. We describe spacetime foam entirely in algebraic terms. To implement the correspondence principle we express the classical spacetime manifold of general relativity and the commutative coordinates of its events by means of appropriate limit constructions., Comment: 34 pages, LaTeX2e, the section concerning classical spacetimes in the limit essentially corrected

A quantum causal topology is presented. This is modeled after a non-commutative scheme type of theory for the curved finitary spacetime sheaves of the non-abelian incidence Rota algebras that represent `gravitational quantum causal sets'. The finitary spacetime primitive algebra scheme structures for quantum causal sets proposed here are interpreted as the kinematics of a curved and reticular local quantum causality. Dynamics for quantum causal sets is then represented by appropriate scheme morphisms, thus it has a purely categorical description that is manifestly `gauge-independent'. Hence, a schematic version of the Principle of General Covariance of General Relativity is formulated for the dynamically variable quantum causal sets. We compare our non-commutative scheme-theoretic curved quantum causal topology with some recent $C^{*}$-quantale models for non-abelian generalizations of classical commutative topological spaces or locales, as well as with some relevant recent results obtained from applying sheaf and topos-theoretic ideas to quantum logic proper. Motivated by the latter, we organize our finitary spacetime primitive algebra schemes of curved quantum causal sets into a topos-like structure, coined `quantum topos', and argue that it is a sound model of a structure that Selesnick has anticipated to underlie Finkelstein's reticular and curved quantum causal net. At the end we conjecture that the fundamental quantum time-asymmetry that Penrose has expected to be the main characteristic of the elusive `true quantum gravity' is possibly of a kinematical or structural rather than of a dynamical character, and we also discuss the possibility of a unified description of quantum logic and quantum gravity in quantum topos-theoretic terms., Comment: 65 pages, LaTeX2e,typos corrected

Using the leading vector method, we show that any vector $h\in(C^2)^{\otimes l}$ can be decomposed as a sum of at most (and at least in the generic case) $2^l-l$ product vectors using local bitwise unitary transformations. The method is based on representing the vectors by chains of appropriate simplicial complex. This generalizes the Scmidt decomposition of pure states of a 2-bit register to registers of arbitrary length $l$., Comment: RevTeX, 3 pages, no figures, summary added

A quantum picture of the causal structure of Minkowski space M is presented. The mathematical model employed to this end is a non-classical version of the classical topos {H} of real quaternion algebras used elsewhere to organize the perceptions of spacetime events of a Boolean observer into M. Certain key properties of this new quantum topos are highlighted by contrast against the corresponding ones of its classical counterpart {H} modelling M and are seen to accord with some key features of the algebraically quantized causal set structure., Comment: LaTeX 2.09, 20 pages

A scheme for an algebraic quantization of the causal sets of Sorkin et al. is presented. The suggested scenario is along the lines of a similar algebraization and quantum interpretation of finitary topological spaces due to Zapatrin and this author. To be able to apply the latter procedure to causal sets Sorkin's `semantic switch' from `partially ordered sets as finitary topological spaces' to `partially ordered sets as locally finite causal sets' is employed. The result is the definition of `quantum causal sets'. Such a procedure and its resulting definition is physically justified by a property of quantum causal sets that meets Finkelstein's requirement from `quantum causality' to be an immediate, as well as an algebraically represented, relation between events for discrete locality's sake. The quantum causal sets introduced here are shown to have this property by direct use of a result from the algebraization of finitary topological spaces due to Breslav, Parfionov and Zapatrin., Comment: LaTex 2.09, 9pages

An algebraic quantization procedure for discretized spacetime models is suggested based on the duality between finitary substitutes and their incidence algebras. The provided limiting procedure that yields conventional manifold characteristics of spacetime structures is interpreted in the algebraic quantum framework as a correspondence principle., Comment: 15 pages, no figures, LaTeX 2.09