ALBERT

Notes: Earlier mathematical models that have been developed for use with data acquisition/data reduction systems in close range photogrammetry have not taken into consideration the geometry of video imagery. In this paper, a recently developed multi-application collinearity equation (MACE) model is analysed and assessed. The model is simple to apply and is adaptable to any data acquisition/data reduction system in close range photogrammetry.

Notes: The use of CCD cameras for digital photogrammetry has only slowly been taken up by the photogrammetric community. However, the pace of change is quickening with better understanding and cheaper equipment. Recently a decision was made to start some research into close range three dimensional digital measurement. The overall objective was to develop a low cost system for the automated measurement of small industrial objects. This shorter contribution describes the initial stages of this work.

Notes: This paper presents two mathematical models of the accuracy of digital terrain model (DTM) surfaces which have been linearly constructed from square gridded data, respectively with and without additional feature specific data. In these models, the accuracy of DTM surfaces is expressed as a function of a few parameters, such as slope angle, grid interval or the accuracy of raw data, which are familiar to practitioners in photogrammetry and other mapping related disciplines. These models are in a form similar to the conventional formulae for contour map accuracy and they are compared with experimental test results.

Notes: In 1989, the Council of the Photogrammetric Society decided that some record of the use of analogue photogrammetric equipment in Great Britain should be preserved before it was too late. The author agreed to investigate this matter as a research project funded by a number of survey organisations. Work on the project so far has resulted in the completion of two documents. The first is a Catalogue of Analogue Instruments that list some 230 instruments of 30 different types. The second is an Archive that gives further details of the history and use of the 30 different types of instrument. The entry included in his paper, by way of example, is that concerned with the Zeiss Stereoplanigraph.Readers are encouraged to provide corrections and additional information wherever possible.

Notes: Feature extraction is the first and crucial step of all image analysis procedures. Together with their mutual relations, features form perceptual structures on which hypothesis generation and reconstruction processes are built. The article gives a framework for feature extraction within an interpretation system, discusses a possible object and image model useful for building up perceptual structures and gives examples for the use of various features for standard photogrammetric tasks.〈section xml:id="abs1-2"〉〈title type="main"〉RésuméĽextraction automatique de silhouettes est la première étape fondamentale dans tout procédéďanalyse ďimages.Ces silhouettes constituent, en même temps que leurs relations mutuelles, des structures identifiables à partir desquelles on peut établir des hypothèses et procéder à la re constitution. On donne dans cet article un cadre général pour ľextraction de silhouettes dans un système ďinterprétation et ľon y discute ďune modélisation possible de ľimage et de ľobjet pour obtenir des structures identifiables. On fournit enfin des exemples sur ľemploi de diverses silhouettes d̄objets dans les travaux photogrammétriques courants.〈section xml:id="abs1-3"〉〈title type="main"〉ZusammenfassungDie Kantenextraktion stellt den I. wesentlichen Schritt aller Bild-analyseverfahren dar. Gemeinsam mit ihren wechselseitigen Beziehungen bilden die Kanten wahrnehmbare Strukturen, auf denen Hypothesen und Rekonstruktionsprozesse aufgebaut werden können. Im Artikel werden ein Grundgerüst z.ur Kantenextraktion angegeben, mögliche Objekt- und Bild-modelle zur Entwicklung von Wahrnehmungsstrukturen diskutiert und Beispiele zur Nutzung verschiedener Kanten für photogrammetrische Standardaufgaben angegeben.

Notes: Accurate and rapid planning techniques are of vital importance when competing for business in the deregulated telecommunications market. This requirement, together with the arrival of personal communication networks, demands more detailed spatial data than are available now. This paper examines difficulties in current planning methods and considers the specifications of future data sets. 〈section xml:id="abs1-2"〉〈title type="main"〉Résumé Les méthodes de planification précises et rapides sont ď une importance vitale lorsque ľ on est en concurrence en affaires dans le marché déréglementé des télécommunications. Ce besoin, parallèlement àľ arrivée des réseaux personnels de communication, nécessite une disponibilité de données tridimensionnelles beaucoup plus détailées que celles que ľ on a actuellement. On examine dans cet article les difficultiés rencontrées dans les méthodes actuelles de planification, et ľ on réfléchit sur les spécifications des jeux de données dans ľ avenir. 〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung Genaue und schnelle Planungsverfahren sind beim Geschäftswettbewerb aufdem nicht regulierten Telekommunikations-Markt von auβerordentlicher Bedeutung. Diese Anforderung in Verbindung mit dem Aufkommen von Personal-Kommunikations-Netzwerken erfordert detailliertere Raumdaten ah jetzt verfügbar sind. In dem Artikel werden die Schwierigkeiten der gegen-wärtigen Planungsmethoden geprüft und Anforderungen an künftige Datensätze erörtert.

Notes: The President's Medallist directs the Survey Services Branch of the Technical Services Group of English Heritage. This Branch includes the Photogrammetric Unit which was first established at the Institute of Advanced Architectural Studies of the University of York in 1975, with support from English Heritage's predecessor organisation. The Unit has played a leading part in the development of architectural photogrammetry in the UK, with particular reference to the recording of ancient monuments and historic buildings. The writer discusses some aspects of the application of photogrammetry in this area, illustrating the modern photogrammetric product in comparison with 18th and 19th century pictures and prints from his own collection. 〈section xml:id="abs1-2"〉〈title type="main"〉Résumé C'est le titulaire de la Médaile du Président de notre Société qui dirige le Département de Levés du groupe des services techniques du Patrimoine anglais. Ce Département comprend ľUnité de Photogrammétrie qui était installée initialiement àľInstitut des Études Architecturales Avancées ďYork en 1975, avec le soutien de ľorganisme qui précédait le Patrimoine anglais. Cette Unité a joué un rôle prépondérant dans le développement de la photogrammétrie architecturale au Royaume-Uni, et une mention particulière est à faire en ce qui concerne ľarchivage des constructions anciennes et des monuments historiques. Ľauteur discute de quelques aspects des applications de la photogrammétrie dans ce domaine, illustrant le produit photogrammétrique moderne par comparaison avec des tableaux et des gravures des 18èmeet 19ème siècles issus de sa collection personelle. 〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung Der Träger der Präsidenten-Medaille leitet die Vermessungsabteilung innerhalb des Technischen Dienstes von English Heritage. Diese Abteilung schlieβt die Photogrammetrie-Einheit mit ein, die zunächst im Jahre 1975 im Institut für Moderne Architektur an der Universitat York mit unterstüt-zung der Vorgängerorganisation von English Heritage aufgebaut wurde. Diese Einheit hat eine führende Rolle bei der Entwicklung der Architektur-photogrammetrie im Vereinigten Königreich gespielt, wobei besonders altehrwürdige Denkmäler und historische Bauwerke aufgenommen wurden. Der Schreiber diskutiert einige Aspekte der Anwendung der Photogrammetrie aufdiesem Gebiet und illustriert das moderne photogrammetrische Erzeugnis im Vergleich mit Bildern und Kopien aus dem 18. und 19. Jahrhundert, die der eigenen Sammlung entstammen.

Notes: The reunification of Germany also paved the way for the reunion of Carl Zeiss in Jena and Oberkochen in 1991, after many months of negotiation. For the surveying and photogrammetry divisions, this means that a joint policy for development, production and marketing will be pursued in Jena and Oberkochen in the future. The two photogrammetry divisions in Jena and Oberkochen will continue to exist on the market, with both their advanced aerial survey cameras (LMK 2000 and RMK TOP) being available. While Oberkochen will concentrate on the further development of analytical plotters and aerial photograph rectification systems, Jena will increasingly specialise in software development, which is an expanding field. Due to its longstanding expertise in these fields, Jena will also continue its activities in the fields of interpretation and close range photogrammetry, including the taking and evaluation of images. 〈section xml:id="abs1-2"〉〈title type="main"〉Résumé La ré-unification de ľAllemagne a éalement ouvert la voie à celle des deux Zeiss, léna et Oberkochen, en 1991, après plusieurs mois de négotiation. Pour les divisions des levés et de la photogrammétrie, cela signifie qu ľon mènera dorénavant, à Iéna et à Oberkochen, une politique commune en ce qui concerne les developpements, la production et le marketing. Les deux divisions photogrammétriques, celle ďléna et celle ďOberkochen, continueront ďêtre présentes sur le marché, offrant toutes deux lews chambres photogrammétriques aériennes avancées LMK 2000 et RMK TOP. Tandis qu à Oberkochen on concentre les activités sur lapoursuite du développement des restituteurs analytiques et des systèmes de redressement des photographies aériennes, à léna on se spécialisera dans le développement des logiciels, domaine qui est en expansion. Etant donnée la compétence que ľon a acquise de longue date à Iéna dans le domaine de ľinterprétation et dans celui de la photogrammétrie à courte distance, on γ poursuivra les activités dans ces domaines, qui comprennent également la saisie et ľévaluation des images. 〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung Die Wiedervereinigung Deutschlands bahnte im Jahre 1991 nach vielen Monalen der Verhandlung auch den Weg zur Vereinigung der Firmen von Carl Zeiss in Jena und Oberkochen. Für die Abteilungen Vermessung und Photogrammetrie bedeutet das, daβ in Zukunft für Entwicklung, Produktion und Vertrieb in Jena und Oberkochen eine abtestimmte Politik verfolgt wird. Die beiden photogrammetrischen Abteilungen in Jena and Oberkochen werden weiterhin mil ihren beiden Luftbildmeβkammern (LMK 2000 und RMK TOP) auf dem Markt vertreten sein. Wahrend sich Oberkochen auf die Weiterentwicklung analytischer Auswertegeräte und von Systemen zur Luftbildentzerrung konzentriert, wird sich Jena auf die Software-Entwicklung spezialisieren, was ein stark expandierendes Gebiet ist. Dank langjähriger Erfahrungen wird Jena seine Aktivitäten auf den Gebieten Photointerpretation und Nahbereichsphoto-grammetrie fortsetzen, wobei Bildaufnahme und Bildbewertung einges-chlossen sind.

Notes: The paper outlines the author's background and continuing education in photogrammetry. The potential of low cost instruments for non-specialists is examined from her viewpoint as a research student and as an employee of Leica UK Ltd. 〈section xml:id="abs1-2"〉〈title type="main"〉Résumé Ľarticle retrace les compétences acquises par ľauteur en photogrammétrie et lew prolongement en formation continue. Ľauteur présente également les possibilityé des appareils à bas prix pour les non-spécialistes en so plaçant à un double point de vue, en tant qu'tudiant faisant de la recherche et en tant qu'employee de la Société Leica Ltd. (RU). 〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung In dem Artikel werden die Ausbildung und die fortwährende Weiter-bildung der Autorin in Photogrammetrie skizziert. Der Nutzen preiswerter Gerdte für Nichtspezialisten wird unter ihrem Blickwinkel als Forschungs-student und Mitarbeiterin der Leica UK Ltd. beleuchtet.

Notes: Book Reviewed in this article:MONITORING BUILDING STRUCTURES. Edited by J. F. A. Moore.POSTCODES: THE NEW GEOGRAPHY. By J. F. Raper, D. W. Rhind and J. W. Shepherd.TEST OF TRIANGULATION OF SPOT DATA. Edited by I. J. Dowman.EUROPEAN SPACE DIRECTORY 1992. Seventh edition. Endorsed by Eurospace. Edited by D. Shirvanian.WISSENSBASIERTE METHODEN ZUR FERNERKUNDUNG DER UMWELT. Edited by O. Günther and W.-F. Riekert.PROCEEDINGS OF THE ISPRS AND OEEPE JOINT WORKSHOP ON UPDATING DIGITAL DATA BY PHOTOGRAMMETRIC METHODS, CHRIST CHURCH, OXFORD UNIVERSITY, ENGLAND, 15–17 SEPTEMBER 1991. Edited by P. R. T. Newby and C. N. Thompson.GLIMPSES OF WORLD WAR By W. P. Roe.A GUIDE TO MAP AND IMAGE PROCESSING By M. P. Skrdla.MULTIPHOTO GEOMETRICALLY CONSTRAINED MATCHING. By E. P. Baltsavias.DIGITALE PHOTOGRAMMETRIE IN DER DREIDIMENSIONALEN STRÖMUNGSMESSTECHNIK. By H.-G. Maas.LONDON. AN AEROFILMS GUIDE. By I. Thomson.THE SOUTH DEVON COAST PATH: AN AEROFILMS GUIDE. By D. Hannigan.THE COTSWOLD WAY: AN AEROFILMS GUIDE. By T. Fryer.THE SOUTH DOWNS WAY: AN AEROFILMS GUIDE. By J. Godfrey.SURVEYING. Sixth edition. By A. Bannister, S. Raymond and R. Baker.INSTRUMENT MAKERS TO THE WORLD: A HISTORY OF COOKE, TROUGHTON AND SIMMS. By A. McConnell.TRANSIT CIRCLE–THE STORY OF WILLIAM SIMMS 1793–1860. By E. Mennim.TERRA-1: UNDERSTANDING THE TERRESTRIAL ENVIRONMENT. THE ROLE OF EARTH OBSERVATIONS FROM SPACE. Edited by P. M. Mather.SPACE. By R. Gibson.KATASTER-ABC. Second edition. By O. Kriegel and D. Dresbach.

Notes: The osp(3||2) superalgebra is realized, first by means of two supervectorial operators canonically conjugated, and then by differential operators acting on functions defined on E(3||2). As for the vectorial representation of osp(1||2) the eigenvalue problem yields to the superspherical harmonics.

Notes: Using a matrix realization, generic elements of the supergroups U(m/n) and OSP(m/2n) are obtained through the exponentiation of the corresponding super Lie algebra elements. The emphasis is put on the contribution of the odd part. The application to the factorization problem is given, and the relationship with nonlinear differential superequations is clarified.

Notes: A general procedure for evaluating the induction coefficients (IDCs) or the outer-product reduction coefficients of the Hecke algebra Hf(q) of the Af−1 type is formulated based on the linear equation method. Analytical expressions for the IDC's of Hf(q) are tabulated for f≤4. It is also proved that these IDCs are the Clebsch–Gordan coefficients for the special Gel'fand basis of the quantum group SUq(N).

Notes: In this article Nevanlinna's theorem about Borel summability is improved. Moreover, the Borel summability of the asymptotic expansion in the method of stationary phase in infinitely many dimensions is discussed.

Notes: It is proved that all bounded subsets of Rn with nonempty interior are cutable. Also found is an if and only if classification of when a nontrivial absolute steady state (ASS) of the sandpile-type cellular automata model will exist. The classification helps one to narrow down the geometrical structure of the ASS and is applicable to both Abelian and non-Abelian models. It is important for the future studies of self-organized criticality.

Notes: A generalization of the standard class of solutions in the Kaluza–Klein (4+1) gravity, wherein the spherically symmetric metric depends not only on the radius but also on the extra coordinate, is considered. Two new classes of exact solutions of the empty Kaluza–Klein field equations are given. However, it is known that apparently empty solutions of the (4+1) Kaluza–Klein equations can be interpreted as solutions with effective matter properties of the (3+1) Einstein equations. The physical importance of the new solutions is that in this approach the dependency on the extra coordinate allows us to obtain more general equations of state than before, including ones for radiation, dust, vacuum, and stiff matter.

Notes: Following the spectral methods established by Adler in his treatment of scattering and decay in the quaternionic quantum theory, it is shown that an anti-self-adjoint operator with quaternion defined spectrum in [0,∞) has a symmetrical effective spectrum, from the point of view of functional analysis, under quite general conditions. In the case of an operator with absolutely continuous spectrum in [0,∞), the effective spectrum is absolutely continuous in (−∞,∞), and a canonically conjugate operator exists. If the anti-self-adjoint operator is the generator of motion in time (Hamiltonian), the conjugate operator is a "time operator.'' Moreover, according to a theorem of Misra, Prigogine, and Courbage, such a system may admit a Lyapunov operator, and therefore describe irreversible behavior. It is shown directly that no evident contradiction (as is found in the semibounded case in complex quantum mechanics) arises from the definition of a Lyapunov operator in quaternionic quantum mechanics.

Notes: A general class of exact solutions for the Minkowskian σ model of the hyperbolic Grassmanian manifold, which was considered earlier by Lambert and Piette, are constructed. The energy momentum tensor associated with these solutions can be made finite with a suitable choice of the arbitrary functions involved in their construction.

Notes: The infinite hierarchy of lower Korteweg–de Vries (KdV) equations which corresponds to negative degrees of Olver's operator is constructed. Basic forms of these equations are found. The supersymmetry structure of the Miura's transformations is considered.

Notes: The intrinsic generalized sine–Gordon and wave equations are integrable n-dimensional generalizations of the sine–Gordon and wave equations, respectively. It is shown that their Lie-point symmetry groups are finite-dimensional. They consist only of translations in the first case and translations and dilations in the second. The symmetries are used to obtain classes of exact invariant solutions.

Notes: Generalized expressions are proposed for the supercharges in N=2 supersymmetric quantum mechanics and their implications are analyzed mainly in the one-dimensional context. In fact, these generalizations are exploited in three typical examples admitting the largest numbers of symmetries, i.e., the harmonic oscillator, the hydrogen atom, and their superposition, the so-called "Calogero context,'' successively. Different matrix realizations are exploited in the harmonic oscillator case more particularly. The link between these considerations and quantum deformations must be insisted on, our deformation b parameter referring to a family of physically acceptable supersymmetrization procedures including, in particular, the standard one (when b→1) initially proposed by Witten.

Notes: The sine-Gordon equation is considered in the Hamiltonian framework provided by the Adler–Kostant–Symes theorem. The phase space, a finite dimensional coadjoint orbit in the dual space @Fg* of a loop algebra @Fg, is parameterized by a finite dimensional symplectic vector space W embedded into @Fg* by a moment map. Real quasiperiodic solutions are computed in terms of theta functions using a Liouville generating function which generates a canonical transformation to linear coordinates on the Jacobi variety of a suitable hyperelliptic curve.

Notes: An explicit algebraic description of finite Lorentz transformations of vectors in ten-dimensional Minkowski space is given by means of a parametrization in terms of the octonions. The possible utility of these results for superstring theory is mentioned. Along the way automorphisms of the two highest dimensional normed division algebras, namely, the quaternions and the octonions, are described in terms of conjugation maps. Similar techniques are used to define SO(3) and SO(7) via conjugation, SO(4) via symmetric multiplication, and SO(8) via both symmetric multiplication and one-sided multiplication. The noncommutativity and nonassociativity of these division algebras plays a crucial role in our constructions.

Notes: Degenerations of Lie algebras of meromorphic vector fields on elliptic curves (i.e., complex tori) which are holomorphic outside a certain set of points (markings) are studied. By an algebraic geometric degeneration process certain subalgebras of Lie algebras of meromorphic vector fields on P1, the Riemann sphere, are obtained. In case of some natural choices of the markings these subalgebras are explicitly determined. It is shown that the number of markings can change.

Notes: A matrix Riemann–Hilbert problem associated with the one-dimensional Schrödinger equation is considered, and the existence and uniqueness of its solutions are studied. The solution of this Riemann–Hilbert problem yields the solution of the inverse scattering problem for a larger class of potentials than the usual Faddeev class. Some examples of explicit solutions of the Riemann–Hilbert problem are given, and the connection with ambiguities in the inverse scattering problem is established.

Notes: By use of path integrals in N=2 supersymmetric quantum mechanics, a formula for the kernel of the supersymmetric time evolution operator is found. Structurally it is similar to the Feynman–Kac formula used in the path integrals of quantum mechanics. Crucial to the development of this formula is a generalization of the Wiener measure based upon the free supercharges rather than the free Hamiltonian.

Notes: Spinor, scalar, and vector irreducible representations of generators of the Lorentz transformations in a four-dimensional Grassmann subspace of a d-dimensional Grassmann space are studied. For d=5 there are four spinor representations with the properties of Majorana spinors and two scalar, two three-vector and two four-vector representations with the properties of bosons, offering the possibility of the canonical quantization of not only bosonic but also fermionic fields.

Notes: A formulation for singular Lagrangians with higher derivatives in a previous paper is supplemented and a complete formulation of canonical theory for the systems is presented. It is shown that our previous formulation reproduces all constraints included in the Ostrogradski transformations, by using stationarity conditions of primary constraints. Secondarily, on the effect of adding a total time derivative term to a nonsingular Lagrangian with higher derivatives, the previous paper is revised so as to give a correct formulation for counting the numbers of first and second class constraints. It is concluded that the main part of our previous formulation needs no essential change.

Notes: It is shown that the spin-weighted cylindrical harmonics s Jαm(ρ,φ), which are functions defined on the plane, are related to representation matrices of the group of rigid motions on the plane. It is also shown that the raising and lowering operators of the spin weight and of the z component of the angular momentum are related to two copies of the Lie algebra of the Euclidean group of the plane.

Notes: A continuous time spinflip process, started from an arbitrary state, is proven to yield a Gibbs state after any time t(approximately-greater-than)0 if the noise is high enough. This is done by approximating the process by discrete time processes. The relative energies of those Gibbs states are constructed explicitly via a high-noise cluster expansion.

Notes: It is shown that the discrete spectrum of Schrödinger Hamiltonians of the form H=−Δ+vf may be represented by the semiclassical expression Enl=minr(approximately-greater-than)0 {K(f)nl(r) + vf(r)}. The K functions are found to be invariant with respect to coupling and shifts: K(Af+B)=K(f). For pure power laws, f(r)=sgn(q)rq, and the log potential, they are also invariant with respect to scale, and have the simple forms (Pnl(q)/r)2 and (Lnl/r)2, respectively. K functions are also derived for sech-squared and Hulthén potentials. If f=g(h), where g is a smooth transformation, then the envelope approximation is expressed in terms of K by the relation K(f)(approximately-equal-to)K(h). When the transformation g has definite convexity, then the approximation immediately yields eigenvalue bounds for all n and l. The theory is used to prove the log-power theorem Lnl = Pnl(0), which, in turn, generates a simple eigenvalue formula for the log potential.

Notes: It is shown that the transition probability between eigenstates of a time-dependent two-level system is given by the Fourier transform in the adiabatic parameter of a function determined by the Hamiltonian. Sufficient and necessary conditions for the transition probability between levels to decrease exponentially in the adiabatic limit are established. Our method involves a Paley–Wiener type theorem to study the Fourier transform of functions belonging to a Hardy class on a strip.

Notes: Following constructions in the case of canonical and affine coherent states, a universal propagator for a general Hamiltonian appropriate to a single spin degree of freedom is introduced. The universal propagator is a single function, independent of any particular choice of fiducial vector, which, nonetheless, propagates all coherent state Hilbert space representatives correctly. Furthermore, we explicitly construct the universal propagator for several examples.

Notes: Nonlinear Lorentz invariant field models are constructed in such a way to be integrable by a transformation of a linear free field. The transformation is explicitly found by requiring that the U(1) conserved current keeps the same expression of the free field. This method is applied to the complex scalar and spinor fields in 1+1 dimensions. Localized nondispersive waves exist and interact with each other. A way of introducing nonlinear interactions between different fields is also given with the property of preserving integrability.

Notes: A new (non-Noetherian) symmetry transformation for the small oscillations problem is exhibited. A novel way to solve this classical problem, based on conservation laws constructed using the new symmetry, is presented. Applications of these results to other fields of physics are outlined.

Notes: The Boltzmann equation has been used in the literature to show that the entropy differential consists of a Pfaffian form in the space of conserved and nonconserved variables (moments) and a term related to the energy dissipation due to the irreversible processes in the system. The said Pfaffian form is called the compensation differential. In this paper, the integrability of the compensation differential is examined by means of the theory of differential forms. The integrability conditions turn out to be generalized forms of the Maxwell relations in equilibrium thermodynamics. It is also shown that the generalized form of the Gibbs–Duhem relation can be seen as an equivalent of the integrability conditions. This conclusion is drawn by using the notion of homotopy operator. The Caratheodory principle is also applied to make the connection with the second law of thermodynamics more intimate than the direct but mathematically more abstract approach using the integrability conditions. The meaning of the integrating factor is clarified by using the notion of contact temperature since the integrating factor, a mathematical function, is endowed with a thermodynamically operational meaning when it is identified with the inverse local absolute temperature through the notion of contact temperature, and the compensation differential is thereby made a thermodynamically meaningful equation governing nonequilibrium processes. Through this study it is shown that in irreversible thermodynamics the compensation differential can play the role parallel to the equilibrium Gibbs relation for the entropy in equilibrium thermodynamics and thus can serve as the foundation on which to formulate a theory of irreversible processes.

Notes: The eigenvalue correlation functions of the Laguerre ensembles (exponential ensembles) of real symmetric and self-dual quaternion random matrices are calculated. It is shown that in certain intervals of the spectrum the renormalized correlations are not universal. A physical example in the transmission spectrum of a disordered conductor where such nonuniversality occurs is given.

Notes: The classical motion and the quantum evolution determined by linear Hamiltonians on coherent state manifolds of compact and noncompact Hermitian symmetric space structure are considered. A matrix Riccati equation, with opposite signs of the quadratic terms in the compact and noncompact cases, determines the classical motion and the quantum evolution. The geometric meaning of equations of motion as flow on a complex Grassmann manifold and his noncompact dual is emphasized. Possibilities of generalizing the results to flag manifolds are pointed out.

Notes: Lattice soliton solutions that have doubly periodic array of the localized structures in the plane are presented to the Kadomtsev–Petviashvili equation with the positive dispersion by using the superposition of the rational functions. Each structure is similar to the rational soliton. The existence condition and some properties are also presented.

Notes: In this paper is presented a new class of spherically symmetric analytic solutions in canonical coordinates in general relativity which corresponds to causal models of perfect fluid balls. Tolman's IV solution is a member of this class. Two new solutions belonging to this class have been studied here in detail. These correspond to positive, monotonically decreasing expressions for pressure and density and realistic equations of state within causal fluid balls.

Notes: It is shown that, since the finite dimensional representations of the super-rotation algebra are characterized by the superspin j and the parity λ of the representation space, all features of the Racah–Wigner calculus: Clebsch–Gordan coefficients, recoupling coefficients as well as the Wigner and Racah symbols depend on both j and λ. However, it is noticed that the dependence on the parities of the Wigner and Racah symbols can be factorized out into phases so that one can define parity-independent super S3−j and S6−j symbols. The properties of these symbols are analyzed, in particular, it is shown that the S6−j symbols possess a symmetry similar to the Regge symmetry satisfied by the rotation 6−j symbols. Analytical and numerical tables of the symbols are given for the lowest values of their arguments.

Notes: The sech2 solitary wave solution of the regularized long wave equation is reobtained via the inverse scattering transform. The wave function of the eigenvalue problem of the relevant Schrödinger equation is proved reflectionless with sech2 potential of arbitrary amplitude. Moreover, the nonexistence of N-solitary wave solution (sech2 form and N≥2) is confirmed for the regularized long wave (RLW) equation and the method provides the possibility of solving some incompletely integrable equations via the inverse scattering transform.

Notes: Axially symmetric gauge field configurations of a certain type are studied in generalized Yang–Mills theory in 4p dimensions (p=2,3,...). For all p, finite action solutions of the generalized self-duality equations with topological charge n=1,2,... are found. It is also shown that the self-duality equations are overdetermined, when other types of axial symmetry are imposed.

Notes: On a Fock superspace, a linear canonical supertransformation is implemented by an inner-product-preserving operator which is necessarily not continuous.

Notes: We give a new proof of a theorem locally classifying classical dynamical systems with constraints, and discuss possible applications of the result to quantization of such systems, in particular of General Relativity.

Notes: Using the known realizations of the quantum algebras Aq(n), Dq(n) in terms of quantum Clifford algebra and folding procedure of Lie algebras we construct some new spinor representations of Bq(n), Cq(n), Gq(2).

Notes: A noncanonical parametrization for the graded group U(1/1) is introduced similar to the Euler angles for the ordinary group SU(2). Two differential representations for the underlying algebra of U(1/1) are constructed on the full and the restricted, i.e., coset, parameter space, respectively. A space of functions living on the latter is found exhibiting close formal similarities to a Hilbert space. Remarkably, the indices of those functions and thus the orthogonality and completeness relations involve anticommuting variables. Using this Hilbert space a representation of U(1/1) is evaluated which shows analogies to the Wigner functions for SU(2).

Notes: A one-parameter group inserting the spectral parameter into the nonparametric Lax pair of the nonhomogeneous, nonlinear Schrödinger system is presented and discussed. Being a nonlocal extension of a Lie point transformation this group is probably not contained in any known class of symmetries.

Notes: Some results of a systematic study of the coupled Einstein–Yang–Mills (EYM) equations for arbitrary gauge groups are presented herein. In a first step a group theoretical analysis of spherically symmetric EYM fields is given. This will lead to a concise description of a large class of principal bundles for which the spherically symmetric solutions of the EYM equations have to be static (generalized Birkhoff theorem) and the metric must be of the Reissner–Nordstrøm-type.

Notes: In terms of three signs associated to two vectors and to a 2-plane, a formula for the signature of any four-dimensional metric is given. In the process, a simple expression for the sign of the Lorentzian metric signature is obtained. The relationship between these results and those already known are commented upon.

Notes: The usual quantum mechanics methods of defining angular momentum and spherical harmonics in the ordinary space E(3) are generalized to the real commuting super-space E(3||2). The super angular momentum, generators of the super-rotation, is realized as differential operators acting on functions defined on E(3||2). By solving the eigenvalue problem in spherical coordinates, a set of pseudo-orthonormalized functions is constructed, the super-spherical harmonics Ylmj, whose main properties are given.

Notes: It is shown how using the graded Faddeev–Reshetikhin–Takhtajan (FRT) formalism, one can multiparametrize the quantum superalgebra slq(n||m). Since the naive Serre–Chevalley relations are not adequate for the presentation of this algebra, the quantization is carried out in the Cartan–Weyl basis. The final results can be generalized to any finite dimensional superalgebra with a symmetrizable Cartan matrix.

Notes: The q-deformed three-dimensional harmonic oscillator is defined in terms of the q-bosons corresponding to the spherical components of a nondeformed three-dimensional oscillator. It is shown that the dynamical algebra is spq(6,R). Two important subalgebra chains are identified: spq(6,R)&supuline;suq(3)&supuline;soq(3) and spq(6,R) &supuline; spq2(2,R) ⊕ soq(3). The basis states of the q-deformed oscillator are classified according to these subalgebras. Finally, the Hamiltonian eigenvalues are discussed.

Notes: A systematic method of finding nontrivial generalized variational symmetries are presented herein and the corresponding first integrals for nonlinear dynamical system with two degrees of freedom are derived. Then, a method is applied to a Toda lattice, given by the potential V= ex−y+c1e2y+c2ey+c3e−2x+c4 e−x, where c1, c2, c3, and c4 are parameters. The symmetries and constants of motion are presented explicitly.

Notes: Many completely integrable equations have been shown to be reductions of the Yang–Mills equations in a flat space. In this paper two-dimensional symmetry reductions of the self-dual Yang–Mills equations, together with the corresponding linear equations, in an arbitrary self-dual space are classified. The reduced equations are given in different forms. Some new equations are found, in particular one which generalizes the sine-Gordon equation and the Liouville equation.

Notes: The recent result that there is no Bekenstein black hole at D(approximately-greater-than)4 is confirmed by the direct integration of the field equations of the self-gravitating conformal scalar field. The case D=3 where the result turns out to be same is also solved. The curvature singularities of the obtained solutions are analyzed and it is shown that there do exist special (critical) space-times which, however, are of the black hole type only at D=4.

Notes: A sequence of infinitely generated p-adic Banach superalgebras and their locally convex projective and inductive limits were constructed; their spectral properties are characterized herein and by studying the algebras of matrices constructed with these superalgebras an exponential map and its inverse can be defined. Some examples of superalgebra with pathological spectral properties are discussed.

Notes: Let s≥2 and let α=〈a1,a2,...,as(approximately-greater-than), where s, ai ∈ Z and a1≥a2≥...≥as≥1. Let 1≤t≤s. The method of undetermined coefficients was used to investigate natural transformations Ω:Λα → Λa1 ⊗ Λa2 ⊗... ⊗ Λas and Ωt:Λα → Λ〈a1,...,aˆt,...,as(approximately-greater-than) ⊗ Λat. Necessary and sufficient conditions were found on the coefficients defining Ω and sufficient conditions on the coefficients defining Ωt. Explicit formulas for Ω1 and Ωs are given herein.

Notes: A frequency expansion method is formulated for the one-variable Fokker–Planck equation. It is shown that this method is valid for potentials U(x) which diverge faster than log||x|| or converge to a limit faster than any powers of x as x→±∞. The analysis is based on the study of the reflection coefficients for the Fokker–Planck equation. Expressions for the expansion coefficients of the reflection coefficients are also derived.

Notes: A solution of the Einstein field equations for a rotating radiating body was presented by Carmeli and Kaye in 1977 [Ann. Phys. 103, 97 (1977)]. Their paper presented a new line element along with the corresponding energy-momentum tensor for the radiation field. The Newman–Penrose formalism [J. Math. Phys. 3, 566 (1962)] was used to solve and analyze the new solution which lead to the tetrad components of the Ricci, Maxwell, and Weyl tensors. However, no attempt was made to solve the geodesic equation in this new metric. The present paper starts with a linearized form of the Carmeli and Kaye line element and solves the resulting differential equations to a particular order for a null geodesic. The deflection angle is found from the equations of motion and compared in detail for the two cases of a rotating, highly luminous quasar and our sun.

Notes: An ultralocal structure discovered in a derivative nonlinear Schrödinger model enables us to investigate the complete integrability of the classical system through action-angle variables and solve exactly the quantum model by quantum inverse scattering transformation. Exploiting the underlying quantum group related Sklyanin-like algebra an exact lattice version of the model is constructed, and also the deep-rooted reason for the existence of such an ultralocal model is revealed.

Notes: The theory of the algebras of Ignat'ev and Kuz'min and of Greenberg and Mohapatra is developed. These are algebras of inhibited parafermionic oscillators that describe possible small violations of the Pauli exclusion principle. The cyclic modules of the algebras are constructed so as to exhibit the connection of their fundamental properties, as uniqueness and statistical signature, with the relations in the algebras. The formalism is presented in such a way suited for further mathematical and physical developments, and has a natural continuation in a forthcoming paper.

Notes: Continuous Hahn polynomials Sn(x) appear in a formulation of quantum mechanics on a discrete time lattice, where they form a natural basis for the state vectors. In this paper we derive some of their generating functions, the expression of the raising and lowering operators and give a lower bound for the largest root of the equation Sn(x)=0.

Notes: Coherent states associated with the m-sheeted covering group of SU(1,1) are considered and such properties as overcompleteness and resolution of the identity are studied. They are used to define analytic representations in which the transformations of the m-sheeted covering group of SU(1,1) are implemented as extended Mobius conformal mappings. Applications of this formalism in the study of Hamiltonians that describe m-particle clustering are also considered.

Notes: A closed form of the perfect matching number for the cubic lattice of 2×3×n was obtained in terms of the linear combination of determinants, which was derived from its recursive expression. This is the first analytic solution for dimer statistics on three-dimensional, cubic-cell lattices. It is potentially applicable to the analytic solution of the Ising model for the general l×m×n lattice.

Notes: Tensor product decompositions of the N=2 superconformal algebras are examined to determine when they are finite or multiplicity-free, using various combinatorial identities. Using similar techniques, branching rules for the winding subalgebras of the N=2 superconformal algebras are investigated.

Notes: Power series are introduced that are simultaneously convergent for all real and all p-adic numbers. Our expansions are in some aspects similar to those of exponential, trigonometric, and hyperbolic functions. Starting from these series and using their factorial structure new and summable series with rational sums are obtained. For arguments x∈Q adeles of series are constructed. Possible applications at the Planck scale are also considered.

Notes: The explicit expressions of the 2N commuting conserved quantities of the N-particle Davey–Stewartson I (DSI) system are given. The conserved quantities are also expressed in terms of field operators, thus the infinitely many conservation laws are obtained.

Notes: A method of calculation of translation normalizers is described which can be used in any dimension and which consists in three steps: (i) In the first step is determined the continuous part of the normalizer, which is a linear envelope of a stability space of the point group and of the continuous part of translation subgroup of the Euclidean group in question. (ii) In the second step are determined translation normalizers of irreducible components. (iii) And finally there is a theorem which gives an upper and lower limit for the translation normalizer of a reducible arithmetic class in terms of normalizers of its irreducible components. A check of coset representatives in resolution of the upper limit with respect to the lower limit concludes the calculation. The method is used to derive a table of translation normalizers of discrete, continuous, and semicontinous Euclidean groups up to three dimensions; all these groups are of interest in material physics. The consideration of translation normalizers is preceded by an extension and amendment of concepts used in crystallography which aim to specify the groups for which the tables are finally given.

Notes: The quantum sphere is covered by two quantum complex planes. Quantum line bundle on S2 is introduced by specifying a transition function. The structure of the quantum complex planes enables one to construct representations of SUq(2) acting in spaces of holomorphic sections. In the quantum case, dressing transformation replaces coadjoint action.

Notes: Starting with a two-cocycle globally defined on a Lie group G, the Lagrangian system on G is constructed whose Lagrangian is quasi-invariant under a right translation canonically lifted to the tangent bundle TG. A direct consequence of the quasi-invariance is the appearance of central extensions in the Noether-symmetry algebra. Then, it is shown that the kinematical sector of such a model realizes, through a symplectic reduction, the Kirillov–Kostant symplectic structure on the coadjoint orbit of the Lie group with the central extensions. The generalization to a "higher-dimensional'' theory of the Wess–Zumino–Witten-type is also discussed herein.

Notes: The asymptotic estimations, as period tends to infinity, for negative spectral bands of three-dimensional periodical Schrödinger operator who's potential was generated by shifting along the cubic lattice of a negative function with compact support are described herein. The problem is of primary importance for solid state physics.

Notes: The vacuum configuration for the inflationary superstring theory is established. It is argued that the basic physical contents of the inflationary universe are characterized by the Novikov higher signature. Finally it is shown, with respect to the splitting of Paper II, that the index of the Dirac operator defined in our inflated universe, and in the parallel shadow one, is indeed h-cobordant.

Notes: The hypothesis is formulated that our universe and the parallel "shadow'' one, as predicted by superstring theory, are connected by a ring of global gravitational instantons. This amounts to a favored conceptual model in which the universe would have split into two equal parts around the Planck scale. In the present formulation, the ten-dimensional space-time in which superstring theory is defined is inflated in two symmetrical directions, reaching a maximal full-fledged representation of E'8 ⊗ E8. The radius of the inflationary universe is determined and arguments are made that strongly restrict the number of interactions between our universe and the parallel one. This yields an explanation as to why the other entity is shadow to ours.

Notes: Equation (14) of the paper by Aguilera-Navarro and Guardiola on the nonsingular spiked harmonic oscillator [J. Math. Phys. 32, 2135 (1991)] appears not to be applicable when α=2. By carrying out simple transformations on this equation it is shown that it can be evaluated in closed form in the limiting case α→2.

Notes: Finite dimensional irreducible representations of the quantum supergroup Uq(gl(m/n)) at both generic q and q being a root of unity are investigated systematically within the framework of the induced module construction. The representation theory is rather similar to that of gl(m/n) at generic q, but drastically different when q is a root of unity. In the latter case, atypicality conditions of highest weight irreducible representations (irreps) are substantially altered, and such finite-dimensional irreps arise that do not have highest weight and/or lowest weight vectors. As concrete examples, the irreps of Uq(gl(2/1)) are classified.

Notes: Conditions are formulated under which a representation of an intrinsic C*- algebra of (often quasilocal) observables of an infinite system is appropriate to describe measurement-type processes: such a representation should allow for the description of robust experiments, it should be separable, and the pointer observable should be in its weak closure. If the pointer values are discrete the existence of such a measurement representation can be proven. If the pointer can take continuously many values, then the existence can only be proven under the additional assumptions of having an asymptotically Abelian system or dealing with type I representations. In the constructed measurement representations the pointer observable turns out to be classical. The structure of the representation suggests that spontaneous symmetry breaking might be a physical explanation of the emergence of the classical pointer.

Notes: Techniques of Erickson and Yennie [Ann. Phys. N.Y. 35, 271 (1965)] are applied in an investigation of a simple Taylor's series which involves products of commutators and nested commutators and which represents the self-energy operator Σ of an electron in an external potential. A technique of "moving the parameter integrals to the mass'' leads to a relatively simple representation of all terms as a one-dimensional integral over a common virtual electron mass squared parameter. Although not a true expansion in powers of the external field, the method provides a window on the higher-order terms in the self-energy operator. Mass eigenfunction expansion concepts are shown to be quite useful in a discussion of infrared divergences.

Notes: It is shown that SU(∞) self-dual Yang–Mills equations on a four-dimensional flat Euclidean background are equivalent to Plebanski's second heavenly equation for the self-dual gravitational background, which is associated with the cotangent space of a suitable Riemann surface. The Riemann surface in question is nothing but the internal two-dimensional surface required in the formulation of SU(∞) self-dual Yang–Mills theories. Such equivalence only occurs when a dimensional reduction has taken place. After a discussion of Euclidean supersymmetries the results are generalized to self-dual super-Yang–Mills theories and self-dual supergravity.

Notes: On the basis of the Yang–Baxter equation and the initial condition of R matrix, Yang Baxterization of representations of braid group and that of Temperley–Lieb algebra are discussed systematically. It is shown that there exist many Yang-Baxterization formulas beyond the knowns. New developments of Yang-Baxterization approach are presented. As examples some new solutions of the Yang–Baxter equation are given.

Notes: A direct method is proposed for determining the conditions under which the Lorenz system is integrable. This method uses only leading singular behavior to recover all known integrals of motion together with a few new ones.

Notes: The superconformal structure of the (2,0) supergeometry of two-dimensional superspace is characterized by the Beltrami differentials. The superdiffeomorphism transformations of the Beltrami variables are given and the superfields with arbitrary conformal weights are studied.

Notes: Several solvable n-body problems on the line are exhibited. They are characterized by equations of motion of Newtonian type, mjx¨j=Fj, j=1,...,n, with the "forces'' Fj given as explicit functions of the "particle coordinates'' xk and their velocities x(overdot)k, k=1,...,n; the forces Fj generally also depend on some free parameters, so that in each case there is actually a class of solvable models. In this paper the emphasis is on translation-invariant models, and on values of n sufficiently small ("few-body problems'') to allow the display of the solution of the initial-value problem in completely explicit form.

Notes: An analysis of a Green's function approach (based on wave splitting) to the one-dimensional electromagnetic inverse problem is given. The Green's functions refer to split components of the fundamental solution. The linear system of equations for the Green's functions is shown to be well-posed for the inverse problem and stability estimates, theorems on existence, uniqueness, local correctness, and convergence are given.

Notes: For a class of self-interacting multicomponent scalar field theories with a global discrete non-Abelian symmetry group, mixed order–disorder correlation functions are defined in terms of Euclidean functional integrals. These correlation functions satisfy Osterwalder–Schrader positivity. From a representation of the correlation functions in terms of the transfer matrix, the dual algebra at fixed time is derived. This algebra implies parafermion operators showing non-Abelian braid group statistics. In a pure phase of spontaneous symmetry breaking for a related class of order–disorder correlation functions a convergent polymer representation is developed, emerging from a combined low- and high-temperature-type expansion. The infinite volume correlation functions of this class show exponential clustering in the disorder fields.

Notes: An exact spherically symmetric thin shell solution to the Einstein equations is obtained in the Weyl–Dirac theory with an oscillating scalar field. Boundary conditions are chosen such that the conformal invariance is broken in the interior space and the bubble has a scalar-field-induced surface stress-energy tensor that causes the thin shell to oscillate.

Notes: The problem of constructing representations of the quantum group SU(3)q is considered. This algebra may be constructed from two sets of SU(2)q generators, oriented at an angle of 120° from each other. If one makes use of known results for SU(2)q, the problem reduces to finding the transformation matrices between two sets of basis vectors, in each of which the Casimir operator for one of the SU(2)q's is diagonal. Two simple prescriptions are given for finding these matrices in a general case, using a convenient set of variables. An exact solution is given for all irreducible representations in which the maximum weight multiplicity is 2.

Notes: A brief introduction to the parabolic and hyperbolic complex number systems is given. Parabolic and hyperbolic phase transformations are shown to be equivalent to two-dimensional Galilean and Lorentzian relativity transformations, respectively. Basic properties and definitions for the hyperbolic complex numbers are given, and they are then applied to special relativistic physics and the Dirac equation in 1+1 dimensions. Turning to string theory, it is briefly shown that for Minkowski signature the string world sheet possesses an integrable almost hyperbolic complex structure, with a metric that is generalized Hermitian. Next, the hyperbolic complex numbers are applied to the formalism of Dirac spinors in 3+1 dimensions. It is shown that a four-component Dirac spinor is equivalent to a two-component hyperbolic complex spinor, that the Lorentz group is equivalent to a generalized SU(2), and that the Dirac adjoint corresponds to a generalized Hermitian adjoint. A complete formalism is presented for the hyperbolic complex two-component spinors. Lastly, it is shown that the operations of C, P, and T on Dirac spinors are closely related to the three types of complex conjugation that exist when both hyperbolic and ordinary imaginary units are present.

Notes: Let Uq(Gˆ) denote the quantized affine Lie algebra and Uq(G(1)) the quantized nontwisted affine Lie algebra. Let Ofin be the category defined in Sec. III. It is shown that when the deformation parameter q is not a root of unit all integrable representations of Uq(Gˆ) in the category Ofin are completely reducible and that every integrable irreducible highest weight module over Uq(G(1)) corresponding to q(approximately-greater-than)0 is equivalent to a unitary module.

Notes: The definition of quantum characteristic exponents for a general quantum dynamical map is given. The basic properties of this notion and illustrative models are presented.

Notes: Degree 2 and 6 Casimir operators of G2 are calculated in explicit form, with the generators of G2 written in terms of the subalgebra A2.

Notes: Superdifferential operators of order 2n+1 which are covariant with respect to superconformal changes of coordinates on a compact super-Riemann surface are studied herein. It is shown that all such operators arise from super-Möbius covariant ones. A canonical matrix representation is presented and applications to classical super-W-algebras are discussed.

Notes: The complete reduction of simple Clifford algebras is spanned by matrix bases. Considered as generators of an algebra the basis elements pij satisfy the product relations pijpkl= δjkpil. It follows that subsets of pij with subscript or superscript fixed span minimal left or right ideals. The expression of the basis elements pij in the usual multivector basis is derived from isomorphic matrix algebras. Spinors and Hermitian conjugate spinors are defined as members of minimal left and right ideals. Spinor scalar products are defined, these scalar products are in fact scalar multiples of an idempotent.