ALBERT

Notes: Users of photogrammetric data have often been puzzled by conflicting advice given by companies regarding achievable height accuracies which could be expected from varying scales of aerial photography. The author chairs the Photogrammetry Group which is convened under the auspices of the Council of the UK Land and Hydrographic Survey Association (TSA). The first aim of this Group was to establish some common ground among TSA member companies with regard to achievable heighting accuracies. A series of tests was carried out on the most commonly used photographic scales and guidelines were published to clients based on these tests. 〈section xml:id="abs1-2"〉〈title type="main"〉Résumé Les utilisateurs des données photogrammétriques ont été souvent déconcertés par les avis contraires émanant des entreprises concernant les précisions altimétriques accessibles à partir de diverses échelles des photographies aériennes. Dans ce contexte, le Conseil de ľ‘Association britannique des levées terrestres et hydrographiques (TSA) a mis sur pied un Groupe “Photogrammétrie” et en a confié la présidence àľauteur. Le premier objectif de ce Groupe a consistéàétablir des bases communes aux entreprises membres de la TSA, en ce qui concerne les précisions altimétriques réalisables. On a effectué un ensemble ď essais sur les échelles des photographies aériennes les plus communément utilisées et ľ on a publicé, à partir de ces essais un opuscule destinéà guider les clients. 〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung Die Nutzer photogrammetrischer Daten werden oft durch widerspriich-liche Ratschläge der Firmen in bezug auf die bei verschiedenen Bildmaβstäben erreichbare Höhengenauigkeit verwirrt. Der Autor leitet den photogrammetrischen Ausschuβ, der unter Aufsicht des Council des UK Land and Hydrographic Survey Association (TSA) einberufen wurde. Erstes Ziel dieses Gremiums war es, eine gemeinsame Grundlage unter den TSA-Mitgliedsgesellschaften im Hinblick auf erreichbare Höhengenauigkeit zu schaffen. Es wurde eine Reihe von Versuchen zu den am meisten verwendeten Bildmaβstäben durchgeführt, undes wurden daraufberuhende Richtlinien für Kunden veröffentlicht.

Notes: Cartography andphotogrammetry have always been strongly linked and this paper emphasises a view of cartography as a process of spatial data representation, rather than as a method of creating a product, a map. Of course, the definition of cartography must change with developments in its practice. An outline of the historical background to the subject is presented, followed by a consideration of the relationship between the cartographer and the user of the product. The basic problems involved in the transfer of information from the complex real world to a generalized cartographic product are analysed and technological aspects of the discipline are described. Contemporary applications of cartography include hazard mapping, links with GIS, tactile products and electronic hydrographic representation, all of which represent customer driven progress. It is concluded that the future of cartography is, overall, dependent on a graphical representation of high quality being supplied to the end user, rather than on the methods used to attain this objective. 〈section xml:id="abs1-2"〉〈title type="main"〉Résumé La cartographie et la photogramétrie ont toujours été liées étroitement ľune àľautre, et ľ‘on souligne dans cet article ľ‘idée que la cartographie est un systéme de représentation des données à trois dimensions, plutôt qu‘un ensemble de procédés de fabrication ďun produit, la carte. II est évident que la définition de la cartographie doit changer au fur et à mesure que des évolutions apparaissent dans sa pratique. Aprés un aperçu du contexte historique de ce thème, on examine les relations qui se développent entre le cartographe et ľutilisateur du produit. On analyse les problèmes fondamentaux du transfert ďinformations du monde réel, complexe, à un produit cartographique géneralisé et ľ‘on décrit la technologie qu'utilise cette discipline. Les applications récentes de la cartographie concernent la cartographie des risques naturels, les relations avec les SIG, les cartes qui se lisent par le toucher, et la représentation electronique de ľ‘hydrographic produits qui tous sont issus de cheminements pilotés en aval, par le client. On en déduit que la cartographie est, àľ'avenir, par dessus tout fonction ďune represséntation graphique de haute qualityé fournie à un utilisateur final, plutôt que de processus utilisé pour parvenir à cet objectif.〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung Kartographie und Photogrammetrie waren immer eng verbunden, und dieser Beitrag betont starker eine Auffassung von der Kartographie als einem Prozeβ der Darstellung von Raumdaten als ein Verfahren zur Herstellung eines Erzeugnisses, der Karte. Natürlich muβ sich die Definition der Kartographie bei Entwicklungen in ihrer Praxis ändern. Es erfolgt eine Skizzierung des historischen Hintergrundes des Gegenstands, gefolgt von Betrachtungen zur Beziehung zwischen Kartographen und dem Nutzer des Erzeugnisses. Die mit der Übertragung von Informationen aus der komplexen realen Welt in ein generalisiertes kartographisches System verbundenen grundlegenden Probleme werden analysiert, und es werden technologische Aspekte der Disziplin beschrieben. Zeitgenössische Anwendungen der Kartographie schlieβen die Kartierung von Katastrophen, Verbindungen zu GPS, gefühlsmäβge Erzeugnisse und elektronische hydrographische Darstellungen ein, die alle den vom Nutzer beeinflufiten Fortschritt darstellen. Es wird geschluffolgert, daft die Zukunft der Kartographie vor allem von der graphischen Darstellung hoher Qualität, die dem Endnutzer geliefert wird, abhängig ist undnicht so sehr von den Methoden, me dieses Ziel erreicht wird.

Notes: Since its foundation in 1982, the Institut Cartogràfic de Catalunya has been dealing with digital images. Though digital images and techniques were initially confined to remote sensing, they have been gradually transferred to and combined with photogrammetry. In addition to the development of remote sensing and digital orthophotography systems, this expertise has facilitated the introduction of commercially available digital photogrammetric systems (scanning and stereocompilation) which are now in full operation. The paper describes the experience in scanning, orthophotography production and stereocompilation, draws some conclusions and gives an insight of the future of digital photogrammetry. 〈section xml:id="abs1-2"〉〈title type="main"〉Résumé L‘Institut Cartographique de Catalogue s‘est intéressé au traitement des images numériques depuis le moment où il a été crée, en 1982. Bien qu'au début les images numériques et leur technologie aient été réservées à la télédétection, elles se sont trouvées peu à peu transférées et intégrées à la photogrammétrie. Ce savoir-faire a permis le développement des systèmes de télédétection et ď orthophotographie numérique et a, de plus, facilité la mise à disposition sur le marché de systèmes de photogrammétrie numérique qui sont maintenant pleinement opérationnels. On présente dans cet article ľ expérience acquise dans le domaine du balayage, de la production des orthophotographies et de la stéréocompilation, on en tire quelques conclusions et ľ on termine par un aperçu sur ľ‘avenir de la photogrammétrie numérique. 〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung Seit seiner Gründung im Jahre 1982 hat sich das Institut Cartogràfic de Catalunya mit digitalen Bildern beschäftigt. Obwohl digitate Bilder und Verfahren ursprünglich auf die Fernerkundung begrenzt waren, wurden sie schrittweise auf die Photogrammetrie übertragen bzw. mit ihr kombiniert. Zusätzlich zur Entwicklung von Fernerkundung und digitaler Orthophoto-Systeme hat diese Erfahrung die Einführung kommerziell verfügbarer digitaler photogrammetrischer Systeme erleichtert, die jetzt überall in der Praxis angewendet werden. Der Beitrag beschreibt die Erfahrung bei der Abtastung, der Herstellung von Orthophotos und bei der Stereoauswertung und gibt einige Schluβfolgerungen und Aussichten in bezug auf die Digitalphotogrammetrie wieder.

Notes: The paper entitled The Photogrammetric Society Analogue Instrument Project was published in The Photogrammetric Record, 14(82): 565–582 (October 1993) and contained, as Appendix A, an extract from the Archive concerned with the history of the Zeiss Stereoplanigraph in Great Britain. In The Photogrammetric Record, 14(83): 769–782 (April 1994), three further Appendices were published which dealt with the Williamson Multiplex equipment and other instruments. In this third extract, the history of five other instruments of the Multiplex type are described. This brings to a close that section of the Archive dealing with instruments using direct optical projection.〈section xml:id="abs1-2"〉〈title type="main"〉Résumé On a publicé dans le numéro 14(82) (pp 565–582) ďoctobre 1993 du“Photogrammetric Record”ľarticle intitulé“le projet de la Société (britannique) de Photogrammétrie sur les instruments analogiques” et inséré dans ľannexe A un extrait des Archives relatif a ľhistorie du Stereoplan-igraphe Zeiss en Grande-Bretagne. On a publicé dans le numéro 14 (83) ďavril 1994 du“Photogrammetric Record”(pp 769–782) trois annexes supplémentaires qui traitaient de ľappareil Multiplex de Williamson et ďautres instruments. Dans ce troisième extrait, on présente ľhistorie de cinq autres appareils du type Multiplex. Cette partie des Archives qui traitait des appareils basés sur ľemploi ďuneprojection optique directe, se trouve ainsi achevee.〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung Der Beitrag mit dem Titel“The Photogrammetric Society Analogue Instrument Project”wurde im Photogrammetric Record, 14(82): 565–582 (Oktober 1993) veröffentlicht und enthielt als Anlage A einen Extrakt vom Archiv, das mit der Geschichte des Zeiss Steroplanigraph in Groβbritannien verbunden war. Im Photogrammetric Record, 14(83); 769–782 (April 1984) wurden 3 weitere Anlagen veröffentlicht, die vom Williamson Multiplex und anderen Geräten handelten. In diesem 3. Auszug wird die Geschichte von 5 anderen Geräten des Multiplex-Typs beschrieben. Das führt zum Abschluβ desjenigen Teils des Archivs, das sich mit Geräten mit direkter optischer Projektion befaβt.

Notes: An analysis is given of the modelling of XY stage errors in an analytical plotter by polynomial functions. This is discussed as an alternative to demanding strict mechanical accuracy in the stage construction. In an era of plentiful computational resources, it is concluded that apportioning more of the responsibility to the mathematical co-ordinate refinement is an effective strategy. A second degree polynomial is used for each of the two stages in an analytical plotter, modelling the systematic stage errors to within the variability of a single observation. A requirement for instrument stability is emphasized. Such techniques have application for other instruments such as image scanners. 〈section xml:id="abs1-2"〉〈title type="main"〉Résumé On étudie la modélisation des erreurs des chariots XY ďun restituteur analytique en utilisant des fonctions polynomials. II s‘agit en fait de trouver une alternative à des exigences de prêcision mêcanique très strictes dans la réalisation de ces chariots. A une époque oùľon dispose ďune abondance de ressources en moyens de calcul, il apparait en effet relever ďune stratégie efficace de s'appuyer davantage sur les traitements mathématiques pour améliorer les valeurs des coordonnées. On utilise, pour chacun des deux chariots, un polynome du second degrè pour modéliser les erreurs systématiques des chariots, et les rendre inférieures aux erreurs propres à une observation unique. On insiste sur la nécessitéďune grande stabilityé instrumental. Ces processus peuvent également s'appliquer àďautres instruments comme les scanneurs ďimages. 〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung Es wird die Modellierung von Fehlern des xy-Wagens bei analytischen Auswertegeräten mit Hilfe von Polynomen analysiert. Des wird als Alternative zur Forderung nach einer strengen mechanischen Genauigkeit bei der Bildwagenkonstruktion diskutiert. Im Zeitalter hervorragender Computer-Ressourcen, so wird geschluβfolgert, ist es eine effektive Strategie, sich mehr der mathematischen Koordinatenverbesserung zuzuwenden. Ein Polynom 2. Grades wird für jeden der 2 Bildträger eines analytischen Auswertegerätes genutzt, um die systematischen Fehler im Bereich der Variation von einzelnen Beobachtungen zu modellieren. Bezüglich der Stabilität des Instruments wird eine Anforderung formuliert. Solche Verfahren kännen auch bei anderen Geräten, wie zum Beispiel Bildscanner angewendet werden.

Notes: Photogrammetry is used extensively in British Coal Opencast to provide surface profiles of both new sites and existing sites. This paper will discuss the development of GPS within Opencast and its deployment as a tool to provide the ground control required for mapping. Comparisons with conventional equipment will also be made and relative cost effectiveness assessed. 〈section xml:id="abs1-2"〉〈title type="main"〉Résumé On utilise largement la photogrammétrie dans British Coal Opencast, pour déterminer des profiles de surface, à la fois dans les sites nouveaux et dans les sites en cours. On examine dans cet article ľ emploi du GPS dans ces mines et son développement comme outil permettant ď obtenir les points ď appui sur le terrain nécessaires à la cartographie. On se propose defaire les comparaisons avec les équipements classiques et ďévaluer les avantages relatifs coût/efficacité. 〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung Die Photogrammetrie wird bei den Britischen Kohlen-Tagebauen extensiv zur Beschaffung von Oberflächenprofilen sowohl bei Neuaufschlüssen als auch bei bestehenden Gruben eingesetzt. Im Artikel wird die Entwicklung von GPS und seine Entwicklung als Hilfsmittel zur Beschaffung der für die Kartenherstellung benötigten Paβpunkte diskutiert. Es erfolgen auch Vergleiche mit der konventionellen Ausrüstung und in bezug auf die relative Kosteneffektivität.

Notes: An investigation has been made into the feasibility of using a low cost GPS receiver fitted to a light fixed wing aircraft as an aid to flying parallel flight lines for block aerial vertical photography. Two cases were examined: (I) the pilot controlled the heading of the aircraft according to the indications on the course deviation indicator of the GPS receiver; (2) the autopilot, being connected to the GPS receiver, controlled the heading of the aircraft. The results obtained for both sidelap and endlap were equal to or superior to those obtained with a CCTV system. 〈section xml:id="abs1-2"〉〈title type="main"〉RésuméOn a mené une recherche sur la faisabilité du GPS comme aide au maintien du parallélisme des axes de prises de vues d'un bloc de photographies aériennes verticales, en utilisant un récepteur GPS bon marché fié a un avion léger. On a examiné deux cas: (i) celui ou le pilote a maintenu le cap de l'avion en suivant les indications de l'index de changement de route du récepteur GPS; (ii) celui ou c'est le pilote automatique qui, par une liaison au récepteur GPS, a maintenu le cap de l'uvion. Les résultats obtenus pour le recouvrement latéral et l'extrgmité des bandes se sont révélés égaux ou supérieurs ii ceux obtenus avec un systPme CCTV.〈section xml:id="abs1-3"〉〈title type="main"〉ZusammenfassungEs wurde eine Untersuchung über die mögliche Nutzung billiger GPS-Empfanger, die in ein Kleinjlugzeug als Hilfsmittel zum Flug paralleler Bildstreifen für die blockweise Luftbildaufnahme eingebaut wurden, durchgefuhrt. Dabei wurden 2 Falle geprüft: (1) Der Pilot steuerte das Flugzeug entsprechend der Anzeige auf dem Kursindikator des GPS-Empfangers. (2) Der mit dem GPS-Empfänger verbundene Autopilot steuerte das Flugzeug. Die Ergebnisse sowohl für Querals auch Längsüberdeckung waren gleich oder besser als die mit einem CCTV-System erhaltenen.

Notes: The authors provide an account of current Directorate of Military Survey approaches to the provision of photocontrol by use of GPS and they pay special attention to the requirements of height control. 〈section xml:id="abs1-2"〉〈title type="main"〉SummaryGPS now provides us with a rapid method for determining plan and height for photogrammetric control directly over the points required by the photogrammetrist. By utilizing the slope of the geoid model, orthometric heights can be obtained to an accuracy of 0.15 m to 0.2 m. Surveys can be carried out quickly in all weather conditions and with none of the traditional terrestrial survey skills. Time, cost and manpower are greatly reduced. The plan accuracies are well within any requirements of the photogrammetrist and heights can be achieved to within 0.15 m to 0.2 m on test areas of UK. As photogrammetric height accuracy requires at best 1 m for plotting a 10 metre contour interval, heighting by GPS and the use of a geoid model solves the heighting problem.〈section xml:id="abs1-3"〉〈title type="main"〉Résumé Les auteurs fournissent un compte-rendu de la façon dont le Directorate of Military Survey aborde actuellement le problème de la fourniture de points ďappui photos avec ľemploi du GPS et prêterit une attention particulière à celui des points ďappui altimétriques.〈section xml:id="abs1-4"〉〈title type="main"〉Zusammenfassung Die Autoren geben einen Bericht über die gegenwärtigen Versuche beim Directorate of Military Survey zur Paβpunktbestimmung mit GPS, wobei sie den Anforderungen an die Höhenbestimmung besondere Aufmerksamkeit zollen.

Notes: The operational situation of precise airborne GPS positioning of camera air stations and its use in combined block adjustment for aerial triangulation is reviewed. The acceptance of systematic GPS errors, which occur especially with biased phase ambiguity solutions after loss of lock of signals in-flight, has motivated the method of additional linear GPS drift parameters which are to be solved in the combined block adjustment. It also implies that no absolute GPS co-ordinates are required and that the datum transformation is provided via a few ground control points. A great number of practical examples demonstrates that the method has been successfully applied within a wide range of photoscales and of block sizes. It has proved to be fully operational under various practical conditions. The high accuracy performance of GPS blocks and the economic benefit have been confirmed. It is particularly noticeable that the stationary receiver can be at a great distance, up to 500 km or more from the mission area. The conclusion is drawn that GPS supported aerial triangulation is ready to become standard. The potential improvement from dual frequency receivers and fast ambiguity solutions only enhances that conclusion. 〈section xml:id="abs1-2"〉〈title type="main"〉Résumé On examine, dans cet article, le degréďopérationnalité atteint par le GPS aéroporté dans le positionnement précis des sommets de prises de vues des caméras aériennes, et son insertion dans la compensation par blocs en aérotriangulation. La possibilityéďerreurs systématiques dans le GPS, en particulier en cas de perte des signaux en vol, qui se manifeste par des ambiguïtés de phase et des biais dans les résultats, à conduit a introduire dans la compensation en bloc des inconnues supplémental s, traduisant une dérive linéaire du GPS. II en résulte qu'il n'est pas nécessaire de disposer de coordonnées GPS absolues et que quelques points ďappui sur le terrain suffisent à assurer les changements de coordonnées. On a utilisé avec succès cette méthode dans de nombreux essaispratiques comportant un large eventail ďéchelles de prises de vues et de tallies des blocs. On a pu ainsi mettre en évidence ľentière opérationnalité de cette méthode, dans des conditions réelles différentes. On a également confirmé la haute précision des résultats des blocs GPS et les avantages économiques de ce système. En particulier, il est tout à fait remarquable que le récepteur fixe puisse se trouver à une grande distance de la zone du lasé, 500 km ou plus. On peut en conclure que ľaérotriangulation appuyée sur le GPS est en train de devenir une méthode standard. Les améliorations possibles que peuvent apporter les récepteurs à deux fréquences et la résolution rapide des ambiguïtés, ne font que renforcer cette conclusion. 〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung Die hochgenaue Kamera Positionierung durch GPS und ihre Anwendung für die Aerotriangulation wird zusammengefaβt dargestellt. Die Existenz systematischer GPS Fehler, wie sie inbesondere durch nur genäherte Phasen-Mehrdeutigkeitslösungen nach Signalabrissen im Flug verursacht werden, haben zu der speziellen Methode zusätzlicher linearer GPS Drift Parameter geführt, die in der kombinierten Blockausgleichung bestimmt werden. Ferner wird dabei die Datumtransformation mit Hilfe einiger weniger Paβpunkte bestimmt, d.h. es werden keine absoluten GPS Koordinaten vorausgesetzt. Mit einer groβen Zahl praktischer Beispiele wird demonstriert, daβ die Methode in einem groβen Bereich von Bildmaβstäben und Blockgröβen bereits erfolgreich angewendet wird. Sie hat sich unter den verschiedensten praktischen Bedingungen als voll operationell erwiesen. Dabei wurde die theoretisch erwartete hohe Genauigkeit der GPS Blöcke und ihre Wirtschaftlichkeit bestätigt. Besonders bedeutsam ist der empirische Nachweis, daβ der stationäre GPS Empfänger sehr weit, bis 500 km oder mehr, von dem Fluggebiet entfernt sein kann. Als Ergebnis wird empfohlen, die GPS gestützte Aerotriangulation als verfügbares photogrammetrisches Standardverfahren allgemein zu verwenden. Diese Schluβfolgerung wird durch die neuesten Entwicklungen von Zwei-Frequenz Empfängern und schnellen Mehrdeutigkeitslösungen, in die hohe Erwartungen gesetzt werden, nur noch bekräftigt.

Notes: A description is given of a heighting accuracy test carried out at Cartographical Services Limited on 1:2000 scale aerial photography. Premarked points, with ground surveyed heights, were established on a dual carriageway road near Worcester and photography was flown using a Leica RC20 aerial camera with EMC. Photogrammetric height measurement was carried out with both a Zeiss P3 analytical plotter and a Cartographic Engineering Limited/Carto Instruments AP190 plotter; respective r.m.s.e. accuracies of ± 25 mm and ± 30 mm were obtained. The new Cartographic Engineering Limited CP2 analytical plotter has subsequently been used for height measurement on similar photography and comparable accuracy has been achieved. 〈section xml:id="abs1-2"〉〈title type="main"〉Résumé On présente un essai de précision altimétrique, effectué par la Société“Cartographical Services” de Southampton, sur des photographies aérienne àľéchelledu 1:2000. On a utilisé des points pré-balisés, repartis sur une route à double voie près de Worcester, et dont ľaltitude a été déterminé sur le terrain. La prise de vue a été fait avec une caméra Leica RC20 munie ďun système de compensation du filé longitudinal. On a effectué les déterminations photogrammétriques des altitudes en utilisant à la fois un restituteur analytique Zeiss P3 et une restituteur AP190 de la Société“Cartographic Engineering/Carto Instruments.” On a obtenu comme erreurs moyennes quadratique les valeurs de ± 25 mm et ± 30 mm respectivement. On a enfin utilisé le noveau restituteur analytique CP2 de la Société“Cartographic Engineering” pour ces déterminations altimétriques et ľon a obtenu une précision comparable. 〈section xml:id="abs1-3"〉〈title type="main"〉Zusammenfassung Es wird ein Versuch zur Bestimmung der Höhengenauigkeit, der beim Cartographical Services Ltd. in Southampton für den Bildmaβstab 1:2000 durchgeführt wurde beschrieben. Signalisierte Punkte, deren Höhen im Gelände bestimmt wurden, wurden aufeiner Doppelfahrbahn in der Nähe von Worcester angeordnet undmit einerLeica RC20 mit EMC aufgenommen. Die photogrammetrische Höhenbestimmung erfolgte mit einem analytischen Auswertegerät P3 von Zeiss und einem AP 190 Plotter von Cartographic Engineering Ltd./Carto Instruments. Dabei wurden mittlere Fehler von ± 25 mm bzw. ± 30 mm erhalten. Das neue analytische Auswertegerät CP 2 von Cartographic Engineering Ltd. wurde danach für Höhenmessungen bei ähnlichen Luftbildern genutzt, wobei vergleichbare Genauigkeiten erreicht wurden.

Notes: This paper describes a method for modelling the SPOT satellite imaging process using a single Ground Control Point (GCP) and for extracting ground planimetric information. Satellite position is derived using the collinearity condition equations. Initial orbit is obtained from the given ephemeris data and refinement is carried out using an iterative least squares solution. Attitude variations are taken into account by fitting a third order polynomial in time to the attitudes derived from the ephemeris by integration. After refining the satellite position, planimetric co-ordinates can be derived using the collinearity equations with an accuracy of up to 38 m in latitude and 52 m in longitude (average of the r.m.s. errors from two tests) when the GCP used for modelling is obtained from the 1:50 000 scale map. These figures include the measurement errors as well as the modelling error. It has been found that the method is not sensitive to the location of the GCP used for modelling. Résumé On présente dans cet article une méthode de modélisation du traitement des images du satellite SPOT ne faisant appel qu'à un seul point ďappui au sol et permettant ďextraire des informations planimétriques. On obtient la position du satellite à partir des équations traduisant la condition de colinéarité. On part ďune orbite initiale issue des données fournies par les éphémerides, et on ľameliore par une résolution itèrative, basée sur les moindres carrés. On tient compte des variations ďattitude par ľintermédiaire ďun polynome du troisième ordre fonction du temps et correspondant aux attitudes dérivées des éphémérides par intégration. Une fois obtenue la position définitive du satellite, on peut tirer les coordonnees planimétriques des équations de colinéarité avec une précision qui atteint 38 m en latitude et 52 m en longitude (valeurs moyennes des erreurs moyennes quadratiques rèsultant de deux essais), lorsque le point ďappui au sol utilisé dans la modélisation provident de la carte au 1:50 000. Ces chiffres incluent les erreurs de mesure ainsi que ľerreur de modélisation. On a trouvé que la méthode nétait pas du tout sensible à la position qu'occupait le point ďappui au sol utilisé dans le modélisation. 〈section xml:id="abs1-2"〉〈title type="main"〉Zusammenfassung Im Artikel wird ein Verfahren zur Geländemodellierung des SPOT-Abbildungsprozesses unter Verwendung eines einzelnen Geländepaβpunktes (GCP) und zur Ableitung planimetrischer Informational beschrieben. Die Satellitenposition wird mit Hilfe der Kollinearitätsgleichungen beschrieben. Die Näherungswerte für die Bahnkurve werden aus gegebenen Ephemeriden bestimmt, die durch einen iterativen Prozeβ nach der Methode der kleinsten Quadrate verbessert werden. Veränderungen der Bahn werden rechtzeitig durch Anpassung mit einem Polynom 3. Grades an die aus den Ephemeriden abgeleitete Bahn durch Integration angepaβt. Nach Verbesserung der Satelliten-position können die Lagekoordinaten mit Hilfe der Kollinearitätsgleichungen mit einer Genauigkeit bis zu 38 m in der Breite und 52 m in der Länge (Mittelwert der mittleren Fehler aus 2 Versuchen) bestimmt werden, wenn der Paβpunkt (GCP) aus einer Karte 1:50 000 entnommen wurde. Dabei ist der Meβfehler ebenso wie der Modellierungsfehler enthalten. Es zeigte sich, daβ das Verfahren nicht empfindlich ist in bezug aufdie Lage des zur Modellierung benutzten Paβpunktes.

Notes: Two-point and four-point graphs appearing in a three-loop approximation in the field theory renormalization group scheme are calculated in general dimensions. Combining Feynman parameterization and direct integration loop integrals are represented in the form of the expressions, depending on the space dimension d as a parameter. The expressions obtained enables one to consider renormalization group functions directly at noninteger d.

Notes: It is observed that the existence of an attracting set in the class of solutions to the stochastic Lagrangian variational principle leads to a natural problem of convergence of diffusions in the Carlen class. It is then shown how dynamical properties enable one to prove some convergence results in the two-dimensional Gaussian case.

Notes: A complete classification of natural transformations of finite order of Lagrangians into energies over n-dimensional manifolds, where n≥2, is given herein. The classification of natural transformations of finite order of vector fields and Lagrangians into energies and constants of the motion is also obtained for n≥3. By using these results, a classification of Lagrangians into Poincaré–Cartan one-forms and Legendre transformations is obtained.

Notes: The quantum net model differs from other quantizations and discretizations of space-time in being founded upon quantum-logical principles. Fermions are envisaged as arising from defects in the net. Such defects are investigated and it is found that the attempt to vary them infinitesimally gives rise to a reticular version of the Dirac operator, which now has a simple interpretation in terms of quantum mechanical infinitesimal parallel transport. The analog of the usual massless Weyl equations are found to hold on the net, and algebraic solutions are deduced by exploiting its inherent logical structure. A correspondence principle then applies to show that these equations and their solutions correspond precisely to the ones expected in the continuum. (In the course of this, new basis-independent expressions for the complex Minkowski-space Dirac matrices are found.) Finally, an attempt is made to introduce gravity into the net, and the flat space Dirac Lagrangian is modified accordingly.

Notes: Exact analytic solutions are provided for a class of equations describing a composite scalar particle in constant electric and magnetic fields, and also in plane electromagnetic fields. The external fields are taken in both classical and quantum cases. A detailed physical analysis of the results is given.

Notes: It is well known that quantum physics forbids the simultaneous measurement of two noncommuting projection operators. However, sometimes there exists a third projection (mirror projection) which may be measured together with the second one and whose outcomes are completely correlated, in the Einstein–Podolsky–Rosen sense, with the outcomes of the first projection. In this article the states for which such a mirror projection exists are characterized and a procedure for constructing the mirror is given. In particular, it is shown that for two noncommuting projections there are always states for which the mirror projection exists, but also states for which it does not exist.

Notes: A Cauchy problem is formulated with data that describe the scattering of two vortices in a superconductor, and the following is proven: First, a unique global finite-energy solution exists. Second, the symmetry of the solution rules out all cases other than 0°, 90°, or 180° scattering. Third, a local analytic solution exists near the origin. The leading terms of this solution show that, of the three cases, 90° scattering is realized.

Notes: A modified Korteweg–de Vries (mKdV)-type bilinear equation that has been shown to pass the three-soliton solution and the Painlevé tests [see, J. Math. Phys. 2094, 28 (1987); 2572, 31 (1990)] is considered herein. In this article, a Bäcklund transformation and a nonlinear superposition formula are proven rigorously. As an application of the obtained results, N-soliton solutions are obtained. Therefore, the integrability of this mKdV-type bilinear equation is confirmed.

Notes: In this paper all the potential constraints (with or without first-order partial derivatives) of the KP system, from which the associated linear problem can be restricted into a (1+1)-dimensional Hamiltonian equation, are obtained by using the sufficient and necessary condition for a nonlinear equation to be a Hamiltonian system. Some well-known integrable systems, such as (1+1)-dimensional AKNS system, generalized NS system, and several new Hamiltonian equations, are deduced as particular examples.

Notes: A more simple form than the one given by Vaklev [J. Math. Phys. 33, 4111 (1992)] for the soliton solutions related to the quadratic bundle [E. A. Kuznetzov and A. V. Mikhailov, Teor. Mat. Fiz. 30, 303 (1977); D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 798 (1978); V. S. Gerdjikov, M. I. Ivanov, and P. P. Kulish, Teor. Mat. Fiz. 44, 342 (1980); V. S. Gerdjikov and M. I. Ivanov, Bulg. J. Phys. 10, 13 (1983); 10, 2, 129 (1983)] is obtained. The corresponding solutions for the gauge equivalent case are obtained as well. Among them one can find the N-soliton solutions for the derivative nonlinear Schrödinger equation, their gauge equivalents, etc. The method used in the article allows one to express some integrals by means of appropriate determinants and thus the result is obtained without integration.

Notes: For an asymptotically flat initial-data set in general relativity, the total mass-momentum may be interpreted as a Hermitian quadratic form on the complex, two-dimensional vector space of "asymptotic spinors.'' A generalization to an arbitrary initial-data set is obtained. The mass-momentum is retained as a Hermitian quadratic form, but the space of "asymptotic spinors'' on which it is a function is modified. Indeed, the dimension of this space may range from zero to infinity, depending on the initial data. There is given a variety of examples and general properties of this generalized mass-momentum.

Notes: The spectral function (also known as the Plancherel measure), which gives the spectral distribution of the eigenvalues of the Laplace–Beltrami operator, is calculated for a field of arbitrary integer spin (i.e., for a symmetric traceless and divergence-free tensor field) on the N-dimensional real hyperbolic space (HN). In odd dimensions the spectral function μ(λ) is analytic in the complex λ plane, while in even dimensions it is a meromorphic function with simple poles on the imaginary axis, as in the scalar case. For N even a simple relation between the residues of μ(λ) at these poles and the (discrete) degeneracies of the Laplacian on the N sphere (SN) is established. A similar relation between μ(λ) at discrete imaginary values of λ and the degeneracies on SN is found to hold for N odd. These relations are generalizations of known results for the scalar field. The zeta functions for fields of integer spin on HN are written down. Then a relation between the integer-spin zeta functions on HN and SN is obtained. Applications of the zeta functions presented here to quantum field theory of integer spin in anti-de Sitter space–time are pointed out.

Notes: In this article the q-deformed gamma function and q-deformed beta function are discussed in detail. Three definitions for the q-gamma function are obtained and the q-deformed duplication and multiplication formulas are proven. The q-beta function is defined and its useful properties are derived.

Notes: "Classical'' topological entropy is one of the main numerical invariants in topological dynamics on compact spaces. Here, the author's recent development of a noncommutative generalization of topological entropy, in the natural setting of general C*-algebras as the noncommutative counterpart of continuous function algebras on compact spaces, is presented in a slightly modified and improved form. This includes both a survey of earlier results with some important corrections, and also new general results in response to (and inspired by) a more recent counterproposal for a noncommutative topological entropy by Thomsen. Finally, some partially new examples for the calculation of the defined topological entropy are shown. The rather self-evident physical interpretation in the framework of (operator-algebraic) quantum statistical mechanics and of "chaotic'' quantum dynamical systems is briefly touched upon.

Notes: For the fundamental boson realization of SU(N) in its principal SO(3) basis, the generators of SU(N) are SO(3) tensor operators constructed out of one single type of boson, namely a boson with SO(3) multiplet label (N−1)/2. Here, the adjoint boson realization of SU(N) is given: this time the SO(3) tensor operators are constructed by means of N−1 types of bosons [those with SO(3) labels 1,2,..., N−1]. The validity of this realization depends upon a new identity between 6j coefficients. From this realization, or equivalently from this new identity, one can deduce the existence of a family of structural zeros of 6j coefficients.

Notes: This article presents a theory of scalar quantum fields based on nonstandard analysis. Free scalar momentum fields and free Klein–Gordon fields are treated first. It is shown that the nonrigorous formalism of the standard theory can be replaced by a mathematically rigorous nonstandard theory. The second quantized energy–momentum operators are derived and the free quantum fields are diagonalized. Interacting scalar fields are considered and a nonstandard scattering operator is derived. A nonstandard counterpart to Wick's theorem is developed and the φ4 interaction is briefly discussed.

Notes: General adiabatic evolutions associated to Hamiltonians, which admit a holomorphic extension with respect to the time variable in a complex strip, and whose spectrum satisfies a gap condition are studied. An explicit rate of exponential decay is given, which is related to simple geometric quantities associated to the spectrum of the Hamiltonian, for the transition probability between the two parts of the spectrum when the evolution is taken from −∞ to +∞.

Notes: The classical phase space structure of N SU(n+1) non-Abelian Chern-Simons (NACS) particles is investigated by first constructing the product space of associated SU(n+1) bundle with CPn as the fiber. The Poisson bracket is calculated using the symplectic structure on the associated bundle and it is found that the minimal substitution in the presence of external gauge fields is equivalent to the modification of symplectic structure by the addition of field strength two form. Then, a direct product of the associated bundle is taken by the space of all connections and a specific connection is chosen by the condition of vanishing momentum map corresponding to the gauge transformation, thus recovering the quantum mechanical model of NACS particles in Lee and Oh [Phys. Rev. Lett. 72, 1141 (1994)].

Notes: A model is proposed for calculating the magnetic susceptibility of isotropic classical spin chains showing nearest neighbor correlated cationic distributions. It allows one to obtain a general closed-form expression: This expression formally looks like previous ones obtained in the case of random chains for which the neighboring cationic species are not correlated; but here the previous intervening scalar parameters are replaced by vectorial and matricial expressions.

Notes: Recently a doubly periodic solution of the Kadomtsev–Petviashvili equation was deduced by summing over component solitons. The same solution was derived directly by the Hirota bilinear method. This alternate route enables one to obtain a new solution. Such a mechanism can also be applied to wavepacket dynamics, e.g., the Davey–Stewartson equation.

Notes: The mass of gravitational sources differs by a factor of 2 when computed by the Komar superpotential and when computed in the gravitational weak field limit. The difference has been recognized for some time since both the linearized theory and the Einstein pseudotensor yield the same value, different by a factor of 2 from the Komar mass. The great advantage of the Komar superpotential over other methods is its tensorial formulation, however it fails to produce a Noether quantity. A different tensorial method is suggested that is free of the mass anomaly, namely, the zero and first Taub numbers [Phys. Rev. D 47, 474 (1993)] which are indeed Noether quantities. The zeroth Taub number is obtained from the Penrose–Goldberg superpotential [Phys. Rev. D 41, 410 (1990)].

Notes: The power density estimation problem with a priori information about spectrum is considered. The problem is reduced to a power density estimation problem with spectral gaps. The solution of the problem is studied in the context of the trigonometric moment problem with gaps. The necessary and sufficient conditions are established for the existence of solution and all solutions are described in indeterminate cases. A new definition of the entropy functional is given. The effective algorithm to construct maximum entropy solution is proposed. A number of explicit illustrations are presented.

Notes: A realization of the Heisenberg q-algebra whose generators are first-order difference operators on the full real line is discussed herein. The eigenfunctions of the corresponding q-oscillator Hamiltonian are given explicitly in terms of the q−1-Hermite polynomials. The nonuniqueness of the measure for these q-oscillator states is also studied.

Notes: The osp(1,2) Gaudin algebra is defined and integrable models described by it are considered. The models include the osp(1,2) Gaudin magnet and the Dicke model related to it. Detailed discussion of the simplest cases of these models is presented. The effect of the presence of fermions on the separation of variables is indicated.

Notes: In this article, a system of finite-dimensional involutive functions is presented and proven to be integrable in the Liouville sense. By using the nonlinearization method, the C. Neumann system associated with the modified Korteweg–de Vries (mKdV) hierarchy is obtained. Thus, the C. Neumann system is shown to be completely integrable via a gauge transformation between it and an integrable Hamiltonian system. Finally, the solution of a stationary mKdV equation and the involutive solutions of the mKdV hierarchy are secured. As two examples, the involutive solutions are given for the mKdV equation: vt+1/4vxxx−3/2v2vx=0 and the 5th mKdV equation vt−1/16vxxxxx+5/8v2vxxx +5/2vvxvxx +5/8v3x−3/40v4vx=0.

Notes: An algebraically special solution of the Regge–Wheeler–Zerilli equation that has not been noticed is examined. This solution is proved to be related to Chandrasekhar's solution but has a simpler analytical expression.

Notes: The vacuum fluctuations for an impulsive spherical gravitational wave due to the snapping of a "rotating'' cosmic string are calculated herein. This is complementary to the previous calculation due to a snapping string, which had the null result. Although the mathematical expressions are different at the intermediate steps, the same null result is found.

Notes: Vector coherent state theory is used to give an algorithm for constructing the so(5) irreps of type [N,0] and [N,N] in the physical so(3) basis. The construction is remarkably simple and permits the matrices of the so(5) operators, for irreps of low-dimension and/or small multiplicities, to be computed by hand. The construction is applied to the [2,0] and [2,2] irreps to illustrate the process.

Notes: An analytical method to calculate invariants is developed, based on the theory of complete systems of simultaneous linear first order partial differential equations. The application of this algorithm to the general case, in which the invariants are not polynomials is included. For the case of Lie algebras with one invariant the procedure of finding the invariant can be computerized. As an example of this procedure, the algorithm is applied in the calculation of the invariants of the group SA(n,R). The explicit construction of its invariants for n=2, 3, 4 is given. A theorem determining the order of a polynomial invariant of a Lie group is proven allowing us to show that the invariant of SA(n,R) is a homogeneous polynomial of order 1/2n(n+1) in the generators. From this fact an immediate consequence is that the order in the generators of translations is n. The method is also applied in the calculation of invariants of the dimension six solvable Lie algebras, and dimension seven nilpotent.

Notes: The structure of generalized supersymmetry with mutually commuting involutions for the meromorphic supersymmetric quantum mechanics on the compact Riemann surface is introduced. For this purpose a Riemann surface is considered as a self-conjugate complex manifold. On this basis the Witten index for the new involution is calculated.

Notes: The connection between supersymmetric quantum mechanics and the Korteweg–de Vries (KdV) equation is discussed, with particular emphasis on the KdV conservation laws. It is shown that supersymmetric quantum mechanics aids in the derivation of the conservation laws, and gives some insight into the Miura transformation that converts the KdV equation into the modified KdV equation. The construction of the τ function by means of supersymmetric quantum mechanics is discussed.

Notes: A nonprincipal value prescription is used to define the spurious singularities of Yang–Mills theory in the temporal gauge. Typical one-loop dimensionally regularized temporal-gauge integrals in the prescription are explicitly calculated, and a regularization for the spurious gauge divergences is introduced. The divergent part of the one-loop self-energy is shown to be local and has the same form as that in the spatial axial gauge with the principal-value prescription. The renormalization of the theory is also briefly mentioned.

Notes: In this paper, the classical two-fluid model for superconductors is used to determine the time-dependent partial differential equations (PDEs) which govern wave propagation in superconducting media. These equations are then solved exactly in both two and three dimensions using localized wave solutions rather than the traditional eigenfunction solutions. We have applied these localized wave solutions to the problem of a symmetric superconducting slab, neglecting the normally conducting current density, and found that the resultant focus wave mode magnetic field solution expels flux from the interior of the slab and can regain its initial amplitude as it travels along the surface of the slab. A comparison of the transverse part of the localized wave solution with the transverse part of the more usual plane wave solution shows remarkable agreement.

Notes: The Dirac equation is rewritten in the framework of a Clifford algebra, revealing some new features, of which the most interesting is the U(1)×SU(2) symmetry of the corresponding Lagrangian.

Notes: The field fluctuations are studied for the free and imperfect Bose gas. In the phase where there is condensation, these field fluctuations are abnormal, with critical exponents depending on the way the symmetry-breaking external field tends to zero in the thermodynamic limit. On the critical line there is also abnormal behavior, but the dependence on the way the field disappears, is slightly different. For densities below the critical density, the fluctuations are normal and independent of the way the external field goes to zero.

Notes: Using the inverse strong symmetry of the Korteweg–de Vries (KdV) equation on the trivial symmetry and τ0 symmetry, one gets four new sets of symmetries of the KdV equation. These symmetries are expressed explicitly by the multi-integrations of the Jost function of the KdV equation and constitute an infinite dimensional Lie algebra together with two hierarchies of the known symmetries. Contrary to the general belief, the time-independent symmetry groups of the KdV and mKdV equations are non-Abelian and the infinite dimensional Lie algebras of the KdV and mKdV equations are nonisomorphic though two equations are related by the Miura transformation. Starting from these sets of symmetries, four hierarchies of the integrodifferential KdV equations, which can be solved by the Schrödinger inverse scattering transformation method, are obtained. Some of these hierarchies enjoy a common strong symmetry and/or same local conserved densities.

Notes: From a bi-Hamiltonian viewpoint the equivalence of two supersymmetric Korteweg–deVries theories, introduced by Manin–Radul and Laberge–Mathieu, is discussed herein. It is shown that the transformation connecting the two theories (proposed recently in the literature) preserves the bi-Hamiltonian structures; moreover, another derivation of this transformation, stemming from bi-Hamiltonian reduction theory and strongly emphasizing the geometrical meaning of the above equivalence is presented.

Notes: In this comment we obtain two types of solutions of the graded Yang–Baxter equation whose symmetry group is GLqs(1||1).

Notes: The osp(1||2) algebra is first realized through one boson and two fermion pairs of creation annihilation operators and then both osp(1||2) and osp(3||2) are realized in terms of three boson and two fermion pairs. All these realizations are reducible, and the reduced states are labeled by the total occupation number.

Notes: Algebras that represent a generalization of both quantum groups, quantum supergroups, and braided groups are defined. They are given by a pair of solutions of the Yang–Baxter equation that satisfy some additional conditions. Several examples are presented.

Notes: An addition law is introduced for the usual quantum matrices A(R) by means of a coaddition Δ(underbar)t=t⊗1+1⊗t. It supplements the usual comultiplication Δt=t⊗t and together they obey a codistributivity condition. The coaddition does not form a usual Hopf algebra but a braided one. The same remarks apply for rectangular m×n quantum matrices. As an application, left-invariant vector fields are constructed on A(R) and other quantum spaces. They close in the form of a braided Lie algebra. As another application, the wave functions in the lattice approximation of Kac–Moody algebras and other lattice fields can be added and functionally differentiated.

Notes: Representations of the n-braid group where generators are given by matrices whose elements belong to a noncommutative algebra are discussed. Representation of this algebra in a Hilbert space generalizes the Burau representation. Two algebras which are closely related and are physically indistinguishable provided that a particular eigenvalue is fine tuned are discussed. It is shown that the generalized oscillator system given by one of the algebras generates a hydrogenlike spectrum.

Notes: The Wigner–Poisson (WP) system (or quantum Vlasov–Poisson system) is modified to include dissipative terms in the Hamiltonian. By utilizing the equivalence of the WP system to the Schrödinger–Poisson system, global existence and uniqueness are proved and regularity properties are deduced. The proof differs somewhat from that for the nondissipative case treated previously by Brezzi–Markowich and Illner et al.; in particular the Hille–Yosida Theorem is used since the linear evolution is not unitary, and a Liapunov function is introduced to replace the energy, which is not conserved.

Notes: A quantum logic formulation of quantum measurements is introduced using Boolean powers of orthomodular σ-posets to the description of the coupled system. A comparison with the traditional Hilbert space approach to quantum measurements and a discussion of some important classes of measurements is given.

Notes: Nonlinear wave equations describing forced vibrations of a string subject to random perturbations of intensity ε are studied herein. Two kinds of random perturbations are considered; one is Gaussian in the form of initial white noise and the other is a non-Gaussian random forcing which involves the formal derivative of a Brownian sheet. The solutions uε(t,x) of the complete forced equation are a stochastic process in space and time which has a unique stochastic equilibrium. The time variable t varies in a finite interval I:=[0, T], where T is a fixed positive time, and the space variable x varies in an interval J. Assuming that the initial conditions are of sufficient regularity, the fluctuations uε(t,x) from u0(t,x), the unperturbed solution, are analyzed as the intensity of perturbation ε↓0.

Notes: Study of radial and circular geodesics including those of tachyons in two spherically symmetric vacuum five-dimensional space–time is done. One of them is of Schwarzschild type and the other the so-called soliton solution. We obtain results which are different from the usual analogous four-dimensional (4D) ones. It is found that all the circular geodesics irrespective of whether they are timelike, null, or spacelike will be stable in the region r(approximately-greater-than)6M. It has also been shown that in a homogeneous cosmological model where the isotropic 3 space expands but the extra space contracts with time, the existence of tachyon is possible. This is in contradiction to the analogous 4D result obtained previously.

Notes: It is shown that Bäcklund transformations (BTs) and zero-curvature representations (ZCRs) of systems of partial differential equations (PDEs) are closely related. The connection is established by nonlinear representations of the symmetry group underlying the ZCR which induce gauge transformations relating different BTs. This connection is used to construct BTs from ZCRs (and vice versa). Furthermore a procedure is outlined which allows a systematic search for ZCRs of a given system of PDEs.

Notes: Using the supersymplectic framework of Berezin, Kostant, and others, two types of supersymmetric extensions of the Schrödinger algebra (itself a conformal extension of the Galilei algebra) were constructed. An ‘I-type' extension exists in any space dimension, and for any pair of integers N+ and N−. It yields an N=N++N− superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin-1/2 particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, ‘exotic' or ‘IJ-type' extensions arise for each pair of integers ν+ and ν−, yielding an N=2(ν++ν−) superalgebra of the type discovered recently by Leblanc et al. in nonrelativistic Chern–Simons theory. For the magnetic monopole the symmetry reduces to o(3)×osp(1/1), and for the magnetic vortex it reduces to o(2)×osp(1/2).

Notes: In the present paper all typical finite-dimensional representations of the quantum Lie superalgebra Uq[gl(2/2)] are constructed at generic deformation parameter q. As in the nondeformed case, the finite-dimensional Uq[gl(2/2)]-module Wq obtained is irreducible and can be decomposed into finite-dimensional irreducible Uq[gl(2)⊕gl(2)]-submodules Vqk.

Notes: The geometric phase has been defined as the holonomy of two bundles with different base spaces and different connections. In this article we establish the relation between the Berry–Simon and the Anandan–Aharonov definitions for a large class of cyclic Hamiltonians (with N-fold degeneracy) using the classifying theorem for principal fiber bundles.

Notes: It is shown that the polynomials introduced recently by Aldaya, Bisquert, and Navarro-Salas [Phys. Lett. A 156, 381 (1991)] in connection with a relativistic generalization of the quantum harmonic oscillator can be expressed in terms of Gegenbauer polynomials. This fact is useful in the investigation of the properties of the corresponding wave function. Some examples are given, in particular, related to the asymptotic behavior and to the distribution of zeros of the polynomials for large quantum numbers.

Notes: The eigenvalue problem for the radial Schrödinger equation with an "almost'' Coulomb potential is considered. This problem provides the simplest example of a system whose classical trajectories have singularities. Thus the standard semiclassical quantization procedure (in the sense of Einstein–Brillouin–Keller–Maslov) cannot be applied straightforwardly to this situation. The equation under consideration has two transition points on the half axis 0〈x〈∞: a regular singular point at the origin and a turning point for some x0(approximately-greater-than)0; these points coalesce at low energy levels. The well-known comparison equation method provides uniform asymptotics for equations of this kind, but has the following shortcomings: it does not appeal to the corresponding classical mechanical problem and the formulas for the eigenvalues and eigenfunctions contain phase integrals which are not analytic with respect to the parameter characterizing the closeness of the transition points. In the present article we derive new formulas for the eigenfunctions in the form of inverse Fourier transforms of rapidly oscillating exponentials multiplied by some powers of x, together with simple formulas for the eigenvalues. These formulas are directly connected with the classical trajectories and do not have the shortcomings of the comparison equation method, i.e., all the functions entering the asymptotic expansions are analytic in x and energy. As a by-product, an asymptotic expansion of integrals having two stationary phase points which coalesce for some value of the parameter and tend to infinity when the parameter tends to zero is obtained.

Notes: Using the formulation in terms of principal fiber bundles, it is shown that the external symmetry implies the conservation of the energy-momentum tensor for a general gauge theory. When applying this theory to the case of Dirac fields, one is naturally forced to replace the dependence on the metric by the dependence on the underlying spin structure in the variational problems.

Notes: A study of the superconformal covariantization of superdifferential operators defined on (1||1) superspace is presented. It is shown that a superdifferential operator with a particular type of constraint can be covariantized only when it is of odd order. In such a case, the action of superconformal transformation on the superdifferential operator is nothing but a Hamiltonian flow defined by the corresponding supersymmetric second Gelfand–Dickey bracket. The covariant form of a superdifferential operator of odd order is given.

Notes: Inversion formulas are obtained for the restrictions of the Hartley and Hilbert transforms to R+. Regularity results are derived, and an illustrative example presented.

Notes: The noncommutative geometry corresponding to arbitrary braidings is studied in a purely algebraical way. The new and original proposal of the geometry is given in terms of the theory of braided monoidal categories. The quantum oscillator is considered as an example of application of our geometry.

Notes: Hamiltonians with strong Wigner–von Neumann potentials are known to possess bound states in the continuous spectrum. For smaller couplings these bound states become resonances, which appear as singularities of the m-function. These singularities are intimately connected with the decay of the corresponding subordinate eigenfunction. For Coulomb-type potentials with Schrödinger operators this has recently be shown by Atkinson, the author, and Hinton, Klaus and Shaw. Here these results are extended to a general class of Wigner–von Neumann potentials for Dirac and Schrödinger operators.

Notes: A study is made of the real-energy limits of approximate solutions of the Chandler–Gibson equations, as well as the real-energy limits of the approximate equations themselves. It is proved that (1) the approximate time-independent transition operator Tπ(z) and an auxiliary operator Mπ(z), when restricted to finite energy intervals, are trace class operators and have limits in trace norm for almost all values of the real energy; (2) the basic dynamical equation that determines the operator Mπ(z), when restricted to the space of trace class operators, has a real-energy limit in trace norm for almost all values of the real energy; (3) the real-energy limit of Mπ(z) is a solution of the real-energy limit equation; (4) the diagonal (on-shell) elements of the kernels of the real-energy limit of Tπ(z) and of all solutions of the real-energy limit equation exactly equal the on-shell transition operator, implying that the real-energy limit equation uniquely determines the physical transition amplitude; and (5) a sequence of approximate on-shell transition operators converges strongly to the exact on-shell transition operator. These mathematically rigorous results are believed to be the most general of their type for nonrelativistic N-body quantum scattering theories.

Notes: In his classic book on group representations and special functions Vilenkin studied the matrix elements of irreducible representations of the Euclidean and oscillator Lie algebras with respect to countable bases of eigenfunctions of the Cartan subalgebras, and he computed the summation identities for Bessel functions and Laguerre polynomials associated with the addition theorems for these matrix elements. He also studied matrix elements of the pseudo-Euclidean and pseudo-oscillator algebras with respect to the continuum bases of generalized eigenfunctions of the Cartan subalgebras of these Lie algebras and this resulted in realizations of the addition theorems for the matrix elements as integral transform identities for Bessel functions and for confluent hypergeometric functions. Here we work out q analogs of these results in which the usual exponential function mapping from the Lie algebra to the Lie group is replaced by the q-exponential mappings Eq and eq. This study of representations of the Euclidean quantum algebra and the q-oscillator algebra (not a quantum algebra) leads to summation, integral transform, and q-integral transform identities for q analogs of the Bessel and confluent hypergeometric functions, extending the results of Vilenkin for the q=1 case.

Notes: The problem of the correspondence between symmetries and conservation laws for partial differential equations is considered. For Lagrangian systems the set of Noether (variational) symmetries can be shown to lead to the set of all local conservation laws. For partial differential equations without well-defined Lagrangian functions there is no universal correspondence between symmetries and conservation laws. In this article it is shown that for a large class of differential equations there is a natural way to associate conservation laws with symmetries. The class consists of many interesting equations, e.g., Korteweg–de Vries equation, Kadomtsev–Petviashvili equation, Boussinesq equation, nonlinear diffusion equation, Monge–Ampère equation, regularized long-wave equation, and Navier–Stokes equations. Characteristics of the corresponding conservation laws are calculated and examples are given. For Lagrangian systems the consistency of the approach with the standard Noether results is discussed.

Notes: A rigorous derivation of the semiclassical Liouville equation for electrons which move in a crystal lattice (without the influence of an external field) is presented herein. The approach is based on carrying out the semiclassical limit in the band-structure Wigner equation. The semiclassical macroscopic densities are also obtained as limits of the corresponding quantum quantities.

Notes: A formulation of the complete dynamics of electrons and nuclei is presented. The dynamical equations are derived using the time-dependent variational principle (TDVP). The approximate electronic state vectors are antisymmetrized geminal power (AGP) states parameterized as projected coherent states, while the nuclei are treated as classical point particles. This leads to a formulation of time-dependent AGP theory that generalizes time-dependent Hartree–Fock (TDHF) theory and explicitly includes the dynamics of the nuclei. The linear approximation to the evolution equations (the classical harmonic approximation) which corresponds to a generalized random phase approximation (RPA) based on an AGP electronic reference state and which explicitly includes the dynamics of the nuclei, is studied and presented in this paper. The equations are formulated in terms of the primitive nonorthogonal electronic atomic basis thus avoiding any transformation to orthonormal molecular orbitals during the evolution.

Notes: Chaos in adiabatic Hamiltonian systems is a recent discovery and a pervasive phenomenon in physics. In this work, a geometric criterion is discussed based on the theory of action from classical mechanics to detect the existence of Smale horseshoe chaos in adiabatic systems. It is used to show that generic adiabatic planar Hamiltonian systems exhibit stochastic dynamics in large regions of phase space. To illustrate the method, results are obtained for three problems concerning relativistic particle dynamics, fluid mechanics, and passage through resonance, results which either could not be obtained with existing methods, or which were difficult and analytically impractical to obtain with them.

Notes: The complete scheme of the application of one- and two-dimensional subspaces and the subgroups method to five-dimensional gravity with a G3 group of motion are presented here in space–time and in potential space formalisms. From this method one obtains the Kramer, Belinsky–Ruffini, Dobiasch–Maison, Clément, Gross–Perry–Sorkin solutions, etc., as special cases.

Notes: The following inequality is established: ||Pn(cos cursive-theta)||〈 [(square root of)1+(π4/16)(n+1/2)4 sin4 cursive-theta]−1, 0〈cursive-theta〈π, n=1,2,..., where Pn(x) denotes the Legendre polynomial of degree n. The relation P2n(cos cursive-theta) + (4/π2)× Q2n(cos cursive-theta) 〈 [(square root of)1+(π4/16)(n+1/2)4 sin4 cursive-theta]−1, n=1,2,..., on [θn1,θn,n+1], is proven where Qn(x) denotes the Legendre function of second kind, cos θn1 the largest zero of Qn(x), and cos θn,n+1=−cos θn1. Similarly we obtain the inequalities ||J0(x)|| 〈 [(square root of)1+(π4/16)x4]−1, x≠0, and J20(x) + Y20(x)〈 [(square root of)1+(π4/16)x4]−1, x≥y1, where y1=0.893577... is the first positive zero of Y0(x), and J0(x), Y0(x) denote the Bessel functions of the first and second kind, respectively. The results of the present paper arise out of some problems of nuclear and particle physics.

Notes: A new family of asymptotic solutions of the Helmholtz equation can be derived. It is shown that new solutions are regular at caustics and turning points and do not require any special functions.

Notes: Operators of the form Tβf(x)=∑Jj=1 cursive-phij(βx)f(σj +ρjx) are considered herein where β is a real parameter; σj and ρj are real numbers, and {cursive-phij} are holomorphic functions which satisfy certain additional conditions. Transfer operators of this form appear in statistical physics: the leading eigenvalue may be interpreted as the free energy of a lattice of interacting particles, as a function of inverse temperature β. A method for obtaining Taylor series expansions in β of the leading and second leading eigenvalues, and the leading eigenfunction are given. The method may be extended to find series expansions for other eigenvalues and eigenfunctions. The radii of convergence of these series have not been found. These results generalize the author's findings in another article [J. Math. Phys. 32, 2718 (1991)].

Notes: A description of the Fock space of a parabose oscillator of order p based on commutation relations bilinear in creation and annihilation operators rather than trilinear ones is developed. A new statement of the well-known Green's ansatz is introduced to help understand the significance of the bilinear commutation relations. It is also applied to show how all parabose oscillators of even order (respectively, odd order) can be obtained as irreducible constituents of the Fock space associated with a Green's ansatz of two (three) terms. Analogs of the form in which the parabose Green's ansatz is presented are provided for one parafermi oscillator and for systems of many parabose and parafermi oscillators of the same order. Methods used in the paper allow various new results for q-deformed parabose oscillators to be derived.

Notes: The time independent Bateman model which describes the damped harmonic oscillator coupled to its dual is studied using the path integral approach. The related propagator is calculated by means of the matrix representation.

Notes: A study of the existence of some integrable systems with nonlinear constants of motion is presented using the approach of the theory of generalized (dynamical or hidden) symmetries. Two Lagrangians are considered, both obtained by modifying the Toda Lagrangian. First a two-particle system is studied and then the results are generalized to a three-particle system. It is shown that in both cases the Lagrangians possess nonlinear constants of motion in involution and, thus, they are integrable.

Notes: A finite subgroup of the conformal group SL(2,C) can be related to invariant polynomials on a hypersurface in C3. The latter then carries a simple singularity, which resolves by a finite iteration of basic cycles of deprojections. The homological intersection graph of these cycles is the Dynkin graph of an ADE Lie group, i.e., a Lie group from the cartan series A, D, or E. The deformation of the simple singularity corresponds to ADE symmetry breaking. A (3+1)-dimensional topological model of observation is constructed, transforming consistently under SL(2,C), as an evolving three-dimensional system of world tubes, which connect "possible points of observation.'' The existence of an initial singularity for the four-dimensional space–time is related to its global topological structure. Associating the geometry of ADE singularities to the vertex structure of the topological model puts forward the conjecture on a likewise relation of inner symmetries of elementary particles to local space–time structure.

Notes: An exact asymptotically flat axisymmetric solution of the Einstein–Maxwell equations representing the exterior field of two arbitrary Kerr–Newman masses located on the symmetry axis is constructed in explicit form. In a particular case, when the solution describes two identical Kerr–Newman sources, simple analytical formulas are obtained which allow a straightforward analysis of the equilibrium of two charged rotating masses. Some arguments are given in favor of the possibility of the balance of stationary black holes due to their gravitational spin–spin interaction. © 1994 American Institute of Physics.

Notes: Let @Fg be a simple complex (finite dimensional) Lie algebra, and let R be the ring of regular functions on a compact complex algebraic curve with a finite number of points removed. Lie algebras of the form @Fg⊗CR are considered; these generalize Kac–Moody loop algebras since for a curve of genus zero with two punctures R(approximately-equal-to)C[t,t−1]. The universal central extension of @Fg⊗R is analogous to an untwisted affine Kac–Moody algebra. By Kassel's theorem the kernel of the universal central extension is linearly isomorphic to the Kähler differentials of R modulo exact differentials. The dimension of the kernel for any R is determined first. Restricting to hyperelliptic curves with 2, 3, or 4 special points removed, a basis for the kernel is determined. Restricting further to an elliptic curve with punctures at two points (of orders one and two in the group law) we explicitly determine the cocycles which give the commutation relations for the universal central extension. The results involve Pollaczek polynomials, which are a genus-one generalization of ultraspherical (Gegenbauer) polynomials. © 1994 American Institute of Physics.

Notes: It is noted that the hypothesis of independent random complex elements for the off diagonal blocks (say, M2 and M3) of the transfer matrix describing conductance in a mesoscopic wire allows the eigenvalue distribution of the matrix product M3°M3 (or M2°M2) to be computed exactly. Exact expressions, in terms of double Wronskian and Toeplitz determinants, are derived for the distribution of the smallest and second smallest eigenvalue of this and similar random matrix products. © 1994 American Institute of Physics.

Notes: The one-dimensional Schrödinger equation is considered when the potential and its first moment are absolutely integrable. When the potential has support contained on the left (right) half-line, it is uniquely constructed by using only the reflection coefficient from the right (left). The bound state norming constants determine whether the potential has support contained on a half-line or on the full-line. The bound state energies and the unique set of norming constants yielding the potential with support contained on the left (right) half-line are completely determined by the reflection coefficient from the right (left). An explicit example is provided. © 1994 American Institute of Physics.

Notes: An axiomatic approach to the problem of reconstruction of a dynamics from the Euclidean multi-time Green functions is proposed. The existence of a W*-algebra on which the reconstructed dynamics acts as the canonical group of modular automorphisms is shown. A condition under which the reconstruction from the one-time and multi-time Green functions are equivalent is formulated. © 1994 American Institute of Physics.

Notes: Gell-Mann and Hartle have proposed a significant generalization of quantum theory with a scheme whose basic ingredients are "histories'' and decoherence functionals. Within this scheme it is natural to identify the space UP of propositions about histories with an orthoalgebra or lattice. This raises the important problem of classifying the decoherence functionals in the case where UP is the lattice of projectors P(V) in some Hilbert space V; in effect, one seeks the history analog of Gleason's famous theorem in standard quantum theory. In the present article the solution to this problem for the case where V is finite dimensional is presented. In particular, it is shown that every decoherence functional d(α,β), α,β∈P(V) can be written in the form d(α,β)=trV⊗V(α⊗βX) for some operator X on the tensor-product space V⊗V. © 1994 American Institute of Physics.

Notes: Properties of a relativistic plasma dispersion function (PDF) that is required for the description of waves propagating perpendicular to a magnetic field in a fully relativistic, magnetized, Maxwellian plasma, are presented. Series, asymptotic series, recurrence relations, integral representations, derivatives, generating functions, approximations, differential equations, and connections with standard transcendental functions are discussed, as are the PDF's analytic properties. © 1994 American Institute of Physics.

Notes: What will be shown is the first-order perturbation of closed orbits for two-body problems using approximate symmetry theory. A complete group classification with respect to approximate symmetries of the first-order perturbation of the closed orbit equations corresponding to Kepler's law and Hooke's law is made. A discussion then follows of the connection between closed orbits and their stable symmetries. © 1994 American Institute of Physics.

Notes: A study of a three-particle system that possesses, in addition to the energy function, two nonlinear constants of motion is presented herein. The system is obtained by modification of the Lagrangian of the Toda lattice. The explicit expression of the two integrals is obtained and it is proven that this new three-particle Lagrangian is integrable. Finally, the possible generalization to the case of n particles is discussed. © 1994 American Institute of Physics.

Notes: An extension of the Dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in Dirac's original proposal. These issues play an important role especially in the context of nonlinear, diffeomorphism invariant theories such as general relativity. Recently, an extension of the required type was proposed using algebraic quantization methods. In this paper, the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples. The choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity. However, prior knowledge of general relativity is not assumed in the main discussion. Indeed, the methods introduced and conclusions arrived at are applicable to any system with first class constraints. In particular, they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems. © 1994 American Institute of Physics.

Notes: Probabilistic (recurrent) renewal theory is used herein to describe the geometry of a new class of self-similar distributions in a "skewed'' multiplicative mass splitting procedure. It is shown how this model class can account for a deterministic and conservative treelike structure, in connection with recent developments on "anomalous'' multiplicative multifractals. Coexistence of both micro- and macroscopic scales in such objects is a consequence of the underlying renewal phenomena being transient, as is shown in a second part of the article. Various "entropylike'' measures of the intrinsic disorder prevailing in these structures are proposed. © 1994 American Institute of Physics.