Relatively Prime Polynomials and Nonsingular Hankel Matrices over Finite Fields

The probability for two monic polynomials of a positive degree n with coefficients in the finite field F_q to be relatively prime turns out to be identical with the probability for an n x n Hankel matrix over F_q to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is the classical notion of Bezoutian of two polynomials. Moreover, we give simpler and direct proofs of the general formulae for the number of m-tuples of relatively prime polynomials over F_q of given degrees and for the number of n x n Hankel matrices over F_q of a given rankComment: 10 pages; to appear in the Journal of Combinatorial Theory, Series

A combinatorial proof of the Rogers-Ramanujan and Schur identities

We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson's rank. These symmetries are established by direct bijections.Comment: 12 pages, 5 figures; incorporated referee suggestions, simplified definition of (k,m)-rank, to appear in JCT(A

The cyst-motile stage relationship of three Protoperidinium species from south-east coast of Iran

Resting cyst and motile thecate cell stages of three heterotrophic Protoperidinium oblongum, Protoperidinium sp. and Protoperidinium claudicans were assessed. Cysts were isolated from sediment collected from southeast coast of Iran. Individual live cysts were incubated under optimal conditions for germination. Results showed that Protoperidinium oblongum cysts were pentagonal smooth walled and three cysts of this species were identified from Iranian sediment. The germinated cell of P. oblongum was colourless and elongated pentagonal in shape. P. claudican theca cell has a pointed antapical horn with a four sided apical plate. Protoperidinium sp. had dark brown pentagonal cyst. Its germinated cell differed from other Protoperidinium species. Two divergent distally antapical horns formed 90˚ between the horn and Posterio-lateral of main body edge, thus, comprising a unique species in the Protoperidinium genus. This kind of antapical horn has not been previously reported among Protoperidinium spp

Synthesis of crystalline microporous Mo−V−Bi oxide for selective (Amm)oxidation of light alkanes

Bismuth (Bi) was successfully introduced into the crystalline orthorhombic Mo3VOx (MoVO) structure for the first time by using the ethylammonium cation (EtNH3+) as a structure-directing agent in hydrothermal synthesis, and the catalytic activities of MoVO-containing Bi (MoVBiO) for selective oxidation of ethane and ammoxidation of propane were compared with those of ternary MoVO. Bi and EtNH3+ were located in hexagonal and heptagonal channels in the MoVO structure, respectively. EtNH3+ could be removed without collapse of the crystal structure by appropriate heat treatment, leaving the heptagonal channels empty. The introduction of Bi had only a little effect on the catalytic activity for selective oxidation of ethane. On the other hand, the conversion of propane was significantly enhanced in propane ammoxidation. Acrylonitrile selectivity was also enhanced by the introduction of Bi, especially at high temperatures (>440 °C)

The historical topography of a provincial Byzantine city in Thrace: Vize (Bizye)

This is a topographical study of an ancient settlement in southern Thrace. Located in the provincial district of the city Kırklareli in Turkey today, Vize (Bizye) was the residential capital of the Thracian kings from the 3rd century BC on, a prosperous Roman city under Trajan and Hadrian and a Late Antique and early Christian city with fortifications repaired and modified throughout the following centuries of the Byzantine period. Becoming a garrison regarding the Bulgarian threat especially from the 9th century on, the city survived, as proved by the 9th century church of Hagia Sophia. The city held strong in the years of intensive attacks in the 13th –14th centuries, until it was taken by the Ottomans in 1453. The Ottoman settlement lay mainly on the old Byzantine topography and extended with residential quarters on a small scale. After the Russian-Ottoman and Balkan Wars of the 19th and the early 20th centuries, the modern settlement of the Turkish Republic started developing towards the south on the plain, leaving the old acropolis to the small local houses and the ghosts of the Roman and Byzantine monuments. Although the archaeological significance of the site was known already at the beginning of the 20th century, neither a well-documented systematic excavation nor a comprehensive topographical study of the acropolis was made. Hitting the road with this motivation, we have undertaken a topographical survey throughout two summer campaigns in 2011 and 2012 on the Acropolis of Vize, which is the core of the settlement from the Roman until the end of the Ottoman period. The main purpose of the survey was primarily to document the fortifications in detail in order to produce a digital topographical plan. Other monuments on site have also been measured and inserted into this plan. This thesis discusses the results gained through the comprehensive analysis of the fortifications — their material, masonry and typology of the structural elements comparing with parallel examples from particularly the same administrative territory of Thrace. The presumable circuit of the walls half of which is missing and the extension and the changes of the urban structure are discussed while considering the construction periods of the walls together with the other remaining monuments and archaeological rests on site.Die vorliegende topographische Studie befasst sich mit einer antiken Siedlung im südlichen Thrakien. Vize (Bizye), das heute in der türkischen Provinz Kırklareli liegt, diente seit dem 3. Jahrhundert v. Chr. den Thrakerkönigen als Residenz. In römischer Zeit erlebte die Stadt unter den Kaisern Trajan und Hadrian eine Blüte. In der Spätantike wurde die Stadt ummauert. Diese Stadtbefestigung wurde in den darauffolgenden Jahrhunderten mehrfach repariert und verändert. Angesichts der bulgarischen Bedrohung, sprich besonders seit dem 9. Jahrhundert n. Chr. war in Vize eine byzantinische Garnison stationiert. Dieser Umstand sicherte auch das Fortbestehen der Siedlung, was wiederum durch die Errichtung der Sophienkirche im 9. Jahrhundert bestätigt wird. Die Siedlung überstand auch die Zeiten schwerer Angriffe im 13. und 14. Jahrhundert bis sie schließlich 1453 durch die Osmanen eingenommen wurde. Die osmanische Siedlung überlagert weitgehend den byzantinischen Siedlungsbereich und erweiterte diesen um Viertel, die von einfacher Wohnbebauung gekennzeichnet waren. Nach dem Krieg zwischen Russland und osmanischen Reich im 19. bzw. dem Balkankrieg im frühen 20. Jahrhundert entwickelte sich eine moderne türkische Siedlung in der Ebene südlich des historischen Siedlungshügels. Die Akropolis von Vize mit ihren römischen und byzantinischen Baudenkmälern überließ man dabei der lokalen Bevölkerung, die dort einfache Häuser errichteten. Obwohl die archäologische Bedeutung der Stätte bereits zu Beginn des 20. Jahrhunderts erkannt worden war, wurde sie weder systematisch ergraben und dokumentiert noch war der Bereich der Akropolis bislang Gegenstand einer umfassenden topographischen Studie. Um diese Forschungslücke zu schließen, wurde in den Jahren 2011 und 2012 auf dem Burgberg von Vize, der mit dem Siedlungskern von römischer bis osmanischer Zeit nahezu identisch ist, ein topographischer Survey durchgeführt. Hauptziel des Surveys war eine detaillierte Dokumentation der Befestigungsanlage um darauf aufbauend einen digitalen topographischen Plan erstellen zu können. Dabei wurden auch weitere Bauten im Surveybereich vermessen und in den Plan eingefügt. Die vorliegende Dissertation legt die Ergebnisse der umfassenden bauhistorischen Untersuchung der Befestigungsanlagen vor, wobei ein besonderes Augenmerk auf das verwendete Baumaterial, die Beschaffenheit des Mauerwerks und die typologische Analyse der Bauelemente gelegt wird. Hierfür werden die Baudenkmäler mit weiteren Verteidigungsanlagen verglichen, die hauptsächlich im selben Verwaltungsbezirk, also in Thrakien, liegen. Der mutmaßliche Verlauf der nur ungefähr zur Hälfte erhaltenen Akropolismauer werden ebenso diskutiert, wie die Erweiterung und die Veränderung der Siedlungsstruktur. Dabei werden die einzelnen Bauabschnitte der Befestigungsmauer in Bezug zu den übrigen Bauwerken und archäologischen Zeugnissen vor Ort gesetzt

Efficient implementation of the Hardy-Ramanujan-Rademacher formula

We describe how the Hardy-Ramanujan-Rademacher formula can be implemented to allow the partition function $p(n)$ to be computed with softly optimal complexity $O(n^{1/2+o(1)})$ and very little overhead. A new implementation based on these techniques achieves speedups in excess of a factor 500 over previously published software and has been used by the author to calculate $p(10^{19})$, an exponent twice as large as in previously reported computations. We also investigate performance for multi-evaluation of $p(n)$, where our implementation of the Hardy-Ramanujan-Rademacher formula becomes superior to power series methods on far denser sets of indices than previous implementations. As an application, we determine over 22 billion new congruences for the partition function, extending Weaver's tabulation of 76,065 congruences.Comment: updated version containing an unconditional complexity proof; accepted for publication in LMS Journal of Computation and Mathematic