21 research outputs found
Meromorphic Jacobi Forms of Half-Integral Index and Umbral Moonshine Modules
No full textIn this work we consider an association of meromorphic Jacobi forms of half-integral index to the pure D-type cases of umbral moonshine, and solve the module problem for four of these cases by constructing vertex operator superalgebras that realise the corresponding meromorphic Jacobi forms as graded traces. We also present a general discussion of meromorphic Jacobi forms with half-integral index and their relationship to mock modular forms
Optimal mock Jacobi theta functions
No full textWe classify the optimal mock Jacobi forms of weight one with rational coefficients. The space they span is thirty-four-dimensional, and admits a distinguished basis parameterized by genus zero groups of isometries of the hyperbolic plane. We show that their Fourier coefficients can be expressed explicitly in terms of singular moduli, and obtain positivity conditions which distinguish the optimal mock Jacobi forms that appear in umbral moonshine. We find that many of Ramanujan's mock theta functions can be expressed simply in terms of the optimal mock Jacobi forms with rational coefficients
Advanced Flight Control Design and Evaluation: An application of time delayed Incremental Backstepping
No full textThe sensor-based approach of Incremental Backstepping is applied to flight control law design in this research project. It allows the usage of the same control law on different types of aircraft without the need for redesign. Apart from full state availability, the derivation of Incremental Backstepping assumes instantaneous control action. Due to actuator lags and delays, the implementation of control commands cannot necessarily be considered instantaneous. This mitigates the stability guarantee provided by Lyapunov theory. Therefore, a novel technique to estimate the time delay margins of the Incremental Backstepping controlled systems is proposed in the thesis. This provides an important stability measure for possible certification and widens the application range of Incremental Backstepping. This simple, yet effective, Lyapunov-based control technique shows positive robustness properties with respect to model uncertainties, unknown parameters, external disturbances and time delay effects. It is applied to the DA 42 aircraft as a (pilot-in-the-loop) rate controller in the scope of this thesis. The implementation requires measurements of the aircrafts angular accelerations and control surface deflections. If the latter is not available, it is shown that filters can still be used in the control system. However, the usage of filters mitigates the highly favorable robustness properties of the closed-loop system. Moreover, a controller evaluation strategy is proposed. It rates the performance and stability properties of the Incremental Backstepping controlled system in terms of the flight control system requirements. Evaluation of the Incremental Backstepping controller shows allowable input multiplicative uncertainties of up to 40% of the nominal value at the worst-case excitation frequency for a controller update rate of 100Hz. When no reference shaping is applied, the handling qualities of the incremental rate controller show to be less desirable than that of a conventional linear controller designed specifically for the DA 42. However, it is possible to improve handling characteristics by reference shaping. Furthermore, the handling characteristics of the incremental controller remain fairly constant along the flight envelope and in adverse flight conditions.Guidance, Control & NavigationControl and Operations (C&O) - Control and Simulation Division (C&S)Aerospace Engineerin
Aflsuiting Grevelingen: Noordelijk Sluitgat: Rapport modelonderzoek
No full textIn een detailmodel is een vergelijkend onderzoek naar de vormgeving van de sluitgaten uitgevoerd, waarbij van een bepaalde keuze voor de lengte van de verdediging is uitgegaan, n.l. aan weerszijden 80 m gemeten uit de drempelas. De ontgronding is hierbij als criterium genomen. Het onderzoek is uitgevoerd voor eb, daar hierbij de grootste snelheden in de sluitgaten zijn te verwachten.Of de uit het onderzoek gevolgde gunstigste oplossing bij de gekozen verdedigingslengte aanvaardbaar iss zal afhangen van de in het prototype te verwachten ontgrondingen. Hiertoe is het noodzakelijk een voorspelling te doen. voor de tijdschaal van de ontgronding. Aan de hand van een vergelijking tussen de in het Veerse G-at opgetreden ontgrondingen en die in een model hiervan, is getracht voor een niet samengetrokken drie-dimensionaal model enig inzicht in de, aan de hand van een twee dimensionaal onderzoek bepaalde9 formule voor de tijdschaal te verkrijgen. Tenslotte is voor verschillende drempelcombinaties in de beide sluitgaten een onderzoek ingesteld naar de aantasting van de grindbestorting, die als afdekking van de verdediging op de Krammerplaat is toegepast.Deltawerken, Grevelinge
Umbral moonshine
No full textWe describe surprising relationships between automorphic forms of various kinds, imaginary quadratic number fields and a certain system of six finite groups that are parameterized naturally by the divisors of 12. The Mathieu group correspondence recently discovered by Eguchi-Ooguri-Tachikawa is recovered as a special case. We introduce a notion of extremal Jacobi form and prove that it characterizes the Jacobi forms arising by establishing a connection to critical values of Dirichlet series attached to modular forms of weight 2. These extremal Jacobi forms are closely related to certain vector-valued mock modular forms studied recently by Dabholkar-Murthy-Zagier in connection with the physics of quantum black holes in string theory. In a manner similar to monstrous moonshine the automorphic forms we identify constitute evidence for the existence of infinite-dimensional graded modules for the six groups in our system. We formulate an Umbral moonshine conjecture that is in direct analogy with the monstrous moonshine conjecture of Conway-Norton. Curiously, we find a number of Ramanujan’s mock theta functions appearing as McKay-Thompson series. A new feature not apparent in the monstrous case is a property which allows us to predict the fields of definition of certain homogeneous submodules for the groups involved. For four of the groups in our system we find analogues of both the classical McKay correspondence and McKay’s monstrous Dynkin diagram observation manifesting simultaneously and compatibly
Class numbers, cyclic simple groups, and arithmetic
No full textHere, we initiate a program to study relationships between finite groups and arithmetic–geometric invariants in a systematic way. To do this, we first introduce a notion of optimal module for a finite group in the setting of holomorphic mock Jacobi forms. Then, we classify optimal modules for the cyclic groups of prime order, in the special case of weight 2 and index 1, where class numbers of imaginary quadratic fields play an important role. Finally, we exhibit a connection between the classification we establish and the arithmetic geometry of imaginary quadratic twists of modular curves of prime level.</p
Weight one Jacobi forms and umbral moonshine
No full textWe analyze holomorphic Jacobi forms of weight one with level. One such form plays an important role in umbral moonshine, leading to simplifications of the statements of the umbral moonshine conjectures. We prove that nonzero holomorphic Jacobi forms of weight one do not exist for many combinations of index and level, and use this to establish a characterization of the McKay-Thompson series of umbral moonshine in terms of Rademacher sums
Rademacher Sums and Rademacher Series
No full textWe exposit the construction of Rademacher sums in arbitrary weights and describe their relationship to mock modular forms. We introduce the notion of Rademacher series and describe several applications, including the determination of coefficients of Rademacher sums and a very general form of Zagier duality. We then review the application of Rademacher sums and series to moonshine both monstrous and umbral and highlight several open problems. We conclude with a discussion of the interpretation of Rademacher sums in physics
Equivariant K3 invariants
No full textIn this note, we describe a connection between the enumerative geometry of curves in K3 surfaces and the chiral ring of an auxiliary superconformai field theory. We consider the invariants calculated by Yau-Zaslow (capturing the Euler characters of the moduli spaces of D2-branes on curves of given genus), together with their refinements to carry additional quantum numbers by Katz-Klemm-Vafa (KKV), and Katz-Klemm-Pandharipande (KKP). We observe that these invariants can be reproduced by studying the Ramond ground states of an auxiliary chiral superconformal field theory which has recently been used to establish mock modular moonshine for a variety of sporadic simple groups that are subgroups of Conway's group. This observation leads us to conjectural descriptions of equivariant versions of the KKV and KKP invariants. A K3 sigma model is specified by a choice of 4-plane in the K3 D-brane charge lattice. Symmetries of K3 sigma models are naturally identified with 4-plane preserving subgroups of the Conway group, according to the work of Gaberdiel-Hohenegger-Volpato, and one may consider corresponding equivariant refined K3 Gopakumar-Vafa invariants. The same symmetries naturally arise in the auxiliary CFT state space, affording a suggestive alternative view of the same computation. We comment on a lift of this story to the generating function of elliptic genera of symmetric products of K3 surfaces, and the connection to work of Oberdieck-Pandharipande on curve counts for the product of a K3 surface with an elliptic curve.</p
Rademacher Sums and Rademacher Series
No full textWe exposit the construction of Rademacher sums in arbitrary weights and describe their relationship to mock modular forms. We introduce the notion of Rademacher series and describe several applications, including the determination of coefficients of Rademacher sums and a very general form of Zagier duality. We then review the application of Rademacher sums and series to moonshine both monstrous and umbral and highlight several open problems. We conclude with a discussion of the interpretation of Rademacher sums in physics
