Academic literature on the topic 'Generalized cusps'

It is known that Nambu–Goto extended objects present some pathological structures, such as cusps and kinks, during their evolution. In this paper, we propose a model through the generalized Raychaudhuri (Rh) equation for membranes to determine if there are cusps and kinks in the worldsheet. We extend the generalized Rh equation for membranes to allow the study of the effect of higher order curvature terms in the action on the issue of cusps and kinks, using it as a tool for determining when a Nambu–Goto string generates cusps or kinks in its evolution. Furthermore, we present three examples where we test graphically this approach.

ln this paper, a generalized cusp is a properly convex manifold with strictly convex boundary that is diffeomorphic to M × [ 0 , ∞ ) M\times [0,\infty ) where M M is a closed Euclidean manifold. These are classified by Ballas, Cooper, and Leitner [J. Topol. 13 (2020), pp. 1455-1496]. The marked moduli space is homeomorphic to a subspace of the space of conjugacy classes of representations of π 1 M \pi _1M . It has one description as a generalization of a trace-variety, and another description involving weight data that is similar to that used to describe semi-simple Lie groups. It is also a bundle over the space of Euclidean similarity (conformally flat) structures on M M , and the fiber is a closed cone in the space of cubic differentials. For 3 3 -dimensional orientable generalized cusps, the fiber is homeomorphic to a cone on a solid torus.

Let $X$ be a smooth geometrically connected projective curve over the field of fractions of a discrete valuation ring $R$ , and $\mathfrak {m}$ a modulus on $X$ , given by a closed subscheme of $X$ which is geometrically reduced. The generalized Jacobian $J_\mathfrak {m}$ of $X$ with respect to $\mathfrak {m}$ is then an extension of the Jacobian of $X$ by a torus. We describe its Néron model, together with the character and component groups of the special fibre, in terms of a regular model of $X$ over $R$ . This generalizes Raynaud's well-known description for the usual Jacobian. We also give some computations for generalized Jacobians of modular curves $X_0(N)$ with moduli supported on the cusps.

AbstractMotivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifoldMto be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3–manifold is diffeo morphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3–manifolds that cannot arise as a cusp cross-section of a 1–cusped nonsingular Hilbert modular surface.

Abstract Radula in all melongeninae species is rather uniform and characterized by bicuspid lateral teeth with strongly curved cusps and sub rectangular rachidians, bearing usually 3 cusps. The aim of the present study was to describe the radula of 2 Pugilina cochlidium populations using SEM. The radula in 2 species proves itself as a rachiglossate type showing the radular formula of 1 + R + 1. The first population hasthe central tooth wide with sharp cusps equal in length, emanate from posterior margin of tooth base. The lateral teeth have 2 cusps and are long, sharp, pointed and bent towards the rachidian tooth. Whereas the second population, the central tooth is narrow with sharp cusps equal in length, emanate from posterior margin of tooth base. The lateral teeth have 2 cusps and are broad, longer, sharp, pointed and bent towards the rachidian tooth. They are typically sickle shaped with broad strong base. In both populations the rachidian tooth is subquadrate with 3 big cusps in the middle, but in the second population the base of the rachidian is concave while in the first population it is straight. In the present study the median rachidian of the second population, has a broad basal region when compared to first. This similar observation has been made in Chicoreus virgineus ponderosus and Siratus virgineus ponderosus. In the present study, since 2 populations exhibit the same generalized rachiglossate pattern it does not offer much scope for systematic diagnosis below generic level.

We show that every Fricke-invariant meromorphic modular form for $\unicode[STIX]{x1D6E4}_{0}(N)$ whose divisor on $X_{0}(N)$ is defined over $\mathbb{Q}$ and supported on Heegner divisors and the cusps is a generalized Borcherds product associated to a harmonic Maass form of weight $1/2$. Further, we derive a criterion for the finiteness of the multiplier systems of generalized Borcherds products in terms of the vanishing of the central derivatives of $L$-functions of certain weight $2$ newforms. We also prove similar results for twisted Borcherds products.

In this thesis we study the Laplace operator Δ acting on p-forms, defined on an n dimensional manifold with generalized cusps. Such a manifold consists of a compact piece and a noncompact one. The noncompact piece is isometric to the generalized cusp. A generalized cusp [1,∞) x N is an n dimensional noncompact manifold equipped with the warped product metric dx{2}+x{-2a}h, where N is a compact oriented manifold, h is a metric on N and a > 0 is a fixed constant. First we regard the cusp separately, where by using separation of variables we determine the spectral properties of the Laplacian and we determine explicitly the structure of the continuous part of the spectral theorem. Using this result, we meromorphically continue the resolvent of the Laplace operator to a certain Riemann surface, which we determine. By standard gluing techniques, the resolvent of the Laplace operator Δ on the manifold with cusp is meromorphically continued to the same Riemann surface. This enables us to construct the generalized eigenforms for the original manifold without boundary. That describes the continuous spectral decomposition of Δ and determines some of its important properties, like analyticity and the existence of a functional equation. We also define the stationary scattering matrix and find its analytic properties and its functional equation.

Cette thèse est consacrée à l’étude des orbivariétés projectives convexes géométriquement finies, et fait suite aux travaux de Ballas, Cooper, Crampon, Leitner, Long, Marquis et Tillmann sur le sujet. Une orbivariété projective convexe est le quotient d’un ouvert convexe et borné d’une carte affine de l’espace projectif réel (appelé aussi ouvert proprement convexe) par un groupe discret de transformations projectives préservant cet ouvert. S’il n’y a pas de segment dans le bord du convexe, on dit que l’orbivariété est strictement convexe, et si de plus il y a un unique hyperplan de support en chaque point du bord, on dit qu’elle est ronde. Suivant Cooper-Long-Tillmann et Crampon-Marquis, on dit qu’une orbivariété strictement convexe est géométriquement finie si son cœur convexe est l’union d’un compact et d’un nombre fini de bouts, appelés pointes, où le rayon d’injectivité est inférieur à une constante ne dépendant que de la dimension. Comprendre la géométrie des pointes est primordial pour l’étude des orbivariétés géométriquement finies. Dans le cas strictement convexe, la seule restriction connue sur l’holonomie des pointes vient d’une généralisation du lemme de Margulis due à Cooper-Long-Tillmann et Crampon-Marquis, qui implique que cette holonomie est virtuellement nilpotente. On donne dans cette thèse une caractérisation de l’holonomie des pointes des orbivariétés strictement convexes et des orbivariétés rondes. En généralisant la méthode de Cooper, qui a produit le seul exemple connu jusqu’ici d’une pointe de variété strictement convexe dont l’holonomie n’est pas virtuellement abélienne, on construit des pointes de variétés strictement convexes et de variétés rondes dont l’holonomie est isomorphe à n’importe quel groupe nilpotent sans torsion de type fini. En collaboration avec M. Islam et F. Zhu, on démontre que dans le cas des groupes relativement hyperboliques sans torsion, les représentations relativement P1-anosoviennes (au sens de Kapovich-Leeb, Zhu et Zhu-Zimmer) qui préservent un ouvert proprement convexe sont exactement les holonomies des variétés rondes géométriquement finies.Dans le cas des orbivariétés projectives convexes non strictement convexes, il n’y a pas pour l’instant de définition satisfaisante de la finitude géométrique. Toutefois, Cooper-Long-Tillmann puis Ballas-Cooper-Leitner ont proposé une définition de pointe généralisée dans ce contexte. Bien qu’ils demandent que l’holonomie des pointes généralisées soit virtuellement nilpotente, tous les exemples connus jusqu’à présent avaient une holonomie virtuellement abélienne. On construit des exemples de pointes généralisées dont l’holonomie peut être n’importe quel groupe nilpotent sans torsion de type fini. On s’autorise également à modifier la définition originale de Cooper-Long-Tillmann en affaiblissant l’hypothèse de nilpotence en une hypothèse naturelle de résolubilité, ce qui nous permet de construire de nouveaux exemples dont l’holonomie n’est pas virtuellement nilpotente.Une orbivariété géométriquement finie qui n’a pas de pointes, c’est-à-dire dont le cœur convexe est compact, est dite convexe cocompacte. On dispose par les travaux de Danciger-Guéritaud-Kassel d’une définition de la convexe cocompacité pour les orbivariétés projectives convexes sans hypothèse de stricte convexité, contrairement au cas géométriquement fini. Ils démontrent que l’holonomie d’une orbivariété projective convexe convexe cocompacte est Gromov hyperbolique si et seulement si la représentation associée est P1-anosovienne. À l’aide de ce résultat, de la théorie de Vinberg et des travaux d’Agol et Haglund-Wise sur les groupes hyperboliques cubulés, on construit en collaboration avec S. Douba, T. Weisman et F. Zhu des représentations P1-anosoviennes pour tout groupe hyperbolique cubulé. Ceci fournit de nouveaux exemples de groupes hyperboliques admettant des représentations anosoviennes
This thesis is devoted to the study of geometrically finite convex projective orbifolds, following work of Ballas, Cooper, Crampon, Leitner, Long, Marquis and Tillmann. A convex projective orbifold is the quotient of a bounded, convex and open subset of an affine chart of real projective space (called a properly convex domain) by a discrete group of projective transformations that preserve it. We say that a convex projective orbifold is strictly convex if there are no non-trivial segments in the boundary of the convex subset, and round if in addition there is a unique supporting hyperplane at each boundary point. Following work of Cooper-Long-Tillmann and Crampon-Marquis, we say that a strictly convex orbifold is geometrically finite if its convex core decomposes as the union of a compact subset and of finitely many ends, called cusps, all of whose points have an injectivity radius smaller than a constant depending only on the dimension. Understanding what types of cusps may occur is crucial for the study of geometrically finite orbifolds. In the strictly convex case, the only known restriction on cusp holonomies, imposed by a generalization of the celebrated Margulis lemma proven by Cooper-Long-Tillmann and Crampon-Marquis, is that the holonomy of a cusp has to be virtually nilpotent. We give a complete characterization of the holonomies of cusps of strictly convex orbifolds and of those of round orbifolds. By generalizing a method of Cooper, which gave the only previously known example of a cusp of a strictly convex manifold with non virtually abelian holonomy, we build examples of cusps of strictly convex manifolds and round manifolds whose holonomy can be any finitely generated torsion-free nilpotent group. In joint work with M. Islam and F. Zhu, we also prove that for torsion-free relatively hyperbolic groups, relative P1-Anosov representations (in the sense of Kapovich-Leeb, Zhu and Zhu-Zimmer) that preserve a properly convex domain are exactly the holonomies of geometrically finite round manifolds.In the general case of non strictly convex projective orbifolds, no satisfactory definition of geometric finiteness is known at the moment. However, Cooper-Long-Tillmann, followed by Ballas-Cooper-Leitner, introduced a notion of generalized cusps in this context. Although they only require that the holonomy be virtually nilpotent, all previously known examples had virtually abelian holonomy. We build examples of generalized cusps whose holonomy can be any finitely generated torsion-free nilpotent group. We also allow ourselves to weaken Cooper-Long-Tillmann’s original definition by assuming only that the holonomy be virtually solvable, and this enables us to construct new examples whose holonomy is not virtually nilpotent.When a geometrically finite orbifold has no cusps, i.e. when its convex core is compact, we say that the orbifold is convex cocompact. Danciger-Guéritaud-Kassel provided a good definition of convex cocompactness for convex projective orbifolds that are not necessarily strictly convex. They proved that the holonomy of a convex cocompact convex projective orbifold is Gromov hyperbolic if and only if the associated representation is P1-Anosov. Using these results, Vinberg’s theory and work of Agol and Haglund-Wise about cubulated hyperbolic groups, we construct, in collaboration with S. Douba, T. Weisman and F. Zhu, examples of P1-Anosov representations for any cubulated hyperbolic group. This gives new examples of hyperbolic groups admitting Anosov representations

Grundlage der vorliegenden Arbeit sind mathematische Aspekte des Kapillarflächenproblems. Die Arbeit ist in zwei Teile gegliedert. Im ersten Teil werden existierende glatte Lösungen des klassischen Kapillarflächenproblems betrachtet. Diese sind unbeschränkt, wenn das Definitionsgebiet Spitzen enthält. Es wird eine Vielzahl von asymptotischen Formeln hergeleitet. Der zweite Teil der Arbeit beschäftigt sich mit verallgemeinerten Lösungen des allgemeinen Kapillar- flächenproblems. Es wird die Existenz dieser Lösungen unter sehr schwachen Voraussetzungen bewiesen. Für konstante Gravitationspotentiale und Benetzungsverhalten werden verallgemeinerte Lösungen näher untersucht und z. T. sogar explizit konstruiert. Die Eigenschaften einer speziellen Klasse solcher Lösungen könnte einen Beitrag zur Erklärung des Wasseranstiegs in Bäumen liefern
The present paper is based on mathematical aspects of the capillary surface problem. It is divided into two parts. In the first part we consider the classical capillary surface problem, for which smooth solutions exist. These solutions are unbounded if the domain of definition contains cusps. We prove a large variety of asymptotic formulas. The second part is concerned with generalized solutions of the general capillary problem, for which there is not always a smooth solution. We prove existence of generalized solutions under very weak preconditions. We can construct some generalized solutions for zero-gravity and constant wetting-behaviour explicitly. These solutions have a very restricted geometry and could be of interest for the understanding of water lift in trees

We find out how to maximize the volume covered by disjoint cusp neighborhoods and boundary collars inside a hyperbolic 3-manifold with both geodesic boundary and toric cusps. We generalize Mom-structures to 3-manifolds with boundary components of higher genus and study their relations with triangulations.

Grundlage der vorliegenden Arbeit sind mathematische Aspekte des Kapillarflächenproblems. Die Arbeit ist in zwei Teile gegliedert. Im ersten Teil werden existierende glatte Lösungen des klassischen Kapillarflächenproblems betrachtet. Diese sind unbeschränkt, wenn das Definitionsgebiet Spitzen enthält. Es wird eine Vielzahl von asymptotischen Formeln hergeleitet. Der zweite Teil der Arbeit beschäftigt sich mit verallgemeinerten Lösungen des allgemeinen Kapillar- flächenproblems. Es wird die Existenz dieser Lösungen unter sehr schwachen Voraussetzungen bewiesen. Für konstante Gravitationspotentiale und Benetzungsverhalten werden verallgemeinerte Lösungen näher untersucht und z. T. sogar explizit konstruiert. Die Eigenschaften einer speziellen Klasse solcher Lösungen könnte einen Beitrag zur Erklärung des Wasseranstiegs in Bäumen liefern.
The present paper is based on mathematical aspects of the capillary surface problem. It is divided into two parts. In the first part we consider the classical capillary surface problem, for which smooth solutions exist. These solutions are unbounded if the domain of definition contains cusps. We prove a large variety of asymptotic formulas. The second part is concerned with generalized solutions of the general capillary problem, for which there is not always a smooth solution. We prove existence of generalized solutions under very weak preconditions. We can construct some generalized solutions for zero-gravity and constant wetting-behaviour explicitly. These solutions have a very restricted geometry and could be of interest for the understanding of water lift in trees.

This epilogue describes some work on generalizations and applications, as well as a direction of further research outside the context of modular forms. Theorems 14.1.1 and 15.2.1 will certainly be generalized to spaces of cusp forms of arbitrarily varying level and weight. This has already been done for the probabilistic variant of Theorem 14.1.1, in the case of square-free levels (and of level two times a square-free number). Some details and some applications of Bruin's work, as well as a perspective on point counting outside the context of modular forms are described. Deterministic generalizations of the two theorems mentioned above will lead to deterministic applications.

Abstract Several QMC calculations had earlier treated quantum dots of various shapes and sizes by either variational or diffusion methods. This study extended much farther, with ground and low-lying excited states for circular two-dimensional dots with 2 to 13 electrons. The method was fixed-node diffusion QMC with importance sampling. Comparisons were made with Hartree-Fock and density functional calculations at the LSDA level. The system treated was a standard circular quantum dot with N electrons confined in two dimensions by a parabolic potential V = ½kr2 centered at the origin. The masses, dielectric constants, and the potential energy constants were specified as those for a GaAs crystal. Atomic units were replaced by effective atomic units. Anti-symmetry rules were the same as for atomic systems. Trial functions were expressed for each case as sums of one to five Slater determinants from density functional calculations, along with elaborate generalized Jastrow functions. The coefficients required for satisfying cusp conditions were different from those usually found for three-dimensional problems. A total of 23 cases with different numbers of electrons and L, S eigenstates were treated. The first version of the paper contained a few errors. Corrected results (also with lower uncertainties) were reported in a second version.0 In all cases the diffusion QMC energies were lower than the variational QMC energies, which were in turn lower than the HF energies. Fixed-node error was expected to be small. The differences from LSDA density functional energies were small but not negligible.

Oblateness in drops has been shown to produce caustics in the farfield scattering that are more complicated than the ordinary rainbow [1-7]. The complicated caustics generated are examples of "diffraction catastrophes" involving the coalescence of more than the two rays present for the usual Airy caustic [8,9]. Such diffraction catastrophes produced by scattering from oblate drops of water have been referred to as "generalized rainbows" [2,7]. While the analysis of such caustics is relevant to understanding complexity in wave propagation [8-11]. consideration of the scattering by oblate drops is helpful for anticipating whether or not generalized rainbows are likely to be visible in sunlit raindrops. The situation usually considered is the one which has the greatest symmetry: horizontal illumination of an oblate drop having a vertical symmetry axis. Generalized rainbows occur because of vertical focusing within the drop which causes the vertical curvature of the outgoing wavefront to vanish in the equatorial plane. Let D be the diameter of such a drop in the equatorial plane and let H the height. A relevant question is whether the values of the aspect ratio q = D/H required to produce the vertical focusing are sufficiently close to unity so as to be present for naturally falling raindrops. Naturally falling drops are ordinarily only slightly oblate because of the flow of air past the drop [12]. Previous laboratory observations and calculations indicate that for rays having two or three internal chords (corresponding to primary and secondary rainbow mechanisms), the required values of q are so large as to make their natural occurrence extremely rare, if ever [3-7]. It was calculated and observed that for drops having 6 chords, a bright complicated caustic was produced for q near 1.08. It was also predicted that cusps would be present for even smaller q as part of the unfolding of an axial glory caustic (as has been observed for bubbles in water [11]).

A compliant mechanism transmits motion and force by deformation of its flexible members. It has no relative moving parts and thus involves no wear, lubrication, noise, and backlash. Compliant mechanisms aims to maximize flexibility while maintaining sufficient stiffness so that satisfactory output motion can be achieved. When designing compliant mechanisms, the resulting shapes sometimes lead to rigid-body type linkages where compliance and rotation is lumped at a few flexural pivots. These flexural pivots are prone to stress concentration and thus limit compliant mechanisms to applications that only require small-deflected motion. To overcome this problem, a systematic design method is presented to synthesize the shape of a compliant mechanism so that compliance is distributed more uniformly over the mechanism. With a selected topology and load conditions, this method characterizes the free geometric shape of a compliant segment by its rotation and thickness functions. These two are referred as intrinsic functions and they describe the shape continuously within the segment so there is no abrupt change in geometry. Optimization problems can be conveniently formulated with cusps and intersecting loops naturally circumvented. To facilitate the optimization process, a numerical algorithm based on the generalized shooting method will be presented to solve for the deflected shape. Illustrative examples will demonstrate that through the proposed design method, compliant mechanisms with distributed compliance will lessen stress concentration so they can be more robust and have larger deflected range. It is expected that the method can be applied to design compliant mechanisms that have a wide variety of applications from precision instruments to biomedical devices.

Previous laboratory observations with laser1-3 and white light4 illumination of oblate drops of water have shown that the primary rainbow becomes distorted and an additional caustic, having the form of a transverse cusp, can be produced when the drop's aspect ratio is sufficiently large.

We study quantum mechanical behavior of a Morse oscillator coupled to a bath of harmonic oscillators and driven by an infrared laser. The purpose is to compare quantum behavior of such a system with the classical and semiclassical solutions of a driven damped Morse oscillator. The general picture coming out of classical and semiclassical studies is that the total energy of the Morse oscillator in steady states or steady oscillations show jump and hysteretic behavior, as expected in a cusp catastrophe, when laser intensity or frequency is varied. Along with the bistable behavior we have also found period-doubling bifurcations leading to chaos on one or both branches. In the quantum treatment a generalized master equation is derived from the Liouville equation, in which the Hamiltonian of the Morse oscillator is expressed in terms of generators of an SU(2) algebra. This algebra and the Markoffian approximation allows us to derive the equation of motion of the reduced density matrix, which is solved numerically. Bistable behavior does not exist in this quantum treatment, but solutions do show jump behavior and bifurcation phenomenon. Experimental implications of these studies are also discussed.

The physics of the film cooling process for shaped, streamwise-injected, inclined jets is studied for blowing ratio (M = 1.25, 1.88), density ratio (DR = 1.6), and length-to-diameter ratio (L/D = 4) parameters typical of gas turbine operations. A previously documented computational methodology is applied for the study of five distinct film cooling configurations: (1) cylindrical film hole (reference case); (2) forward-diffused film hole; (3) laterally-diffused film hole; (4) inlet shaped film hole, and (5) cusp-shaped film hole. The effects of various film hole geometries on both flow and thermal field characteristics is isolated, and the dominant mechanisms responsible for differences in these characteristics are documented. Special consideration is given to explaining crucial flow mechanisms from a vorticity point of view. It is found that vorticity analysis of the flow exiting the film hole can aid substantially in explaining the flow behavior downstream of the film hole. Results indicate that changes in the film hole shape can significantly alter the distribution of the exit-plane variables, therefore strongly affecting the downstream behavior of the film. Computational solutions of the steady, Reynolds-averaged Navier-Stokes equations are obtained using an unstructured/adaptive, fully implicit, pressure-correction solver. Turbulence closure is obtained via the high Reynolds number k-ε model with generalized wall functions. Detailed field results as well as surface phenomena involving adiabatic film effectiveness (η) and heat transfer coefficient (h) are presented. When possible, computational results are validated against corresponding experimental cases from data found in the open literature. Detailed comparisons are made between surface and field results of the film hole shapes investigated in this work; design criteria for optimizing downstream heat transfer characteristics are then suggested.