A previous paper [M. F. Atiyah and N. S. Manton, arXiv:1609.02816 ] modeled atoms and their isotopes by complex algebraic surfaces with the projective plane modeling Hydrogen. In this paper, models of the stable isotopes Helium-4 and Helium-3 are constructed.
Inspired by soliton models, we propose a description of static particles in terms of Riemannian 4-manifolds with self-dual Weyl tensor. For electrically charged particles, the 4-manifolds are non-compact and asymptotically fibred by circles over physical 3-space. This is akin to the Kaluza-Klein description of electromagnetism, except that we exchange the roles of magnetic and electric fields, and only assume the bundle structure asymptotically, away from the core of the particle in question. We identify the Chern class of the circle bundle at infinity with minus the electric charge and, at least provisionally, the signature of the 4-manifold with the baryon number. Electrically neutral particles are described by compact 4-manifolds. We illustrate our approach by studying the Taub-Newman, Unti, Tamburino (Taub-NUT) manifold as a model for the electron, the Atiyah-Hitchin manifold as a model for the proton, CP² with the Fubini-Study metric as a model for the neutron and S⁴ with its standard metric as a model for the neutrino.
We compute the prepotential of N = 2 supersymmetric gauge theories in four dimensions obtained by toroidal compactifications of gauge theories from 6 dimensions, as a function of Kahler and complex moduli of T-2. We use three different methods to obtain this: matrix models, geometric engineering and instanton calculus. Matrix model approach involves summing up planar diagrams of an associated gauge theory on T-2. Geometric engineering involves considering F-theory on elliptic threefolds, and using topological vertex to sum up worldsheet instantons. Instanton calculus involves computation of elliptic genera of instanton moduli spaces on R-4. We study the compactifications of N = 2* theory in detail and establish equivalence of all these three approaches in this case. As a byproduct we geometrically engineer theories with massive adjoint fields. As one application, we show that the moduli space of mass deformed M5-branes wrapped on T-2 combines the Kahler and complex moduli of T-2 and the mass parameter into the period matrix of a genus 2 curve.
The paper focuses on the photogrammetric investigation of geometric models for different types of optical fisheye constructions (equidistant, equisolid-angle, sterographic and orthographic projection). These models were implemented and thoroughly tested in a spatial resection and a self-calibrating bundle adjustment. For this purpose, fisheye images were taken with a Nikkor 8 mm fisheye lens on a Kodak DSC 14n Pro digital camera in a hemispherical calibration room. Both, the spatial resection and the bundle adjustment resulted in a standard deviation of unit weight of 1/10 pixel with a suitable set of simultaneous calibration parameters introduced into the camera model. The camera-lens combination was treated with all of the four basic models mentioned above. Using the same set of additional lens distortion parameters, the differences between the models can largely be compensated, delivering almost the same precision parameters. The relative object space precision obtained from the bundle adjustment was ca. 1:10 000 of the object dimensions. This value can be considered as a very satisfying result, as fisheye images generally have a lower geometric resolution as a consequence of their large field of view and also have a inferior imaging quality in comparison to most central perspective lenses.
This paper proposes a fast target indexing method based on target geometric models and further conceives a forward method based coarse-to-fine hierarchical synthetic aperture radar (SAR) automatic target recognition (ATR) system. The proposed fast SAR target indexing method is employed for coarse classification before the implementation of fine classification and recognition, resulting in significant acceleration of target recognition. Furthermore, the fast SAR target indexing method is purely based on target geometric models without resorting to any time-consuming electromagnetic (EM) computation, which enables the online and real-time classification. The binary target region and shadow region are selected as the indexing features, and a forward feature prediction method based on optical visibility is developed to predict the features from the target geometric model in real time. Then, the features extracted from the testing SAR image are aligned with the predicted features to establish correlation ship and a fused similarity based on four types of complementary similarity criteria is designed to further implement target indexing. The effectiveness of the proposed fast SAR target indexing method is demonstrated by using the MSTAR dataset.
We show that the “geometric models of matter” approach proposed by the first author can be used to construct models of anyon quasiparticles with fractional quantum numbers, using 4-dimensional edge-cone orbifold geometries with orbifold singularities along embedded 2-dimensional surfaces. The anyon states arise through the braid representation of surface braids wrapped around the orbifold singularities, coming from multisections of the orbifold normal bundle of the embedded surface. We show that the resulting braid representations can give rise to a universal quantum computer.
Fisher's geometric model has been widely used to study the effects of pleiotropy and organismic complexity on phenotypic adaptation. Here, we study a version of Fisher's model in which a population adapts to a gradually moving optimum. Key parameters are the rate of environmental change, the dimensionality of phenotype space, and the patterns of mutational and selectional correlations. We focus on the distribution of adaptive substitutions, that is, the multivariate distribution of the phenotypic effects of fixed beneficial mutations. Our main results are based on an "adaptive-walk approximation," which is checked against individual-based simulations. We find that (1) the distribution of adaptive substitutions is strongly affected by the ecological dynamics and largely depends on a single composite parameter γ, which scales the rate of environmental change by the "adaptive potential" of the population; (2) the distribution of adaptive substitution reflects the shape of the fitness landscape if the environment changes slowly, whereas it mirrors the distribution of new mutations if the environment changes fast; (3) in contrast to classical models of adaptation assuming a constant optimum, with a moving optimum, more complex organisms evolve via larger adaptive steps.
The experimental data on proton–proton elastic and inelastic scattering emerging from the measurements at the Large Hadron Collider, calls for an efficient model to fit the data. We have examined the optical, geometrical picture and we have found the simplest, linear dependence of this model parameters on the logarithm of the interaction energy with the significant change of the respective slopes at one point corresponding to the energy of about 300 GeV. The logarithmic dependence observed at high energies allows one to extrapolate the proton–proton elastic, total (and inelastic) cross sections to ultra high energies seen in cosmic rays events which makes a solid justification of the extrapolation to very high energy domain of cosmic rays and could help us to interpret the data from an astrophysical and a high energy physics point of view.
Spacer fabrics are very attractive nowadays for use as technical fabrics. Our interest in this study is to give geometrical models for weft-knitted spacer fabrics which can be used in related engineering software. Models of two commonly used weft-knitted spacer fabrics are created here based on Kurbak’s 1998 plain knit model and are drawn to scale using the 3DS-MAX computer graphical program. It is observed that similar shapes to the real fabrics are obtained by the models.