arXiv subjects
Publications and source records attributed to Charles Wang.
In this report we use our methods in arXiv:2204.03548 and its upcoming generalization to complete the construction of canonical mirror models for all cominuscule homogeneous spaces, by considering the maximal orthogonal Grassmannians $\mathrm{OG}(n + 1, 2n + 2)$.
We present projective Landau-Ginzburg models for the exceptional cominuscule homogeneous spaces $\mathbb{OP}^2 = E_6^\mathrm{sc}/P_6$ and $E_7^\mathrm{sc}/P_7$, known respectively as the Cayley plane and the Freudenthal variety. These models are defined on the complement $X^\vee_\mathrm{can}$ of an anti-canonical divisor of the "Langlands dual homogeneous spaces" $\mathbb{X}^\vee = P^\vee\backslash G^\vee$ in terms of generalized Plücker coordinates, analogous to the canonical models defined for Grassmannians, quadrics and Lagrangian Grassmannians in arXiv:1307.1085, arXiv:1404.4844, arXiv:1304.4958. We prove that these models for the exceptional family are isomorphic to the Lie-theoretic mirror models defined in arXiv:math/0511124 using a restriction to an algebraic torus, also known as the Lusztig torus, as proven in arXiv:1912.09122. We also give a cluster structure on $\mathbb{C}[\mathbb{X}^\vee]$, prove that the Plücker coordinates form a Khovanskii basis for a valuation defined using the Lusztig torus, and compute the Newton-Okounkov body associated to this valuation. Although we present our methods for the exceptional types, they generalize immediately to the members of other cominuscule families.
We illustrate the relation between supermoduli integration and picture changing operators (PCOs) particularly concerning the role of vertical integration, in the context of superstring vacuum amplitudes, by an explicit comparison of different parameterizations of the supermoduli space of genus two super Riemann surfaces.
We establish the equivalence between two formulations of superstring perturbation theory, one based on integration over the supermoduli space of super Riemann surfaces (SRS), the other based on integration over the bosonic moduli space with insertions of picture changing operators (PCO) on the worldsheet and the vertical integration prescription, by showing how the latter arises from a specific construction of the supermoduli integration contour.
Vulnerable road users (VRUs) such as pedestrians, cyclists and motorcyclists are at the highest risk in the road traffic environment. Globally, over half of road traffic deaths are vulnerable road users. Although substantial efforts are being made to improve VRU safety from engineering solutions to law enforcement, the death toll of VRUs' continues to rise. The emerging technology, Cooperative Intelligent Transportation System (C-ITS), has the proven potential to enhance road safety by enabling wireless communication to exchange information among road users. Such exchanged information is utilized for creating situational awareness and detecting any potential collisions in advance to take necessary measures to avoid any possible road casualties. The current state-of-the-art solutions of C-ITS for VRU safety, however, are limited to unidirectional communication where VRUs are only responsible for alerting their presence to drivers with the intention of avoiding collisions. This one-way interaction is substantially limiting the enormous potential of C-ITS which otherwise can be employed to devise a more effective solution for the VRU safety where VRU can be equipped with bidirectional communication with full C-ITS functionalities. To address such problems and to explore better C-ITS solution suggestions for VRU, this paper reviewed and evaluated the current technologies and safety methods proposed for VRU safety over the period 2007-2020. Later, it presents the design considerations for a cellular-based Vehicle-to-VRU (V2VRU) communication system along with potential challenges of a cellular-based approach to provide necessary recommendations.
In [RW19] Rietsch and Williams relate cluster structures and mirror symmetry for type A Grassmannians Gr(k, n), and use this interaction to construct Newton-Okounkov bodies and associated toric degenerations. In this article we define a cluster seed for the Lagrangian Grassmannian, and prove that the associated Newton-Okounkov body agrees up to unimodular equivalence with a polytope obtained from the superpotential defined by Pech and Rietsch on the mirror Orthogonal Grassmannian in [PR13].
Time synchronization in communication networks provides a common time frame among all nodes, thussupporting various network functions such as message transmission, channel scheduling and resourcesharing in real-time and in correct order. In vehicular ad-hoc networks (VANETs) for connected andautomated vehicles, network nodes must be synchronized to exchange time-critical vehicle locationsand warning messages for various road safety applications. However, this is challenging as the datacommunication systems need to achieve low latency and high reliability under the conditions ofhigh user dynamics and density. While many synchronization techniques have been developed forgeneral communications networks, it is necessary to understand the requirements for VANET timesynchronization and the applicability of existing time synchronization techniques in VANET applications.This paper provides a survey on theory and practice of time synchronization in VANETs. It addressessome key factors in VANET time synchronization such as requirements analysis, precision, accuracy,availability, scalability and compatibility, and highlights the advantages of Global Navigation SatelliteSystem (GNSS) in VANET time synchronization. Through this survey, some insights are developed intoexisting and emerging protocols for time synchronization in VANETs.
Unitary Representations corresponding to local shifts in reference frames are a key topic for progress in Quantum Gravity. The fibre bundle construct defined in our previous work in which quantum fields become liftings of; or sections through; a fibre bundle with base space curved space-time, continues to be the context for this paper. We investigate in more depth the subgroup T of the Poincare group consisting of translations of space-time as a gauge group of automorphisms. We define to be a representation of T as such automorphisms of the local fibre algebra A(x) which we assume to be isomorphic to a von Neumann algebra with trivial centre acting on a separable Hilbert space. Provided this group representation is weakly measurable, then we have previously proved that it is also norm continuous, and is implemented by a norm, hence weakly and strongly, continuous unitary representation. We discuss the dependence of this key result on the strange properties of Stonean spaces and extend our exploitation of Mackey Theory to give a more straightforward group theoretic proof. From such unitary representations a minimal form of Supersymmetry naturally emerges, which predicts the existence of both the mass zero graviton, and its gravitino partner.
A graphical model encodes conditional independence relations via the Markov properties. For an undirected graph these conditional independence relations can be represented by a simple polytope known as the graph associahedron, which can be constructed as a Minkowski sum of standard simplices. There is an analogous polytope for conditional independence relations coming from a regular Gaussian model, and it can be defined using multiinformation or relative entropy. For directed acyclic graphical models and also for mixed graphical models containing undirected, directed and bidirected edges, we give a construction of this polytope, up to equivalence of normal fans, as a Minkowski sum of matroid polytopes. Finally, we apply this geometric insight to construct a new ordering-based search algorithm for causal inference via directed acyclic graphical models.
The toric h-numbers of a dual hypersimplex and the Chow Betti numbers of the normal fan of a hypersimplex are the ranks of intersection cohomology and Chow cohomology respectively of the torus orbit closure of a generic point in the Grassmannian. We give explicit formulas for these numbers. We also show that similar formulas hold for the coordinator numbers of type A^* lattices.
There is a critical need for standard approaches to assess, report, and compare the technical performance of genome-scale differential gene expression experiments. We assess technical performance with a proposed "standard" dashboard of metrics derived from analysis of external spike-in RNA control ratio mixtures. These control ratio mixtures with defined abundance ratios enable assessment of diagnostic performance of differentially expressed transcript lists, limit of detection of ratio (LODR) estimates, and expression ratio variability and measurement bias. The performance metrics suite is applicable to analysis of a typical experiment, and here we also apply these metrics to evaluate technical performance among laboratories. An interlaboratory study using identical samples shared amongst 12 laboratories with three different measurement processes demonstrated generally consistent diagnostic power across 11 laboratories. Ratio measurement variability and bias were also comparable amongst laboratories for the same measurement process. Different biases were observed for measurement processes using different mRNA enrichment protocols.
We propose an experiment for the measurement of gravitational effect on cold atoms by applying a one-dimensional vertically sinusoidal oscillation to the magneto-optical trap; and observe the signature of low quantum energy shift of quantum bound states as a consequence of gravitational fluctuation. To this end, we present brief details of the experiment on a BEC, and a simplistic calculation of the Gross-Pitaevskii solution using Thomas-Fermi approximation with focus on the density of the BEC, for the time-dependent perturbation. Moreover, we calculate the power induced by quantum gravity on a generic atomic ensemble. We also address the possible challenges for the measurement of the expected results. And finally, we discuss the prospect of further developing this experiment towards measuring the effect of quantum spacetime fluctuations on cold atoms.
We present and derive a technique for the introduction of defects into a beam model based on the Cosserat theory of rods. The technique is designed for the derivation of component models of non-ideal rods for use in MEMS devices. We also present a worked through example of blob/nick defects (where the rod has an area with an excess/lack of material) and a guide for a model with random pits and blobs along the length of the beam. Finally we present a component level model of a beam with a defect and compare it to results from a Finite Element Analysis simulation. We test the Cosserat model for two cases without any defect and four with a defect. Results are in good agreement with a maximum 0.5% difference for the ideal case and under 1% differences for all but one of the defective cases, the exception being a 2% error in an extreme case for which the model is expected to break down. Overall, the Cosserat model with and without defects provides an accurate way of modelling long slender beams. In addition, simulation times are greatly reduced through this approach and further development for both component level models as well as as FEA components is important for practical yet accurate modelling of MEMS both for prediction and comparison.
The derivation of the recently proposed nonlinear quantum evolution of gravity from an action principle is considered in this brief note. It is shown to be possible if a set of consistency conditions are satisfied that are analogous to the Dirac relations for the super-Hamiltonian and momenta in classical canonical gravity.
A generalization of the recently formulated nonlinear quantization of a parameterized theory is presented in the context of quantum gravity. The parametric quantization of a Friedmann universe with a massless scalar field is then considered in terms of analytic solutions of the resulting evolution equations.
The Mathisson-Papapetrou-Dixon equations for a massive spinning test particle in plane gravitational waves are analysed and explicit solutions constructed in terms of solutions of certain linear ordinary differential equations. For harmonic waves this system reduces to a single equation of Mathieu-Hill type. In this case spinning particles may exhibit parametric excitation by gravitational fields. For a spinning test particle scattered by a gravitational wave pulse, the final energy-momentum of the particle may be related to the width, height, polarisation of the wave and spin orientation of the particle.
A discussion is given of the quantisation of a physical system with finite degrees of freedom subject to a Hamiltonian constraint by treating time as a constrained classical variable interacting with an unconstrained quantum state. This leads to a quantisation scheme that yields a Schrodinger-type equation which is in general nonlinear in evolution. Nevertheless it is compatible with a probabilistic interpretation of quantum mechanics and in particular the construction of a Hilbert space with a Euclidean norm is possible. The new scheme is applied to the quantisation of a Friedmann Universe with a massive scalar field whose dynamical behaviour is investigated numerically.
A simple modification to Einstein's theory of gravity in terms of a non-Riemannian connection is examined. A new tensor-variational approach yields field equations that possess a covariance similar to the gauge covariance of electromagnetism. These equations are shown to possess solutions analogous to those found in the Einstein-Maxwell system. In particular one finds gravi-electric and gravi-magnetic charges contributing to a spherically symmetric static Reissner-Nordström metric. Such Weyl ``charges'' provide a source for the non-Riemannian torsion and metric gradient fields instead of the electromagnetic field. The theory suggests that matter may be endowed with gravitational charges that couple to gravity in a manner analogous to electromagnetic couplings in an electromagnetic field. The nature of gravitational coupling to spinor matter in this theory is also investigated and a solution exhibiting a plane-symmetric gravitational metric wave coupled via non-Riemannian waves to a propagating spinor field is presented.