Unfolding conformal geometry

Conformal geometry is studied using the unfolded formulation à la Vasiliev. Analyzing the first-order consistency of the unfolded equations, we identify the content of zero-forms as the spin-two off-shell Fradkin-Tseytlin module of $$ \mathfrak{so}\left(2,d\right) $$ so 2 d . We sketch the nonlinear structure of the equations and explain how Weyl invariant densities, which Type-B Weyl anomaly consist of, could be systematically computed within the unfolded formulation. The unfolded equation for conformal geometry is also shown to be reduced to various on-shell gravitational systems by requiring additional algebraic constraints.

Optimal Control Applied to Vaccination and Testing Policies for COVID-19

In this paper, several policies for controlling the spread of SARS-CoV-2 are determined under the assumption that a limited number of effective COVID-19 vaccines and tests are available. These policies are calculated for different vaccination scenarios representing vaccine supply and administration restrictions, plus their impacts on the disease transmission are analyzed. The policies are determined by solving optimal control problems of a compartmental epidemic model, in which the control variables are the vaccination rate and the testing rate for the detection of asymptomatic infected people. A combination of the proportion of threatened and deceased people together with the cost of vaccination of susceptible people, and detection of asymptomatic infected people, is taken as the objective functional to be minimized, whereas different types of algebraic constraints are considered to represent several vaccination scenarios. A direct transcription method is employed to solve these optimal control problems. More specifically, the Hermite–Simpson collocation technique is used. The results of the numerical experiments show that the optimal control approach offers healthcare system managers a helpful resource for designing vaccination programs and testing plans to prevent COVID-19 transmission.

A General Numerical Scheme for the Optimal Control of Fractional Birkhoffian Systems

This paper deals with the optimal control of fractional Birkhof-fian systems based on the numerical method of variational integrators. Firstly, the fractional forced Birkhoff equations within Riemann–Liouville fractional derivatives are derived from the fractional Pfaff–Birkhoff–d'Alembert principle. Secondly, by directly discretizing the fractional Pfaff–Birkhoff–d'Alembert principle, we develop the equivalent discrete fractional forced Birkhoff equations, which are served as the equality constraints of the optimization problem. Together with the initial and final state constraints on the configuration space, the original optimal control problem is converted into a nonlinear optimization problem subjected to a system of algebraic constraints, which can be solved by the existing methods such as sequential quadratic programming. Finally, an example is given to show the efficiency and simplicity of the proposed method.

Hybrid Probabilistic Inference with Logical and Algebraic Constraints: a Survey

Real world decision making problems often involve both discrete and continuous variables and require a combination of probabilistic and deterministic knowledge. Stimulated by recent advances in automated reasoning technology, hybrid (discrete+continuous) probabilistic reasoning with constraints has emerged as a lively and fast growing research field. In this paper we provide a survey of existing techniques for hybrid probabilistic inference with logic and algebraic constraints. We leverage weighted model integration as a unifying formalism and discuss the different paradigms that have been used as well as the expressivity-efficiency trade-offs that have been investigated. We conclude the survey with a comparative overview of existing implementations and a critical discussion of open challenges and promising research directions.

Uncertainty Quantification of Differential Algebraic Equations Using Polynomial Chaos

The focus of this paper is on the use of Polynomial Chaos for developing surrogate models for Differential Algebraic Equations with time-invariant uncertainties. Intrusive and non-intrusive approaches to synthesize Polynomial Chaos surrogate models are presented including the use of Lagrange interpolation polynomials as basis functions. Unlike ordinary differential equations, if the algebraic constraints are a function of the stochastic variable, some initial conditions of the differential algebraic equations are also random. A benchmark RLC circuit which is used as a benchmark for linear models is used to illustrate the development of a Polynomial Chaos based surrogate model. A nonlinear example of a simple pendulum also serves as a benchmark to illustrate the potential of the proposed approach. Statistics of the results of the Polynomial Chaos models are validated using Monte Carlo simulations in addition to estimating the evolving PDFs of the states of the pendulum.

General consistency of strong discontinuity kinematics in Embedded Finite Element Method (E-FEM) formulations

This paper discusses the consistency of the theoretical basis behind the kinematic models of strong discontinuity methods for local fracture simulations using the Embedded Finite Element Method (E-FEM). A brief review is made on the elemental enhancement functions from the current E-FEM literature and how previous works managed to model mode I (normal) and mode II (parallel) fracture kinematics in multiple dimensions. Further analysis is made on how these approaches also introduce unintended mesh dependencies and basic kinematic inconsistencies of the fracture model with respect to its hosting element. Notable work from a few authors discussing some of these issues and their contributions to resolve them is reviewed as well. Based on this analysis, a new proposal of strong discontinuity enhancement functions is introduced to ensure a broader kinematic coherence within the element and to avoid the observed theoretical faults. This is done by making a more extensive use of the flexibility granted by the Hu-Washizu variational principle and by introducing new algebraic constraints that will ensure more correct fracture kinematics without compromising the acknowledged simplicity of the whole E-FEM framework. Element-level simulations are done to compare the outputs within a group of selected formulation approaches, including the novel proposal. Simulations show that the new element formulation grants a wider level of basic kinematic coherence between the local fracture outputs and element kinematics themselves, demonstrating an increase in robustness that might drive the usefulness and competitiveness of E-FEM techniques for fracture simulations to a higher level.

Explicitly solvable systems of two autonomous first-order Ordinary Differential Equations with homogeneous quadratic right-hand sides

After tersely reviewing the various meanings that can be given to the property of a system of nonlinear ODEs to be solvable, we identify a special case of the system of two first-order ODEs with homogeneous quadratic right-hand sides which is explicitly solvable. It is identified by 2 explicit algebraic constraints on the 6 a priori arbitrary parameters that characterize this system. Simple extensions of this model to cases with nonhomogeneous quadratic right-hand sides are also identified, including isochronous cases

Centric Allometry: Studying Growth Using Landmark Data

The geometric morphometric (GMM) construction of Procrustes shape coordinates from a data set of homologous landmark configurations puts exact algebraic constraints on position, orientation, and geometric scale. While position as digitized is not ordinarily a biologically meaningful quantity, and orientation is relevant mainly when some organismal function interacts with a Cartesian positional gradient such as horizontality, size per se is a crucially important biometric concept, especially in contexts like growth, biomechanics, or bioenergetics. “Normalizing” or “standardizing” size (usually by dividing the square root of the summed squared distances from the centroid out of all the Cartesian coordinates specimen by specimen), while associated with the elegant symmetries of the Mardia–Dryden distribution in shape space, nevertheless can substantially impeach the validity of any organismal inferences that ensue. This paper adapts two variants of standard morphometric least-squares, principal components and uniform strains, to circumvent size standardization while still accommodating an analytic toolkit for studies of differential growth that supports landmark-by-landmark graphics and thin-plate splines. Standardization of position and orientation but not size yields the coordinates Franz Boas first discussed in 1905. In studies of growth, a first principal component of these coordinates often appears to involve most landmarks shifting almost directly away from their centroid, hence the proposed model’s name, “centric allometry.” There is also a joint standardization of shear and dilation resulting in a variant of standard GMM’s “nonaffine shape coordinates” where scale information is subsumed in the affine term. Studies of growth allometry should go better in the Boas system than in the Procrustes shape space that is the current conventional workbench for GMM analyses. I demonstrate two examples of this revised approach (one developmental, one phylogenetic) that retrieve all the findings of a conventional shape-space-based approach while focusing much more closely on the phenomenon of allometric growth per se. A three-part Appendix provides an overview of the algebra, highlighting both similarities to the Procrustes approach and contrasts with it.