Canonical Space Research Articles

Regularity of transition densities and ergodicity for affine jump‐diffusions

AbstractThis paper studies the transition density and exponential ergodicity for affine processes on the canonical state space . Under a Hörmander‐type condition for diffusion components as well as a boundary nonattainment condition, we derive the existence and regularity of the transition densities and then prove the strong Feller property of the associated semigroup. Moreover, we also show that, under an additional subcriticality condition on the drift, the corresponding affine process is exponentially ergodic in the total variation distance.

Towards high-accuracy deep learning inference of compressible flows over aerofoils

The present study investigates the accurate inference of Reynolds-averaged Navier–Stokes solutions for the two-dimensional compressible flow over aerofoils with a deep neural network. Our approach yields networks that learn to generate precise flow fields for varying body-fitted, structured grids by providing them with an encoding of the corresponding mapping to a canonical space for the solutions. We apply the deep neural network model to a benchmark case of incompressible flow at randomly given angles of attack and Reynolds numbers and achieve an improvement of more than an order of magnitude compared to previous work. Further, for transonic flow cases, the deep neural network model accurately predicts complex flow behavior at high Reynolds numbers, such as shock wave/boundary layer interaction, and quantitative distributions like pressure coefficient, skin friction coefficient as well as wake total pressure profiles downstream of aerofoils. The proposed deep learning method significantly speeds up the predictions of flow fields and shows promise for enabling fast aerodynamic designs.

Notes on Peng’s independence in sublinear expectation theory

This paper introduces a new viewpoint of Peng’s independence in sublinear expectation theory via conditional expectation and provides new construction of independent random variables on the canonical product space RN via probability kernels. As an application, the law of large numbers in random type under sublinear expectation is proved.

Current and Future γ-Ray Searches for Dark Matter Annihilation Beyond the Unitarity Limit

For decades, searches for electroweak-scale dark matter (DM) have been performed without a definitive detection. This lack of success may hint that DM searches have focused on the wrong mass range. A proposed candidate beyond the canonical parameter space is ultraheavy DM (UHDM). In this work, we consider indirect UHDM annihilation searches for masses between 30 TeV and 30 PeV—extending well beyond the unitarity limit at ∼100 TeV—and discuss the basic requirements for DM models in this regime. We explore the feasibility of detecting the annihilation signature, and the expected reach for UHDM with current and future very-high-energy (VHE; >100 GeV) γ-ray observatories. Specifically, we focus on three reference instruments: two Imaging Atmospheric Cherenkov Telescope arrays, modeled on VERITAS and CTA-North, and one extended air shower array, motivated by HAWC. With reasonable assumptions on the instrument response functions and background rate, we find a set of UHDM parameters (mass and cross section) for which a γ-ray signature can be detected by the aforementioned observatories. We further compute the expected upper limits for each experiment. With realistic exposure times, the three instruments can probe DM across a wide mass range. At the lower end, it can still have a point-like cross section, while at higher masses the DM could have a geometric cross section, indicative of compositeness.

CVR-MRICloud: An online processing tool for CO2-inhalation and resting-state cerebrovascular reactivity (CVR) MRI data.

Cerebrovascular Reactivity (CVR) provides an assessment of the brain's vascular reserve and has been postulated to be a sensitive marker in cerebrovascular diseases. MRI-based CVR measurement typically employs alterations in arterial carbon dioxide (CO2) level while continuously acquiring Blood-Oxygenation-Level-Dependent (BOLD) images. CO2-inhalation and resting-state methods are two commonly used approaches for CVR MRI. However, processing of CVR MRI data often requires special expertise and may become an obstacle in broad utilization of this promising technique. The aim of this work was to develop CVR-MRICloud, a cloud-based CVR processing pipeline, to enable automated processing of CVR MRI data. The CVR-MRICloud consists of several major steps including extraction of end-tidal CO2 (EtCO2) curve from raw CO2 recording, alignment of EtCO2 curve with BOLD time course, computation of CVR value on a whole-brain, regional, and voxel-wise basis. The pipeline also includes standard BOLD image processing steps such as motion correction, registration between functional and anatomic images, and transformation of the CVR images to canonical space. This paper describes these algorithms and demonstrates the performance of the CVR-MRICloud in lifespan healthy subjects and patients with clinical conditions such as stroke, brain tumor, and Moyamoya disease. CVR-MRICloud has potential to be used as a data processing tool for a variety of basic science and clinical applications.

Parameter identification for Hammerstein nonlinear system with polynomial and state space model

This study investigates a two-stage parameter identification algorithm for the Hammerstein nonlinear system based on special test signals. The studied Hammerstein nonlinear system has a static nonlinear subsystem represented by polynomial basis function and a dynamic linear subsystem described by canonical observable state space model, and special test signals composed of binary signals and random signals are applied to parameter identification separation of the nonlinear subsystem and linear subsystem. The detailed identification procedures consist of two main steps. Firstly, using the characteristics that binary signals do not excite the static nonlinear subsystem, the dynamic linear subsystem parameters are identified through recursive least squares algorithm based on input-output data of binary signals. Secondly, unmeasurable state variables of the identified system are replaced with estimated values, thus the nonlinear subsystem parameters are obtained using recursive least squares algorithm with the help of input-output data of random signals. The efficiency and accuracy of proposed identification scheme are confirmed on experiment results of a numerical simulation and a practical nonlinear process, and experimental simulation results show that the developed two-stage identification algorithm has excellent predictive performance for identifying the Hammerstein nonlinear state space systems.

Linear vs nonlinear transport during chaotic advection in fluid flows.

The goal of this study is explicit demarcation of the region of validity of a linear canonical representation for chaotic advection of Lagrangian fluid parcels in "chaotic seas" in two-dimensional (2D) and three-dimensional (3D) time-periodic fluid flows governed by Hamiltonian mechanics. The concept of lobe dynamics admits exact and unique geometric demarcation of this region and, inherently, distinction of the portions of chaotic seas with essentially linear vs nonlinear Lagrangian transport. This, furthermore, admits explicit establishment of a topological equivalence between the (embedded) Hamiltonian structure of the Lagrangian dynamics in 2D (3D) flows and their canonical form. The linear transport region in physical space encompasses four adjacent subregions that each corresponds to one of the four quadrants in canonical space and may exchange material with their environment in two essentially nonlinear ways. First, exchange between quadrants within the linear transport region and, second, exchange with the exterior of this region. Both forms of exchange can be linked to specific subsets of material elements defined by interacting lobes and combined give rise to circulation through the quadrants of the linear transport region that systematically exchanges the material with the exterior.

Fault Detection and Identification of Furnace Negative Pressure System with CVA and GA-XGBoost

The boiler is an essential energy conversion facility in a thermal power plant. One small malfunction or abnormal event will bring huge economic loss and casualties. Accurate and timely detection of abnormal events in boilers is crucial for the safe and economical operation of complex thermal power plants. Data-driven fault diagnosis methods based on statistical process monitoring technology have prevailed in thermal power plants, whereas the false alarm rates of those methods are relatively high. To work around this, this paper proposes a novel fault detection and identification method for furnace negative pressure system based on canonical variable analysis (CVA) and eXtreme Gradient Boosting improved by genetic algorithms (GA-XGBoost). First, CVA is used to reduce the data redundancy and construct the canonical residuals to measure the prediction ability of the state variables. Then, the fault detection model based on GA-XGBoost is schemed using the constructed canonical residual variables. Specially, GA is introduced to determine the optimal hyperparameters of XGBoost and speed up the convergence. Next, this paper presents a novel fault identification method based on the reconstructed contribution statistics, considering the contribution of state space, residual space and canonical residual space. Besides, the proposed statistics renders different weights to the state vectors, the residual vectors and the canonical residual vectors to improve the sensitivity of faulty variables. Finally, the real industrial data from a boiler furnace negative pressure system of a certain thermal power plant is used to demonstrate the ability of the proposed method. The result demonstrates that this method is accurate and efficient to detect and identify the faults of a true boiler.

Convex rough sets on finite domains

This paper addresses a foundational aspect of imprecision in information and knowledge. It makes a convincing case that convexity can take part in the progress of rough set theory in finite settings. To this purpose we resort to convex geometries, which constitute a special type of coverings that abstract many combinatorial features of convexity. We define convex geometry (cg) approximation spaces on a grand set, and we produce novel cg-upper and cg-lower approximation operators. Their basic properties are presented. Then we show that the model that arises has connections with well-established models in the rough set literature, both from relation and covering-based approaches. We identificate three types of subsets of the grand set that have different behaviors with respect to their cg-approximations, and we refine this classification in some benchmark cases. Finally, we produce a canonical convex geometry approximation space from any covering on a set. Examples illustrate our constructions and main results.

Generalizations of Snyder model to curved spaces

We consider generalizations of the Snyder algebra to a curved spacetime background with de Sitter symmetry. As special cases, we obtain the algebras of the Yang model and of triply special relativity. We discuss the realizations of these algebras in terms of canonical phase space coordinates, up to fourth order in the deformation parameters. In the case of triply special relativity we also find exact realization, exploiting its algebraic relation with the Snyder model.

Nambu mechanics viewed as a Clebsch parameterized Poisson algebra: Toward canonicalization and quantization

Abstract In his pioneering paper [Phys. Rev. E 7, 2405 (1973)], Nambu proposed the idea of multiple Hamiltonian systems. The explicit example examined there is equivalent to the $\mathfrak {so}(3)$ Lie–Poisson system, which represents noncanonical Hamiltonian dynamics with a Casimir; the Casimir corresponds to the second Hamiltonian of Nambu’s formulation. The vortex dynamics of an ideal fluid, while it is infinite dimensional, has a similar structure, in which the Casimir is the helicity. These noncanonical Poisson algebras are derived by the reduction, i.e., restricting the phase space to some submanifold embedded in the canonical phase space. We may reverse the reduction to canonicalize some Nambu dynamics, i.e., view the Nambu dynamics as the subalgebra of a larger canonical Poisson algebra. Then, we can invoke the standard corresponding principle for quantizing the canonicalized system. The inverse of the reduction, i.e., representing the noncanonical variables by some canonical variables may be called “Clebsch parameterization” following the fluid mechanical example.

Intrinsic Riemannian functional data analysis for sparse longitudinal observations

A new framework is developed to intrinsically analyze sparsely observed Riemannian functional data. It features four innovative components: a frame-independent covariance function, a smooth vector bundle termed covariance vector bundle, a parallel transport and a smooth bundle metric on the covariance vector bundle. The introduced intrinsic covariance function links estimation of covariance structure to smoothing problems that involve raw covariance observations derived from sparsely observed Riemannian functional data, while the covariance vector bundle provides a rigorous mathematical foundation for formulating such smoothing problems. The parallel transport and the bundle metric together make it possible to measure fidelity of fit to the covariance function. They also play a critical role in quantifying the quality of estimators for the covariance function. As an illustration, based on the proposed framework, we develop a local linear smoothing estimator for the covariance function, analyze its theoretical properties and provide numerical demonstration via simulated and real data sets. The intrinsic feature of the framework makes it applicable to not only Euclidean submanifolds but also manifolds without a canonical ambient space.

Least square algorithm based on bias compensated principle for parameter estimation of canonical state space model

Due to the existence of system noise and unknown state variables, it is difficult to realize unbiased estimation with minimum variance for the parameter estimation of canonical state space model. This paper presents a new least squares estimator based on bias compensation principle to solve this problem, transforms canonical state space into the form suitable for the least square algorithm, introduces an augmented parameter vector and an auxiliary variable, derives parameter estimation formula based on noise compensation, realizes the unbiased estimation, and gives the specific algorithm. A simulation example is provided to verify the effectiveness of the estimator.

Quantum carpets in a leaky box: Poincaré's recurrences in the continuous spectrum

The freedom to define branch cuts of the complex function is used to derive an integral representation of the quantum carpet, thus producing a generalization of the Poincar\'e recurrence theorem in the case of the continuous spectrum. This approach provides a different way to renormalize resonant states to be both space and time convergent. The coherence of quantum carpets was related to the properties of the Wigner function in the canonical time-frequency phase space. It has been shown that the distortion of the Wigner function shape is directly responsible for the lack of the ability of the dynamics to produce revivals equally as sharp as the initial wave packet.

McKean–Vlasov optimal control: The dynamic programming principle

We study the McKean–Vlasov optimal control problem with common noise which allow the law of the control process to appear in the state dynamics under various formulations: strong and weak ones, Markovian or non-Markovian. By interpreting the controls as probability measures on an appropriate canonical space with two filtrations, we then develop the classical measurable selection, conditioning and concatenation arguments in this new context, and establish the dynamic programming principle under general conditions.

McKean–Vlasov Optimal Control: Limit Theory and Equivalence Between Different Formulations

We study a McKean–Vlasov optimal control problem with common noise in order to establish the corresponding limit theory as well as the equivalence between different formulations, including strong, weak, and relaxed formulations. In contrast to the strong formulation, in which the problem is formulated on a fixed probability space equipped with two Brownian filtrations, the weak formulation is obtained by considering a more general probability space with two filtrations satisfying an (H)-hypothesis type condition from the theory of enlargement of filtrations. When the common noise is uncontrolled, our relaxed formulation is obtained by considering a suitable controlled martingale problem. As for classic optimal control problems, we prove that the set of all relaxed controls is the closure of the set of all strong controls when considered as probability measures on the canonical space. Consequently, we obtain the equivalence of the different formulations of the control problem under additional mild regularity conditions on the reward functions. This is also a crucial technical step to prove the limit theory of the McKean–Vlasov control problem, that is, proving that it consists in the limit of a large population control problem with common noise.

Single-camera 3D head fitting for mixed reality clinical applications

We address the problem of estimating the shape of a person’s head, defined as the geometry of the complete head surface, from a video taken with a single moving camera, and determining the alignment of the fitted 3D head for all video frames, irrespective of the person’s pose. 3D head reconstructions commonly tend to focus on perfecting the face reconstruction, leaving the scalp to a statistical approximation. Our goal is to reconstruct the head model of each person to enable future mixed reality applications. To do this, we recover a dense 3D reconstruction and camera information via structure-from-motion and multi-view stereo. These are then used in a new two-stage fitting process to recover the 3D head shape by iteratively fitting a 3D morphable model of the head with the dense reconstruction in canonical space and fitting it to each person’s head, using both traditional facial landmarks and scalp features extracted from the head’s segmentation mask. Our approach recovers consistent geometry for varying head shapes, from videos taken by different people, with different smartphones, and in a variety of environments from living rooms to outdoor spaces.

Path Integral Quantization of Doubly Supersymmetric Model

The Hamilton-Jacobi formalism is used to discuss the path integral quantization of the double supersymmetric models with the spinning superparticle in the component and superfield form. The equations of motion are obtained as total differential equations in many variables. The equations of motion are integrable, and the path integral is obtained as an integration over the canonical phase space coordinates.

Neural actor

We propose Neural Actor (NA), a new method for high-quality synthesis of humans from arbitrary viewpoints and under arbitrary controllable poses. Our method is developed upon recent neural scene representation and rendering works which learn representations of geometry and appearance from only 2D images. While existing works demonstrated compelling rendering of static scenes and playback of dynamic scenes, photo-realistic reconstruction and rendering of humans with neural implicit methods, in particular under user-controlled novel poses, is still difficult. To address this problem, we utilize a coarse body model as a proxy to unwarp the surrounding 3D space into a canonical pose. A neural radiance field learns pose-dependent geometric deformations and pose- and view-dependent appearance effects in the canonical space from multi-view video input. To synthesize novel views of high-fidelity dynamic geometry and appearance, NA leverages 2D texture maps defined on the body model as latent variables for predicting residual deformations and the dynamic appearance. Experiments demonstrate that our method achieves better quality than the state-of-the-arts on playback as well as novel pose synthesis, and can even generalize well to new poses that starkly differ from the training poses. Furthermore, our method also supports shape control on the free-view synthesis of human actors.

Assigning probabilities to non-Lipschitz mechanical systems.

We present a method for assigning probabilities to the solutions of initial value problems that have a Lipschitz singularity. To illustrate the method, we focus on the following toy example: d2r(t)dt2=ra, r(t=0)=0, and dr(t)dt∣r(t=0)=0, with a∈]0,1[. This example has a physical interpretation as a mass in a uniform gravitational field on a frictionless, rigid dome of a particular shape; the case with a=1/2 is known as Norton's dome. Our approach is based on (1) finite difference equations, which are deterministic; (2) elementary techniques from alpha-theory, a simplified framework for non-standard analysis that allows us to study infinitesimal perturbations; and (3) a uniform prior on the canonical phase space. Our deterministic, hyperfinite grid model allows us to assign probabilities to the solutions of the initial value problem in the original, indeterministic model.