Finite Projection Research Articles

Housing, COVID & Climate: Urban Social Housing Post-COVID Climate Change Mitigation and Redress

This short paper, draws on 40 years of research and advisory work, rather than a single, short-term, finite research project. It examines recent and current examples of urban housing policies, programmes and projects in a wide range of countries in both the Global North and South in its aim order to meet its aim to stimulate and contribute to the search for strategic approaches to the production, maintenance and management of urban low-income group housing; it examines the historical, geographic, cultural/political and economic contexts within which they are set, laying some emphasis on the prevailing existential threats to humanity caused by: (1) global heating, and (2) the pandemic pervasiveness of viral pathogen transmissions and their social, economic and legislative impact on physical settlement planning and management. Emphasis is given to distinguishing between ‘public housing’, for which all governing decisions are unilaterally made by relevant state agencies and institutions, and ‘social housing’, in which non-governmental beneficiary households and enterprises, communities and organisations are engaged in risk- and benefit-sharing partnership with responsible state agencies at all levels of the governance and management of affordable and accessible urban housing. The paper concludes by drawing attention to the urgency of need to build the capability of the extensive range of actors and stake-holders, worldwide and calling for the initiation and dissemination of successful policy approaches and their administration.

Error analysis of a fully discrete projection method for Cahn–Hilliard Inductionless MHD problems

This article investigates the fully discrete finite element approximation and error analysis for a diffuse interface model of the two-phase incompressible inductionless magnetohydrodynamics problem. This model consists of Cahn–Hilliard equations, Navier–Stokes equations and Poisson equations, which are nonlinearly coupled through convection, stresses, and Lorentz forces. To address this highly nonlinear and multi-physics system, we propose a fully discrete energy stable scheme with the finite element projection method for spatial discretization, in which the velocity and pressure are decoupled. The temporal discretization is a combination of the first-order Euler semi-implicit scheme and a convex splitting energy strategy. We show that the proposed scheme is mass-conservative, charge-conservative and unconditional energy stable. The error estimates for the phase variable, chemical potential, velocity, pressure, current density and electric potential are rigorously established. Finally, several three-dimensional numerical experiments are performed to illustrate the features of the proposed numerical method and verify the theoretical conclusions.

Reducing penumbral blur in computed tomography by learning the inverse finite focal spot model

Penumbral blur is one of the major limitations of the high spatial resolution micro-CT, due to a nonideal large focal spot. Penumbral blur hinders the ability to resolve small features that may only be a few pixels in size. Reducing the focal spot size by decreasing the x-ray tube power is a straightforward solution, but it leads to prolonged scan durations. In this paper, we propose to mitigate the penumbral blur by learning the inverse finite focal spot model. First, we derived the finite focal spot model that builds a relationship from the ideal point source projection to the finite focal spot projection. Based on the derived model, we numerically compute a paired projection dataset. Second, we utilized two neural networks—U-net, and convolution modulation-based U-net (CMU-net) —to learn the inverse finite focal spot model. The goal is to estimate the ideal point source projection from the actual finite focal spot projection. CMU-net, which introduces convolution modulation blocks into the contracting path of the U-net, is proposed to boost the robustness of the U-net. Finally, the standard filtered back-projection (FBP) is employed for reconstruction using the estimated ideal point projection. The experiments show that both U-net and CMU-net can effectively reduce the penumbral blur, whereas CMU-net demonstrates better performance on the real data. Experiments on real measured data demonstrate that CMU-net is more robust than U-net and can effectively resolve fine details. This method has great potential in improving the efficiency of micro-CT acquisition. It allows increasing the tube power since our method can computationally compensate for the blur caused by an increased focal spot size.

Feedback between stochastic gene networks and population dynamics enables cellular decision-making.

Phenotypic selection occurs when genetically identical cells are subject to different reproductive abilities due to cellular noise. Such noise arises from fluctuations in reactions synthesizing proteins and plays a crucial role in how cells make decisions and respond to stress or drugs. We propose a general stochastic agent-based model for growing populations capturing the feedback between gene expression and cell division dynamics. We devise a finite state projection approach to analyze gene expression and division distributions and infer selection from single-cell data in mother machines and lineage trees. We use the theory to quantify selection in multi-stable gene expression networks and elucidate that the trade-off between phenotypic switching and selection enables robust decision-making essential for synthetic circuits and developmental lineage decisions. Using live-cell data, we demonstrate that combining theory and inference provides quantitative insights into bet-hedging-like response to DNA damage and adaptation during antibiotic exposure in Escherichia coli.

Regular variation in Hilbert spaces and principal component analysis for functional extremes

Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements X in L2[0,1]. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation X whose L2-norm ‖X‖ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal ‘directions’ of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert–Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.

State Dependent Delay Maps: Numerical Algorithms and Dynamics of Projections

This work concerns the dynamics of a certain class of delay differential equations (DDEs) which we refer to as state dependent delay maps. These maps are generated by delay differential equations where the derivative of the current state depends only on delayed variables, and not on the un-delayed state. However, we allow that the delay is itself a function of the state variable. A delay map with constant delays can be rewritten explicitly as a discrete time dynamical system on an appropriate function space, and a delay map with small state dependent terms can be viewed as a “non-autonomous” perturbation. We develop a fixed point formulation for the Cauchy problem of such perturbations, and under appropriate assumptions obtain the existence of forward iterates of the map. The proof is constructive and leads to numerical procedures which we implement for illustrative examples, including the cubic Ikeda and Mackey-Glass systems with constant and state-dependent delays. After proving a local convergence result for the method, we study more qualitative/global convergence issues using data analytic tools for time series analysis (dimension and topological measures derived from persistent homology). Using these tools we quantify the convergence of the dynamics in the finite dimensional projections to the dynamics of the infinite dimensional system.

Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

This manuscript proposes a class of fractional stochastic integro-differential equations (FSIDEs) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. In order to demonstrate the existence and convergence of approximate solutions, we utilize stochastic analysis theory, fractional calculus, theory of fractional cosine family of linear operators, and fixed point approach. Furthermore, we examine the convergence of the Faedo–Galerkin (F–G) approximate solutions to the mild solution of our given problem. Finally, a concrete example involving a partial differential equation is provided to validate the main abstract results.

Analysis of the chemical diffusion master equation for creation and mutual annihilation reactions

We propose an infinite dimensional generating function method for finding the analytical solution of the so-called chemical diffusion master equation (CDME) for creation and mutual annihilation chemical reactions. CDMEs model by means of an infinite system of coupled Fokker–Planck equations the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles; here, we focus an creation and mutual annihilation chemical reactions combined with Brownian diffusion of the single particles. Using our method we are able to link certain finite dimensional projections of the solution of the CDME to the solution of a single linear fourth order partial differential equation containing as many variables as the dimension of the aforementioned projection space. Our technique extends the one presented in Lanconelli [J. Math. Anal. Appl. 526, 127352 (2023)] and Lanconelli et al. [arXiv:2302.10700 [math.PR] (2023)] which allowed for an explicit representation for the solution of birth-death type CDMEs.

“El Chamizal is ours forever:” Rumor, time, and the law in El Paso’s settler society

This essay contributes to literature on the intersections of white settler colonialisms, racial capitalism, and U.S.-Mexico borderlands history by tracing the web of spatial, temporal, and legal power relations that produced El Paso, Texas’ seemingly legitimate possession of stolen Mexican territory known as “El Chamizal” in the El Paso-Cd. Juárez borderlands. This land theft became the Chamizal Dispute: an international land and boundary conflict between the U.S. and Mexico caused by the meandering Río Grande that defines the “fixed” international border between El Paso, Texas and Cd. Juárez, Chihuahua. In the 1860s, multiple shifts in the Rio Grande “relocated” El Chamizal north of this river/boundary. Soon thereafter, and despite Mexico’s sustained claim to and jurisdiction over this land, recently arrived Anglo American settlers incorporated El Chamizal into the nascent City of El Paso. In 1964, the U.S. and Mexico finally agreed to resolve this conflict by virtue of the landmark Chamizal Treaty, which ceded 630-acres of El Paso to Cd. Juárez as El Chamizal. Contrary to what dominant state accounts and the mainstream historical literature on this settlement would have us believe, however, this ceded land includes only a fraction of the original contested terrain. El Chamizal therefore remains a stolen tract of land nestled within the heart of El Paso. Drawing on oral histories, court testimonies and affidavits, and an array of binational records, this essay demonstrates that this ongoing theft is not a finite or complete project. Rather, the process hinges on a fragile web of spatial, white settler temporal, and legal practices of concealment/denial anchored to a colonial rumor that refuses to open this region to the mystery and wonder of the Río Grande’s “wayward life, beautiful experiment in how to live.”

Industrial computed tomography for three-dimensional cultural relic model reconstruction based on L1-αL2+TV norm minimization

Industrial computed tomography (ICT) has emerged as a valuable tool for the conservation of cultural relics since it enables the extraction of information about the internal structure of artifacts without causing any damage. However, sparse and finite angle projection computed tomography (CT) often generates images with a large number of artifacts caused by insufficient data. This paper introduces a new cone-beam CT hybrid model based on compressed sensing (CS) theory and total variance (TV) denoising using the Split Bregman algorithm to address this problem, for the first time using compressed sensing theory for CT reconstruction of cultural relics. The model combines the L1 and L2 norms and optimizes the total variance of the image as the objective function. To enhance the reconstruction of L1-L2, the paper employs the continuation strategy of α and proposes a new combination function to update α. Consequently, the reconstruction results are compared with those of traditional common methods, and the effect of different α update functions is observed on the reconstruction results. The results from the model and real CT images demonstrate that the proposed L1-αL2 + TV method yields superior reconstruction results compared to ART + TV, SART, and L1/L2 algorithms based on objective evaluation indexes such as RMSE, SSIM, and PSNR, the proposed L1-αL2 + TV method achieves SSIM improvement of at least 15.2 %. Moreover, the newly proposed combined function is better at preserving the grain of the image, resulting in a much-improved reconstruction of the contours, making it highly beneficial for subsequent conservation and restoration of heritage. Overall, this proposed L1-αL2 + TV mixture model provides low-noise and high-quality reconstruction results, while maintaining the grain of the original image, making it a great boon for heritage conservation and restoration.

Experimental and theoretical investigation on vibration of laminated composite conical-cylindrical-combining shells with elastic foundation in hygrothermal environment

The present investigation focuses on the vibration of composite conical-cylindrical-combining shells (CCCCSs) subjected to elastic supports and hygrothermal effects by means of experimental and theoretical investigation. The kinetic energy, potential energy and hygrothermal potential energy are derived via the Donnell shell hypothesis. A set of elastic springs are introduced to the free edges of the CCCCSs aiming to simulate the elastic supports and the interface of the two shell. The dynamic equations of laminated CCCCSs with elastic supports in hygrothermal environment have been obtained in frame of Rayleigh-Ritz method by means of a set of modified orthogonal polynomials as the proper admittance displacement functions. Several experimental and finite element simulation research projects were carried out to validate the effectiveness of the proposed analytical strategy. The unique vibration characteristics of CCCCSs with respect to various temperature, humidity, interfacial bonded stiffness and elastic boundary conditions are thoroughly discussed. The results indicated that the coupling phenomenon performs obviously with the increase of the interface joint stiffness of the structures. Furthermore, the effect of the thermal load on the vibration of CCCCSs become more obviously during some specific temperature range which is determined by the mechanics of the material and manufactured scheme.

Bayesian filtering for model predictive control of stochastic gene expression in single cells

This study describes a method for controlling the production of protein in individual cells using stochastic models of gene expression. By combining modern microscopy platforms with optogenetic gene expression, experimentalists are able to accurately apply light to individual cells, which can induce protein production. Here we use a finite state projection based stochastic model of gene expression, along with Bayesian state estimation to control protein copy numbers within individual cells. We compare this method to previous methods that use population based approaches. We also demonstrate the ability of this control strategy to ameliorate discrepancies between the predictions of a deterministic model and stochastic switching system.

Using Malliavin calculus to solve a chemical diffusion master equation

We propose a novel method to solve a chemical diffusion master equation of birth and death type. This is an infinite system of Fokker-Planck equations where the different components are coupled by reaction dynamics similar in form to a chemical master equation. This system was proposed in [4] for modelling the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles. Using some basic tools and ideas from infinite dimensional Gaussian analysis we are able to reformulate the aforementioned infinite system of Fokker-Planck equations as a single evolution equation solved by a generalized stochastic process and written in terms of Malliavin derivatives and differential second quantization operators. Via this alternative representation we link certain finite dimensional projections of the solution of the original problem to the solution of a single partial differential equations of Ornstein-Uhlenbeck type containing as many variables as the dimension of the aforementioned projection space.

R⁎-algebras and C⁎-uniqueness of pre-C⁎-algebras

In 1967, Kadison raised the following question, if a self-adjoint operator algebra on a Hilbert space is finitely generated (algebraically) and each self-adjoint operator in it has finite spectrum, is it finite dimensional? In 2021, M. Mori connected the Kadison's question to von Neumann's concept of regular rings given in 1930's, and introduced a notion of R⁎-algebras. In this paper, we study R⁎-algebras, the Kadison's question, and a related question of uniqueness of C⁎-norm on pre-C⁎-algebras. Precisely, we study atomic R⁎-algebras, and find a maximal ultramatricial R⁎-subalgebra of an atomic R⁎-algebra, which properly contains the maximal purely atomic R⁎-subalgebra generated by all finite projections of the algebra. We prove that the set of all ⁎-algebras for which the Kadison's question has an affirmative answer is closed under direct sum, tensor product, and crossed product, but it is not closed under free product. From the aforementioned stable properties of the algebra in the Kadison's question, we find plentiful examples of ⁎-algebras for which the Kadison's question has an affirmative answer, for instance, the crossed product ⁎-algebra B⋊αG, where G is a locally finite group and B is one of the following algebras: an algebra of finite rank operators, an abelian algebra, a finite dimensional algebra, a twisted group algebra satisfying a certain condition, a rotation algebra, or a group measure space construction algebra. We characterize uniqueness of C⁎-norm on a pre-C⁎-algebras in terms of its ⁎-representations, weak containment of its ⁎-representations, ideals of its enveloping C⁎-algebra, or the primitive ideal space of the C⁎-algebra. We also prove that a unital pre-C⁎-algebra has a unique C⁎-norm if and only if, for a faithful ⁎-representation of the algebra, invertibility of a self-adjoint element of the algebra, as an operator on the representation Hilbert space, is independent of the choice of the ⁎-representation. These characterizations provide answers to the question raised by M. Mori in 2021 on finding a sufficient and necessary condition for a ⁎-algebra to have a unique C⁎-norm. For discrete groups, we give a characterization for a group algebra to have a unique C⁎-norm in terms of the group's unitary representations, providing an answer to the question raised by Alekseev in 2019 on characterizing C⁎-unique discrete groups.

A general framework for the rigorous computation of invariant densities and the coarse-fine strategy

In this paper we present a general, axiomatical framework for the rigorous approximation of invariant densities and other important statistical features of dynamics. We approximate the system through a finite element reduction, by composing the associated transfer operator with a suitable finite dimensional projection (a discretization scheme) as in the well-known Ulam method.We introduce a general framework based on a list of properties (of the system and of the projection) that need to be verified so that we can take advantage of a so-called “coarse-fine” strategy. This strategy is a novel method in which we exploit information coming from a coarser approximation of the system to get useful information on a finer approximation, speeding up the computation. This coarse-fine strategy allows a precise estimation of invariant densities and also allows to estimate rigorously the speed of mixing of the system by the speed of mixing of a coarse approximation of it, which can easily be estimated by the computer.The estimates obtained her e are rigorous, i.e., they come with exact error bounds that are guaranteed to hold and take into account both the discretization and the approximations induced by finite-precision arithmetic.We apply this framework to several discretization schemes and examples of invariant density computation from previous works, obtaining a remarkable reduction in computation time.We have implemented the numerical methods described here in the Julia programming language, and released our implementation publicly as a Julia package.

Fault-tolerant consensus of leader–following multi-agent systems with jointly connected topologies

The fault-tolerant leader–following consensus problem of multi-agent systems (MASs) with jointly connected topologies is considered in this paper. A control law, which is compensated by adaptive estimation of fault parameters using finite interval projection operator, is designed and enables the MAS with jointly connected topologies to achieve consensus in the presence of actuator faults. Furthermore, compared with the most existing works of fault-tolerant consensus with fixed and switching connected topologies, the results with jointly connected topologies are more general in practical systems. Moreover, the effectiveness of the controller for the considered problem is proved based on the Lyapunov stability theory. At the final part, by a numerical simulation, the validity of the results is verified.

Approximate equivalence in von Neumann algebras

Suppose $\mathcal{A}$ is a separable unital ASH C*-algebra, $\mathcal{R}$ is a sigma-finite II$_{\infty}$ factor von Neumann algebra, and $\pi,\rho :\mathcal{A}\rightarrow\mathcal{R}$ are unital $\ast$-homomorphisms such that, for every $a\in\mathcal{A}$, the range projections of $\pi\left( a\right) $ and $\rho\left( a\right) $ are Murray von Neuman equivalent in $\mathcal{R}% $. We prove that $\pi$ and $\rho$ are approximately unitarily equivalent modulo $\mathcal{K}_{\mathcal{R}}$, where $\mathcal{K}_{\mathcal{R}}$ is the norm closed ideal generated by the finite projections in $\mathcal{R}$. We also prove a very general result concerning approximate equivalence in arbitrary finite von Neumann algebras.

Atomic-Scale Mapping and Quantification of Local Ruddlesden-Popper Phase Variations.

The Ruddlesden-Popper (An+1BnO3n+1) compounds are highly tunable materials whose functional properties can be dramatically impacted by their structural phase n. The negligible differences in formation energies for different n can produce local structural variations arising from small stoichiometric deviations. Here, we present a Python analysis platform to detect, measure, and quantify the presence of different n-phases based on atomic-resolution scanning transmission electron microscopy (STEM) images. We employ image phase analysis to identify horizontal Ruddlesden-Popper faults within the lattice images and quantify the local structure. Our semiautomated technique considers effects of finite projection thickness, limited fields of view, and lateral sampling rates. This method retains real-space distribution of layer variations allowing for spatial mapping of local n-phases to enable quantification of intergrowth occurrence and qualitative description of their distribution suitable for a wide range of layered materials.

Real-Fibered Morphisms of del Pezzo Surfaces and Conic Bundles

It goes back to Ahlfors that a real algebraic curve admits a real-fibered morphism to the projective line if and only if the real part of the curve disconnects its complex part. Inspired by this result, we are interested in characterising real algebraic varieties of dimension n admitting real-fibered morphisms to the n-dimensional projective space. We present a criterion to classify real-fibered morphisms that arise as finite surjective linear projections from an embedded variety which relies on topological linking numbers. We address special attention to real algebraic surfaces. We classify all real-fibered morphisms from real del Pezzo surfaces to the projective plane and determine which such morphisms arise as the composition of a projective embedding with a linear projection. Furthermore, we give some insights in the case of real conic bundles.

Error analysis of a fully discrete projection method for magnetohydrodynamic system

AbstractIn this paper, we develop and analyze a finite element projection method for magnetohydrodynamics equations in Lipschitz domain. A fully discrete scheme based on Euler semi‐implicit method is proposed, in which continuous elements are used to approximate the Navier–Stokes equations and H(curl) conforming Nédélec edge elements are used to approximate the magnetic equation. One key point of the projection method is to be compatible with two different spaces for calculating velocity, which leads one to obtain the pressure by solving a Poisson equation. The results show that the proposed projection scheme meets a discrete energy stability. In addition, with the help of a proper regularity hypothesis for the exact solution, this paper provides a rigorous optimal error analysis of velocity, pressure and magnetic induction. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.