Singular Domains Research Articles

Weak-type estimates of the Bergman projection on a class of singular Reinhardt domains

In this paper, we study the weak-type regularity of the Bergman projection on monomial polyhedra, which is a wide class of bounded singular pseudoconvex Reinhardt domains defined as sublevel sets of holomorphic monomials. We prove that the weak-type estimate of the Bergman projection holds at the upper endpoint of Lp boundedness range, but fails at the lower endpoint. This result generalizes the works of Huo-Wick and Christopherson-Koenig on 2-dimensional Hartogs triangles to a much more general setting, as well as the work of Jing et al. on n-dimensional generalized Hartogs triangle. As a consequence, we also obtain the weak-type regularity on the well-known elementary Reinhardt domains, which has not been considered by others.

OneSLAM to map them all: a generalized approach to SLAM for monocular endoscopic imaging based on tracking any point.

Monocular SLAM algorithms are the key enabling technology for image-based surgical navigation systems for endoscopic procedures. Due to the visual feature scarcity and unique lighting conditions encountered in endoscopy, classical SLAM approaches perform inconsistently. Many of the recent approaches to endoscopic SLAM rely on deep learning models. They show promising results when optimized on singular domains such as arthroscopy, sinus endoscopy, colonoscopy or laparoscopy, but are limited by an inability to generalize to different domains without retraining. To address this generality issue, we propose OneSLAM a monocular SLAM algorithm for surgical endoscopy that works out of the box for several endoscopic domains, including sinus endoscopy, colonoscopy, arthroscopy and laparoscopy. Our pipeline builds upon robust tracking any point (TAP) foundation models to reliably track sparse correspondences across multiple frames and runs local bundle adjustment to jointly optimize camera poses and a sparse 3D reconstruction of the anatomy. We compare the performance of our method against three strong baselines previously proposed for monocular SLAM in endoscopy and general scenes. OneSLAM presents better or comparable performance over existing approaches targeted to that specific data in all four tested domains, generalizing across domains without the need for retraining. OneSLAM benefits from the convincing performance of TAP foundation models but generalizes to endoscopic sequences of different anatomies all while demonstrating better or comparable performance over domain-specific SLAM approaches. Future research on global loop closure will investigate how to reliably detect loops in endoscopic scenes to reduce accumulated drift and enhance long-term navigation capabilities.

Uniqueness of Positive Vorticity Solutions to the 2d Euler Equations on Singular Domains

Uniqueness of Positive Vorticity Solutions to the 2d Euler Equations on Singular Domains

Nonlinear extension of the Kolosov-Muskhelishvili stress function formalism.

The method of stress function in elasticity theory is a powerful analytical tool with applications to a wide range of physical systems, including defective crystals, fluctuating membranes, and more. A complex coordinates formulation of stress function, known as the Kolosov-Muskhelishvili formalism, enabled the analysis of elastic problems with singular domains, particularly cracks, forming the basis for fracture mechanics. A shortcoming of this method is its limitation to linear elasticity, which assumes Hookean energy and linear strain measure. Under finite loads, the linearized strain fails to describe the deformation field adequately, reflecting the onset of geometric nonlinearity. The latter is common in materials experiencing large rotations, such as regions close to the crack tip or elastic metamaterials. While a nonlinear stress function formalism exists, the Kolosov-Muskhelishvili complex representation had not been generalized and remained limited to linear elasticity. This paper develops a Kolosov-Muskhelishvili formalism for the nonlinear stress function. Our formalism allows us to port methods from complex analysis to nonlinear elasticity and to solve nonlinear problems in singular domains. Upon implementing the method to the crack problem, we discover that nonlinear solutions strongly depend on the applied remote loads, excluding a universal form of the solution close to the crack tip and questioning the validity of previous studies of nonlinear crack analysis.

Bergman functions and the equivalence problem of singular domains

Bergman functions and the equivalence problem of singular domains

Characterisation of Singular Domains in Threshold-Dependent Biological Networks

Threshold-dependent networks behave like piecewise smooth system which belongs to a class of systems with discontinuous right hand side. It can be studied with piecewise linear differential equations. At threshold values and their intersections, known as switching boundaries and surfaces, the state of such networks is not defined because of singularity at such points. These states are termed singular states or domains of the network. This study is to characterise, in terms of number, singular domains of any order in a network and the total number of such domains. It shall also propose new definitions for walls which are switching domains of order one. The new definition, which derives from Fillipov’s First Order Theory on characterisation of (sliding) wall, shall be shown to prove better than existing ones as it can reveal intrinsic behaviours in such networks.

Method for solving bang-bang and singular optimal control problems using adaptive Radau collocation

A method is developed for solving bang-bang and singular optimal control problems using adaptive Legendre-Gauss-Radau (LGR) collocation. The method is divided into several parts. First, a structure detection method is developed that identifies switch times in the control and analyzes the corresponding switching function for segments where the solution is either bang-bang or singular. Second, after the structure has been detected, the domain is decomposed into multiple domains such that the multiple-domain formulation includes additional decision variables that represent the switch times in the optimal control. In domains classified as bang-bang, the control is set to either its upper or lower limit. In domains identified as singular, the objective function is augmented with a regularization term to avoid the singular arc. An iterative procedure is then developed for singular domains to obtain a control that lies in close proximity to the singular control. The method is demonstrated on four examples, three of which have either a bang-bang and/or singular optimal control while the fourth has a smooth and nonsingular optimal control. The results demonstrate that the method of this paper provides accurate solutions to problems whose solutions are either bang-bang or singular when compared against previously developed mesh refinement methods that are not tailored for solving nonsmooth and/or singular optimal control problems, and produces results that are equivalent to those obtained using previously developed mesh refinement methods for optimal control problems whose solutions are smooth.

Euler Equations on General Planar Domains

We obtain a general sufficient condition on the geometry of possibly singular planar domains that guarantees global uniqueness for any weak solution to the Euler equations on them whose vorticity is bounded and initially constant near the boundary. While similar existing results require domains that are \(C^{1,1}\) except at finitely many convex corners, our condition involves much less domain smoothness, being only slightly more restrictive than the exclusion of corners with angles greater than \(\pi \). In particular, it is satisfied by all convex domains. The main ingredient in our approach is showing that constancy of the vorticity near the boundary is preserved for all time because Euler particle trajectories on these domains, even for general bounded solutions, cannot reach the boundary in finite time. We then use this to show that no vorticity can be created by the boundary of such possibly singular domains for general bounded solutions. We also show that our condition is essentially sharp in this sense by constructing domains that come arbitrarily close to satisfying it, and on which particle trajectories can reach the boundary in finite time. In addition, when the condition is satisfied, we find sharp bounds on the asymptotic rate of the fastest possible approach of particle trajectories to the boundary.

Memory response on wave propagation in a micropolar magneto-thermo-viscoelastic half-space

The present analysis reports the Kelvin–Voigt-type magneto-thermo-viscoelastic interactions in a thermally conducting unbounded half-space whose surface is subjected to time-harmonic thermal source in the context of micropolar thermoelasticity, being enlightened by memory-dependent derivative (MDD) in the context of three-phase (3P) lag model. The bounding plane of the half-space is stress-free and is subjected to a prescribed temperature distribution. Employing the Laplace transform and Fourier transforms, analytical results for the distributions of the thermophysical quantities have been derived in the transformed domain. The numerical inversions of the respective transforms have been carried out using a suitable numerical scheme based on Fourier series expansion technique. Numerical computations for the stresses, displacement and temperature within the medium have been carried out and also have been demonstrated graphically. Since, micropolar elasticity allows better results to be obtained for microstructural and singular domains as compared to the classical theory of elasticity, therefore, the investigations outline how the micropolarity, magnetic field and viscoelastic parameters influence the thermophysical quantities of the half-space to determine the microstructural changes within the body. Moreover, significant differences on the thermophysical quantities are revealed due to the influence of memory effect and time-delay also, which helps to determine the past effects to the present.

Recent Applications of Magnesium Oxide (MgO) Nanoparticles in various domains

Magnesium oxide nanoparticles have emerged as a potential candidate for meeting ends of various problems due to its unique properties such as biodegradability, non-toxicity, inhibition of biofilm growth and degradation of harmful dyes such as methyl violet and many more. Along with its easy synthesis by methods such as sol-gel technique, precipitation method, and green synthesis, it is widely applicable for toxic waste remediation, antibacterial materials, removal of industrial pollutants and also used in anti-arthritic and anti-cancer activities. Prior reviews have laid focus on singular domains whereas our review clubs three major domains i.e., clinical, agricultural and environmental that are involved in the day to day life of plants as well as animals. Besides the above information, properties, synthesis, nanotoxicity and future perspectives of magnesium oxide nanoparticles have also been elaborated in this review.

The Loewner-Nirenberg problem in singular domains

We study asymptotic behaviors of solutions of the Loewner-Nirenberg problem in singular domains and prove that the solutions are well approximated by the corresponding solutions in tangent cones at singular points on the boundary. The conformal structure of the underlying equation plays an essential role in the derivation of the optimal estimates.

Multi-Scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains

In this paper, we propose multi-scale deep neural networks (MscaleDNNs) using the idea of radial scaling in frequency domain and activation functions with compact support. The radial scaling converts the problem of approximation of high frequency contents of PDEs' solutions to a problem of learning about lower frequency functions, and the compact support activation functions facilitate the separation of frequency contents of the target function to be approximated by corresponding DNNs. As a result, the MscaleDNNs achieve fast uniform convergence over multiple scales. The proposed MscaleDNNs are shown to be superior to traditional fully connected DNNs and be an effective mesh-less numerical method for Poisson-Boltzmann equations with ample frequency contents over complex and singular domains.

Hybrid sensitivity‐correlation regularisation matrix for electrical impedance tomography

In electrical impedance tomography, the primary task of image reconstruction process is to solve a discrete ill-posed inverse problem. The estimated solution is commonly obtained under a regularisation framework so that the noise amplified solution, which occurs during the matrix inversion process, can be avoided. The regularisation framework aims at balancing the model-data fitness while simultaneously constraining the solution space with additional prior (commonly prescribed through the regularisation matrix and solution-norm). In this study, a relationship between two robust regularisation matrices namely Newton one-step error reconstruction and fidelity-embedded regularisation is explicitly highlighted in both spatial and singular value decomposition domains. A hybrid regularisation matrix which encompasses the two prior knowledge, non-uniform sensitivity distribution and array response correlation, is then proposed as an alternative prior. Experimental results along with several evaluated performance parameters highlight the ability of the proposed prior to achieve a well-balanced and robust performance.

Numerical Solution to the 3D Static Maxwell Equations in Axisymmetric Singular Domains with Arbitrary Data

Abstract We propose a numerical method to solve the three-dimensional static Maxwell equations in a singular axisymmetric domain, generated by the rotation of a singular polygon around one of its sides. The mathematical tools and an in-depth study of the problem set in the meridian half-plane are exposed in [F. Assous, P. Ciarlet, Jr., S. Labrunie and J. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, J. Comput. Phys. 191 2003, 1, 147–176] and [P. Ciarlet, Jr. and S. Labrunie, Numerical solution of Maxwell’s equations in axisymmetric domains with the Fourier singular complement method, Differ. Equ. Appl. 3 2011, 1, 113–155]. Here, we derive a variational formulation and the corresponding approximation method. Numerical experiments are proposed, and show that the approach is able to capture the singular part of the solution. This article can also be viewed as a generalization of the Singular Complement Method to three-dimensional axisymmetric problems.

Analysis and boundary value problems on singular domains: An approach via bounded geometry

We prove well-posedness and regularity results for elliptic boundary value problems on certain singular domains that are conformally equivalent to manifolds with boundary and bounded geometry. Our assumptions are satisfied by the domains with a smooth set of singular cuspidal points, and hence our results apply to the class of domains with isolated oscillating conical singularities. In particular, our results generalize the classical L2-well-posedness result of Kondratiev for the Laplacian on domains with conical points. However, our domains and coefficients are too general to allow for singular function expansions of the solutions similar to the ones in Kondratiev's theory. The proofs are based on conformal changes of metric, on the differential geometry of manifolds with boundary and bounded geometry, and on our earlier geometric and analytic results on such manifolds.

Higgs bundles over cell complexes and representations of finitely presented groups

The purpose of this paper is to extend the Donaldson-Corlette theorem to the case of vector bundles over cell complexes. We define the notion of a vector bundle and a Higgs bundle over a complex, and describe the associated Betti, de Rham and Higgs moduli spaces. The main theorem is that the $SL(r, \mathbb{C})$ character variety of a finitely presented group $\Gamma$ is homeomorphic to the moduli space of rank $r$ Higgs bundles over an admissible complex $X$ with $\pi_1(X) = \Gamma$. A key role is played by the theory of harmonic maps defined on singular domains.

Association of Adverse Childhood Experiences with Co-occurring Health Conditions in Early Childhood.

To understand how adverse childhood experiences (ACEs) are associated with co-occurring physical, mental and developmental problems during early childhood. A subsample of 19,957 children aged 2-5 years were selected from the 2011-2012 National Survey for Child Health. Outcomes included 18 health conditions organized in singular condition domains (physical, mental, and developmental), and combinations of condition domains (e.g., physical plus mental, mental plus developmental, etc.). Predictors included 8 ACEs (divorce of a parent, death of a parent, exposure to domestic violence, living with someone with a drug or alcohol abuse problem, household member with a mental illness, parent incarceration, neighborhood violence, discrimination). Multivariable logistic regression was performed controlling for demographic characteristics, having a personal doctor, health insurance coverage, and seeing a health care professional in the previous year. Experiencing 3 or more ACEs before the age of 5 years was associated with increased likelihood of nearly every co-occurring condition combination across 3 domains of health. Most notably, experiencing 3 or more ACEs was also associated with a 2-fold increase in likelihood of having ≥1 physical condition and ≥1 developmental condition, a 9-fold increase in likelihood of having ≥1 mental and ≥1 developmental condition, and a 7-fold increase in likelihood of having ≥1 physical, ≥1 mental, and ≥1 developmental condition. This study demonstrates that we can identify the health effects of adversity quite early in development and that management should include communication between both health care and early childhood education providers.

Boundary behaviors for Liouville's equation in planar singular domains

We study asymptotic behaviors near the boundary of complete metrics of constant curvature in planar singular domains and establish an optimal estimate of these metrics by the corresponding metrics in tangent cones near isolated singular points on boundary. The conformal structure plays an essential role.

Strong well-posedness for a class of dynamic outflow boundary conditions for incompressible Newtonian flows

Based on energy considerations, we derive a class of dynamic outflow boundary conditions for the incompressible Navier–Stokes equations, containing the well-known convective boundary condition but incorporating also the stress at the outlet. As a key building block for the analysis of such problems, we consider the Stokes equations with such dynamic outflow boundary conditions in a halfspace and prove the existence of a strong solution in the appropriate Sobolev–Slobodeckij setting with \({L_p}\) (in time and space) as the base space for the momentum balance. For nonvanishing stress contribution in the boundary condition, the problem is actually shown to have \({L_p}\)-maximal regularity under the natural compatibility conditions. Aiming at an existence theory for problems in weakly singular domains, where different boundary conditions apply on different parts of the boundary such that these surfaces meet orthogonally, we also consider the prototype domain of a wedge with opening angle \({\frac{\pi}{2}}\) and different combinations of boundary conditions: Navier-Slip with Dirichlet and Navier-Slip with the dynamic outflow boundary condition. Again, maximal regularity of the problem is obtained in the appropriate functional analytic setting and with the natural compatibility conditions.

Application of the Micropolar Theory to the Strength Analysis of Bioceramic Materials for Bone Reconstruction

The application of the linear micropolar theory to the strength analysis of bioceramic materials for bone reconstruction is described. Micropolar elasticity allows better results to be obtained for microstructural and singular domains as compared to the classical theory of elasticity. The fundamental equations of the Cosserat continuum are cited. The description of FEM implementation of micropolar elasticity is given. The results of solving selected 3D test problems are presented. Comparison of classical and micropolar solutions is discussed.