Boundary Problem Research Articles

Regularity for one-phase Bernoulli problems with discontinuous weights and applications

We study a one-phase Bernoulli free boundary problem with weight function admitting a discontinuity along a smooth jump interface. In any dimension N ≥ 2 N\ge 2 , we show the C 1 , α C^{1, \alpha } regularity of the free boundary outside of a singular set of Hausdorff dimension at most N − 3 N-3 . In particular, we prove that the free boundaries are C 1 , α C^{1, \alpha } regular in dimension N = 2 N=2 , while in dimension N = 3 N=3 the singular set can contain at most a finite number of points. We use this result to construct singular free boundaries in dimension N = 2 N=2 , which are minimizing for one-phase functionals with weight functions in L ∞ L^\infty that are arbitrarily close to a positive constant.

Некоторые направления совершенствования правоприменительной деятельности в сфере расследования экономических преступлений

The article analyzes some areas of improvement of investigation of economic crimes. In particular, the problem of boundaries of investigator's discretion is viewed as one of important problems in the economic crime investigation sphere. The evidence reliability problem is also studied. Besides, the need to establish special investigative subdivisions in accordance with regional demands is singled out as an organizational problem.

A Penalized Empirical Likelihood Approach for Estimating Population Sizes under the Negative Binomial Regression Model

In capture–recapture experiments, the presence of overdispersion and heterogeneity necessitates the use of the negative binomial regression model for inferring population sizes. However, within this model, existing methods based on likelihood and ratio regression for estimating the dispersion parameter often face boundary and nonidentifiability issues. These problems can result in nonsensically large point estimates and unbounded upper limits of confidence intervals for the population size. We present a penalized empirical likelihood technique for solving these two problems by imposing a half-normal prior on the population size. Based on the proposed approach, a maximum penalized empirical likelihood estimator with asymptotic normality and a penalized empirical likelihood ratio statistic with asymptotic chi-square distribution are derived. To improve numerical performance, we present an effective expectation-maximization (EM) algorithm. In the M-step, optimization for the model parameters could be achieved by fitting a standard negative binomial regression model via the R basic function glm.nb(). This approach ensures the convergence and reliability of the numerical algorithm. Using simulations, we analyze several synthetic datasets to illustrate three advantages of our methods in finite-sample cases: complete mitigation of the boundary problem, more efficient maximum penalized empirical likelihood estimates, and more precise penalized empirical likelihood ratio interval estimates compared to the estimates obtained without penalty. These advantages are further demonstrated in a case study estimating the abundance of black bears (Ursus americanus) at the U.S. Army’s Fort Drum Military Installation in northern New York.

Asymptotic stability of rarefaction wave with non-slip boundary condition for radiative Euler flows

This paper is devoted to studying the initial-boundary value problem for the radiative full Euler equations, which are a fundamental system in the radiative hydrodynamics with many practical applications in astrophysical and nuclear phenomena, with the non-slip boundary condition on an impermeable wall. Due to the difficulty from the disappearance of the velocity on the impermeable boundary, quite few results for compressible Navier-Stokes equations and no result for the radiative Euler equations are available at this moment. So the asymptotic stability of the rarefaction wave proven in this paper is the first rigorous result on the global stability of solutions of the radiative Euler equations with the non-slip boundary condition. It also contributes to our systematical study on the asymptotic behaviors of the rarefaction wave with the radiative effect and different boundary conditions such as the inflow/outflow problem and the impermeable boundary problem in our series papers including [5,6].

Boundary problems of sequential fractional differential equations having a monomial coefficient

The article examines sequential fractional-order differential equations composed of monomial coefficients defined as nonlinear boundary problems. The original problem is converted into an integral equation in an equivalent form. The contraction mapping principle and Krasnoselskii's fixed-point theorem are implemented to obtain two existence conditions. The problem of Ulam-Hyers stability is also investigated. An illustration is presented to showcase practical application of the obtained results.

Multiple solutions to nonlocal Neumann boundary problem with sign‐changing coefficients

We consider the following Neumann boundary problem with a nonlocal Kirchhoff term where is a bounded smooth domain and and are sign‐changing coefficients which play important impact on the study. By using variational method and dividing corresponding Nehari manifold into two disjoint subsets, we study the existence of two positive solutions. To show the multiplicity of positive solutions, we find a set whose elements can be mapped to two subsets of Nehari manifold, which is also the key to proving strong convergence for minimizing sequences. Finally, the asymptotic behavior of solutions is studied as parameter tends to 0. Different from Dirichlet boundary problem, we show that the two solutions of Neumann problem tend to two different positive constants, respectively.

Upper and lower solutions for an integral boundary problem with two different orders (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\left ( p,q\\right ) $\\end{document}-fractional difference

In this paper, a (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\left ( p,q\\right ) $\\end{document}-fractional nonlinear difference equation of different orders is considered and discussed. With the help of (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\left ( p,q\\right ) $\\end{document}-calculus for integrals and derivatives properties, we convert the main integral boundary value problem (IBVP) to an equivalent solution in the form of an integral equation, we use the upper–lower solution technique to prove the existence of positive solutions. We present an example of the IBVP to apply and demonstrate the results of our method.

NUMERICAL SIMULATION OF UNSTEADY SIGNAL TRANSDUCTION DURING THE FORMATION OF INVADOPODIA USING LEVEL SET METHOD

The finger-like protrusions formed by an invasive cancer cell known as invadopodia is actively observed recently because of this formation on the cancer invasion. The signal on the interface which is stimulated upon contact between epidermal growth factor receptor and ligand, is investigated to be the start of invadopodia formation. In this research, a model of invadopodia formation with signal variable is formulated in two dimensions. The plasma membrane is assumed to be free boundary. The signal is in an unsteady state. The equation of signal is represented by heat-like equation with time-dependent boundary condition. Trigonometry types of the boundary condition which are sine and cosine function are tested to identify the most suitable boundary condition represented the free boundary. The plasma membrane is considered as zero level set function. The membrane moves by the velocity of the signal inside the cell. To handle the free boundary problem, the level set method combining features of ghost method, interpolation and extrapolation method is applied to solve the model numerically. Our result shows that the free boundary is moved at different positions and seen to move inward meaning that the boundary has shrunk. Cosine function is discovered to fit the boundary conditions since the boundary solutions are stable across the time. The computation of the signal density profiles displayed highest density on the membrane.

Analysis of Fault Events in Rail Transit Vehicle Traction Systems Based on Knowledge Graph Reasoning

Fault text records provide detailed information on faults and handling steps, which are valuable for fault analysis. However, the different individuals’ recording styles can lead to ambiguities, and it is challenging to uncover potential fault associations in complex systems. To address these issues, this paper proposes a novel method for fault information extraction and analysis. Firstly, to tackle the problem of ambiguous boundaries between entities in fault texts, an integration algorithm is employed to accurately recognize fault entities considering contextual semantic features to establish fault knowledge graph (FKG). Then, a Relational Graph Convolutional Networks (RGCN) is improved with Graph Attention Networks (GAT) for sparse nodes caused by specific types of faults, to dynamically adjust the weight distribution of node learning, inferring potential links within the graph. The proposed method was validated using actual fault records from the traction system of rail transit vehicles, and contributes a reference for the mining and analysis of fault records in complex systems.

Mathematical Framework to Model Electrodeposition/Stripping in Li-Metal Batteries during Cycling

Li-metal batteries, often heralded as the next frontier in energy storage technology, boast remarkable characteristics such as high energy density. Unlike traditional lithium-ion batteries, Li-metal batteries incorporate a lithium-metal anode, which significantly enhances their energy density. This higher energy density translates to a substantial increase in the amount of energy that can be stored in a given volume or weight, paving the way for more powerful and longer-lasting energy solutions[1–3]. However, significant shortcomings do exist, such as uneven Li-Metal deposition, dendrite formation, dead lithium, etc.[4] These phenomena are different than what are usually observed in Li-ion batteries with intercalation chemistry and lead to issues such as poor cycle life, low coulumbic efficiency, and safety concerns.[3]Mathematical modeling is a promising approach to studying the physicochemical processes involved in the Li-metal batteries to understand the morphology changes during cycling. The plating and stripping of the lithium metal on the anode during cycling can be modeled as a moving boundary problem.[5] Since the process involves a shrinking and expanding electrolyte domain, moving boundary problems call for the fastidious implementation of the model equations and boundary conditions.[6]In this talk, we will show the general mathematical framework involved in modeling Li-metal batteries in 1D systems. It expands upon the importance of carefully implementing boundary conditions to tackle the mass conservation issue that arises while cycling the model. It will also touch upon the additional complexities associated with expanding the model to 2D systems with convection in the electrolyte phase.

Weakly supervised semantic segmentation of leukocyte images based on class activation maps.

Leukocytes are an essential component of the human defense system, accurate segmentation of leukocyte images is a crucial step towards automating detection. Most existing methods for leukocyte images segmentation relied on fully supervised semantic segmentation (FSSS) with extensive pixel-level annotations, which are time-consuming and labor-intensive. To address this issue, this paper proposes a weakly supervised semantic segmentation (WSSS) approach for leukocyte images utilizing improved class activation maps (CAMs). Firstly, to alleviate ambiguous boundary problem between leukocytes and background, preprocessing technique is employed to enhance the image quality. Secondly, attention mechanism is added to refine the CAMs generated by improving the matching of local and global features. Random walks, dense conditional random fields and hole filling were leveraged to obtain final pseudo-segmentation labels. Finally, a fully supervised segmentation network is trained with pseudo-segmentation labels. The method is evaluated on BCCD and TMAMD datasets. Experimental results demonstrate that by employing the pseudo segmentation annotations generated through this method can be utilized to train UNet as close as possible to FSSS. This method effectively reduces manual annotation cost while achieving WSSS of leukocyte images.

Maximum and Minimum Results for the Green’s Functions in Delta Fractional Difference Settings

The present paper is dedicated to the examination of maximum and minimum results based on Green’s functions via delta fractional differences for a class of fractional boundary problems. For such a purpose, we built the corresponding Green’s functions based on the falling factorial functions. In addition, using the constructed Green’s function, the positivity of the function and its corresponding delta function are presented. We also verified the occurrence of two distinct functions with the same Green’s function. The maximality and minimality of the Green’s function show a good qualitative agreement. Finally, we considered some special examples to explain the obtained results.

Numerical Analysis of the Cylindrical Shell Pipe with Preformed Holes Subjected to a Compressive Load Using Non-Uniform Rational B-Splines and T-Splines for an Isogeometric Analysis Approach

In this paper, we implement the finite detail technique primarily based on T-Splines for approximating solutions to the linear elasticity equations in the connected and bounded Lipschitz domain. Both theoretical and numerical analyses of the Dirichlet and Neumann boundary problems are presented. The Reissner–Mindlin (RM) hypothesis is considered for the investigation of the mechanical performance of a 3D cylindrical shell pipe without and with preformed hole problems under concentrated and compression loading in the linear elastic behavior for trimmed and untrimmed surfaces in structural engineering problems. Bézier extraction from T-Splines is integrated for an isogeometric analysis (IGA) approach. The numerical results obtained, particularly for the displacement and von Mises stress, are compared with and validated against the literature results, particularly with those for Non-Uniform Rational B-Spline (NURBS) IGA and the finite element method (FEM) Abaqus methods. The obtained results show that the computation time of the IGA based on the T-Spline method is shorter than that of the IGA NURBS and FEM Abaqus/CAE (computer-aided engineering) methods. Furthermore, the highlighted results confirm that the IGA approach based on the T-Spline method shows more success than numerical reference methods. We observed that the NURBS IGA method is very limited for studying trimmed surfaces. The T-Spline method shows its power and capability in computing trimmed and untrimmed surfaces.

PlaneSeg: Building a Plug-In for Boosting Planar Region Segmentation.

Existing methods in planar region segmentation suffer the problems of vague boundaries and failure to detect small-sized regions. To address these, this study presents an end-to-end framework, named PlaneSeg, which can be easily integrated into various plane segmentation models. Specifically, PlaneSeg contains three modules, namely, the edge feature extraction module, the multiscale module, and the resolution-adaptation module. First, the edge feature extraction module produces edge-aware feature maps for finer segmentation boundaries. The learned edge information acts as a constraint to mitigate inaccurate boundaries. Second, the multiscale module combines feature maps of different layers to harvest spatial and semantic information from planar objects. The multiformity of object information can help recognize small-sized objects to produce more accurate segmentation results. Third, the resolution-adaptation module fuses the feature maps produced by the two aforementioned modules. For this module, a pairwise feature fusion is adopted to resample the dropped pixels and extract more detailed features. Extensive experiments demonstrate that PlaneSeg outperforms other state-of-the-art approaches on three downstream tasks, including plane segmentation, 3-D plane reconstruction, and depth prediction. Code is available at https://github.com/nku-zhichengzhang/PlaneSeg.

NeighborNet: Learning Intra- and Inter-Image Pixel Neighbor Representation for Breast Lesion Segmentation.

Breast lesion segmentation from ultrasound images is essential in computer-aided breast cancer diagnosis. To alleviate the problems of blurry lesion boundaries and irregular morphologies, common practices combine CNN and attention to integrate global and local information. However, previous methods use two independent modules to extract global and local features separately, such feature-wise inflexible integration ignores the semantic gap between them, resulting in representation redundancy/insufficiency and undesirable restrictions in clinic practices. Moreover, medical images are highly similar to each other due to the imaging methods and human tissues, but the captured global information by transformer-based methods in the medical domain is limited within images, the semantic relations and common knowledge across images are largely ignored. To alleviate the above problems, in the neighbor view, this paper develops a pixel neighbor representation learning method (NeighborNet) to flexibly integrate global and local context within and across images for lesion morphology and boundary modeling. Concretely, we design two neighbor layers to investigate two properties (i.e., number and distribution) of neighbors. The neighbor number for each pixel is not fixed but determined by itself. The neighbor distribution is extended from one image to all images in the datasets. With the two properties, for each pixel at each feature level, the proposed NeighborNet can evolve into the transformer or degenerate into the CNN for adaptive context representation learning to cope with the irregular lesion morphologies and blurry boundaries. The state-of-the-art performances on three ultrasound datasets prove the effectiveness of the proposed NeighborNet.

Classical solvability to the free boundary problem for a foam drainage equation. II. From the interface to the bottom

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. in 1988 in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region below an interface to the bottom in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle. Moreover, the existence of its steady solution and its stability are shown.

Classical solvability to the free boundary problem for a foam drainage equation. I. From the top to the interface

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. [Izv. Akad. Nauk SSSR Mekh. Ghidk. Gaza 2, 103–108 (1988)] in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region from the top to the interface in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle and the comparison theorem. Moreover, the existence of its steady solution and its stability are discussed.

Optimal drift rate control and two-sided impulse control for a Brownian system with the long-run average criterion

Abstract In this paper, we consider a joint drift rate control and two-sided impulse control problem in which the system manager adjusts the drift rate as well as the instantaneous relocation for a Brownian motion, with the objective of minimizing the total average state-related cost and control cost. The system state can be negative. Assuming that instantaneous upward and downward relocations take a different cost structure, which consists of both a setup cost and a variable cost, we prove that the optimal control policy takes an $\left\{ {\!\left( {{s^{\ast}},{q^{\ast}},{Q^{\ast}},{S^{\ast}}} \right),\!\left\{ {{\mu ^{\ast}}(x)\,:\,x \in [ {{s^{\ast}},{S^{\ast}}}]} \right\}} \right\}$ form. Specifically, the optimal impulse control policy is characterized by a quadruple $\left( {{s^{\ast}},{q^{\ast}},{Q^{\ast}},{S^{\ast}}} \right)$ , under which the system state will be immediately relocated upwardly to ${q^{\ast}}$ once it drops to ${s^{\ast}}$ and be immediately relocated downwardly to ${Q^{\ast}}$ once it rises to ${S^{\ast}}$ ; the optimal drift rate control policy will depend solely on the current system state, which is characterized by a function ${\mu ^{\ast}}\!\left( \cdot \right)$ for the system state staying in $[ {{s^{\ast}},{S^{\ast}}}]$ . By analyzing an associated free boundary problem consisting of an ordinary differential equation and several free boundary conditions, we obtain these optimal policy parameters and show the optimality of the proposed policy using a lower-bound approach. Finally, we investigate the effect of the system parameters on the optimal policy parameters as well as the optimal system’s long-run average cost numerically.

A finite volume formulation for one-dimensional Stefan problems

This paper presents a numerical formulation that combines the finite-volume method with a time-dependent boundary-fitted coordinate system to tackle the moving boundary problems for the one-dimensional (1-D) heat equation. Following the formulation process, an algorithm was created with the help of the linear algebra package Eigen to generate the data for the numerical solution. Four benchmark cases of heat conduction were investigated by this approach and compared to the analytical solution: a fixed boundary case, the liquid phase of a melting process, the solid phase of a melting process, and an ablating process. The results demonstrate the capability of the presented numerical formulation in efficiently solving the general 1-D heat problems, especially the one-phase Stefan problems, as well as the consistency of the numerical and analytical solutions. Future works can strive to extend the formulations to Stefan problems of two phases or higher dimensions. The work can then be extended to real-life applications, such as the spacecraft re-entry ablation process, and the process of iceberg melting, demonstrating its critical value in solving practical problems.

Self-Supervised Contrastive Learning for Automated Segmentation of Brain Tumor MRI Images in Schizophrenia

Schizophrenic patients’ brain tumor magnetic resonance imaging (MRI) images are important references for doctors to diagnose and treat schizophrenia. However, automatic segmentation of these images is a professional and tedious task. Existing methods suffer from problems such as large model parameters, long computation time, and inadequate image processing. To achieve more accurate segmentation of brain tumors, we propose brain tumor MRI images for automatic segmentation using self-supervised contrastive learning in schizophrenia patients (BTCSSSP). First, a denoising algorithm based on progressive principal component analysis approximation and adaptive clustering is designed to process the noisy MRI images. Second, a brightness-aware image enhancement algorithm is developed to address the problems of non-uniformity, unclear boundaries, and poor spatial resolution of the MRI images. Finally, a cross-scale U-Net network with selective feature fusion attention module is designed based on self-supervised contrastive learning to achieve automatic segmentation of brain tumor MRI images. The results show that the BTCSSSP method yields higher Recall and Precision than existing methods. The maximum recall is 95%, and the image segmentation precision is 95%, thus indicating good practical applicability.