Singular Functions Research Articles

Dependence of GaN Exciton Energy on Temperature

In this paper, we investigate the relationship between GaN exciton energy and temperature by using high-quality, strain-free GaN epilayers. Traditional models, such as Varshni’s model and the Bose–Einstein model, are primarily based on empirical fitting and give little or no consideration to electron–phonon interactions, which prevents them from accurately calculating GaN exciton energy over a wide temperature range. Considering the interaction of electrons and phonons, we use singular functions, linear functions and power functions to express the phonon density of GaN, and then 2BE, singular-linear, power-law-delta, and power-law-v models are proposed. All of them provide results that are more consistent with actual measurements compared to traditional models. Among them, the singular-linear model summarizes the contributions of acoustic and optical phonons. The error associated with the singular-linear model is smaller than that of the 1BE and Varshni models across nearly the entire temperature range. Therefore, the singular-linear model is a better choice.

Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part I: Formulation and Odd-Mode Perturbations

This article is Part I of our series of full papers on a gauge-invariant “linear” perturbation theory on the Schwarzschild background spacetime which was briefly reported in our short papers by the present author in 2021. We first review our general framework of the gauge-invariant perturbation theory, which can be easily extended to the “higher-order” perturbation theory. When we apply this general framework to perturbations on the Schwarzschild background spacetime, gauge-invariant treatments of l=0,1 mode perturbations are required. On the other hand, in the current consensus on the perturbations of the Schwarzschild spacetime, gauge-invariant treatments for l=0,1 modes are difficult if we keep the reconstruction of the original metric perturbations in our mind. Due to this situation, we propose a strategy of a gauge-invariant treatment of l=0,1 mode perturbations through the decomposition of the metric perturbations by singular harmonic functions at once and the regularization of these singularities through the imposition of the boundary conditions to the Einstein equations. Following this proposal, we derive the linearized Einstein equations for any modes of l≥0 in a gauge-invariant manner. We discuss the solutions to the odd-mode perturbation equations in the linearized Einstein equations and show that these perturbations include the Kerr parameter perturbation in these odd-mode perturbations, which is physically reasonable. In the Part II and Part III papers of this series of papers, we will show that the even-mode solutions to the linearized Einstein equations obtained through our proposal are also physically reasonable. Then, we conclude that our proposal of a gauge-invariant treatment for l=0,1-mode perturbations is also physically reasonable.

Correspondences Among Inner Functions, Functions with Non-Negative Real Parts and Conformal Mappings

We study an interesting family of dynamical systems on the set of the singular inner functions (defined on the unit disk). Starting with an inner function \( S_0(z) \), we obtain new singular inner functions \( S_1(z), S_2(z), \ldots \). This sequence converges to a holomorphic self-map of the unit disk which we call \( S \). The convergence is proved with the aid of a fixed-point theorem, a special case of the Earle-Hamilton Theorem. The function \( S \) itself is not a singular inner function as \( zS(z) \) is a conformal map. This conformal map has the surprising property that its inverse (which is a priori defined on a proper subset of the disk) extends to the entire disk. The motivating question for this research is whether \( z \) times a singular inner function can have an omitted value in the unit disk. This question appears within a book on the Krzyz problem written by the author. This question is still open.

An efficient method for identifying surface damage in hydraulic concrete buildings

Traditional hydraulic structures rely on manual visual inspection for apparent integrity, which is not only time-consuming and labour-intensive but also inefficient. The efficacy of deep learning models is frequently constrained by the size of available data, resulting in limited scalability and flexibility. Furthermore, the paucity of data diversity leads to a singular function of the model that cannot provide comprehensive decision support for improving maintenance measures. This paper proposes an efficacious methodology for identifying diverse apparent damages in hydraulic structures to address the limitations of existing technologies. The advanced features of apparent damage in hydraulic structures were elucidated by fine-tuning the top-level parameters of the lightweight pre-trained model, thereby mitigating the data dependency issue inherent in the model. Ensemble learning algorithms are employed to classify high-dimensional samples to enhance the accuracy and stability of the classification. However, ensemble learning algorithms are subject to time consuming issues when applied to high-dimensional datasets. To this end, we propose a robust discriminative feature selection model to identify the most salient features, thereby enhancing the performance of apparent damage recognition in hydraulic structures while concurrently reducing the inference time. The results demonstrated that the accuracies of this method in identifying crack, fracture, hole and normal structures were 87.65%, 87.82%, 96.99%, and 95.25%, respectively. This methodology exhibits significant applicability and practical value for the intelligent inspection of hydraulic structures.

Unveiling the effects of landscape-fire interactions on functional diversity in a Southern European mountain.

Climate and land-use changes are contributing to impacts on global ecosystem functioning. These effects are particularly severe in areas undergoing land abandonment and extreme wildfire events, such as the Mediterranean regions of the Iberian Peninsula. Previous studies have evaluated the impacts of land management on fire mitigation and biodiversity (species distribution and species richness), but how such strategies influence functional diversity remains unexplored. This study investigates how alternative land-fire management strategies may affect functional diversity. We modeled for 2050 for the Transboundary Biosphere Reserve Gerês-Xurés (Portugal-Spain). Land-use scenarios simulated processes of land abandonment ("business-as-usual"-BAU) and the implementation of EU rural policies ("high nature value farmlands"-HNVf), and were combined with three fire suppression levels. Species distribution models (102 vertebrates) were projected to each scenario, and functional diversity indices were consequently calculated. The highest functional richness was predicted for BAU scenarios, probably representing the benefits to unique species that deliver singular functions. The HNVf scenarios provided the highest functional divergence, probably indicating a high niche differentiation and low resource competition amongst agricultural communities. HNVf was the most beneficial scenario for ecosystem functioning, while fire suppression did not affect functional diversity. Despite the proneness to burn of our study area and the effects of firefighting on its fire regime, land-use policies are expected to have greater influence than fire suppression effects on functional diversity. These findings suggest that different facets of functional diversity will be unevenly influenced by fire-landscape dynamics driven by the land-use policies to be implemented in the upcoming decades.

SVTR-SRNet: A Deep Learning Model for Scene Text Recognition via SVTR Framework and Spatial Reduction Mechanism

Most deep learning models suffer from the problems of large computational complexity and insufficient feature extraction. To achieve a dynamic balance and tradeoff between computational complexity and performance, an enhanced SVTR-based scene text recognition model (SVTR-SRNet) was designed in this paper. In the SVTR-SRNet, we first created a bottom-up jump connection network that increases the number of information transfer pathways between the top and bottom features and improves the accuracy of information extraction. Second, we modified the attention mechanism by adding a new intermediate parameter called SR(Q) (Spatial Reduction (Q)), which finds a suitable compromise between the representational power and computing efficiency. In contrast to the conventional attention mechanism, the novel technique maintains the ability to model the global context while also enhancing efficiency. Ultimately, we developed a novel adaptive hybrid loss function to mitigate the shortcomings of a singular loss function’s inadequate generalization capacity and enhance the model’s resilience in handling a variety of challenging scenarios. Our technique outperforms existing standard models in terms of recognition performance on both the English and Chinese datasets, which deal with a high number of similar characters. As the model possesses great efficiency and outstanding cross-linguistic adaptability, it has a wide range of practical applications.

Periodic Boundary Value Problems On Thermoelastic Star Graphs And Their Solutions By Use General Function Method

This paper presents a comprehensive study on periodic boundary value problems (BVPs) in thermoelastic star graphs, utilizing the method of generalized functions. The research focuses on the behavior of rod structures subjected to thermal heating and cooling, providing a unified approach to solving various boundary value problems relevant to practical applications. We derive integral representations of generalized solutions that facilitate the determination of displacements, deformations, stress, temperature, and heat fluxes across each element of the graph. The study also addresses the modeling of force and heat sources through both regular and singular generalized functions under diverse boundary conditions. By establishing continuity and Kirchhoff conditions at the common node of the graph, we formulate the governing equations for the amplitudes of displacement and temperature. The findings highlight the versatility of the proposed method, which can be extended to a wide range of network structures, distinguishing it from existing techniques. This work contributes to the understanding of thermoelastic behavior in complex systems, with implications for engineering applications in fields such as mechanical design, materials science, and structural analysis.

Playful framings of social robots in dementia care: reconsidering the principle of transparency in interactions with robot animals

Abstract Research on social robots in dementia care has focused on their effects, for example in relation to the patients’ wellbeing or the care-givers’ working environment. Such approaches to social robots treat them as stable objects with a singular function. Combining social gerontology with social studies of science, the current study offers a new angle by asking: How do patients and care-givers in care homes for older people establish a shared definition of the situation in interactions involving robot animals? Drawing on ethnography and multimodal conversation analysis of 211 minutes of video recordings in two care homes in Sweden, we demonstrate the embodied work by which participants in interactions establish activities with robot animals. In contrast to the ideal of transparency in social robotics, we show that a central affordance of the robots is their vagueness, which allows for their inclusion in playful interactions. Playful framings of the robots highlight their social functions and downplay care-giver–patient asymmetries. However, situations where patients resist a playful frame actualise a dilemma of social inclusion, on the one hand, and the right to not participate in play, on the other. Showing this, the article contributes to knowledge on how people age with technology; in particular, it draws attention to the limits of an ideal of transparency when social robots are included in dementia care.

Time-fractional of cubic-quartic Schrödinger and cubic-quartic resonant Schrödinger equations with parabolic law: various optical solutions

Abstract Schrödinger's nonlinear equation is a fundamental model in fiber optics and many other areas of science. Using the Jacobi elliptic expansion function method, the time-fractional cubic-quartic nonlinear Schrödinger equation and cubic-quartic resonant nonlinear Schrödinger equation are investigated. By applying the effective Jacobi elliptic expansion function method, optical soliton solutions such as bright, dark, singular, periodic singular, exponential, and Jacobi elliptic function solutions have been obtained. The effect of the time-fractional derivative on the solutions is also revealed. Graphical representations are illustrated to showcase the physical properties of raised solutions, providing a comprehensive understanding of the solutions’ functionality.

A novel dimension reduction model based on POD and two-grid Crank–Nicolson mixed finite element methods for 3D nonlinear elastodynamic sine–Gordon problem

Earthquake, petroleum and gas extraction, and underground nuclear test all could cause nonlinear tectonic deformation. It is of great significance to effectively predict and evaluate the disasters and impacts of these geological structure deformation. In this end, a three-dimension (3D) nonlinear elastodynamic sine–Gordon model that can be used to describe nonlinear tectonic deformation including 3D displacement vector, 3 × 3 symmetric stress tensor matrix, nonlinear sin term, and singular initial value functions is first proposed. Then, a new time semi-discrete mixed Crank–Nicolson (TSDMCN) scheme for the 3D nonlinear elastodynamic sine–Gordon model is developed, and the existence, stability, and error estimates of the TSDMCN solutions are proved. Next, a new two-grid mixed finite element Crank–Nicolson (TGMFECN) method with unconditional stability for the 3D nonlinear elastodynamic sine–Gordon model is developed, and the existence, stability, and error estimates of the TGMFECN solutions are proved. Thenceforth, it is the most important thing is that a novel reduced-dimensional iterative TGMFECN (RDITGMFECN) method in matrix form is established by resorting to proper orthogonal decomposition only to lower the unknown TGMFECN solution coefficient vectors and keep TGMFECN basis functions unchanged, which can ensure that the RDITGMFECN method has the same accuracy as the usual TGMFECN method, but can greatly lower the dimension of the unknown TGMFECN solution coefficient vectors so as to mitigate calculated workload, save CPU operating-time, improve computing efficiency, and improve real-time calculating accuracy. In theory, the existence, stability, and error estimates of RDITGMFECN solutions are demonstrated by matrix analysis such that the theoretical analysis becomes very intuitive and easy to be understood by the public, which is a new attempt of theoretical analysis. In application, two numerical examples are used to simulate the 3D nonlinear tectonic deformation caused by earthquake and to verify the correctness of our theoretical results and the effectiveness of the RDITGMFECN method.

Numerical simulation of the two-dimensional fractional Schrödinger equation for describing the quantum dynamics on a comb with the absorbing boundary conditions

This paper mainly contributes to investigating a distinctive fractional Schrödinger equation (FSE) for describing the quantum dynamics on a comb. The special characteristic of the comb is that the diffusion versus the x-direction only occurs on the x-axis. We replace the infinite boundary with the absorbing boundary conditions (ABCs), which can be obtained using the Laplace transform. We utilize the integration property to deal with the singularity function in the FSE and propose the finite difference method. The stability and convergence of this scheme are investigated. The L1 scheme to discretize the Caputo fractional derivative needs a particularly expensive computational and storage cost. To increase the calculation speed, a fast algorithm is used to improve the operation efficiency. By introducing a source term, the exact solution and the numerical solution are compared. The comparison of the CPU time for the L1 scheme and fast scheme is given. The comparison between the FSE on a comb and the classical one is also analyzed and discussed. The transport process versus the x-axis and the y-axis with various parameters are shown. Finally, the mean square displacement (MSD) is investigated. We find that the solution with the ABCs has a better agreement with the exact expression compared to the solution with truncated zero boundary conditions.

Identification and Spatiotemporal Evolution Analysis of the Urban–Rural Fringe in Polycentric Cities Based on K-Means Clustering and Multi-Source Data: A Case Study of Chengdu City

Urban fringe areas, serving as transitional zones between urban and rural landscapes, are characterized by their transitional nature, high dynamics, and spatial heterogeneity. Identifying the extent of an urban–rural fringe (URF) and analyzing its evolutionary characteristics are crucial for urban planning and development. However, limited research exists regarding the identification of a URF and the analysis of its spatiotemporal evolution in polycentric cities. Using Chengdu as a case study, this research employed the K-means clustering method to identify the spatial extent and evolution patterns of the URF in Chengdu from 2010 to 2020 based on the spatiotemporal characteristics of multi-source data. The results indicate that (1) the K-means clustering method can reasonably and efficiently identify URF in polycentric cities; (2) Chengdu exhibited a polycentric urban structure with a “main center-subcenter” pattern, where the URF was adjacent to the main and subcenters, assuming an overall annular wedge shape; (3) there was a significant expansion of the URF in the northeast–southwest direction from 2010 to 2020, accompanied by substantial land use changes. The evolution of the URF was driven by the dual mechanisms of urban suburbanization and rural urbanization, exhibiting characteristics such as singular urban functions, dispersed and chaotic land use, fragmented landscapes, and increasing complexity. This study extended the research on URFs, aiding in the understanding of urban spatial growth patterns and providing decision support for the integrated development of urban and rural areas.

Hausdorff dimension of limsup sets of isotropic rectangles in Heisenberg groups

A formula for the Hausdorff dimension of typical limsup sets generated by randomly distributed isotropic rectangles in Heisenberg groups is derived in terms of directed singular value functions.

An anthropomorphic diagnosis system of pulmonary nodules using weak annotation-based deep learning

The accurate categorization of lung nodules in CT scans is an essential aspect in the prompt detection and diagnosis of lung cancer. The categorization of grade and texture for nodules is particularly significant since it can aid radiologists and clinicians to make better-informed decisions concerning the management of nodules. However, currently existing nodule classification techniques have a singular function of nodule classification and rely on an extensive amount of high-quality annotation data, which does not meet the requirements of clinical practice. To address this issue, we develop an anthropomorphic diagnosis system of pulmonary nodules (PN) based on deep learning (DL) that is trained by weak annotation data and has comparable performance to full-annotation based diagnosis systems. The proposed system uses DL models to classify PNs (benign vs. malignant) with weak annotations, which eliminates the need for time-consuming and labor-intensive manual annotations of PNs. Moreover, the PN classification networks, augmented with handcrafted shape features acquired through the ball-scale transform technique, demonstrate capability to differentiate PNs with diverse labels, including pure ground-glass opacities, part-solid nodules, and solid nodules. Through 5-fold cross-validation on two datasets, the system achieved the following results: (1) an Area Under Curve (AUC) of 0.938 for PN localization and an AUC of 0.912 for PN differential diagnosis on the LIDC-IDRI dataset of 814 testing cases, (2) an AUC of 0.943 for PN localization and an AUC of 0.815 for PN differential diagnosis on the in-house dataset of 822 testing cases. In summary, our system demonstrates efficient localization and differential diagnosis of PNs in a resource limited environment, and thus could be translated into clinical use in the future.

Effects of Arthrospira platensis on Human Umbilical Vein Endothelial Cells

Atherosclerosis is initiated by injury or damage to the vascular endothelial cell monolayer. Therefore, the early repair of the damaged vascular endothelium by a proliferation of neighbouring endothelial cells is important to prevent atherosclerosis and thrombotic events. Arthrospira platensis (AP) has been used as a dietary supplement, mainly due to its high content of vitamins, minerals, amino acids, and pigments such as chlorophylls, carotenoids, and phycocyanin, ingredients with antioxidant, anti-inflammatory, and anti-thrombotic properties. Therefore, in this prospective, placebo-controlled, data-driven, sample-size-estimated in vitro study, we tested whether an aqueous extract of AP at different concentrations (50, 100, and 200 µg/mL) had an effect on the different cellular parameters of human umbilical vein endothelial cells. Therefore, cell impedance measurement and cell proliferation were measured to investigate the monolayer formation. In addition, cell viability, integrity, and metabolism were analysed to evaluate singular cellular functions, especially the antithrombotic state. Furthermore, cell–cell and cell–substrate interactions were observed. The highest proliferation was achieved after the addition of 100 µg/mL. This was consistently confirmed by two independent optical experiments in cell cultures 48 h and 85 h after seeding and additionally by an indirect test. At this concentration, the activation or dysfunction of HUVECs was completely prevented, as confirmed by prostacyclin and interleukin-6 levels. In conclusion, in this study, AP induced a significant increase in HUVEC proliferation without inducing an inflammatory response but altered the hemostasiological balance in favour of prostacyclin over thromboxane, thereby creating an antithrombotic state. Thus, APE could be applied in the future as an accelerator of endothelial cell proliferation after, e.g., stent placement or atherosclerosis.

Thermal Buckling and Postbuckling Analysis of Cracked FG-GPL RC Plates Using a Phase-Field Crack Model

A phase-field crack model is developed for numerical analysis of thermal buckling and postbuckling behavior of a functionally graded (FG) graphene platelet-reinforced composite (FG-GPLRC) plate with a central crack. The inclined central crack is represented according to a stable, effective phase-field formulation (PFF) by introducing a virtual crack rotation. The problem is formulated using first-order shear deformation theory (SDT) incorporated with von Kármán geometric nonlinearity. And it is approximated by combining regular Laplace interpolation functions and crack-tip singular functions in the framework of the 2D extended natural element method (XNEM). Troublesome shear locking is suppressed by applying the concept of the MITC (mixed-interpolated tensorial components)3+ shell element to the present numerical method. The results demonstrate the effectiveness of this method in accurately predicting the critical buckling temperature rise (CBTR) and the thermal postbuckling path. In addition, the parametric results reveal that the CBTR and postbuckling path of the FG-GPLRC plate are distinct from those of the FG carbon nanotube-reinforced composite (FG-CNTRC) plate and remarkably affected by the parameters associated with the crack and graphene platelet (GPL).

Study on the load capacity of prefabricated buckling-restrained braces with ductile steel castings based on singular function

Prefabricated buckling-restrained braces with ductile steel castings were used in the special concentrically braced frames to enable brace yield without buckling and prevent brittle failure of brace joints. This study presents a simplified three-segment calculation model subjected to axial force, based on the failure mode of the bracing system, utilizing singular function to derive formulas for the ultimate capacity of the braces. The accuracy of the computational formula was validated through numerical simulations and experimental findings. The outcomes indicate that the core unit of the brace undergoes third-order buckling deformation under axial force. The ultimate capacity of the braces is observed to increase with the cross-sectional area of the core unit but decrease with the length of the ductile steel castings. The theoretical values of the ultimate bearing capacity of the bracing system closely align with numerical simulations and experimental results, with error margins of 2.9 % to 9.3 % and 3.2 % to 4.6 %, respectively. These findings offer a theoretical foundation for the stability assessment of such brace configurations.

Edge-featured multi-hop attention graph neural network for intrusion detection system

With the development of the Internet, the application of computer technology has rapidly become widespread, driving the progress of Internet of Things (IoT) technology. The attacks present on networks have become more complex and stealthy. However, traditional network intrusion detection systems with singular functions are no longer sufficient to meet current demands. While some machine learning-based network intrusion detection systems have emerged, traditional machine learning methods cannot effectively respond to the complex and dynamic nature of network attacks. Intrusion detection systems utilizing deep learning can better enhance detection capabilities through diverse data learning and training. To capture the topological relationships in network data, using graph neural networks (GNNs) is most suitable. Most existing GNNs for intrusion detection use multi-layer network training, which may lead to over-smoothing issues. Additionally, current intrusion detection solutions often lack efficiency. To mitigate the issues mentioned above, this paper proposes an Edge-featured Multi-hop Attention Graph Neural Network for Intrusion Detection System (EMA-IDS), aiming to improve detection performance by capturing more features from data flows. Our method enhances computational efficiency through attention propagation and integrates node and edge features, fully leveraging data characteristics. We carried out experiments on four public datasets, which are NF-CSE-CIC-IDS2018-v2, NF-UNSW-NB15-v2, NF-BoT-IoT, and NF-ToN-IoT. Compared with existing models, our method demonstrated superior performance.

Optical soliton solutions for the nonlinear Schrödinger equation with higher-order dispersion arise in nonlinear optics

Optical solitons and traveling wave solutions for the higher-order dispersive extended nonlinear Schrödinger equation are studied. Ultrashort pulse propagation in optical communication networks is described by this equation. To find exact solutions to the model, the unified Riccati equation expansion method and the Jacobi elliptic function expansion method are successfully applied. The optical solutions includes various solitary wave solutions, such as dark, bright, combined dark-bright, singular, combined periodic, periodic, Jacobian elliptic, and rational functions. Three-dimensional and two-dimensional graphs of solutions are presented. Also, the dynamical behavior of waves and the impact of time on solutions by selecting appropriate parameters are illustrated.

Green’s Tensor of BIOT’s Equations by Stationary Oscillations

Various mathematical models of deformable solids mechanics are used in the study of seismic processes in the earth's crust. The processes of waves propagation are most studied in elastic media. But these models do not take into account many real properties of the ambient array. These are, for example, the presence of groundwater, which complicates the construction and operation of surface and underground structures, affect the magnitude and distribution of stresses. Models, which take into account the water saturation form the earth's crust structures, the presence of gas bubbles, etc., are multi-component medium. A variety of multicomponent media, the complexity of the processes associated with their deformation, lead to a large difference in the methods of analysis and modelling used in the solution of wave problems. In this paper the processes of wave propagation in a two-component Biot medium under the action of periodic forces of various forms are considered. Using the Fourier transform of generalized functions, fundamental solutions are constructed - the Green's tensor of the Biot equations and its properties are studied. This tensor describes the process of propagation of harmonic waves of a fixed frequency in spaces of dimension N = 1,2,3 under the action of power sources concentrated at the origin of coordinates, described by a singular delta function. On its basis, generalized solutions of these equations are constructed under the action of various sources of periodic disturbances, which are described by both regular and singular generalized functions. For regularly acting forces, integral representations of solutions are given, which can be used to calculate the stress-strain state of a porous water-saturated medium