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In a previous paper (Feng et al., J. Differential Equations (2023)), we studied a seasonally interactive model between closed seasons and open seasons with Michaelis-Menten type harvesting, in which we assumed that the harvesting quantity was relatively large (0 < κ = l G c E < 1) and obtained a length threshold of the closed season T ¯ ⁎ depending on the harvesting parameter and the seasonal fluctuation period. It was shown that the origin is globally asymptotically stable if and only if T ¯ ≤ T ¯ ⁎ , and there exists a unique globally asymptotically stable T -periodic solution if and only if T ¯ > T ¯ ⁎. In this paper, we continue to investigate the periodic dynamics of this model when the harvesting quantity is relatively small (κ = l G c E ≥ 1). By finding another smaller length threshold T ¯ ⁎ ∈ (0 , T ¯ ⁎) , we determine the number of periodic solutions and study their stability, which imply the occurrence of bifurcation of periodic solutions. Our results demonstrate that designing the closed harvesting season properly can prevent the species from extinction. Moreover, by comparing with the continuous harvesting model and combining the results in our previous paper and this article, we provide a complete understanding on the global dynamics for the periodic switching model. Some numerical simulations are also carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the sharp estimate for the lowest positive eigenvalue for the Camassa-Holm equation y ″ = 1 4 y + λ m (t) y , with the Neumann-Dirichlet boundary conditions, where potential m admits to change sign. We first study the optimal lower bound for the smallest positive eigenvalue in the measure differential equations. Then based on the relationship between the minimization problem of the smallest positive eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest positive eigenvalue for this indefinite Camassa-Holm equation. [ABSTRACT FROM AUTHOR]

In this paper, we construct higher-order modular differential equations for elliptic modular forms, holomorphic Jacobi forms, and skew-holomorphic Jacobi forms by using the Rankin-Cohen brackets extended to the Eisenstein series of weight two. Rankin-Cohen brackets are typical tools in the construction of modular forms by differential operators. We introduce the extended Rankin-Cohen brackets for holomorphic or skew-holomorphic Jacobi forms paired with elliptic modular forms. They provide a unified method of construction of higher-order modular differential equations for the modular forms above. In particular, with these extended Rankin-Cohen brackets, we can reproduce the modular differential equations of previous studies. [ABSTRACT FROM AUTHOR]

This paper aims to present a comprehensive framework for coupling tumor-bone remodeling processes in a 2-dimensional geometry. This is achieved by introducing a bio-inspired damage that represents the growing tumor, which subsequently affects the main populations involved in the remodeling process, namely, osteoclasts, osteoblasts, and bone tissue. The model is constructed using a set of differential equations based on the Komarova's and Ayati's models, modified to incorporate the bio-inspired damage that may result in tumor mass formation. Three distinct models were developed. The first two models are based on the Komarova's governing equations, with one demonstrating an osteolytic behavior and the second one an osteoblastic model. The third model is a variation of Ayati's model, where the bio-inspired damage is induced through the paracrine and autocrine parameters, exhibiting an osteolytic behavior. The obtained results are consistent with existing literature, leading us to believe that our in-silico experiments will serve as a cornerstone for paving the way towards targeted interventions and personalized treatment strategies, ultimately improving the quality of life for those affected by these conditions. [ABSTRACT FROM AUTHOR]

The famous J.C.P. Miller formula provides a recurrence algorithm for the composition B a ∘ f , where B a is the formal binomial series and f is a formal power series, however it requires that f has to be a nonunit. In this paper we provide the general J.C.P. Miller formula which eliminates the requirement of nonunitness of f and, instead, we establish a necessary and sufficient condition for the existence of the composition B a ∘ f. We also provide the general J.C.P. Miller recurrence algorithm for computing the coefficients of that composition, if B a ∘ f is well defined, obviously. Our generalizations cover both the case in which f is a one–variable formal power series and the case in which f is a multivariable formal power series. In the central part of this article we state, using some combinatorial techniques, the explicit form of the general J.C.P. Miller formula for one-variable case. As applications of these results we provide an explicit formula for the inverses of polynomials and formal power series for which the inverses exist, obviously. We also use our results to investigation of approximate solution to a differential equation which cannot be solved in an explicit way. [ABSTRACT FROM AUTHOR]

The paper aims to study the geometry and physics of the Raychaudhuri equation (RE) in the background of a homogeneous and anisotropic space–time described by Kantowski–Sachs (KS) metric. Role of anisotropy/shear in the context of convergence and possible avoidance of singularity has been analyzed subject to a physically motivated constraint. Moreover, using a suitable transformation the first order RE has been converted to a second order differential equation analogous to a Harmonic Oscillator and criterion for convergence has been shown to be associated with the time varying frequency of the Oscillator. Finally, the paper points out a geometric and physical notion of anisotropy along with their corresponding behavior towards convergence. [ABSTRACT FROM AUTHOR]

In this paper we present a number of examples of exact solutions for the Friedmann cosmological equation for metric F (R) gravity model. Emphasis was placed on the possibility of obtaining exact time dependences of the main cosmological physical quantities: scale factor, scalar curvature, Hubble rate and function F (R). For this purpose an ansatz was used to reduce the Friedmann equation to an ordinary differential equation for function F = F (H 2). This made it possible to obtain a number of exact solutions, both already known and new. • Exact solutions for the Friedmann cosmological equation for metric F (R) gravity. • Friedmann–Lemaître–Robertson–Walker (FLRW) metric interval. • An ansatz used to reduce the Friedmann equation to an ordinary differential equation. • Cosmological interpretation of solution obtained. [ABSTRACT FROM AUTHOR]

Recently, there has been a growing interest in the development of intelligent structural damage identification methods that utilize machine learning techniques. However, many existing machine learning-based damage identification methods rely on labeled datasets, which can be large-scale and therefore costly, time-consuming, and/or unrealistic to obtain in practice. To address this issue, a baseline-free damage identification method for plate structures is proposed in this paper that utilizes physics-informed neural networks (PINNs). PINNs are a unique type of neural network that embed physical laws, frequently represented as differential equations, into their loss function. The method assumes that a plate structure in its pristine state is an isotropic and homogeneous thin plate, which can be well modeled by the Kirchhoff–Love plate theory. The proposed damage identification method aims to isolate and intensify local anomalies caused by damage in flexural guided wavefields of damaged thin plates. To accomplish this goal for such a damaged plate, a PINN is constructed and trained using data from a measured flexural guided wavefield. Such a trained PINN is referred to as a pseudo-pristine model, as it can generate a wavefield approximating that of an isotropic and homogeneous thin plate. This unique functionality arises from penalizing the governing equation of the plate in the loss function of the PINN and the fact that the governing equation does not consider the existence of the damage. Any local anomalies in the measured wavefield can be isolated by comparing it with the wavefield generated by the pseudo-pristine model, and they are then intensified using the Teager energy operator. An accumulative damage index is formulated and the damage can be identified within neighborhoods with high index values. The effectiveness and noise-robustness of the proposed method are demonstrated through numerical and experimental examples. Parameter studies are conducted to investigate the robustness of the PINNs with different hyper-parameters. The results show that the damage can be accurately identified by the proposed method under the different hyper-parameters. • PINNs applied to isotropic and homogeneous thin plates with inverse problems. • Baseline-free damage identification method using PINNs is proposed. • Pseudo-pristine models generate wavefields approximating those of pristine plates. • Local anomalies in measured wavefields are isolated by pseudo-pristine models. • Teager energy of isolated local anomalies used for damage identification. [ABSTRACT FROM AUTHOR]

• We proposed variable linear transformation improved physics-informed neural networks (VLT-PINNs) to solve thin-layer flow forward and inverse problems. • Three principles for determining the VLT parameters were identified for three types of variables in PINNs respectively. • The VLT-PINNs with the principle-suggested parameters show excellent performance over the ten test cases. • Only under the constraints of the VLT principles can we obtain satisfactory results, while traditional linear transformation may yield totally wrong results. • VLT method is an attempt to combine the three advantages of accuracy, universality and simplicity. Physics-informed neural networks (PINNs) have attracted wide attention due to their ability to seamlessly embed the learning process with physical laws and their considerable success in solving forward and inverse differential equation (DE) problems. While most studies are improving the learning process and network architecture of PINNs, less attention has been paid to the modification of the DE system, which may play an important role in addressing some limitations of PINNs. One of the simplest modifications that can be implemented to all DE systems is the variable linear transformation (VLT). Therefore, in this work, we propose the VLT-PINNs that solve the DE systems of the linear-transformed variables instead of the original ones. To clearly illustrate the importance of prior knowledge in determining the VLT parameters, we choose the thin-layer flow problems as our focus. Ten related cases were tested, including the jet flows, wake flows, mixing layers, boundary layers and Kovasznay flows. Based on the principle of normalization and for a better match of the DE system to the preference of NNs, we identify three principles for determining the VLT parameters: magnitude normalization for dependent variables (principle 1), local normalization for independent variables (principle 2), and appropriate scaling for physics-related parameters in inverse problems (principle 3). The VLT-PINNs with the VLT parameters suggested by the proposed principles show excellent performance over all the test cases, while the results are quite poor with the VLT parameters suggested by traditional linear transformations, such as nondimensionalization and global normalization. Comparison studies also show that only under the constraints of the VLT principles can we obtain satisfactory results. Besides, we find tanh is more appropriate as the activation function than sin for thin-layer flow problems, from both posteriori results and priori analyses with physical intuition. We highlight that our VLT method is an attempt to combine the three advantages of accuracy, universality and simplicity, and hope that it can provide new insights into the better integration of prior knowledge, physical intuition and the nature of NNs. The code for this paper is available on https://github.com/CAME-THU/VLT-PINN. [ABSTRACT FROM AUTHOR]

Mathematical models play an increasing role in understanding and predicting machining processes, in particular milling. However, despite the considerable efforts that have been dedicated to this problem, a majority of milling models still rely on simplifying assumptions to calculate the chip thickness. In this paper, the chip thickness is determined without these simplifications, based on a surface function that describes the milled surface and on information about the workpiece boundary. By combining the partial differential equation (PDE) governing the evolution of this surface function with the ordinary differential equations (ODE) governing the tool/machine dynamics, a mixed PDE–ODE formulation is proposed to describe the dynamics of the milling process. The coupled system of differential equations is solved using an algorithm that combines finite difference (ODE) and finite volume (PDE) methods. A case study is presented to compare the proposed approach with the classical delay differential equations (DDE) model formulation for milling processes based on a simplified chip thickness model. The PDE–ODE formulation represents an explicit mathematical model for milling process dynamics; it yields a theoretically exact chip thickness and offers a means to assess the validity of models based on DDE formulation. Moreover, the proposed formulation is capable of simulating transient tool behaviors when the tool is milling the outer region of the workpiece, which is in general neglected by the DDE-based models. • Accurate mathematical description of the milling process. • Evolution of the machined surface around the tool is described. • Chip thickness model affects the accuracy of milling simulation in certain scenarios. • Transient tool behaviors are captured while milling the outer part of the workpiece. [ABSTRACT FROM AUTHOR]