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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]

In this paper we study the permanence and impermanence for continuous-time competitive Kolmogorov systems via the carrying simplex. We first give an extension to attractors of V. Hutson's results on the existence of repellors in continuous-time dynamical systems that have found wide use in the study of permanence via average Liapunov functions. We then give a general criterion for the stability of the boundary of carrying simplex for competitive Kolmogorov systems of differential equations, which determines the permanence and impermanence of such systems. Based on the criterion, we present a complete classification of the permanence and impermanence in terms of inequalities on parameters for all three-dimensional competitive systems which have linearly determined nullclines. The results are applied to many classical models in population dynamics including the Lotka-Volterra system, Gompertz system, Leslie-Gower system and Ricker system. [ABSTRACT FROM AUTHOR]

We prove oscillation and nonoscillation theorems for the second order linear differential equation (E) y″+q(t)y=0, where q(t)&ges;0 and locally integrable on [t&ast;,∞). These results are extensions of earlier results of Huang [J. Math. Anal. Appl. 210 (1997) 712–723]. Furthermore, we show that the oscillation criterion established for Eq. (E) can be extended to the delayed differential equation y″+q(t)y(σ(t))=0, where σ(t)&les;t and limt→∞σ(t)=∞. [Copyright &y& Elsevier]

In the case of oscillatory potentials, we give sufficient conditions for the oscillation of the forced nonlinear second order differential equations with delayed argument in the form x″(t)+q(t)x(τ(t))γsgnx(τ(t))=f(t) in the linear (γ=1) and the superlinear (γ>1) cases. [Copyright &y& Elsevier]

In this paper, we study the existence, multiplicity and dynamics of positive periodic solutions to a generalized Liénard equation with repulsive singularities. The Ambrosetti-Prodi type result is proved in the absence of the so-called anticoercivity condition. Furthermore, with s as a parameter, under some conditions on the function h , it has been shown that for any M > 1 there exists s M ∈ R such that the equation x ″ + f (x) x ′ + h (t , x) = s has two positive T -periodic solutions u 1 (⋅ ; s) and u 2 (⋅ ; s) satisfying min ⁡ { u 1 (t ; s) : t ∈ [ 0 , T ] } > M and min ⁡ { u 2 (t ; s) : t ∈ [ 0 , T ] } < 1 / M for every s < s M. As a by-product of the property, we obtain sufficient conditions to guarantee the existence of positive T -periodic solutions of indefinite differential equations. • Establish a refinement result of the Ambrosetti-Prodi type in the absence of anti-coercivity condition. • Investigate the dynamic behaviors of positive periodic solutions to a generalized Liénard equation. • Provide a new approach to study the existence of positive periodic solutions for indefinite differential equations. • Find a relationship between the strict lower functions and Leray-Schauder degree. [ABSTRACT FROM AUTHOR]

The main aim of this article is to establish an effective criterion for a positive parametric topological entropy to differential equations with multivalued nonautonomous impulses on tori. The crucial tool for this goal consists in developing an appropriate Ivanov-like inequality for the lower estimate of a new kind of entropy by means of the logarithm of asymptotic Nielsen numbers for the compositions of a one-parameter family of multivalued admissible maps. This inequality is then applied to impulsive differential equations on tori via the associated Poincaré translation operators along their trajectories. The obtained results generalize in a significant way those in our recent papers, especially in the one with the same title (whence the indication by II), into a multivalued setting. Another nontrivial generalization can be regarded with respect to their simple reduction into a "nonparametric" case. [ABSTRACT FROM AUTHOR]

In this paper, we study exponentially fast mixing induced/enhanced by gravity and stochastic boundary in the kinetic theory of Vlasov equations. We consider the Vlasov equations with and without a vertical magnetic field inside a horizontally-periodic 3D half-space equipped with a non-isothermal diffusive reflection boundary condition of bounded continuous boundary temperature at the bottom. We construct both stationary solutions and global-in-time dynamical solutions in L ∞. We prove that moments of a dynamical fluctuation around the steady solutions decay exponentially fast in L ∞. As a key of this proof, we establish a uniform bound of so-called residual measures independently of the bouncing number of stochastic characteristics, by constructing a continuous stationary outgoing boundary flux which is strictly positive almost everywhere. [ABSTRACT FROM AUTHOR]

This paper is concerned with the asymptotic expansion of solutions to the initial-value problem of u ″ (t) + A u (t) + b (t) u ′ (t) = 0 in a Hilbert space with a nonnegative selfadjoint operator A and a coefficient b (t) ∼ (1 + t) − β (− 1 < β < 1). In the case b (t) ≡ 1 , it is known that the higher-order asymptotic profiles are determined via a family of first-order differential equations of the form v ′ (t) + A v (t) = F n (t) (Sobajima (2021) [10]). For the time-dependent case, it is only known that the asymptotic behavior of such a solution is given by the one of b (t) v ′ (t) + A v (t) = 0. The result of this paper is to find the equations for all higher-order asymptotic profiles. It is worth noticing that the equation for n -th order profile u ˜ n is given via v ′ (t) + m n (t) A v (t) = F n (t) which coefficient m n (time-scale) differs each other. [ABSTRACT FROM AUTHOR]

This paper is devoted to the smooth and stationary Wong-Zakai approximations for a class of rough differential equations driven by a geometric fractional Brownian rough path ω with Hurst index H ∈ (1 3 , 1 2 ]. We first construct the approximation ω δ of ω by probabilistic arguments, and then use the rough path theory to obtain the Wong-Zakai approximation for the solution on any finite interval. Finally, both the original system and the approximative system generate a continuous random dynamical systems φ and φ δ. As a consequence of the Wong-Zakai approximation of the solution, φ δ converges to φ as δ → 0. [ABSTRACT FROM AUTHOR]

We deal with the following fractional Choquard equation (− Δ) s u + V (x) u = (I μ ⁎ | u | 2 μ , s ⁎ ) | u | 2 μ , s ⁎ − 2 u , x ∈ R N , where I μ (x) is the Riesz potential, s ∈ (0 , 1) , 2 s < N ≠ 4 s , 0 < μ < min ⁡ { N , 4 s } and 2 μ , s ⁎ = 2 N − μ N − 2 s is the fractional critical Hardy-Littlewood-Sobolev exponent. By combining variational methods and the Brouwer degree theory, we investigate the existence and multiplicity of positive bound solutions to this equation when V (x) is a positive potential bounded from below. The results obtained in this paper extend and improve some recent works in the case where the coefficient V (x) vanishes at infinity. [ABSTRACT FROM AUTHOR]

Exact solutions of the SEIR epidemic model are derived, and various properties of solutions are obtained directly from the exact solution. In this paper Abel differential equations play an important role in establishing the exact solution of SEIR differential system, in particular the number of infected individuals can be represented in a simple form by using a positive solution of an Abel differential equation. It is shown that the parametric form of the exact solution satisfies some linear differential system including a positive solution of an Abel differential equation. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of computing valuations of the Dieudonné determinants of matrices over discrete valuation skew fields (DVSFs). Under a reasonable computational model, we propose two algorithms for a class of DVSFs, called split. Our algorithms are extensions of the combinatorial relaxation of Murota (1995) and the matrix expansion by Moriyama and Murota (2013) , both of which are based on combinatorial optimization. While our algorithms require an upper bound on the output, we give an estimation of the bound for skew polynomial matrices and show that the estimation is valid only for skew polynomial matrices. We consider two applications of this problem. The first one is the noncommutative weighted Edmonds' problem (nc-WEP), which is to compute the degree of the Dieudonné determinants of matrices having noncommutative symbols. We show that the presented algorithms reduce the nc-WEP to the unweighted problem in polynomial time. In particular, we show that the nc-WEP over the rational field is solvable in time polynomial in the input bit-length. We also present an application to analyses of degrees of freedom of linear time-varying systems by establishing formulas on the solution spaces of linear differential/difference equations. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, we prove the existence of the C ∞ [ 0 , 1 ] steady-state solution by the method of upper and lower solution, and the uniqueness in both W 1 , 2 (0 , 1) and L ∞ (0 , 1) by a generalized maximum principle. We further prove the global existence and uniqueness of the time-dependent solution in C ([ 0 , 1 ] × [ 0 , + ∞)) ∩ C 2 , 1 ([ 0 , 1 ] × (0 , + ∞)) , which, for any continuous initial value, converges to the steady-state solution in C [ 0 , 1 ] (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species. [ABSTRACT FROM AUTHOR]

This paper is concerned with the invariant measures of non-autonomous stochastic reaction-diffusion equations on unbounded domains. We firstly investigate the existence of invariant measures for random dynamical systems over two parametric spaces as well as periodic invariant measures, and then discuss the convergence of (periodic) invariant measures with respect to some system parameters. Random Liouville type equation is also derived for invariant measures associated with non-autonomous random differential equations. Based on such abstract results, we prove the existence of (periodic) invariant measures for non-autonomous stochastic reaction-diffusion equation on unbounded domains, which are supported on the pullback attractors. And we further show every limit of a sequence of (periodic) invariant measures must be the (periodic) invariant measure of the corresponding limiting equation when the noise intensity varies in finite interval. Moreover, we prove that the invariant measures of the underlying equation satisfy a stochastic Liouville type equation. [ABSTRACT FROM AUTHOR]

It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes are Hölder continuous at 0 provided that ∫ B 1 | u (x) | 3 d x + ∫ B 1 | f (x) | q d x or ∫ B 1 | ∇ u (x) | 2 d x + ∫ B 1 | ∇ u (x) | 2 d x (∫ B 1 | u (x) | d x) 2 + ∫ B 1 | f (x) | q d x with q > 3 is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points is zero. For the boundary case, we also obtain that 0 is regular provided that ∫ B 1 + | u (x) | 3 d x + ∫ B 1 + | f (x) | 3 d x or ∫ B 1 + | ∇ u (x) | 2 d x + ∫ B 1 + | f (x) | 3 d x is sufficiently small. These results improve previous regularity theorems by Dong-Strain ([8] , Indiana Univ. Math. J., 2012), Dong-Gu ([7] , J. Funct. Anal., 2014), and Liu-Wang ([27] , J. Differential Equations, 2018), where either the smallness of the pressure or the smallness of the scaling invariant quantities on all balls is necessary. [ABSTRACT FROM AUTHOR]

In this paper, we consider an eigenvalue problem defined on a two-patch domain. Our first aim is to show that the two-patch eigenvalue problem is equivalent to the eigenvalue problem of a measure differential equation defined on an one-patch domain. Our second aim is to study the existence of principal eigenvalue of the measure differential equation, and we will prove the principal eigenvalue is continuously depending on the weight measure in the weak⁎ topology of the measure space. Our third aim is to solve a minimization problem on principal eigenvalues. Some main results of this paper have interesting relations with population dynamics. We will interpret these results in terms of survival chances and optimal distribution of resources. [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 the half-linear differential equation with one-dimensional p -Laplacian (p = α + 1), (| u ′ | α sgn u ′) ′ = α (λ α + 1 + b (t)) | u | α sgn u , t ≥ t 0 , is considered, where α and λ are positive constants. It is proved that if the function b (t) is absolutely integrable on [ t 0 , ∞) , then the above equation has two nonoscillatory solutions u + (t) and u − (t) such that u ± (t) ∼ c e ± λ t and u ± ′ (t) ∼ ± c λ e ± λ t (t → ∞), and moreover, any nontrivial solution u (t) of the above equation satisfies either u (t) ∼ c e λ t , u ′ (t) ∼ c λ e λ t (t → ∞) or u (t) ∼ c e − λ t , u ′ (t) ∼ − c λ e − λ t (t → ∞). Here, c is a nonzero constant. A weaker result is also shown under the weaker condition that b (t) is conditionally integrable on [ t 0 , ∞). [ABSTRACT FROM AUTHOR]

At the 1900 International Congress of Mathematicians, Hilbert claimed that the Riemann zeta function is not the solution of any algebraic ordinary differential equation its region of analyticity [5]. In 2015, Van Gorder addresses the question of whether the Riemann zeta function satisfies a non -algebraic differential equation and constructs a differential equation of infinite order which zeta satisfies [7]. However, as he notes in the paper, this representation is formal and Van Gorder does not attempt to claim a region or type of convergence. In this paper, we show that Van Gorder's operator applied to the zeta function does not converge pointwise at any point in the complex plane. We also investigate the accuracy of truncations of Van Gorder's operator applied to the zeta function and show that a similar operator applied to zeta and other L -functions does converge. [ABSTRACT FROM AUTHOR]

• The normal form formula of double-Hopf bifurcation is generated for general differential equations with the nonlocality. • The formula is applicable to functional differential equations and partial differential equations with or without the nonlocality. • The algorithm is favourable to reveal complex dynamics, e.g. periodic, quasi-periodic and multi-periodic oscillations. • Stable spatially nonhomogeneous periodic and quasi-periodic solutions are found in a nonlocal predator-prey model. This paper is concerned with the calculation of the normal form on the center manifold up to the third-order term at the double-Hopf singularity for general partial functional differential equations with the nonlocal effect. We derive explicit formulas of the normal form, that can be applied for both functional differential equations and partial differential equations with or without nonlocal effects in a bounded spatial domain. This provides an effective tool to establish existences of multi-periodic and quasi-periodic oscillations of a double-Hopf singularity for such equations. As an example, the Holling-Tanner predator-prey model with the nonlocal intraspecific competition of prey is considered. It turns out that double-Hopf bifurcation will occur because of the nonlocal interaction. Many spatio-temporal dynamics are found, including stable spatially homogeneous or nonhomogeneous periodic solutions, and stable spatially nonhomogeneous quasi-periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper we associate to a linear differential equation with coefficients in the field of Laurent formal power series a new geometric object, a spectrum in the sense of Berkovich. We compute this spectrum and show that it contains interesting information about the equation. [ABSTRACT FROM AUTHOR]

We generalize the definition of overconvergent p -adic multiple polylogarithms and of p -adic cyclotomic multiple zeta values and we prove a bound on their norm. A byproduct of the proof is a characterization of these objects in terms of certain regularized p -adic iterated integrals. The generalization of the definition consists in replacing the underlying Frobenius structure by its iterations. The bound on the norms of overconvergent p -adic multiple polylogarithms that we obtain is a prerequisite for our subsequent papers on p -adic cyclotomic multiple zeta values. For a video summary of this paper, please visit https://youtu.be/Uxk_qbJxQo. [ABSTRACT FROM AUTHOR]

• Philip Maini has made multiple significant contributions to the field of Mathematical Enzyme Kinetics. • Philip Maini has also contributed to the field by mentoring young researchers. • The Quasi-Steady State Assumption has now been expanded to include just about every possible initial condition. Modelling enzyme kinetics has a long history, as one of the earliest areas where mathematics was utilized to understand biological (in this case, biochemical) phenomena. By using differential equations and the law of mass action, many researchers have discovered information about the mechanisms of enzyme workings in the more than 100 years since Leonor Michaelis and Maud Menten published their classic paper in 1913. This work uses he laws obeyed by molecules during a reaction, translates them into mathematical equations, and then compares the results predicted by the equations to data obtained in laboratory conditions. This can inform whether the mechanism used to create the mathematical model is consistent with experimental results. Philip K. Maini has contributed to several significant results in the field of mathematical modelling in enzyme kinetics. Along with his collaborators, he has published results analyzing and extending the Quasi-Steady State Assumption, applying it to special enzyme reactions, and considering special conditions. This review paper will provide an overview of the contributions made by PK Maini to the field of modelling enzyme kinetics, summarize the main results and discuss their significance. [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]

Projection methods are applicable in many fields. It is a natural and practical approach to devise the invariant-preserving schemes for conservative systems. The idea is to project the solution of any underlying numerical scheme onto the manifold determined by the invariant, and this process will be referred to as the projection integrator. Generally, the projection integrator chooses the gradient of invariant as its projection direction and has the same order as the underlying method. In this paper, we propose a different projection direction to construct a new projection integrator whose order is higher than the underlying method. According to this novel direction, we further summarize high-order projection integrators with superconvergence and rigorously prove the truncation error by utilizing the linear integral method as a central tool. Apart from the invariant-preserving property, symmetry is an important geometric property for reversible differential equations. The design and analysis of another high-order projection integrators with symmetry and superconvergence are also presented in this paper. Numerical experiments are provided to verify our theoretical results and illustrate that our proposed projection integrators have superior behaviors in a long time numerical simulation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain results on topological properties of flows for nonnegative time in the framework of generalized ODEs. We define the concept of generalized semiflow and we present some recursive properties as minimality and recurrence. Using correspondence theorems, we translate our results to measure differential equations which are known to be special cases of generalized ODEs. Some examples are presented to illustrate the theory. [ABSTRACT FROM AUTHOR]

In [3] André showed that the minimal differential equations of Э-functions and E -functions have a basis of holomorphic solutions at every point except for zero and infinity. In addition, in [5] he observed that if f 1 (z) is an E -function with rational coefficients such that f 1 (1) = 0 , then f 1 (z) / (z − 1) is also an E -function. With this additional result André derived transcendence results for the values of E -functions. These results were further applied by Beukers [7] to obtain a strengthened Siegel-Shidlovskii theorem for E -functions. The arguments of André and Beukers should aid in obtaining a Siegel-Shidlovskii type theorem for Э-functions provided that one can show that if f − 1 (z) is a Э-function with rational coefficients such that the 1-summation F − 1 (z) vanishes at z = 1 , then f − 1 (z) / (z − 1) is also a Э-function. In this paper, we investigate this problem and derive several criteria to check its validity. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the averaging principle for a fast-slow system of rough differential equations driven by mixed fractional Brownian rough path. The fast component is driven by Brownian motion, while the slow component is driven by fractional Brownian motion with Hurst index H (1 / 3 < H ≤ 1 / 2). Combining the fractional calculus approach to rough path theory and Khasminskii's classical time discretization method, we prove that the slow component strongly converges to the solution of the corresponding averaged equation in the L 1 -sense. The averaging principle for a fast-slow system in the framework of rough path theory seems new. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in investigating stability results for generalized ordinary differential equations (generalized ODEs in short), and their applications to measure differential equations and dynamic equations on time scales. First, we establish stability, asymptotic and exponential stability for the trivial solution of generalized ODEs. Secondly, we use the well known correspondence between solutions of generalized ODEs and solutions of measure differential equations, obtaining analogues results for the last equations. Finally, we apply some of these results for dynamic equations on time scales. [ABSTRACT FROM AUTHOR]

This paper considers compound operators on a general vector space. Explicit representations for these operators are obtained in various contexts with emphasis on the compound differential equations corresponding to a given linear differential equation. Applications to estimation of eigenvalues of self-adjoint boundary value problems and of the codimension of the stable manifold of a linear differential equation in a Banach space are given. [ABSTRACT FROM AUTHOR]

In this paper we aim to find the highest number of critical periods in a class of planar systems of polynomial differential equations for fixed degree having a center. We fix our attention to lower bounds of local criticality for low degree planar polynomial centers. The main technique is the study of perturbations of reversible holomorphic (isochronous) centers, inside the reversible centers class. More concretely, we study the Taylor developments of the period constants with respect to the perturbation parameters. First, we see that there are systems of degree 3 ≤ n ≤ 16 for which up to first order at least (n 2 + n − 4) / 2 critical periods bifurcate from the center. Second, we improve this number for centers with degree from 3 to 9. In particular, we obtain 6 and 10 critical periods for cubic and quartic degree systems, respectively. [ABSTRACT FROM AUTHOR]

Orthogonal and biorthogonal (multi)wavelets on the real line have been extensively studied and employed in applications with success. On the other hand, a lot of problems in applications such as images and solutions of differential equations are defined on bounded intervals or domains. Therefore, it is important in both theory and application to construct all possible wavelets on intervals with some desired properties from (bi)orthogonal (multi)wavelets on the real line. Then wavelets on rectangular domains such as [ 0 , 1 ] d can be obtained through tensor product. Vanishing moments of compactly supported wavelets are the key property for sparse wavelet representations and are closely linked to polynomial reproduction of their underlying refinable (vector) functions. Boundary wavelets with low order vanishing moments often lead to undesired boundary artifacts as well as reduced sparsity and approximation orders near boundaries in applications. Scalar orthogonal wavelets and spline biorthogonal wavelets on the interval [ 0 , 1 ] have been extensively studied in the literature. Though multiwavelets enjoy some desired properties over scalar wavelets such as high vanishing moments and relatively short support, except a few concrete examples, there is currently no systematic method for constructing (bi)orthogonal multiwavelets on bounded intervals. In contrast to current literature on constructing particular wavelets on intervals from special (bi)orthogonal (multi)wavelets, from any arbitrarily given compactly supported (bi)orthogonal multiwavelet on the real line, in this paper we propose two different approaches to construct/derive all possible locally supported (bi)orthogonal (multi)wavelets on [ 0 , ∞) or [ 0 , 1 ] with or without prescribed vanishing moments, polynomial reproduction, and/or homogeneous boundary conditions. The first approach generalizes the classical approach from scalar wavelets to multiwavelets, while the second approach is direct without explicitly involving any dual refinable functions and dual multiwavelets. We shall also address wavelets on intervals satisfying general homogeneous boundary conditions. Though constructing orthogonal (multi)wavelets on intervals is much easier than their biorthogonal counterparts, we show that some boundary orthogonal wavelets cannot have any vanishing moments if these orthogonal (multi)wavelets on intervals satisfy the homogeneous Dirichlet boundary condition. In comparison with the classical approach, our proposed direct approach makes the construction of all possible locally supported (multi)wavelets on intervals easy. Seven examples of orthogonal and biorthogonal multiwavelets on the interval [ 0 , 1 ] will be provided to illustrate our construction approaches and proposed algorithms. [ABSTRACT FROM AUTHOR]

It has been shown by Chen that if all masses are in the same proper vector subspace of R d (d ≥ 2) on one of the free boundaries, local minimizers connecting the two free boundaries have no collision on that boundary. However, not much is known for other types of boundary configurations. In this paper, we consider minimizers connecting two free boundaries and study possible quadruple collisions under the following four symmetric configurations: isosceles trapezoid configuration, rectangular configuration, double isosceles configuration and diamond configuration. Whenever the boundary configuration of four bodies coincides with one of the four symmetric configurations, we show that these four bodies are free of quadruple collision on that boundary set. New ideas and detailed analysis are introduced in order to construct deformed paths in some of the cases and estimate their action values. As its applications, we show the existence of several sets of periodic orbits in the planar four-body and five-body problems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain converse Lyapunov theorems for measure functional differential equations which are known to encompass dynamic equations on time scales, impulsive differential equations and functional differential equations (see [7,8]). Our main tool is to consider the framework of generalized ordinary differential equations, introduced by Jaroslav Kurzweil, and to manage all other equations in this setting which has the advantage to handle well both highly oscillating functions and functions with many discontinuities. [ABSTRACT FROM AUTHOR]

To characterize the regularity of distribution-path dependent SDEs in the initial distribution which varies as probability measure on the path space, we introduce the intrinsic and Lions derivatives for probability measures on Banach spaces, and prove the chain rule of the Lions derivative for the distribution of Banach-valued random variables. By using Malliavin calculus, we establish the Bismut type formula for the Lions derivatives of functional solutions to SDEs with distribution-path dependent drifts. When the noise term is also path dependent so that the Bismut formula is invalid, we establish the asymptotic Bismut formula. Both non-degenerate and degenerate noises are considered. The main results of this paper generalize and improve the corresponding ones derived recently in the literature for the classical SDEs with memory and McKean-Vlasov SDEs without memory. [ABSTRACT FROM AUTHOR]

Acoustic black hole (ABH), a thin wedge-shaped structure tapered according to the power-law of power (m) greater than or equal to two, has received much attention to the researchers in wave dynamics due to its potential as a light and effective absorber of elastic waves propagating in beams or plates. In this paper, the Euler-Bernoulli equation for the ABH is reformulated into the form of a generalized hypergeometric differential equation for m > 2. Then, the exact solution is derived in terms of generalized hypergeometric functions (p F q) where p = 0 and q = 3 by dividing the power m into four cases. The derived exact solution is in linearly independent and regular form for arbitrary m. In addition, by using the exact solution, the displacement field of a uniform beam with an ABH and the reflection coefficient from an ABH are calculated to demonstrate the applicability of the present exact solution. This paper aims at providing a mathematical and theoretical basis for the study of the ABHs. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study an implicit version of the sweeping process. Based on methods of convex analysis, we prove the equivalence of the implicit sweeping process with a differential equation, which enables us to show the existence and uniqueness of the solution to the implicit sweeping process in a very general framework. Moreover, this equivalence allows us to give a characterization of nonsmooth Lyapunov pairs and invariance for implicit sweeping processes. The results of the paper are illustrated with two applications to quasistatic evolution variational inequalities and electrical circuits. [ABSTRACT FROM AUTHOR]

Abstract Spatially localized patterns have been observed in numerous physical contexts, and their bifurcation diagrams often exhibit similar snaking behavior: symmetric solution branches, connected by bifurcating asymmetric solution branches, wind back and forth in an appropriate parameter. Previous papers have addressed existence of such solutions; here we address their stability, taking the necessary first step of unifying existence and uniqueness proofs for symmetric and asymmetric solutions. We then show that, under appropriate assumptions, temporal eigenvalues of the front and back underlying a localized solution are added with multiplicity in the right half plane. In a companion paper, we analyze the behavior of eigenvalues at λ = 0 and inside the essential spectrum. Our results show that localized snaking solutions are stable if, and only if, the underlying fronts and backs are stable: unlike localized non-oscillatory solutions, no interaction eigenvalues are present. We use the planar Swift–Hohenberg system to illustrate our results. [ABSTRACT FROM AUTHOR]

Abstract This paper considers the inverse problem of scattering of time-harmonic acoustic and electromagnetic plane waves by a bounded, inhomogeneous, penetrable obstacle with embedded objects inside. A new method is proposed to prove that the inhomogeneous penetrable obstacle can be uniquely determined from the far-field pattern at a fixed frequency, disregarding its contents. Our method is based on constructing a well-posed interior transmission problem in a small domain associated with the Helmholtz or modified Helmholtz equation and the Maxwell or modified Maxwell equations. A key role is played by the smallness of the domain which ensures that the lowest transmission eigenvalue is large so that a given wave number k is not an eigenvalue of the interior transmission problem. Another ingredient in our proofs is a priori estimates of solutions to the transmission scattering problems with data in L p (1 < p < 2), which are established in this paper by using the integral equation method. A main feature of the new method is that it can deal with the acoustic and electromagnetic cases in a unified way and can be easily applied to deal with inverse scattering by unbounded rough interfaces. [ABSTRACT FROM AUTHOR]

Abstract The goal of this paper is to develop the Equivariant Degree based method for studying relative periodic solutions in the setiings with lack of smoothness and/or genericity. In this paper, we consider an equivariant Hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting Γ × S 1 -spatial symmetries. The existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted Γ × S 1 -degree with one free parameter. As a case study, we consider a delay differential model of coupled identical passively mode-locked semiconductor lasers with the dihedral symmetry group Γ = D 8 ; and, a system of hysterestic electro-mechanical oscillators coupled in the same symmetric fashion. [ABSTRACT FROM AUTHOR]

Abstract There is the paper by H. Tietze published in 1905 on differential transcendence of solutions of difference Riccati equations. In this paper, we clarify the essence of Tietze's treatment and make it purely algebraic. As an application, the q -Airy equation is studied. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global well-posedness and decay rates of strong solutions for the three-dimensional compressible Oldroyd-B model. We prove that this set of equations admits a unique global strong solution provided the initial data are close to the constant equilibrium state in H 2 -framework. Moreover, if the initial data belong to L 1 , the convergence rate of the solutions in L p -norm with 2 ⩽ p ⩽ 6 and convergence rates of their spatial derivatives in L 2 -norm are obtained. It is noteworthy that the smallness restriction on the coupling constant ω is not necessary in this paper. [ABSTRACT FROM AUTHOR]

An important point in looking for period solutions of the Camassa–Holm equation is to understand the associated spectral problem y ″ = 1 4 y + λ m ( t ) y . The first aim of this paper is to study the dependence of eigenvalues for the periodic Camassa–Holm Equation on potentials as an infinitely dimensional parameter. To be precise, we prove that as nonlinear functionals of potentials, eigenvalues for the periodic Camassa–Holm Equation are continuous in potentials with respect to the weak topologies in the L p Lebesgue spaces. The second aim of this paper is to find the optimal lower bound of the lowest eigenvalue for the periodic Camassa–Holm Equation when the L 1 norm of potentials are given. In order to make our results more applicable, we will find the optimal lower bound for the lowest eigenvalue in the more general setting of measure differential equations. [ABSTRACT FROM AUTHOR]

This paper deals with a two-species and a three-species predator–prey reaction diffusion systems where the diffusion coefficients are density dependent and time-delays are involved in the reaction functions. The diffusion terms are of porous medium type which are degenerate and the boundary conditions are of Neumann type. The aim of the paper is to investigate the asymptotic behavior of the time-dependent solution in relation to various steady-state solutions of the system. This includes the existence of a unique positive classical solution, global attraction of a steady-state solution, and stability or instability of various semitrivial solutions and the positive steady-state solution. [ABSTRACT FROM AUTHOR]