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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]

This paper deals with the chemotaxis–haptotaxis system with signal-dependent sensitivity { u t = Δ u − ∇ ⋅ ( χ ( v ) u ∇ v ) − ξ ∇ ⋅ ( u ∇ w ) + μ u ( 1 − u − w ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 under homogeneous Neumann boundary conditions and initial conditions, where Ω ⊂ R n ( n ≥ 3 ) is a bounded domain with smooth boundary, ξ , μ > 0 are constants and χ is a function satisfying some conditions. In the case that χ is a constant it is known that the above system possesses a global classical solution under some conditions (Cao [4] , Tao [10] , Tao and Winkler [11] ); however, in the case that χ is a function, the above system has not been studied. The purpose of this paper is to establish global existence and boundedness in the above system. [ABSTRACT FROM AUTHOR]

In this paper we consider a second order nonlinear hyperbolic equation of Monge-Ampère type arizing in the Light Curve Inversion problem [7,8,4] when dealing with objects with regions with negative gaussian curvature. The above works imply that such regions are of practical importance in the Light Curve Inversion problem. It is shown here that in such regions existence and uniqueness of solutions to the original equation can be obtained by solving a system of four first order quasi-linear differential equations supplemented by suitable initial and boundary conditions. Furthermore, these solutions can be found by integrating a system of ordinary differential equations. The numerical methods in [14] can be applied to actually find these solutions. We present here the completed part of the overall project, which establishes the necessary basis for efficient numerical solution. It is expected that the part of the project dealing with numerics will be addressed in a separate paper. This paper is written in a form accessible to physicists, engineers and applied mathematicians working on the Light Curve Inversion problem and design of reflectors. [ABSTRACT FROM AUTHOR]

In this paper, we prove that ζ cannot be a solution to any nontrivial algebraic differential equation whose coefficients are polynomials in Γ , Γ (n) and Γ (ℓ n) over the ring of polynomials in C , where ℓ , n ≥ 1 are positive integers. [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]

• Dynamical model used for long-distance vibration conveyors is proposed. • The four exciters split-driving synchronization theory scheme is constructed. • Synchronization, stability and Sommerfeld effect are analyzed. • Machine functions in the different resonant regions are discussed. • Experimental prototype is designed to verify the correctness of the theory. The previous studies on the vibrating system with multiple exciters less considered the Sommerfeld effect, which were mainly focused on synchronization and stability problems of the system in the whole resonant region. A dynamical model is proposed in this paper to study the synchronization, stability and the Sommerfeld effect of four exciters in the multi-resonant types vibrating system (MRTVS). In order to improve the power of the system, the vibrating system with two rigid frames is driven by four especially distributed exciters. The motion differential equations of the system are given by Lagrange's equations, and the dimensionless coupling equations and synchronization criterion are constructed as well by the average method. The theory condition for stability of four exciters in synchronous states is derived from the Hamilton principle. The coupling dynamic characteristics, stability and Sommerfeld effect of the system are discussed numerically. The whole resonant region of the system is divided into three segments by natural frequencies, and the synchronous and stable states in the different resonant types are analyzed respectively. The diversity phenomenon of the nonlinear system is observed, in other words, the system exists multiple stable equilibrium solutions at a certain particular resonant region. Then the Sommerfeld effect around the natural frequencies (NFs) is further revealed. The correctness of theoretical analyses is examined by simulation and the experiment. It's shown that the reasonable working point in engineering should be selected in the sub-resonant region with respect to the natural frequency of the main vibrating system. The present work can provide theoretical guidance for designing some new types of vibrating machines with high driving power. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop an analytical approach to obtain the sharp estimates for the lowest positive periodic eigenvalue and all Dirichlet eigenvalues of a general Sturm-Liouville problem y ″ = q (t) y + λ m (t) y , where q is a nonnegative potential and another potential m admits to change sign. A typical example of such problems is the well-known Camassa-Holm equations with indefinite potentials, which corresponds to the case q (t) ≡ 1 4. It is shown that the solution of the minimization problems of the lowest positive periodic eigenvalues and Dirichlet eigenvalues will lead to more general distributions of potentials which have no densities with respect to the Lebesgue measure. As a result, it is very natural to choose the general setting of the measure differential equations d y • = y (t) d μ (t) + λ y d ν (t) , to understand the eigenvalues and their minimization, where μ and ν are two suitable measures. The variational characterization of lowest positive eigenvalues, together with a strong continuous dependence of eigenvalues on the potentials, will play crucial roles in our analysis. Different from the periodic case, we are able to obtain the optimal bounds for higher Dirichlet eigenvalues. [ABSTRACT FROM AUTHOR]

A flower graph consists of a half line and N symmetric loops connected at a single vertex with N ≥ 2 (it is called the tadpole graph if N = 1). We consider positive single-lobe states on the flower graph in the framework of the cubic nonlinear Schrödinger equation. The main novelty of our paper is a rigorous application of the period function for second-order differential equations towards understanding the symmetries and bifurcations of standing waves on metric graphs. We show that the positive single-lobe symmetric state (which is the ground state of energy for small fixed mass) undergoes exactly one bifurcation for larger mass, at which point (N − 1) branches of other positive single-lobe states appear: each branch has K larger components and (N − K) smaller components, where 1 ≤ K ≤ N − 1. We show that only the branch with K = 1 represents a local minimizer of energy for large fixed mass, however, the ground state of energy is not attained for large fixed mass if N ≥ 2. Analytical results obtained from the period function are illustrated numerically. [ABSTRACT FROM AUTHOR]

We show a general method allowing the solution calculation, in the form of a power series, for a very large class of nonlinear Ordinary Differential Equations (ODEs), namely the real analytic σπ -ODEs (and, more in general, the real analytic σπ - reducible ODEs) in many indeterminates, characterized by an ODE-function given by generalized polynomials of the indeterminates and their derivatives, i.e. functions formally polynomial with exponents, though the exponent can be any real number, and whose coefficients are analytic time functions. The solution method consists in reducing the ODE to a certain canonical homogeneous quadratic ODE, named driver-type Riccati equation with a larger number of indeterminates, whose solutions include, as sub-solutions, the original solutions, and for which a recursive formula is shown, giving all coefficients of the solution power series directly from the ODE parameters. The reduction method is named exact quadratization and was formerly introduced in another our article, where we considered explicit ODEs only. In the present paper, which is self-contained to a large extent, we review and complete the theory of exact quadratization by solving issues, such as for instance the existence of a piecewise-quadratization, that had remained open, and also extend it to the more general case of an implicitly defined ODE. Finally, we argue that the result can be seen as a partial solution of a differential version of the 22nd Hilbert's problem. [ABSTRACT FROM AUTHOR]

This paper is concerned with the following planar Schrödinger-Poisson system { − Δ u + V (x) u + ϕ u = f (x , u) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where V ∈ C (R 2 , [ 0 , ∞)) is axially symmetric and f ∈ C (R 2 × R , R) is of subcritical or critical exponential growth in the sense of Trudinger-Moser. We obtain the existence of a nontrivial solution or a ground state solution of Nehari-type and infinitely many solutions to the above system under weak assumptions on V and f. Our theorems extend the results of Cingolani and Weth [Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016) 169-197] and of Du and Weth [Nonlinearity, 30 (2017) 3492-3515] and Chen and Tang [J. Differential Equations, 268 (2020) 945-976], where f (x , u) has polynomial growth on u. In particular, some new tricks and approaches are introduced to overcome the double difficulties resulting from the appearance of both the convolution ϕ 2 , u (x) with sign-changing and unbounded logarithmic integral kernel and the critical growth nonlinearity f (x , u). [ABSTRACT FROM AUTHOR]

In this paper, we consider eigenvalue problems of a second-order measure differential equation. We first study some general theories regarding the completely continuity of eigenvalues, the variational characterizations of eigenvalues, and the oscillating properties of eigenfunctions. Then, we solve the following minimization problem: when the m -th Neumann eigenvalue is given, to find explicitly what measures will have the minimal total variation. As applications of this minimization problem, we will solve some extremal problems of eigenvalues and construct some optimal classes of non-degenerate measures for the second-order measure differential equations. [ABSTRACT FROM AUTHOR]

In this paper we study a free boundary problem modeling the growth of vascularized tumors. The model is a modification to the Byrne–Chaplain tumor model that has been intensively studied during the past two decades. The modification is made by replacing the Dirichlet boundary value condition with the Robin condition, which causes some new difficulties in making rigorous analysis of the model, particularly on existence and uniqueness of a radial stationary solution. In this paper we successfully solve this problem. We prove that this free boundary problem has a unique radial stationary solution which is asymptotically stable for large surface tension coefficient, whereas unstable for small surface tension coefficient. Tools used in this analysis are the geometric theory of abstract parabolic differential equations in Banach spaces and spectral analysis of the linearized operator. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge–Ampère equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric polynomials and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation. [ABSTRACT FROM AUTHOR]

Stieltjes differential equations, which contain equations with impulses and equations on time scales as particular cases, simply consist on replacing usual derivatives by derivatives with respect to a nondecreasing function. In this paper we prove new existence results for functional and discontinuous Stieltjes differential equations and we show that such general results have real world applications. Specifically, we show that Stieltjes differential equations are specially suitable to study populations which exhibit dormant states and/or very short (impulsive) periods of reproduction. In particular, we construct two mathematical models for the evolution of a silkworm population. Our first model can be explicitly solved, as it consists on a linear Stieltjes equation. Our second model, more realistic, is nonlinear, discontinuous and functional, and we deduce the existence of solutions by means of a result proven in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and asymptotic behavior of ground state sign-changing solutions to a class of Schrödinger–Poisson systems { − △ u + V ( x ) u + μ ϕ u = λ f ( x ) u + | u | 2 u , x ∈ R 3 , − △ ϕ = u 2 , x ∈ R 3 , where V is a smooth function, f is nonnegative, μ > 0 , λ < λ 1 and λ 1 is the first eigenvalue of the problem − △ u + V ( x ) u = λ f ( x ) u in H . With the help of the sign-changing Nehari manifold, we obtain that the Schrödinger–Poisson system possesses at least one ground state sign-changing solution u μ for all μ > 0 and each λ < λ 1 . Moreover, we prove that its energy is strictly larger than twice that of ground state solutions. Besides, we give a convergence property of u μ as μ ↘ 0 . This paper can be regarded as the complementary work of Shuai and Wang [23] , Wang and Zhou [24] . [ABSTRACT FROM AUTHOR]

The temperature-dependent incompressible nematic liquid crystal flows in a bounded domain Ω ⊂ R N ( N = 2 , 3 ) are studied in this paper. Following Danchin's method in [7] , we use a localization argument to recover the maximal regularity of Stokes equation with variable viscosity, by which we first prove the local existence of a unique strong solution, then extend it to a global one provided that the initial data is a sufficiently small perturbation around the trivial equilibrium state. This paper also generalizes Hu–Wang's result in [21] to the non-isothermal case. [ABSTRACT FROM AUTHOR]

The paper is aimed at the methodological development of ecological monitoring in discrete-time dynamic models. In earlier papers, in the framework of continuous-time models, we have shown how a systems-theoretical methodology can be applied to the monitoring of the state process of a system of interacting populations, also estimating certain abiotic environmental changes such as pollution, climatic or seasonal changes. In practice, however, there may be good reasons to use discrete-time models. (For instance, there may be discrete cycles in the development of the populations, or observations can be made only at discrete time steps.) Therefore the present paper is devoted to the development of the monitoring methodology in the framework of discrete-time models of population ecology. By monitoring we mean that, observing only certain component(s) of the system, we reconstruct the whole state process. This may be necessary, e.g., when in a complex ecosystem the observation of the densities of certain species is impossible, or too expensive. For the first presentation of the offered methodology, we have chosen a discrete-time version of the classical Lotka–Volterra prey–predator model. This is a minimal but not trivial system where the methodology can still be presented. We also show how this methodology can be applied to estimate the effect of an abiotic environmental change, using a component of the population system as an environmental indicator. Although this approach is illustrated in a simplest possible case, it can be easily extended to larger ecosystems with several interacting populations and different types of abiotic environmental effects. [ABSTRACT FROM AUTHOR]

In this paper we study a general n -dimensional mixed type differential equation with state dependence. Many known works give the existence of solutions with the so-called Return Condition. In this paper, without requiring the Return Condition, we prove a non-local existence of solutions by using a technique of compactification and the dependence of the maximum of functions on the size of their domains. We apply our result to differential equations with iterates. [ABSTRACT FROM AUTHOR]

In this paper, we prove that for a random differential equation with the driving noise constructed from a Q -Wiener process and the Wiener shift, there exists a local center, unstable, stable, center-unstable, center-stable manifold, and a local stable foliation, an unstable foliation on the center-unstable manifold, and a stable foliation on the center-stable manifold, the smoothness of which depend on the vector fields of the equation. Also we show that any two arbitrarily local center manifolds constructed as above are conjugate. [ABSTRACT FROM AUTHOR]

In this paper we consider a spectral problem for ordinary differential equations of fourth order with the spectral parameter contained in three of the boundary conditions. We study the oscillatory properties of the eigenfunctions and, using these properties, we obtain sufficient conditions for the system of eigenfunctions of the problem in question to form a basis in the space L p (0 , 1) , 1 < p < ∞ , after removing three functions. [ABSTRACT FROM AUTHOR]

In this paper, we take advantage of the averaging theory to investigate a torus bifurcation in two-parameter families of 2 D nonautonomous differential equations. Our strategy consists in looking for generic conditions on the averaged functions that ensure the existence of a curve in the parameter space characterized by a Neimark-Sacker bifurcation in the corresponding Poincaré map. A Neimark-Sacker bifurcation for planar maps consists in the birth of an invariant closed curve from a fixed point, as the fixed point changes stability. In addition, we apply our results to study a torus bifurcation in a family of 3 D vector fields. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Littlewood's problem for the second order differential equation x ¨ + x 2 n + 1 + q (t) x l = p (t) , l < n , where the functions p (t) and q (t) do not need to be periodic. Using the theory of non-periodic twist mappings which is developed by Kunze and Ortega in [3–7] , we obtained longtime closeness results on the bounded and unbounded solutions of the equation. [ABSTRACT FROM AUTHOR]

Numerical simulations suggest that average velocity of a biological cell depends largely on attachment dynamics and less on the forces exerted by the cell. We determine the relationship between two models of cell motion, one based on finite spring constants modeling attachment properties (a randomly switched differential equation) and a limiting case (a centroid model-a generalized random walk) where spring constants are infinite. We prove the main result of this paper, the Expected Velocity Relationship theorem. This result shows that the expected value of the difference between cell locations in the differential equation model at the initial time and at some elapsed time is proportional to the elapsed time. We also show that the relationship is time invariant. Numerical results show the model is consistent with experimental data. [ABSTRACT FROM AUTHOR]

The Λ-Ω systems are the real planar polynomial differential equations of degree m x ˙ = − y (1 + Λ) + x Ω , y ˙ = x (1 + Λ) + y Ω , where Λ = Λ (x , y) and Ω = Ω (x , y) are polynomials of degree at most m − 1 such that Λ (0 , 0) = Ω (0 , 0) = 0. We study the center problem for these Λ-Ω systems. Any planar vector field with linear type center can be written as a Λ-Ω system if and only if the Poincaré-Liapunov first integral is of the form F = 1 2 (x 2 + y 2) (1 + O (x , y)). These kind of linear type centers are called weak centers, they contain the class of centers studied by Alwash and Lloyd [1] , and also contain the uniform isochronous centers, and the holomorphic isochronous centers, but they do not coincide with the all class of isochronous centers. The main objective of this paper is to study the center problem for two particular classes of Λ-Ω systems of degree m. First if Λ = μ (a 2 x − a 1 y) , and Ω = a 1 x + a 2 y + Ω m − 1 , where μ , a 1 , a 2 are constants and Ω m − 1 = Ω m − 1 (x , y) is a homogenous polynomial of degree m − 1 , then we prove the following results. (i) These Λ-Ω systems have a weak center at the origin if and only if (μ + m − 2) (a 1 2 + a 2 2) = 0 , and ∫ 0 2 π Ω m − 1 (cos ⁡ t , sin ⁡ t) d t = 0 ; (ii) If m = 2 , 3 , 4 , 5 , 6 and (μ + m − 2) (a 1 2 + a 2 2) ≠ 0 , then the given Λ–Ω systems have a weak center at the origin if and only if these systems after a linear change of variables (x , y) ⟶ (X , Y) are invariant under the transformations (X , Y , t) ⟶ (− X , Y , − t). Second if Λ = a 1 x + a 2 y , and Ω = Ω m − 1 , where a 1 , a 2 are constants and Ω m − 1 = Ω m − 1 (x , y) is a homogenous polynomial of degree m − 1 , then we prove the following results. (i) These Λ-Ω systems have a weak center at the origin if and only if a 1 = a 2 = 0 , and ∫ 0 2 π Ω m − 1 (cos ⁡ t , sin ⁡ t) d t = 0 ; (ii) If m = 2 , 3 , 4 , 5 and a 1 2 + a 2 2 ≠ 0 , then the given Λ–Ω systems have a weak center at the origin if and only if these systems after a linear change of variables (x , y) ⟶ (X , Y) are invariant under the transformations (X , Y , t) ⟶ (− X , Y , − t). We observe that the main difficulty to prove results (ii) for m > 6 is related with the huge computations necessary for proving them. [ABSTRACT FROM AUTHOR]

Riesz wavelets in the Sobolev space H m (R) with m ∈ N ∪ { 0 } , whose m th-order derivatives are orthogonal among different levels, are of particular interest and importance in computational mathematics, due to their many desirable properties such as small condition numbers and sparse stiffness matrices. We call such Riesz wavelets in the Sobolev space H m (R) as m th-order derivative-orthogonal Riesz wavelets. In this paper we shall comprehensively study and completely characterize all compactly supported m th-order derivative-orthogonal Riesz wavelets in the Sobolev space H m (R). More precisely, from any given compactly supported refinable vector function ϕ = (ϕ 1 , ... , ϕ r) T in H m (R) satisfying the refinement equation ϕ ˆ (2 ξ) = a ˆ (ξ) ϕ ˆ (ξ) for some r × r matrix a ˆ of 2 π -periodic trigonometric polynomials, we prove that there exists a compactly supported m th-order derivative-orthogonal Riesz wavelet in H m (R) , which is derived from ϕ through the refinable structure, if and only if the refinable vector function ϕ has stable integer shifts and the filter a has at least order 2 m sum rules. This double order of sum rules over the smoothness order m is surprising but is necessary for constructing m th-order derivative-orthogonal Riesz wavelets in H m (R). Then we shall present several examples of such derivative-orthogonal spline Riesz wavelets with short support derived from B-splines and Hermite splines. To illustrate the developed theory and its potential usefulness, we shall apply our constructed such m th-order derivative-orthogonal Riesz wavelets for the numerical solutions of differential equations such as Sturm–Liouville equations and biharmonic equations. Our constructed derivative-orthogonal spline Riesz wavelets on the interval [ 0 , 1 ] have a simple structure with only one boundary wavelet at each endpoint and can easily handle different types of boundary conditions. The resulting coefficient matrices are sparse and have very small condition numbers with some examples even having the optimal condition number one. [ABSTRACT FROM AUTHOR]

The Laguerre–Sobolev polynomials form an orthogonal polynomial system with respect to a Sobolev-type inner product associated with the Laguerre measure on the positive half-axis and two point masses M , N > 0 at the origin involving functions and derivatives. These polynomials have attracted much interest over the last two decades, since they became known to satisfy, for any value of the Laguerre parameter α ∈ N 0 , a spectral differential equation of finite order 4 α + 10. In this paper we establish a new explicit representation of the corresponding differential operator which consists of a number of elementary components depending on α , M , N. Their interaction reveals a rich structure both being useful for applications and as a model for further investigations in the field. In particular, the Laguerre–Sobolev differential operator is shown to be symmetric with respect to the Sobolev-type inner product. [ABSTRACT FROM AUTHOR]

In this paper, we prove the stability results for measure differential equations, considering more general conditions under the Lyapunov functionals and concerning the functions f and g. Moreover, we prove these stability results for the dynamic equations on time scales, using the correspondence between the solutions of these last equations and the solutions of the measure differential equations. [ABSTRACT FROM AUTHOR]

At the beginning of paper [1] there is an error that spreads along the rest of the work and the conclusions are not correct in their present form. Precisely, in Section 2, page 783, there is a contradiction related to the scaling. In the paragraph before formula (6) it is said that t → ε 3 t but Hamiltonian (6) is not scaled accordingly. We have fixed the problem and, after performing due changes, the conclusions are obtained. The existence of the manifolds at infinity is guaranteed (Theorem 3.1) and the transversal intersection of them is concluded in Theorem 5.1. The applications in Section 6 are also valid after adapting them to the new version of the theorems. [ABSTRACT FROM AUTHOR]

• A new biological layer detection strategy based on nonlinear vibration is proposed. • Vibration of Piezoelectrically driven biosensor is studied via couple stress theory. • Biological layer absorption effect on the sensor size-dependent frequency is studied. • By studying the half-stable region, the jump phenomenon in amplitude is investigated. In this paper, the nonlinear forced vibration of a piezoelectrically driven micro/nano cantilever-based biosensor is investigated via couple stress theory. The origin of nonlinear equations is the curvature, which causes cubic nonlinearities in the differential equation. Due to the dimensions of the structure and its size-dependent behavior, the governing equation is derived based on couple stress theory. In order to obtain the frequency-response equation, the differential equation is solved with the assumption of small displacement, damping coefficient, and excitation amplitude by the multiple scales method. Then, the amplitude of the response and the nonlinear resonance frequency based on the classical and couple stress theory are presented and the effect of the absorption of the biological layer on the frequency variation of the two theories is compared with each other. Also, to achieve high sensitivity resulting from optimal dimensions, the variation of beam frequency for various dimensions has been studied. Finally, by studying the bifurcation points and half-stable region, the jump phenomenon in amplitude caused by biologic layer absorption as a strategy for biological detection has been investigated and the results validated qualitatively with previous studies. [ABSTRACT FROM AUTHOR]

Summation-by-parts (SBP) operators allow us to systematically develop energy-stable and high-order accurate numerical methods for time-dependent differential equations. Until recently, the main idea behind existing SBP operators was that polynomials can accurately approximate the solution, and SBP operators should thus be exact for them. However, polynomials do not provide the best approximation for some problems, with other approximation spaces being more appropriate. We recently addressed this issue and developed a theory for one-dimensional SBP operators based on general function spaces, coined function-space SBP (FSBP) operators. In this paper, we extend the theory of FSBP operators to multiple dimensions. We focus on their existence, connection to quadratures, construction, and mimetic properties. A more exhaustive numerical demonstration of multi-dimensional FSBP (MFSBP) operators and their application will be provided in future works. Similar to the one-dimensional case, we demonstrate that most of the established results for polynomial-based multi-dimensional SBP (MSBP) operators carry over to the more general class of MFSBP operators. Our findings imply that the concept of SBP operators can be applied to a significantly larger class of methods than is currently done. This can increase the accuracy of the numerical solutions and/or provide stability to the methods. • Multi-dimensional summation-by-parts operators for general (non-polynomial) function spaces. • Existence proof that connects them to quadratures. • Construction procedure for general multi-dimensional domains/elements/blocks. • Proof of discrete conservation and energy-boundedness. • Numerical comparison demonstrating that non-polynomial function spaces can be of advantage. [ABSTRACT FROM AUTHOR]

In this paper, we propose a cusp-capturing physics-informed neural network (PINN) to solve discontinuous-coefficient elliptic interface problems whose solution is continuous but has discontinuous first derivatives on the interface. To find such a solution using neural network representation, we introduce a cusp-enforced level set function as an additional feature input to the network to retain the inherent solution properties; that is, capturing the solution cusps (where the derivatives are discontinuous) sharply. In addition, the proposed neural network has the advantage of being mesh-free, so it can easily handle problems in irregular domains. We train the network using the physics-informed framework in which the loss function comprises the residual of the differential equation together with certain interface and boundary conditions. We conduct a series of numerical experiments to demonstrate the effectiveness of the cusp-capturing technique and the accuracy of the present network model. Numerical results show that even using a one-hidden-layer (shallow) network with a moderate number of neurons and sufficient training data points, the present network model can achieve prediction accuracy comparable with traditional methods. Besides, if the solution is discontinuous across the interface, we can simply incorporate an additional supervised learning task for solution jump approximation into the present network without much difficulty. • Cusp-capturing physics informed neural network for elliptic interface problems. • Our network can present continuous solutions that inherently have discontinuous first derivatives on interfaces. • A mesh-free approach for solving PDEs with interfaces is presented. • Our network can achieve high prediction accuracy with large contrast discontinuous coefficients. [ABSTRACT FROM AUTHOR]

The paper deals with a suitable transformation related to the metric scalar of the hyper-surface so that the Raychaudhuri Equation (RE) can be written as a second order nonlinear differential equation. A first integral of this second order differential equation gives a possible analytic solution of the RE. Also, it is shown that construction of a Lagrangian (and hence a Hamiltonian) is possible, from which the RE can be derived. Wheeler-Dewitt equation has been formulated in canonical quantization scheme and norm of its solution (wave function of the universe) is shown to affect the singularity analysis in the quantum regime for any spatially homogeneous and isotropic cosmology. Finally Bohmian trajectories are formulated with causal interpretation and these quantum trajectories unlike classical geodesics obliterate the initial big-bang singularity when the quantum potential is included. [ABSTRACT FROM AUTHOR]

In this paper we study the space-homogeneous Boltzmann equation in a spatially flat FLRW spacetime. We consider Israel particles, which are the relativistic counterpart of the Maxwellian particles, and obtain global-in-time existence and the asymptotic behaviour of solutions. The main argument of the paper is to use the energy method of Guo, and we observe that the method can be applied to study small solutions in a cosmological case. It is the first result of this type where a physically well-motivated scattering kernel is considered for the general relativistic Boltzmann equation. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the investigation of various properties of transition fronts in one-dimensional nonlocal equations in heterogeneous media of ignition type, whose existence has been established by the authors of the present paper in a previous work. It is first shown that transition fronts are continuously differentiable in space with uniformly bounded and uniformly Lipschitz continuous space partial derivative. This is the first time that space regularity of transition fronts in nonlocal equations is ever studied. It is then shown that transition fronts are uniformly steep. Finally, asymptotic stability, in the sense of exponentially attracting front-like initial data, of transition fronts is studied. [ABSTRACT FROM AUTHOR]

This paper presents a new gear dynamic model for flexibly supported gear sets aiming to improve the accuracy of gear fault diagnostic methods. In the model, the operating gear centre distance, which can affect the gear design parameters, like the gear mesh stiffness, has been selected as the iteration criteria because it will significantly deviate from its nominal value for a flexible supported gearset when it is operating. The FEA method was developed for calculation of the gear mesh stiffnesses with varying gear centre distance, which can then be incorporated by iteration into the gear dynamic model. The dynamic simulation results from previous models that neglect the operating gear centre distance change and those from the new model that incorporate the operating gear centre distance change were obtained by numerical integration of the differential equations of motion using the Newmark method. Some common diagnostic tools were utilized to investigate the difference and comparison of the fault diagnostic results between the two models. The results of this paper indicate that the major difference between the two diagnostic results for the cracked tooth exists in the extended duration of the crack event and in changes to the phase modulation of the coherent time synchronous averaged signal even though other notable differences from other diagnostic results can also be observed. [ABSTRACT FROM AUTHOR]

This paper is a continuation of a previous paper by the authors. We are interested in the asymptotic behavior of eigenpairs on one dimensional linearized eigenvalue problem for Allen–Cahn equations as the diffusion coefficient tends to zero. We obtain the asymptotic profiles of all eigenfunctions by using the asymptotic formulas of corresponding eigenvalues, which have been obtained in the previous paper. Our results lead us to the concept of the classification of limiting eigenfunctions. In the case of Allen–Cahn equation it is provided by three special eigenfunctions, which correspond to the solutions of rescaled spectral problems on the whole line. [ABSTRACT FROM AUTHOR]

In this paper we continue to deal with the initial–boundary value problem for the coupled Keller–Segel–Stokes system { n t + u ⋅ ∇ n = Δ n − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , ( x , t ) ∈ Ω × ( 0 , T ) , c t + u ⋅ ∇ c = Δ c − c + n , ( x , t ) ∈ Ω × ( 0 , T ) , u t + ∇ P = Δ u + n ∇ ϕ , ( x , t ) ∈ Ω × ( 0 , T ) , ∇ ⋅ u = 0 , ( x , t ) ∈ Ω × ( 0 , T ) , where Ω ⊂ R d is a bounded domain with smooth boundary and the chemotactic sensitivity S is not a scalar function but rather attains values in R d × d , and satisfies | S ( x , n , c ) | ≤ C S ( 1 + n ) − α with some C S > 0 and α > 0 . When d = 2 , our previous work (J. Differential Equations, 2015) has established the existence of global bounded classical solutions under the subcritical assumption α > 0 , which is consistent with the corresponding results of the fluid-free system, but the method seems to be invalid in the three-dimensional setting. In this paper, for the case d = 3 , we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption α > 1 2 , which is slightly stronger than the corresponding subcritical assumption α > 1 3 on the fluid-free system, where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions. The key idea here is to establish the general L p regularity of u from a rather low L p regularity of n , which will be obtained by a new combinational functional. [ABSTRACT FROM AUTHOR]

In this paper we will find optimal lower bound for the first eigenvalue of the fourth order equation with integrable potentials when the L 1 norm of potentials is known. We establish the minimization characterization for the first eigenvalue of the measure differential equation, which plays an important role in the extremal problem of ordinary differential equation. The conclusion of this paper will illustrate a new and very interesting phenomenon that the minimizing measures will no longer be located at the center of the interval when the norm is large enough. [ABSTRACT FROM AUTHOR]

Highlights • A HSLD stiffness vibration isolator with NSS and SLS is constructed. • The proposed NSS can be accurately described by quadratic polynomial. • The MT-IHBM is extended to capture multiple harmonics in vibration simultaneously. • The NSS can improve the isolation capacity of asymmetric supporting systems effectively. Abstract In this paper, a high-static-low-dynamic stiffness (HSLDS) vibration isolator with a novel parabolic-cam-roller negative stiffness mechanism is proposed to study the effectiveness of negative stiffness on asymmetric spring supporting structures. Based on the physical model, the intrinsic geometrical nonlinearity of the isolator is analyzed and the differential equation of motion is derived by the generalized Lagrange's equation. Here, the multi-term incremental harmonic balance method (MT-IHBM) is employed and extended to capture both primary and subharmonic resonances in nonlinear response. By means of arc-length increment, the adaptability and reliability of this method are enhanced under strong nonlinear vibration. Vibration transmissibility and frequency response relationships of various frequency components are demonstrated in order to better explain the properties of asymmetry vibration. The analysis results indicate that the performance of the isolation system is affected by different parametric excitation. Under some parameters, the introduction of negative stiffness will cause obvious 1/2 subharmonic resonance. The existence of the bias term in response makes the peak and frequency of resonances shift vertically and horizontally, respectively. Due to the asymmetry, the proposed HSLDS isolator does not present quasi-zero stiffness at the static equilibrium position, but it has superior ability to suppress vibration. Meanwhile, comparisons show that adding negative stiffness mechanism can still significantly improve the vibration isolation performance when a supporting structure has the characteristic of asymmetric stiffness. [ABSTRACT FROM AUTHOR]

Abstract This paper presents a nested adaptive super-twisting sliding mode (NASTSM) control scheme for a vehicle Steer-by-Wire (SbW) system. Firstly, the plant model of the SbW system is expressed as a second-order differential equation from the steering motor input voltage to the front wheel steering angle. Specifically, the model of the self-aligning torque is elaborated in detail and compared with a simplified one. Next, an NASTSM controller is designed for the SbW system, which adopts a nested adaptive law to promote tracking accuracy by dealing with complex time-varying external disturbances and a super-twisting sliding mode (STSM) control component to guarantee strong robustness while alleviating chattering phenomenon. The stability of the NASTSM control system is verified in the sense of Lyapunov. Finally, experiments are carried out under various conditions. The experimental results show that the proposed NASTSM controller owns superiority in terms of not only higher tracking precision and stronger robustness, but also less dependence on the information of plant models compared with a conventional adaptive sliding mode (CASM) controller. [ABSTRACT FROM AUTHOR]

Abstract In this paper, the performance of vehicle suspension system under time-delayed optimal control is investigated. The effect of time delay on control stability of the active suspension system is discussed. The mathematical simulation is used to verify the correctness of the stable interval obtained by differential equation theory for linear systems with constant coefficients and time delay. In order to keep the stability of the system, time-delayed optimal control is designed through the method of state transformation and optimal control theory. The results show that the control strategy could not only guarantee the stability of the system regardless of the variation on control time delay, but also improve the performance of the suspension system. Additionally, influence of the designed active time delay on the amplitude of sprung mass acceleration, suspension deflection, rode holding are analyzed to provide guidance for its further practical engineering application. Vibration control experiments of the active suspension system under harmonic excitation with time-delayed optimal control are figured to compare with simulations. It is seen that the designed time-delayed optimal control strategy has the effectiveness and advantage. [ABSTRACT FROM AUTHOR]

Abstract In the present paper we generalize the Eulerian numbers (also of the second and third orders). The generalization is related to an autonomous first-order differential equation, solutions of which are used to obtain integral representations of some numbers, including the Bernoulli numbers. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we present a general mathematical framework for integrating smooth vector fields in the vicinity of a fixed point with a spiral saddle. We restrict our study to the three-dimensional setting, where the stable manifold is of spiral type (and thus two-dimensional), and the unstable manifold is one-dimensional. The aim is to produce a general purpose set of bounds that can be applied to any system of this type. The existence (and explicit computation) of such bounds is important when integrating along the flow near the spiral saddle fixed point. As an application, we apply our work to a concrete situation: the cubic Chua's equations. Here, we present a computer assisted proof of the existence of a trapping region for the flow. [ABSTRACT FROM AUTHOR]

Abstract This article deals with the thermodynamic equilibrium recovery mechanisms in two-phase flows and their numerical modeling. The two phases, initially at different pressures, temperatures and chemical potentials, are supposed to be driven towards equilibrium conditions by three relaxation processes. First, a mechanical process applies to relax phasic pressures, then a thermal process, to allow the sensible heat transfer between the phases at different temperatures, and, lastly, a chemical process that is responsible for the mass transfer. The two-phase flow model is composed of six partial differential equations with source terms that allow the description of mixtures at full thermodynamic disequilibrium. Its homogeneous portion is hyperbolic and it is solved by a second-order accurate finite volume scheme that uses a HLLC-type approximate Riemann solver. The source terms modeling the relaxation processes are separately integrated as three systems of ordinary differential equations. The main contributions of this paper are: the capability of describing the possibly non-instantaneous time delay of equilibrium recover in a novel way, the equation of state independence of the numerical scheme, and the possibility to take into account the morphology of the flow pattern by using the interfacial area between phases. Highlights • Development of non-instantaneous relaxation techniques for the equilibrium recovery of the two-phase flows. • Equation of State (EoS) independence of the numerical techniques proposed. • Comparison of an accurate EoS to the Stiffened Gas one. • Analysis of the thermal and chemical disequilibrium using the IAPWS-IF97 equation of state. • Validation of the quasi-instantaneous simulations against the Homogeneous Equilibrium Model (HEM). [ABSTRACT FROM AUTHOR]

Abstract This paper focuses on analysis of a friction-induced mechanical oscillator with cubic nonlinearity and an applied dynamical model of dry friction. The studied model is based on the classical LuGre approach to friction force modelling. Lyapunov exponents spectrum of the frictional oscillator is calculated by means of a recently implemented method of their estimation for non-smooth systems. Results of the numerical research, observations of the nature of the oscillator's response and interaction of the applied dynamic friction model with the oscillator are reported. Highlights • The dynamics of stick-slip oscillator with cubic nonlinearity is demonstrated. • Modified LuGre friction model is applied. • The stability of the system is proven by conditional Lyapunov exponents. • The correlation (synchrony) between oscillator response and dynamics of friction force is confirmed. [ABSTRACT FROM AUTHOR]

Abstract An analytical vibration response in the time domain for an axially translating and laterally vibrating string with mixed boundary conditions is considered in this paper. The domain of the string is a constant, dependent upon the general initial conditions. The translating tensioned strings possess different types of mixed boundary conditions, such as fixed_dashpot, fixed_spring-dashpot, fixed_mass-spring-dashpot. An analytical solution using a reflected wave superposition method is presented for a finite translating string. Firstly, the cycle of boundary reflection for strings is provided, which is dependent upon the string length. Each cycle is divided into three time intervals according to the travelling speed and direction of the string. Applying D'Alembert's principle and the reflection properties, expressions for the reflected waves under three different non-classical boundary conditions are derived. Then, the vibrational response of the axially translating string is solved for three time intervals by using a reflected wave superposition method. The accuracy and efficiency of the proposed method are confirmed numerically by comparison to simulations produced using a Newmark- β method solution. The energy expressions for a travelling string with a fixed_dashpot boundary condition is obtained and the time domain curves for the total energy and the change of energy at the boundaries are given. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we show the existence of the first nontrivial family of analytic stationary patch solutions of the SQG and gSQG equations. This answers an open problem in F. de la Hoz et al. (2016) [13]. [ABSTRACT FROM AUTHOR]

Abstract In this paper we obtain a very general Gauss–Green formula for weakly differentiable functions and sets of finite perimeter. This result is obtained by revisiting Anzellotti's pairing theory and by characterizing the measure pairing (A , D u) when A is a bounded divergence measure vector field and u is a bounded function of bounded variation. [ABSTRACT FROM AUTHOR]