Search

Abstract: In this paper we mainly study the existence of analytic normalization and the normal form of finite dimensional complete analytic integrable dynamical systems. More details, we will prove that any complete analytic integrable diffeomorphism in with B having eigenvalues not modulus 1 and is locally analytically conjugate to its normal form. Meanwhile, we also prove that any complete analytic integrable differential system in with A having nonzero eigenvalues and is locally analytically conjugate to its normal form. Furthermore we will prove that any complete analytic integrable diffeomorphism defined on an analytic manifold can be embedded in a complete analytic integrable flow. We note that parts of our results are the improvement of Moserʼs one in J. Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math. 9 (1956) 673–692 and of Poincaréʼs one in H. Poincaré, Sur lʼintégration des équations différentielles du premier order et du premier degré, II, Rend. Circ. Mat. Palermo 11 (1897) 193–239. These results also improve the ones in Xiang Zhang, Analytic normalization of analytic integrable systems and the embedding flows, J. Differential Equations 244 (2008) 1080–1092 in the sense that the linear part of the systems can be nonhyperbolic, and the one in N.T. Zung, Convergence versus integrability in Poincaré–Dulac normal form, Math. Res. Lett. 9 (2002) 217–228 in the way that our paper presents the concrete expression of the normal form in a restricted case. [Copyright &y& Elsevier]

This paper concerns steady states of the resistive incompressible Hall magnetohydrodynamics (Hall-MHD) system, with emphasis on the effect of the Hall term on the solutions. Existence of H 1 weak solutions is proved for a general domain, and an infinite number of weak solutions exist if the domain has holes. Existence of H 2 weak solutions is proved for either small data or small Hall parameter μ . The asymptotic behavior of the solutions in the vanishing Hall parameter limit is examined, and the asymptotic stability of the steady states is also investigated. [ABSTRACT FROM AUTHOR]

The efficient conditions guaranteeing the existence of a T -periodic solution to the second order differential equation u ″ = h ( t ) g ( u ) are established in the paper. Here, g is a positive and decreasing function which has a strong singularity at the origin, and the weight h ∈ L ( R / T Z ) is a sign-changing function. The obtained results have the form of relation between the multiplicities of the zeroes of the weight function h and the order of the singularity of the nonlinear term. The approach is based on Leray–Schauder degree theory, proving that no T -periodic solution of a certain homotopy appears on the boundary of an unbounded open set during the deformation to an autonomous problem. [ABSTRACT FROM AUTHOR]

The geometric Lorenz attractor is an attractor set constructed in such a way that it satisfies the main qualitative properties evidenced on the Lorenz system equations, particularly the fact that this attractor is a robustly transitive set. In this paper we prove the C 1 -robust transitivity by using geometric properties for singular hyperbolic sets and without the assumption of the uniformly linearizing coordinates around the singularity. [ABSTRACT FROM AUTHOR]

This paper is devoted to pointwise second-order necessary optimality conditions for the Mayer problem arising in optimal control theory. We first show that with every optimal trajectory it is possible to associate a solution p ( ⋅ ) of the adjoint system (as in the Pontryagin maximum principle) and a matrix solution W ( ⋅ ) of an adjoint matrix differential equation that satisfy a second-order transversality condition and a second-order maximality condition. These conditions seem to be a natural second-order extension of the maximum principle. We then prove a Jacobson like necessary optimality condition for general control systems and measurable optimal controls that may be only “partially singular” and may take values on the boundary of control constraints. Finally we investigate the second-order sensitivity relations along optimal trajectories involving both p ( ⋅ ) and W ( ⋅ ) . [ABSTRACT FROM AUTHOR]

This paper concerns the study of the asymptotic behavior of solutions to reaction–diffusion systems modeling multi-components reversible chemistry with spatial diffusion. By solution, we understand any limit of adequate approximate solutions. It is proved in any space dimension that, as time tends to infinity, the solution converges exponentially to the unique homogeneous stationary solution. We adapt and extend to any number of components, the entropy decay estimates which have been exploited for some particular 3 × 3 and 4 × 4 systems. [ABSTRACT FROM AUTHOR]

As the battle continues against hospital-acquired infections and the concurrent rise in antibiotic resistance among many of the major causative pathogens, there is a dire need to conduct controlled experiments, in order to compare proposed control strategies. However, cost, time, and ethical considerations make this evaluation strategy either impractical or impossible to implement with living patients. This paper presents a multi-scale model that offers promise as the basis for a tool to simulate these (and other) controlled experiments. This is a “unified” model in two important ways: (i) It combines inter-host and in-host dynamics into a single model, and (ii) it links two very different modeling approaches - agent-based modeling and differential equations - into a single model. The potential of this model as an instrument to combat antibiotic resistance in hospitals is demonstrated with numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a non-autonomous SIR model with diffusion. We first prove that the ( L 2 , L 2 ) pullback attractor attracts any bounded subset of L 2 in the topology of L 2 + δ for any δ ⩾ 0 . Then we show that the ( L 2 , L 2 ) pullback attractor is indeed a ( L 2 , L 2 + δ ) pullback attractor. [ABSTRACT FROM AUTHOR]

The theory of differential equations has an arithmetic analogue in which derivatives of functions are replaced by Fermat quotients of numbers. Many classical differential equations (Riccati, Weierstrass, Painlevé, etc.) were previously shown to possess arithmetic analogues. The paper introduces and studies an arithmetic analogue of the Euler differential equations for the rigid body. [ABSTRACT FROM AUTHOR]

In this paper, we define the function R q p ( α , β ; z ) and obtain its properties including analytic properties (type and order), contiguous relations, differential property and simple integrals. We also establish its relation with other well known special functions. [ABSTRACT FROM AUTHOR]

This paper studies the global (in time) regularity and large time behavior of solutions to the 2D micropolar equations with only angular viscosity dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. When there is no kinematic viscosity in the momentum equation, the global regularity problem is not easy due to the lack of suitable bounds on the derivatives. The idea here is to fully exploit the structure of the system and control the vorticity via the evolution equation of a combined quantity of the vorticity and the micro-rotation angular velocity. To understand the large time behavior, we overcome two main difficulties, the lack of kinematic viscosity and the presence of linear terms. Classical tools such as the Fourier splitting method of Schonbek and Kato's approach for the decay of small solutions do not apply here. We introduce a diagonalization process to eliminate the linear terms and rely on the uniform bounds for the first derivatives of the solutions to generate suitable decay rates. [ABSTRACT FROM AUTHOR]

In this paper, the nonlinear dynamics of a parametrically excited coupled vehicle-bridge vibration system (CVBVS) is investigated, and the coupled system is subjected to a time-dependent transverse load including a constant value together with a harmonic time-variant component. The dynamic equations of the CVBVS are established by using the generalized Lagrange's equation. With the Galerkin truncation method, a set of nonlinear ordinary differential equations are derived by discretizing the continuous governing equation. The influences of parametric excitation with nonlinear support stiffness, mass ratio, excitation amplitude and position relation on the dynamic behaviors are studied for the interaction between vehicle and the bridge. The analysis results indicate that the nonlinear dynamic characteristics are strongly attributed to the interaction of the coupled system. Nonlinear support stiffness of foundation and mass ratio can lead to complex dynamic behaviors such as jump discontinuous phenomenon, periodic, quasi-periodic and chaotic motions. Vibration amplitude increases depending on the position, where the maximum vibration displacement does not occur at the center of the bridge. The excitation amplitude has an obvious influence on the nonlinear dynamic behaviors and the increase of the excitation amplitude makes the vibration strengthen. The bifurcation diagram and 3-D frequency spectrum are used to analyze the complex nonlinear dynamic behaviors of the CVBVS. The presented results can provide an insight to the understanding of the vibration characteristics of the coupled vehicle-bridge vibration system in engineering. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial-boundary value problem for a generalized Kelvin–Voight equation with p -Laplacian and a damping term: v → t + ( v → ⋅ ∇ ) v → + ∇ P ( x , t ) = ϰ div ( ∇ v → t ) + ν div ( | ∇ v → | p − 2 ∇ v → ) + γ | v → | m − 2 v → , div v → = 0 . Here v → ( x , t ) is the velocity field, P ( x , t ) is the pressure, ν is the viscosity kinematic coefficient, and ϰ is the viscosity relaxation coefficient (is a length scale parameter characterizing the elasticity of the fluid). The coefficient γ and the exponents p , m are given constants. Under appropriate conditions on the data, we prove the existence and uniqueness of the global and local weak solutions. Under several assumptions on the exponents p , m , the coefficients ν , ϰ , and specified initial data, a finite time blow-up and the behavior of the solutions for large times are also established. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic behavior of solutions of a class of non-autonomous delay differential equations. These equations have important practical applications and generalize those on which Bernfeld and Haddock conjectured that each solution of the equations tends to a constant. It is shown that every solution of the equations is bounded and tends to a constant as t → + ∞ , which improves and extends some existing ones. [ABSTRACT FROM AUTHOR]

Exponential integrators have been introduced as an efficient alternative to explicit and implicit methods for integrating large stiff systems of differential equations. Over the past decades these methods have been studied theoretically and their performance was evaluated using a range of test problems. While the results of these investigations showed that exponential integrators can provide significant computational savings, the research on validating this hypothesis for large scale systems and understanding what classes of problems can particularly benefit from the use of the new techniques is in its initial stages. Resistive magnetohydrodynamic (MHD) modeling is widely used in studying large scale behavior of laboratory and astrophysical plasmas. In many problems numerical solution of MHD equations is a challenging task due to the temporal stiffness of this system in the parameter regimes of interest. In this paper we evaluate the performance of exponential integrators on large MHD problems and compare them to a state-of-the-art implicit time integrator. Both the variable and constant time step exponential methods of EPIRK-type are used to simulate magnetic reconnection and the Kevin–Helmholtz instability in plasma. Performance of these methods, which are part of the EPIC software package, is compared to the variable time step variable order BDF scheme included in the CVODE (part of SUNDIALS) library. We study performance of the methods on parallel architectures and with respect to magnitudes of important parameters such as Reynolds, Lundquist, and Prandtl numbers. We find that the exponential integrators provide superior or equal performance in most circumstances and conclude that further development of exponential methods for MHD problems is warranted and can lead to significant computational advantages for large scale stiff systems of differential equations such as MHD. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the global existence and uniqueness of strong solutions to the initial boundary value problem for the 3D compressible Navier–Stokes equations without heat conductivity in a bounded domain with slip boundary. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in H 2 ( Ω ) . Furthermore, the exponential convergence rates of the pressure and velocity are also proved by delicate energy methods. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the behavior of certain solutions to the Swift–Hohenberg equation on a one-dimensional torus T . Combining results from Γ-convergence and ODE theory, it is shown that solutions corresponding to initial data that is L 1 -close to a jump function v , remain close to v for large time. This can be achieved by regarding the equation as the L 2 -gradient flow of a second order energy functional, and obtaining asymptotic lower bounds on this energy in terms of the number of jumps of v . [ABSTRACT FROM AUTHOR]

This paper deals with an accelerated iterative procedure for a coupled system of partial differential equations arising from a catalytic converter model. Based on the main theorem, a finite difference based numerical method is developed. The monotone property as well as the convergence analysis and the error estimate of the proposed finite difference scheme are proved theoretically. The efficiency of the proposed scheme is illustrated by providing a comparative numerical study with the existing method. The proposed iterative procedure reduces the number of iterations at least by forty percent for certain benchmarks available in the literature. [ABSTRACT FROM AUTHOR]

This paper is concerned with 1-parameter families of periodic solutions of piecewise smooth planar vector fields, when they behave like a center of smooth vector fields. We are interested in finding a separation boundary for a given pair of smooth systems in such a way that the discontinuous system, formed by the pair of smooth systems, has a continuum of periodic orbits. In this case we call the separation boundary as a center boundary . We prove that given a pair of systems that share a hyperbolic focus singularity p 0 , with the same orientation and opposite stability, and a ray Σ 0 with endpoint at the singularity p 0 , we can find a smooth manifold Ω such that Σ 0 ∪ { p 0 } ∪ Ω is a center boundary. The maximum number of such manifolds satisfying these conditions is five. Moreover, this upper bound is reached. [ABSTRACT FROM AUTHOR]

Abstract: This paper extends the classical result of Fisher (1937) from the case of two alleles to the case of multiple alleles. Consider a population living in a homogeneous one-dimensional infinite habitat. Individuals in this population carry a gene that occurs in k forms, called alleles. Under the joint action of migration and selection and some additional conditions, the frequencies of the alleles, , satisfy a system of differential equations of the form (1.2). In this paper, we first show that under the conditions is the most fit among the homozygotes, (1.2) is cooperative, the state that only allele A 1 is present in the population is stable, and the state that allele A 1 is absent and all other alleles are present in the population is unstable, then there exists a positive constant, c ⁎, such that allele A 1 propagates asymptotically with speed c ⁎ in the population as . We then show that traveling wave solutions connecting these two states exist for . Finally, we show that under certain additional conditions, there exists an explicit formula for c ⁎. These results allow us to estimate how fast an advantageous gene propagates in a population under selection and migration forces as . Selection is one of the major evolutionary forces and understanding how it works will help predict the genetic makeup of a population in the long run. [Copyright &y& Elsevier]

In this paper a comprehensive and structured analysis of the stabilizing properties linked to the combination of different chatter suppression techniques in milling was carried out. Specifically, the spindle speed variation, the stiffness variation and the usage of tools with variable pitch and variable helixangles were considered. For this purpose, a unique, general and efficient frequency domain formulation (extended harmonic solution) was developed. It is suitable for dealing with differential delayed equations characterized by multiple and distributed time dependent delays and time periodic dynamics. It was demonstrated that the developed solution converges to the reference cases, which were studied in literature through semi-discretization and its extensions. Moreover, by selecting a proper number of considered harmonics, the computation time was reduced of one order of magnitude, in the face of a slight reduction of accuracy with respect to time domain approaches. The computed stability lobe diagrams demonstrated that the combination of multi-chatter suppression techniques allows exploiting the best features of each methodology. The simultaneous use of the spindle speed variation and the stiffness variation was studied for the first time. The absolute limit of stability increased up to 204%, much more than by adopting the techniques in a separate manner. If the enlargement of the stability region is considered as the reference target, the best results were granted by the combination of all the three strategies. Indeed, the improvement with respect to the use of only two techniques was at least 10%. • Stability of multiple point, distributed, time varying delay differential equation. • Frequency domain formulation for complex milling applications. • Investigation of novel chatter suppression technique combinations. • Improvement of chatter stability with respect to previous results in literature. [ABSTRACT FROM AUTHOR]

• Eigenproperties of a multi-cracked Euler-Bernoulli inextensible circular arch. • Cracks are accounted for in governing equation by means of the distribution theory. • Governing equation defined over a unique integration domain regardless n° cracks. • Integration via Laplace transform is adopted to infer closed form mode shapes. • Both validations and parametric analyses are presented. Despite the numerous explicit solutions of free vibration of arches with regular cross sections, in case of concentrated defects such as cracks, no procedure is available to analyse arch vibrations without sub-division of the integration domain. As a result, curved sub-elements comprised between crack and external constraints, or successive cracks, are considered. In this paper a distributional approach is adopted to provide a formulation of the free vibration differential governing equations of circular inextensible arches over a unique integration domain in presence of multiple concentrated open (non-breathing) cracks. Discontinuities due to the presence of an arbitrary number of cracks are modelled by means of Dirac's deltas. An integration procedure is devised to offer closed form solutions of the relevant vibration modes together with the relevant frequency determinantal equation. Natural frequencies and mode shapes of damaged arches with different damage and restraint configurations have been evaluated and compared with experimental results available in the literature as well as finite element numerical simulations. The presented closed form solutions are also employed for two parametric studies to evaluate the influence of an increasing number of along axis concentrated cracks as well as of the location of cracks along the arch span. [ABSTRACT FROM AUTHOR]

This paper analyzes the mechanical and mathematical models in “Ritto et al. (2013) [1]”. The results are that: (1) the mechanical model is obviously incorrect; (2) the mathematical model is not complete; (3) the differential equation is obviously incorrect; (4) the finite element equation is obviously not discretized from the corresponding mathematical model above, and is obviously incorrect. A mathematical model of dynamics should include the differential equations, the boundary conditions and the initial conditions. [ABSTRACT FROM AUTHOR]

In this paper, the formation of singularities for the nonlocal Whitham-type equations is studied. It is shown that if the lowest slope of flows can be controlled by its highest value with the bounded Whitham-type integral kernel initially, then the finite-time blow-up will occur in the form of wave-breaking. This refined wave-breaking result is established by analyzing the monotonicity and continuity properties of a new system of the Riccati-type differential inequalities involving the extremal slopes of flows. Our theory is illustrated via the Whitham equation, Camassa–Holm equation, Degasperis–Procesi equation, and their μ -versions as well as hyperelastic rod equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the cost of null controllability of a fourth-order parabolic equation. When the control time is large enough, we prove that this cost decreases exponentially to zero as the diffusion coefficient of the equation vanishes. When the control time is small, on the contrary, we prove that this cost increases exponentially to infinity. [ABSTRACT FROM AUTHOR]

In this paper we consider an energy critical wave equation ( 3 ≤ d ≤ 5 , ζ = ± 1 ) ∂ t 2 u − Δ u = ζ ϕ ( x ) | u | 4 / ( d − 2 ) u , ( x , t ) ∈ R d × R with initial data ( u , ∂ t u ) | t = 0 = ( u 0 , u 1 ) ∈ H ˙ 1 × L 2 ( R d ) . Here ϕ ∈ C ( R d ; ( 0 , 1 ] ) converges as | x | → ∞ and satisfies certain technical conditions. We generalize Kenig and Merle's results on the Cauchy problem of the equation ∂ t 2 u − Δ u = | u | 4 / ( d − 2 ) u . Following a similar compactness-rigidity argument we prove that any solution with a finite energy must scatter in the defocusing case ζ = − 1 . While in the focusing case ζ = 1 we give a criterion for global behaviour of the solutions, either scattering or finite-time blow-up when the energy is smaller than a certain threshold. As an application we give a similar criterion on the global behaviour of radial solutions to the focusing, energy critical shifted wave equation ∂ t 2 v − ( Δ H 3 + 1 ) v = | v | 4 v on the hyperbolic space H 3 . [ABSTRACT FROM AUTHOR]

In this paper, we provide two-sided estimates and uniform asymptotics for the solution of d -dimensional critical fractal Burgers equation u t − Δ α / 2 u + b ⋅ ∇ ( u | u | q ) = 0 , α ∈ ( 1 , 2 ) , b ∈ R d for q = ( α − 1 ) / d and u 0 ∈ L 1 ( R d ) . We consider also q > ( α − 1 ) / d under additional condition u 0 ∈ L ∞ ( R d ) . In both cases we assume u 0 ≥ 0 , which implies that the solution is non-negative. The estimates are given in the terms of the function P t u 0 , where P t denotes the semigroup for the operator ∂ t − Δ α / 2 . [ABSTRACT FROM AUTHOR]

In this paper, we prove the solution map of the Cauchy problem of Camassa–Holm type equations depends continuously on the initial data in nonhomogeneous Besov spaces in the sense of Hadamard by using the Littlewood–Paley theory and the method introduced by Kato [37] and Danchin [21] . [ABSTRACT FROM AUTHOR]

The inverse problem of determining the location and size of a point mass attached on a simply supported, isotropic and homogeneous rectangular plate from minimal natural frequency data is considered in this paper. Under the assumption that the size of the mass is small compared to the total mass of the plate, we show that the problem can be formulated and solved in closed form in terms of point mass-induced changes on the first three natural frequencies. Numerical simulations indicate that the method allows for accurate identification, provided that measurement/modelling errors are smaller than eigenfrequency changes. [ABSTRACT FROM AUTHOR]

This paper investigated the free vibration analysis of functionally graded piezoelectric materials (FGPMs) nanoscale plates based on the Eringen's nonlocal Kirchhoff plate theory under simply supported edge conditions. The material properties vary continuously along the thickness direction based on the power-law distribution in terms of the volume fractions of the constituents. The material compositions are selected from the PZT family. Using Hamilton's principle, the governing differential equations are derived and the Navier's solution is used to attain the natural frequencies. The accuracy of the method is validated by comparing the results with the previous studies. Finally, the effects of the nonlocal parameter, various gradient indexes, mode numbers, aspect ratio and side-to-thickness ratio on natural frequencies are also studied. [ABSTRACT FROM AUTHOR]

As an application of a general left-definite spectral theory, Everitt, Littlejohn and Wellman, in 2002, developed the left-definite theory associated with the classical Legendre self-adjoint second-order differential operator A in L 2 ( − 1 , 1 ) which has the Legendre polynomials { P n } n = 0 ∞ as eigenfunctions. As a consequence, they explicitly determined the domain D ( A 2 ) of the self-adjoint operator A 2 . However, this domain, in their characterization, does not contain boundary conditions. In fact, this is a general feature of the left-definite approach developed by Littlejohn and Wellman. Yet, the square of the second-order Legendre expression is in the limit-4 case at each end point x = ± 1 in L 2 ( − 1 , 1 ) so D ( A 2 ) should exhibit four boundary conditions. In this paper, we show that this domain can, in fact, be expressed using four separated boundary conditions using the classical GKN (Glazman–Krein–Naimark) theory. In addition, we determine a new characterization of D ( A 2 ) that involves four non-GKN boundary conditions. These new boundary conditions are surprisingly simple – and natural – and are equivalent to the boundary conditions obtained from the GKN theory. [ABSTRACT FROM AUTHOR]

Devices that emit structure-borne sound are commonly decoupled by elastic components to shield the environment from acoustical noise and vibrations. The elastic elements often have a hysteretic behavior that is typically neglected. In order to take hysteretic behavior into account, Bouc developed a differential equation for such materials, especially joints made of rubber or equipped with dampers. In this work, the Bouc model is solved by means of the Gaussian closure technique based on the Kolmogorov equation. Kolmogorov developed a method to derive probability density functions for arbitrary explicit first-order vector differential equations under white noise excitation using a partial differential equation of a multivariate conditional probability distribution. Up to now no analytical solution of the Kolmogorov equation in conjunction with the Bouc model exists. Therefore a wide range of approximate solutions, especially the statistical linearization, were developed. Using the Gaussian closure technique that is an approximation to the Kolmogorov equation assuming a multivariate Gaussian distribution an analytic solution is derived in this paper for the Bouc model. For the stationary case the two methods yield equivalent results, however, in contrast to statistical linearization the presented solution allows to calculate the transient behavior explicitly. Further, stationary case leads to an implicit set of equations that can be solved iteratively with a small number of iterations and without instabilities for specific parameter sets. [ABSTRACT FROM AUTHOR]

In this paper we study the dynamics of parabolic semilinear differential equations with homogeneous Dirichlet boundary conditions via the discretization of finite element method. We provide an appropriate functional setting to treat this problem and, as a first step, we show the continuity of the set of equilibria and of its linear unstable manifolds. [ABSTRACT FROM AUTHOR]

Recently, many new classes of differentially 4-uniform permutations have been constructed. However, it is difficult to decide whether they are CCZ-inequivalent or not. In this paper, we propose a new notion called “Projected Differential Spectrum”. By considering the properties of the projected differential spectrum, we find several relations that should be satisfied by CCZ-equivalent functions. Based on these results, we mathematically prove that any differentially 4-uniform permutation constructed in [11] by C. Carlet, D. Tang, X. Tang, et al., is CCZ-inequivalent to the inverse function. We also get two interesting results with the help of computer experiments. The first one is a proof that any permutation constructed in [11] is CCZ-inequivalent to a function which is the summation of the inverse function and any Boolean function on F 2 2 k when 4 ≤ k ≤ 7 . The second one is a differentially 4-uniform permutation on F 2 6 which is CCZ-inequivalent to any function in the aforementioned two classes. [ABSTRACT FROM AUTHOR]

A.M. Legendre noted that Euler's pentagonal number theorem implies that the number of partitions of n into an even number of distinct parts almost always equals the number of partitions of n into an odd number of distinct parts (the exceptions occur when n is a pentagonal number). Subsequently other classes of partitions, including overpartitions, have yielded related Legendre theorems. In this paper, we examine four subclasses of overpartitions that have surprising Legendre theorems. [ABSTRACT FROM AUTHOR]

In 1941, Claude Shannon introduced the General Purpose Analog Computer (GPAC) as a mathematical model of Differential Analysers, that is to say as a model of continuous-time analog (mechanical, and later on electronic) machines of that time. Following Shannon’s arguments, functions generated by the GPAC must satisfy a polynomial differential algebraic equation (DAE). As it is known that some computable functions like Euler’s Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t or Riemann’s Zeta function ζ ( x ) = ∑ k = 0 ∞ 1 k x do not satisfy any polynomial DAE, this argument has often been used to demonstrate in the past that the GPAC is less powerful than digital computation. It was proved in Bournez et al. (2007), that if a more modern notion of computation is considered, i.e. in particular if computability is not restricted to real-time generation of functions, the GPAC is actually equivalent to Turing machines. Our purpose is first to discuss the robustness of the notion of computation involved in Bournez et al. (2007), by establishing that many natural variants of the notion of computation from this paper lead to the same computability result. Second, to go from these computability results towards considerations about (time) complexity: we explore several natural variants for measuring time/space complexity of a computation. Quite surprisingly, whereas defining a robust time complexity for general continuous time systems is a well known open problem, we prove that all variants are actually equivalent even at the complexity level. As a consequence, it seems that a robust and well defined notion of time complexity exists for the GPAC, or equivalently for computations by polynomial ordinary differential equations. Another side effect of our proof is also that we show in some way that polynomial ordinary differential equations can actually be used as a kind of programming model, and that there is a rather nice and robust notion of ordinary differential equation (ODE) programming. [ABSTRACT FROM AUTHOR]

Classical theories predict that solutions of differential equations will leave any neighborhood of a stable limit cycle, if white noise is added to the system. In reality, many engineering systems modeled by second order differential equations, like the van der Pol oscillator, show incredible robustness against noise perturbations, and the perturbed trajectories remain in the neighborhood of a stable limit cycle for all times of practical interest. In this paper, we propose a new model of noise to bridge this apparent discrepancy between theory and practice. Restricting to perturbations from within this new class of noise, we consider stochastic perturbations of second order differential systems that –in the unperturbed case– admit asymptotically stable limit cycles. We show that the perturbed solutions are globally bounded and remain in a tubular neighborhood of the underlying deterministic periodic orbit. We also define stochastic Poincaré map(s), and further derive partial differential equations for the transition density function. [ABSTRACT FROM AUTHOR]

By sub-diffusion systems we mean to replace the first order time derivative of normal diffusion system with a Riemann–Liouville time fractional derivative of order α ∈ ( 0 , 1 ) . In this paper, we first introduce a new definition of regional gradient controllability for the sub-diffusion system with the control inputs appearing in the differential equations as distributed inputs, which recovers the usual definition of regional gradient controllability as α → 1 . The sufficient and necessary conditions on actuators to achieve the gradient controllability in a given subregion of the whole domain are presented. An approach to guarantee the regional gradient controllability of the problems studied within the considered subregion using minimum energy control effort is presented. Several examples are presented in the end to illustrate the effectiveness of our results, where zone actuators, pointwise actuators or filament actuators are respectively considered. [ABSTRACT FROM AUTHOR]

In this paper, we study existence of solutions for the following elliptic problem, related to mean-field games systems: { − div ( M ( x ) ∇ ζ ) + ζ − div ( ζ A ( x ) ∇ u ) = f in Ω , − div ( M ( x ) ∇ u ) + u + θ A ( x ) ∇ u ⋅ ∇ u = ζ p in Ω , ζ = 0 = u on ∂ Ω , where p > 0 , 0 < θ < 1 , and f ≥ 0 is a function in some Lebesgue space. [ABSTRACT FROM AUTHOR]

In this paper we consider the elastic energy for open curves in Euclidean space subject to clamped boundary conditions and obtain the Łojasiewicz–Simon gradient inequality for this energy functional. Thanks to this inequality we can prove that a (suitably reparametrized) solution to the associated L 2 -gradient flow converges for large time to an elastica, that is to a critical point of the functional. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some new estimates for the trace and inverse trace of Steklov eigenvalues. The estimates generalize some previous results of Hersch–Payne–Schiffer [13] , Brock [2] , Raulot–Savo [21] and Dittmar [5] . [ABSTRACT FROM AUTHOR]

Abstract: This paper considers stochastic population dynamics driven by Lévy noise. The contributions of this paper lie in that: (a) Using the Khasminskii–Mao theorem, we show that the stochastic differential equation associated with our model has a unique global positive solution; (b) Applying an exponential martingale inequality with jumps, we discuss the asymptotic pathwise estimation of such a model. [Copyright &y& Elsevier]

Abstract: A continuous ordinary vector differential equation in Euclidean space has a funnel of solutions through each initial condition. Its cross-section at time t is a continuum. Many continua are known to be funnel sections: For instance the circle is a cross-section of a continuous ODE where y is a variable in the plane, but it is not known whether every Jordan curve J is a planar funnel section. In this paper we give sufficient conditions that imply J is a planar funnel section – “pierceability.” We show that pierceability is not generic when we put a fairly interesting complete metric on the space of Jordan curves. We also give proofs of several statements in the first authorʼs paper on funnel sections that appeared in the JDE in 1975. [Copyright &y& Elsevier]

Abstract: In this paper we study a free boundary problem modeling solid avascular tumor growth. The model is based on the reaction–diffusion dynamics and mass conservation law. The model is considered with time delays in proliferation process. The quasi-steady-state (i.e., ) is studied by Foryś and Bodnar [see U. Foryś, M. Bodnar, Time delays in proliferation process for solid avascular tumour, Math. Comput. Modelling 37 (2003) 1201–1209]. In this paper we mainly consider the case . In the case considered by Foryś and Bodnar, the model is reduced to an ordinary differential equation with time delay, but in the case the model cannot be reduced to an ordinary differential equation with time delay. By theory of parabolic equations and the Banach fixed point theorem, we prove the existence and uniqueness of a local solutions and apply the continuation method to get the existence and uniqueness of a global solution. We also study the long time asymptotic behavior of the solutions under some conditions. [Copyright &y& Elsevier]

Abstract: This paper deals with the existence and stability of periodic pyramidal traveling fronts for reaction–diffusion equations with bistable time-periodic nonlinearity in with . It is well known that two-dimensional periodic traveling curved fronts exist and are stable. In this paper, by constructing various of supersolutions and subsolutions, we first show that there exist three-dimensional periodic pyramidal traveling fronts, and then we prove that such periodic pyramidal traveling fronts are asymptotically stable. Finally, we further prove that our existence result holds for with . [Copyright &y& Elsevier]

Abstract: This paper concerns the Cauchy problem for homogeneous weakly hyperbolic equations with time depending analytic coefficients. We give a sufficient condition for the -well-posedness which is also necessary if the space dimension is equal to one. The main point of the paper consists in expressing our condition only in terms of the coefficients of the operator, without needing to know the behavior of the characteristic roots. This is made possible by using the so-called standard symmetrizer of a companion hyperbolic matrix. [Copyright &y& Elsevier]

Abstract: This paper deals with a reinforcement problem for a plane domain whose boundary is a deterministic or random “mixture” of self-similar Koch curves. We construct an ε-thin polygonal 2-dimensional fiber , , , around pre-fractal approximating domains and related suitable energy functionals. The aim of this paper is to study the asymptotic behavior of the reinforced energy functionals while, simultaneously, the thickness of the fibers and the conductivity of the functionals on the fibers converges to 0 as . [Copyright &y& Elsevier]

Abstract: The functions of computable analysis are defined by enhancing normal Turing machines to deal with real number inputs. We consider characterizations of these functions using function algebras, known as real recursive functions. Since there are numerous incompatible models of computation over the reals, it is interesting to find that the two very different models we consider can be set up to yield exactly the same functions. Bournez and Hainry used a function algebra to characterize computable analysis, restricted to the twice continuously differentiable functions with compact domains. In our earlier paper, we found a different (and apparently more natural) function algebra that also yields computable analysis, with the same restriction. In this paper we improve earlier work, finding three function algebras characterizing computable analysis, removing the restriction to twice continuously differentiable functions and allowing unbounded domains. One of these function algebras is built upon the widely studied real primitive recursive functions. Furthermore, the proof of this paper uses our previously developed method of approximation, whose applicability is further evidenced by this paper. [Copyright &y& Elsevier]

Abstract: In this paper a Human–Vehicle–Road (HVR) model, comprising a quarter-car and a biomechanical representation of the driver, is employed for the analysis. Differential equations are provided to describe the motions of various masses under the influence of a harmonic road excitation. These equations are, subsequently, solved to obtain a closed form mathematical expression for the steady-state vertical acceleration measurable at the vehicle–human interface. The solution makes it possible to find optimal parameters for the vehicle suspension system with respect to a specified ride comfort level. The quantitative definition given in the ISO 2631 standard for the ride comfort level is adopted in this paper for the optimization procedure. Numerical examples, based on actually measured road profiles, are presented to prove the validity of the proposed approach and its suitability for the problem at hand. [Copyright &y& Elsevier]

Abstract: Bovine fertility is the subject of extensive research in animal sciences, especially because fertility of dairy cows has declined during the last decades. The regulation of estrus is controlled by the complex interplay of various organs and hormones. Mathematical modeling of the bovine estrous cycle could help in understanding the dynamics of this complex biological system. In this paper we present a mechanistic mathematical model of the bovine estrous cycle that includes the processes of follicle and corpus luteum development and the key hormones that interact to control these processes. The model generates successive estrous cycles of 21 days, with three waves of follicle growth per cycle. The model contains 12 differential equations and 54 parameters. Focus in this paper is on development of the model, but also some simulation results are presented, showing that a set of equations and parameters is obtained that describes the system consistent with empirical knowledge. Even though the majority of the mechanisms that are included in the model are based on relations that in the literature have only been described qualitatively (i.e. stimulation and inhibition), the output of the model is surprisingly well in line with empirical data. This model of the bovine estrous cycle could be used as a basis for more elaborate models with the ability to study effects of external manipulations and genetic differences. [Copyright &y& Elsevier]