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The aim of this paper is to establish some sufficient conditions ensuring the existence and uniqueness of positive (Bohr) almost periodic solutions to a class of semilinear evolution equations of the form: u′(t)=A(t)u(t)+f(t,u(t)),t∈ℝ$$ {u}^{\prime }(t)=A(t)u(t)+f\left(t,u(t)\right),t\in \mathbb{R} $$. We assume that the family of closed linear operators (A(t))t∈ℝ$$ {\left(A(t)\right)}_{t\in \mathbb{R}} $$ on a Banach lattice X$$ X $$ satisfies the "Acquistapace–Terreni" conditions, so that the associated evolution family is positive and has an exponential dichotomy on ℝ$$ \mathbb{R} $$. The nonlinear term f$$ f $$, acting on certain real interpolation spaces, is assumed to be almost periodic only in a weaker sense (i.e., in Stepanov's sense) with respect to t$$ t $$, and Lipschitzian in bounded sets with respect to the second variable. Moreover, we prove a new composition result for Stepanov almost periodic functions by assuming only continuity of f$$ f $$ with respect to the second variable (see the condition Lemma 1‐(ii)). Finally, we provide an application to a system of Lotka–Volterra predator–prey type model with diffusion and time–dependent parameters in a generalized almost periodic environment. [ABSTRACT FROM AUTHOR]

In this paper, we study a quasilinear parabolic equation with free boundary. A priori estimates of the H¨older norms are established. On the basic of a priori estimates, the existence and uniqueness of the solution is proved. Furthermore, we consider spreading-vanishing dichotomy result with advection term, specifically the species either successfully spreads to the infinity as t → ∞, or it fails to establish and does out in the long run. [ABSTRACT FROM AUTHOR]

For finite-dimensional linear differential systems with bounded coefficients, we prove that their exponential dichotomy on $\mathbb{R}$ is equivalent to their Ulam–Hyers stability on $\mathbb{R}$ with uniqueness. We also consider abstract non-autonomous evolution equations which are exponentially bounded and exponentially dichotomic and prove that Ulam–Hyers stability with uniqueness is maintained when perturbing them with a nonlinear term having a sufficiently small Lipschitz constant.

The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study the asymptotic stability of evolution systems of probability measures of time inhomogeneous transition operators for nonautonomous stochastic systems. Two general theoretical results on this topic are established in a Polish space by establishing some sufficient conditions which can be verified in applications. Our abstract results are applied to a stochastic lattice reaction-diffusion equation driven by a time-dependent nonlinear noise. A time-average argument and an extended Krylov-Bogolyubov method due to Da Prato and Röckner [ Seminar on stochastic analysis, random fields and applications V , Birkhäuser, Basel, 2008] are employed to prove the existence of evolution systems of probability measures. A mild condition on the time-dependent diffusion function is used to prove that the limit of every evolution system of probability measures must be an evolution system of probability measures of the limiting equation. The theoretical results are expected to be applied to various stochastic lattice systems/ODEs/PDEs in the future. [ABSTRACT FROM AUTHOR]

We study linear and semi-linear differential-algebraic equations (DAEs) on the half-line ℝ + . In our strategy, we first show a characterization for the existence of exponential dichotomy for linear DAEs based on the Lyapunov-Perron method. Then, we prove the existence of invariant stable manifolds for semi-linear DAEs in the case that the evolution family corresponding to a linear DAE admits an exponential dichotomy and the non-linear forcing function fulfills the non-uniform φ-Lipschitz condition where the Lipschitz function φ belongs to wide classes of admissible function spaces such as Lp, 1 ≤ p ≤ ∞ , and Lp,q. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic dynamics of a time-periodic parasitic–mutualistic model of mistletoes and birds in heterogeneous environment with the especial concerns over the spreading–vanishing scenarios, in which the Stefan class free boundary is introduced as the spreading frontier. By defining the ecological reproduction number and generalizing it as the spatial-temporal risk index, a considerably universal spreading–vanishing dichotomy and the sharp criteria are first established in birds world in the absence of mistletoes, and some estimates of the asymptotic spreading speed of the free boundary provided that spreading occurs are also obtained. Furthermore, the comprehensive considerations containing the spreading frontiers, asymptotic profiles and estimates of the asymptotic spreading speed are exhibited in mistletoes-birds world by the monotone iteration technique with the proper upper and lower solutions. The results suggest that even for the spreading case, the mistletoes population will eventually persist in long term provided that its own risk index is larger than 1, otherwise it may be eradicated. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of extremals for the Trudinger–Moser functional under the weighted Sobolev norm involved with the trapping potential. Since the trapping potential is non-radial, possible extremals of the Trudinger–Moser functional do not have to be radially symmetric. As a result, one can't expect to apply the symmetrization argument to obtain the existence of extremal functions for these inequalities. Thus, the standard blow-up analysis relying on the symmetrization argument no longer works. In this paper, we will develop a new symmetrization-free approach to establish the existence of an extremal (Theorem 1.1) We first show that if the maximizing sequence does not converge strongly to an extremal function, then it must be a concentration or vanishing sequence through excluding the dichotomy phenomenon (see Lemma 2.9). Since the vanishing or concentration sequence may not be radially symmetric, this causes much challenge to exclude the vanishing or concentration phenomenon of the maximizing sequence. Then the elimination of the vanishing phenomenon can be done by comparing the supremums of the Trudinger–Moser functional under the constraint of the classical Sobolev norm (i.e., S ∞ (α) ) and weighted Sobolev norm associated with the potential (i.e., S (V , α) ) respectively. For the elimination of the concentration phenomenon of the maximizing sequence, we will develop the blow-up analysis procedure for the non-radial maximizing sequence. Furthermore, we also study the existence of extremals for the perturbed Trudinger–Moser inequality in R 2 involved with trapping potential (see Theorem 1.3). While an extremal function of the classical Trudinger–Moser and the perturbed Trudinger–Moser inequality does not exist in some situations, surprisingly, the supremum of the Trudinger–Moser and the perturbed Trudinger–Moser functional involved with the trapping potential can always be achieved. [ABSTRACT FROM AUTHOR]

In a previous work, we proved an index theorem for families of asymptotically hyperbolic discrete dynamical systems and obtained applications to bifurcation theory. A weaker and far more common assumption than asymptotic hyperbolicity is the existence of an exponential dichotomy. In this paper we generalize all our previous results to the latter setting, which requires substantial modifications of our arguments. In addition, we generalize previous results on continuity and differentiability of Nemitski operators for discrete dynamical systems to further weaken the assumptions of our bifurcation theorems. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a new and complete study regarding the detection of nonuniform, uniform and exponential trichotomies of discrete dynamical systems in infinite dimensional spaces, by means of control methods of input-output type. We show for the first time that an admissibility of nonuniform nature implies the existence of a nonuniform trichotomy and we deduce the structures of the trichotomy projections. After that, we prove criteria for uniform trichotomy of discrete variational systems. Next, for p , q ∈ [ 1 , ∞ ] we introduce the notion of uniform admissibility of a (ℓ p , ℓ q) pair and we obtain complete characterizations for uniform exponential trichotomy. Throughout our study, via illustrative examples, we show that the working hypotheses are minimal and cannot be removed. All the criteria are obtained in the most general framework, in which the base space is an arbitrary set and without any additional assumption on the system coefficients. [ABSTRACT FROM AUTHOR]

We provide the necessary and sufficient conditions of Liouvillian integrability for Liénard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Liénard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if the degree of a polynomial responsible for the restoring force is greater than the degree of a polynomial producing the damping, then a generic Liénard differential system is not Liouvillian integrable with the exception of linear Liénard systems. However, for any fixed degrees of the polynomials describing the damping and the restoring force we present subfamilies possessing Liouvillian first integrals. As a by‐product of our results, we find a number of novel Liouvillian integrable subfamilies. In addition, we study the existence of nonautonomous Darboux first integrals and nonautonomous Jacobi last multipliers with a time‐dependent exponential factor. [ABSTRACT FROM AUTHOR]

This paper presents integral charaterizations for nonuniform dichotomy with growth rates and their correspondents for the particular cases of nonuniform exponential dichotomy and nonuniform polynomial dichotomy of skew-evolution cocycles in Banach spaces. The connections between these three concepts are presented. [ABSTRACT FROM AUTHOR]

We will give a general criterion - the existence of an F-obstruction - for showing that a subgroup of PL+I does not embed into Thompson's group F. An immediate consequence is that Cleary's "golden ratio" group Ft does not embed into F. Our results also yield a new proof that Stein's groups Fp,q do not embed into F, a result first established by Lodha using his theory of coherent actions. We develop the basic theory of F-obstructions and show that they exhibit certain rigidity phenomena of independent interest. In the course of establishing the main result of the paper, we prove a dichotomy theorem for subgroups of PL+I. In addition to playing a central role in our proof, it is strong enough to imply both Rubin's Reconstruction Theorem restricted to the class of subgroups of PL+I and also Brin's Ubiquity Theorem. [ABSTRACT FROM AUTHOR]

To any one-sided nonautonomous dynamics obtained from a delay-difference equation, we associate a lifted autonomous delay-difference equation that can be used, together with its solutions and its perturbations, to characterize the hyperbolicity in three different ways: in terms of the hyperbolicity of the corresponding evolution map, which describes the solutions of the lifted autonomous equation; in terms of an admissibility property for this evolutions map, which corresponds to the invertibility of a certain linear operator; and in terms of an admissibility property for the perturbations of the lifted delay-difference equation. We emphasize that the evolution map is necessarily noninvertible in our setting, which led us to introduce a new notion of exponential dichotomy for this map whose existence is equivalent to the existence of an exponential dichotomy for the original nonautonomous dynamics. [ABSTRACT FROM AUTHOR]

We prove the existence and attraction property for admissible invariant unstable and center-unstable manifolds of admissible classes of solutions to the partial neutral functional-differential equation in a Banach space X of the form ∂ ∂ t F u t = A t F u t + f t , u t , t ≥ s , t , s ∈ R , u s = ϕ ∈ C : = C - r , 0 , X under the conditions that the family of linear partial differential operators (A(t))t∈ℝ generates the evolution family (U(t, s))t≥s with exponential dichotomy on the whole line ℝ; the difference operator F : C → X is bounded and linear, and the nonlinear delay operator f satisfies the φ-Lipschitz condition, i.e., ‖ f t , ϕ - f t , ψ ‖ ≤ φ t ‖ ϕ - ψ ‖ C for ϕ, ψ ∈ 풞, where φ(·) belongs to an admissible function space defined on ℝ. We also prove that an unstable manifold of the admissible class attracts all other solutions with exponential rates. Our main method is based on the Lyapunov – Perron equation combined with the admissibility of function spaces. We apply our results to the finite-delay heat equation for a material with memory. [ABSTRACT FROM AUTHOR]

In this article, we investigate the existence and uniqueness of solutions of linear and semilinear second–order equations involving time scales. To obtain such results, we make use of exponential dichotomy and fixed point results. Also, we present some examples and applications to illustrate our main results. [ABSTRACT FROM AUTHOR]

We establish a dichotomy for the rate of the decay of the Cesàro averages of correlations of sufficiently regular functions for typical interval exchange transformations (IET) which are not rigid rotations (for which weak mixing had been previously established in the works of Katok–Stepin, Veech, and Avila–Forni). We show that the rate of decay is either logarithmic or polynomial, according to whether the IET is of rotation class (i.e., it can be obtained as the induced map of a rigid rotation) or not. In the latter case, we also establish that the spectral measures of Lipschitz functions have local dimension bounded away from zero (by a constant depending only on the number of intervals). In our approach, upper bounds are obtained through estimates of twisted Birkhoff sums of Lipschitz functions, while the logarithmic lower bounds are based on the slow deviation of ergodic averages that govern the relation between rigid rotations and their induced maps. [ABSTRACT FROM AUTHOR]

By analyzing the main cable shape of the suspension bridge and the mechanical and geometric relationship between the main cable and the cable saddle, an improved algorithm is established for determining the position of the composite circular curve splay saddle. The geometric conditions of the main cable and saddle are considered to establish the equations, and the position of the composite circular curve splay saddle is obtained by solving only a one-dimensional nonlinear equation with the dichotomy method. The algorithm yields the solution interval of the dichotomy method to achieve the solution of the nonlinear equation, and the calculation process ensures convergence without any initial value. Several examples demonstrate that the improved algorithm can achieve excellent convergence. [ABSTRACT FROM AUTHOR]

In this paper, we study the partial differential equation with piecewise constant argument of the form: xt(t,s)=A(t)x(t,s)+B(t,s)x([t],s)+C(t,s)x(t,[s])+D(t,s)x([t],[s])+f(x(t,[s])),t,s∈IR+=(0,∞),$$ {\displaystyle \begin{array}{cc}\hfill {x}_t\left(t,s\right)=& \kern0.2em A(t)x\left(t,s\right)+B\left(t,s\right)x\left(\left[t\right],s\right)+C\left(t,s\right)x\left(t,\left[s\right]\right)\hfill \\ {}\hfill & +D\left(t,s\right)x\left(\left[t\right],\left[s\right]\right)+f\left(x\left(t,\left[s\right]\right)\right),t,s\in I\kern-0.2em {R}^{+}=\left(0,\infty \right),\hfill \end{array}} $$where A(t)$$ A(t) $$ is a k×k$$ k\times k $$ invertible and continuous matrix function on IR+$$ I\kern-0.2em {R}^{+} $$; B(t,s)$$ B\left(t,s\right) $$, C(t,s)$$ C\left(t,s\right) $$, and D(t,s)$$ D\left(t,s\right) $$ are k×k$$ k\times k $$ continuous and bounded matrix functions on IR+×IR+$$ I\kern-0.2em {R}^{+}\times I\kern-0.2em {R}^{+} $$; [t]$$ \left[t\right] $$ and [s]$$ \left[s\right] $$ are the integral parts of t$$ t $$ and s$$ s $$, respectively; and f:IRk→IRk$$ f:I\kern-0.2em {R}^k\to I\kern-0.2em {R}^k $$ is a continuous function. More precisely, under some conditions on the matrices A(t)$$ A(t) $$, B(t,s)$$ B\left(t,s\right) $$, C(t,s)$$ C\left(t,s\right) $$, and D(t,s)$$ D\left(t,s\right) $$ and the function f$$ f $$, we investigate the asymptotic behavior of the solutions of the above equation. [ABSTRACT FROM AUTHOR]

The paper considers a general concept of dichotomy in Banach spaces for skew-evolution cocycles. As particular cases the properties of exponential dichotomy, polynomial dichotomy and logarithmic dichotomy are obtained. Characterizations of these concepts are presented. [ABSTRACT FROM AUTHOR]

For finite-dimensional linear differential systems with bounded coefficients we prove that their exponential dichotomy on R is equivalent to their Ulam-Hyers stability on R with uniqueness. We also consider abstract non-autonomous evolution equations which are exponentially bounded and exponentially dichotomic and prove that Ulam-Hyers stability with uniqueness is maintained when perturbing them with a nonlinear term having a sufficiently small Lipschitz constant. [ABSTRACT FROM AUTHOR]

This paper is concerned with the tempered exponential dichotomy for random differential systems in Banach spaces. Based on the presentation of bounded solutions for a tempered exponential dichotomous system, we give a bound under which the tempered exponentially dichotomous property of perturbed systems is preserved. Some applications of our results to stochastic partial differential equations are considered. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Bohl–Perron Theorem for linear random dynamical systems. We prove that the tempered exponential stability of a linear co-cycle is equivalent to the boundedness of solutions for inherit difference equation. Paper also proves a similar concept for co-cycle admitting a tempered exponential dichotomy. [ABSTRACT FROM AUTHOR]

In this manuscript, we study the exponential stability and Hyers--Ulam stability of the linear first order impulsive differential system. We prove that the homogeneous impulsive system is exponentially stable if and only if the solution of the corresponding non-homogeneous impulsive system is bounded. Moreover, we prove that the system is Hyers--Ulam stable if and only if it is uniformly exponentially dichotomic. We obtain our results by using the spectral decomposition theorem. To illustrate our theoretical results, at the end we give an example. [ABSTRACT FROM AUTHOR]

We add relevant references about which we learned after the completion of the initial work. We mainly show how the concept of exponential trichotomy can successfully replace the one of exponential dichotomy in some results from the paper in the title.

This paper considers the numerical treatment of singularly perturbed time-dependent convection-diffusion-reaction equation. The diffusion term of the equation is multiplied by a small perturbation parameter (ε), which takes an arbitrary value in the interval (0, 1]. For small values of ε, the solution of the equation exhibits an exponential boundary layer which makes it difficult to solve it analytically or using classical numerical methods. We proposed numerical schemes using the Crank–Nicolson method in time derivative discretization and the nonstandard finite difference method (exact finite difference method) in space derivative discretization on a uniform and piecewise uniform Shishkin mesh. The existence of unique discrete solutions and the stability of the schemes are discussed and proved. Uniform convergence of the schemes is proved. The formulated schemes converge uniformly with linear order of convergence. The method on Shishkin mesh possesses boundary layer resolving property. We validated the methods by considering two numerical examples for different values of ε and mesh length. [ABSTRACT FROM AUTHOR]

In this paper we study the robustness of the exponential dichotomy in nonautonomous linear ordinary differential equations under integrally small perturbations in infinite dimensional Banach spaces. Some applications are obtained to the case of rapidly oscillating perturbations, with arbitrarily small periods, showing that even in this case the stability is robust. These results extend to infinite dimensions some results given in Coppel (Dichotomies in stability theory. Lecture notes in mathematics, Springer, Berlin, 1970). Based in Rodrigues (Invariância para sistemas de equações diferenciais com retardamento e aplicações, Tese de Mestrado, Universidade de São Paulo, São Carlos, 1970) and in Kloeden and Rodrigues (Nonlinear Anal 74:2695–2719, 2011), Rodrigues et al. (Stability problems in non autonomous linear differential equations in infinite dimensions. arXiv:1906.04642, 2019) we use the class of functions that we call Generalized Almost Periodic Functions that extend the usual class of almost periodic functions and are suitable to model these oscillating perturbations. We also present an infinite dimensional example of the previous results. [ABSTRACT FROM AUTHOR]

The recent papers Gajek and Kucinsky (Insur Math Econ 73:1–19, 2017) and Avram et al. (Mathematics 9(9):931, 2021) cost induced dichotomy for optimal dividends in the cramr-lundberg model. Avram et al. (Mathematics 9(9):931, 2021) investigated the control problem of optimizing dividends when limiting capital injections stopped upon bankruptcy. The first paper works under the spectrally negative Lévy model; the second works under the Cramér-Lundberg model with exponential jumps, where the results are considerably more explicit. The current paper has three purposes. First, it illustrates the fact that quite reasonable approximations of the general problem may be obtained using the particular exponential case studied in Avram et al. cost induced dichotomy for optimal dividends in the Cramér-Lundberg model (Avram et al. in Mathematics 9(9):931, 2021). Secondly, it extends the results to the case when a final penalty P is taken into consideration as well besides a proportional cost k > 1 for capital injections. This requires amending the "scale and Gerber-Shiu functions" already introduced in Gajek and Kucinsky (Insur Math Econ 73:1–19, 2017). Thirdly, in the exponential case, the results will be made even more explicit by employing the Lambert-W function. This tool has particular importance in computational aspects and can be employed in theoretical aspects such as asymptotics. [ABSTRACT FROM AUTHOR]

This paper is devoted to the quantitative study of the attractive velocity of compact semi-invariant attracting sets for infinite-dimensional dynamical systems. We introduce the notion of compact φ -attracting set whose attractive speed is characterized by a general non-negative decay function φ , and prove that φ -decay with respect to noncompactness measure is a sufficient condition for a dissipative system to have a compact φ -attracting set. Furthermore, several criteria for φ -decay with respect to noncompactness measure are provided. Finally, as an application, we establish the existence of a compact exponential attracting set and the specific estimate of its attractive velocity for a semilinear wave equation with a critical nonlinearity. [ABSTRACT FROM AUTHOR]

In this work we study the differentiability properties of the conjugation in the Barreira-Valls version of the Hartman-Grobman Theorem for Non-autonomous and Delayed systems. Indeed, we show that the conjugacy in the Barreira-Valls Theorem is a C 1 diffeomorphism if we impose some extra hypothesis related with the decay of the perturbation. [ABSTRACT FROM AUTHOR]

Barreira and Valls (2010) proved in [4] the C 1 dependence of C 0 linearization on parameters for nonuniformly hyperbolic diffeomorphisms, whose nonlinear parts depend on the parameters but linear parts do not. In this paper, we further allow the linear parts to depend on the parameters, where only Hölder dependence is possible and a difficulty arises: the projections in the nonuniform exponential dichotomy do not have explicit expressions. We will overcome the difficulty by giving a new result on the Hölder dependence of nonuniform exponential dichotomy and constructing the conjugacy in a different way. Moreover, under Barreira and Valls's framework we will improve the C 1 dependence to C r , where another difficulty arises: without the non-resonant condition, the systems do not have small derivatives of orders ≥2 in general. We will introduce functional spaces with small bounds to overcome the difficulty. Besides, our different constructing method ensures that the conjugacy is invertible, which was not considered by Barreira and Valls. [ABSTRACT FROM AUTHOR]

In this paper the optical pulses and other solitary wave solutions are extracted in dual-core optical fibers. The decoupled nonlinear Schrodinger equation is taken as a controlling equation. The model is composed of group-velocity dispersion and mismatch, linear coupling coefficient, and nonlinear refractive index. The optical pulses in different shapes like dark, bright, singular, and combined solitons by application of the generalized exponential rational function method and the new extended direct algebraic method. Furthermore, a stack of exponential, hyperbolic, and trigonometric function wave solitary solutions is magnificently constructed by means of the indicated schemes. Some of the acquired wave solutions are characterized graphically in 3D, contour, and 2D shapes to illustrate the dynamical behavior. A comparison of the results of this study with those found in the literature shows that the current wave solutions are innovative and diverse, which boosts the impressive performance of the utilized methodologies. Complicated models in a wide range of physical circumstances may benefit from the strategies presented. We anticipate that this study will be useful to a wide range of engineering modellers. [ABSTRACT FROM AUTHOR]

We establish Poincare type theorems of normal forms for quasi-periodic systems near the invariant torus based on the exponential dichotomy spectra of linear parts at the normal direction. [ABSTRACT FROM AUTHOR]

We introduce a notion of regular separation for solutions of systems of ODEs y' = F(x, y), where F is definable in a polynomially bounded o-minimal structure and y = (y1, y2). Given a pair of solutions with flat contact, we prove that, if one of them has the property of regular separation, the pair is either interlaced or generates a Hardy field. We adapt this result to trajectories of three-dimensional vector fields with definable coefficients. In the particular case of real analytic vector fields, it improves the dichotomy interlaced/separated of certain integral pencils, obtained by F. Cano, R. Moussu and the third author. In this context, we show that the set of trajectories with the regular separation property and asymptotic to a formal invariant curve is never empty and it is represented by a subanalytic set of minimal dimension containing the curve. Finally, we show how to construct examples of formal invariant curves which are transcendental with respect to subanalytic sets, using the so-called (SAT) property, introduced by J.-P. Rolin, R. Shaefke and the third author. [ABSTRACT FROM AUTHOR]

After introducing the notion of weighted shadowing, we give sufficient conditions under which certain nonlinear perturbations of nonautonomous linear delay differential equations exhibit the shadowing property with respect to a given weight function. As a corollary, it is shown that if the unperturbed equation admits a shifted exponential dichotomy and the Lipschitz constant of the nonlinear term is sufficiently small, then the perturbed system has a shadowing property with respect to an exponential weight function. An application to differential equations with small delay is given. The results are new even in the case of the standard shadowing property. [ABSTRACT FROM AUTHOR]

Poincaré profiles are analytically defined invariants, which provide obstructions to the existence of coarse embeddings between metric spaces. We calculate them for all connected unimodular Lie groups, Baumslag–Solitar groups and Thurston geometries, demonstrating two substantially different types of behaviour. For Lie groups, our dichotomy extends both the rank one versus higher rank dichotomy for semisimple Lie groups and the polynomial versus exponential growth dichotomy for solvable unimodular Lie groups. We provide equivalent algebraic, quasi-isometric and coarse geometric formulations of this dichotomy. As a consequence, we deduce that for groups of the form N × S , where N is a connected nilpotent Lie group, and S is a rank one simple Lie group, both the growth exponent of N, and the conformal dimension of S are non-decreasing under coarse embeddings. These results are new even for quasi-isometric embeddings and give obstructions which in many cases improve those previously obtained by Buyalo–Schroeder. [ABSTRACT FROM AUTHOR]

The present paper deals with two of the most significant behaviors in the theory of dynamical systems: the uniform exponential dichotomy and the uniform polynomial dichotomy for evolution operators in Banach spaces. Assuming that the evolution operator has uniform exponential growth, respectively uniform polynomial growth, we give some characterizations for the uniform exponential dichotomy, respectively for the uniform polynomial dichotomy. The proof techniques that we use for the polynomial case are new. In addition, connections between the concepts approached are established. [ABSTRACT FROM AUTHOR]

The Eulerâ€"Poisson (EP) system describes the dynamic behaviour of many important physical flows. In this work, a Riccati system that governs pressureless two-dimensional EP equations is studied. The evolution of divergence is governed by the Riccati type equation with several nonlinear/nonlocal terms. Among these, the vorticity accelerates divergence while others further amplify the blow-up behaviour of a flow. The growth of these blow-up amplifying terms are related to the Riesz transform of density, which lacks a uniform bound makes it difficult to study global solutions of the multi-dimensional EP system. We show that the Riccati system can afford to have global solutions, as long as the growth rate of blow-up amplifying terms is not higher than exponential, and admits global smooth solutions for a large set of initial configurations. To show this, we construct an auxiliary system in 3D space and find an invariant space of the system, then comparison with the original 2D system is performed. Some numerical examples are also presented. [ABSTRACT FROM AUTHOR]

This work is devoted to study a new class of static neural network. We propose a static multidirectional associative memory neural network with time-varying coefficients and delays. We assess the existence and global exponential stability of weighted pseudo almost-periodic solutions for the systems through exponential dichotomy theory, contraction mapping fixed point theorem, differential inequality techniques and weighted pseudo almost-periodic functions. The theoretical results obtained are supported with numerical simulations. To our knowledge, no paper in the literature has investigated the existence and the delay-independent stability of weighted pseudo almost-periodic solution for static delayed MAM neural network. The results obtained are new and complements that found in the literature. [ABSTRACT FROM AUTHOR]

We consider non-autonomous ordinary differential equations in two cases. One is the one-dimensional case that admits a condition of hyperbolicity, and the other one is the higher-dimensional case that admits an exponential dichotomy. For differential equations of this kind, we give a rigorous treatment of the Harmonic Balance Method to look for almost periodic solutions. [ABSTRACT FROM AUTHOR]

We introduce two new tools for studying CAT(0) spaces: curtains , versions of cubical hyperplanes; and the curtain model , a counterpart of the curve graph. These tools shed new light on CAT(0) spaces, allowing us to prove a dichotomy of a rank-rigidity flavour, establish Ivanov-style rigidity theorems for isometries of the curtain model, find isometry-invariant copies of its Gromov boundary in the visual boundary of the underlying CAT(0) space, and characterise rank-one isometries both in terms of their action on the curtain model and in terms of curtains. Finally, we show that the curtain model is universal for WPD actions over all groups acting properly on the CAT(0) space. [ABSTRACT FROM AUTHOR]

Adsorption process is proposed to be determined by such two successively coupling steps as a transport of adsorbate molecules into system by free convection and a formation of collision-complexes by surface complexing, whose kinetic rate equation can be expressed by a product of exponential monotonous with logistic growth function and quite well experimentally validated by three different adsorption systems. [Display omitted] • Non-linearly coupling molecular transport with surface complexing step leads to a new adsorption kinetic equation. • This equation is suitable for three kinds of adsorption system in vacuum basket, dynamic basket and fixed bed. • Its kinetic parameters can be characterized by molecular motions. Inspired by reaction molecular dynamics together with chain reaction theory, physical adsorption process is proposed to be determined by two successively coupling steps, such as a transport of adsorbate molecules into adsorption system in a free convection velocity and a formation of collision-complex between surface and activated adsorbate molecule in a surface complexing velocity. This leads to establish a new kinetic equation by a product of exponential monotonic and logistic growth functions equal to adsorption extent ratio, which can be quite well validated experimentally by both uptake curves in vacuum or dynamic basket and breakthrough curves from fixed bed for adsorption of benzene vapor by micro-porous carbon. Molecular transport step is an ideal flow with a rate constant of the free convection velocity over the system size; whereas, surface complexing step depends not only on a rate constant of the surface complexing velocity over the apparent free path basically around carbon granular size, but also on its initial surface complex density distribution. The activated adsorbate growth probability is less than unity and equal to the ratio of surface complexing over free convection velocity, both of which are almost nearly proportional to the adsorbate concentration relative to adsorption capacity. [ABSTRACT FROM AUTHOR]

Mean hyperbolic systems permit non-hyperbolic behavior at certain moments during the evolution process T (t , s) , owing to non-fixed error hyperbolic degree ε (t , s). Despite the coexistence of compression and expansion behaviors in generalized stable and unstable subspaces, mean hyperbolic systems admit fixed average contraction and expansion rates measured at sufficiently long time. To discuss the relationship between admissibility and mean hyperbolicity for evolution equations, our main goal is to construct admissible function classes and invariant decomposition with generalized stable and unstable subspaces. As applications, the mean hyperbolicity of damped wave equations with variable coefficients and the roughness of mean hyperbolicity are presented. [ABSTRACT FROM AUTHOR]

The hybrid and the dressed metric formalisms for the study of primordial perturbations in Loop Quantum Cosmology lead to dynamical equations for the modes of these perturbations that are of a generalized harmonic-oscillator type, with a mass that depends on the background but is the same for all modes. For quantum background states that are peaked on trajectories of the effective description of Loop Quantum Cosmology, the main difference between the two considered formalisms is found in the expression of this mass. The value of the mass at the bounce is especially important, since it is only in a short interval around this event that the quantum geometry effects on the perturbations are relevant. In a previous article, the properties of this mass were discussed for an inflaton potential of quadratic form, or with similar characteristics. In the present work, we extend this study to other interesting potentials in cosmology, namely the Starobinsky and the exponential potentials. We prove that there exists a finite interval of values of the potential (which includes the zero but typically goes beyond the sector of kinetically dominated inflaton energy density) for which the hybrid mass is positive at the bounce whereas the dressed metric mass is negative. [ABSTRACT FROM AUTHOR]