A vast class of exponential functions are shown to be deterministic. This class includes functions whose exponents are polynomial-like or ``piece-wise'' close to polynomials after differentiation. Many of these functions are proved to be disjoint from the M\"obius function.

Many discrete integrable systems exhibit the Laurent phenomenon. In this paper, we investigate three integrable systems: the Somos-4 recurrence, the Somos-5 recurrence, and a system related to so-called A_1 A_1-system, whose general solutions are derived in terms of Hankel determinant. As a result, we directly confirm that they satisfy the Laurent property. Additionally, it is shown that the Somos-5 recurrence can be viewed as a specified Backlund transformation of the Somos-4 recurrence. Related topics about Somos polynomials are also studied.

We prove we can build (transitive or nontransitive) Anosov flows on closed three-dimensional manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications of this result; for example: 1.We build a closed three-dimensional manifold supporting both a transitive Anosov vector field and a nontransitive Anosov vector field. 2.For any n, we build a closed three-dimensional manifold M supporting at least n pairwise different Anosov vector fields. 3.We build transitive hyperbolic attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive hyperbolic attractors. 4.We build a transitive Anosov vector field admitting infinitely many pairwise nonisotopic transverse tori.

There is much research on the dynamical complexity on irregular sets andlevel sets of ergodic average from the perspective of density in base space, theHausdorff dimension, Lebesgue positive measure, positive or full topological entropy (andtopological pressure), etc. However, this is not the case from the viewpoint of chaos.There are many results on the relationship of positive topological entropy and variouschaos. However, positive topological entropy does not imply a strong version of chaos,called DC1. Therefore, it is non-trivial to study DC1 on irregular sets and level sets. Inthis paper, we will show that, for dynamical systems with specification properties, thereexist uncountable DC1-scrambled subsets in irregular sets and level sets. Meanwhile, weprove that several recurrent level sets of points with different recurrent frequency haveuncountable DC1-scrambled subsets. The major argument in proving the above results isthat there exists uncountable DC1-scrambled subsets in saturated sets.

To derive the hidden dynamics from observed data is one of the fundamental but also challenging problems in many different fields. In this study, we propose a new type of interpretable network called the ordinary differential equation network (ODENet), in which the numerical integration of explicit ordinary differential equations (ODEs) are embedded into the machine learning scheme to build a general framework for revealing the hidden dynamics buried in massive time-series data efficiently and reliably. ODENet takes full advantage of both machine learning algorithms and ODE modeling. On one hand, the embedding of ODEs makes the framework more interpretable benefiting from the mature theories of ODEs. On the other hand, the schemes of machine learning enable data handling, paralleling, and optimization to be easily and efficiently implemented. From classical Lotka-Volterra equations to chaotic Lorenz equations, the ODENet exhibits its remarkable capability in handling time-series data even in the presence of large noise. We further apply the ODENet to real actin aggregation data, which shows an impressive performance as well. These results demonstrate the superiority of ODENet in dealing with noisy data, data with either non-equal spacing or large sampling time steps over other traditional machine learning algorithms.

The molecule solution of an extended discrete LotkaVolterra equation is constructed, from which a new sequence transformation is proposed. A convergence acceleration algorithm for implementing this sequence transformation is found. It is shown that our new sequence transformation accelerates some kinds of linearly convergent sequences and factorially convergent sequences with good numerical stability. Some numerical examples are also presented.

To study arithmetic structures of natural numbers, we introduce a notion of entropy of arithmetic functions, called anqie entropy. This entropy possesses some crucial properties common to both Shannon's and Kolmogorov's entropies. We show that all arithmetic functions with zero anqie entropy form a C*-algebra. Its maximal ideal space defines our arithmetic compactification of natural numbers, which is totally disconnected but not extremely disconnected. We also compute the $K$-groups of the space of all continuous functions on the arithmetic compactification. As an application, we show that any topological dynamical system with topological entropy $\lambda$, can be approximated by symbolic dynamical systems with entropy less than or equal to $\lambda$.

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In this paper, we present a generalized Toeplitz determinant solution for the generalized Schur flow and propose a mixed form of the two known relativistic Toda chains together with its generalized Toeplitz determinant solution. In addition, we also give a Hankel type determinant solution for a nonisospectral Toda lattice. All these results are obtained by technical determinant operations. As a bonus, we finally obtain some new combinatorial numbers based on the moment relations with respect to these semi-discrete integrable systems and give the corresponding combinatorial interpretations by means of the lattice paths.

In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $ LDU $ decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition formula for weighted large Schrder paths.

We examine the convergence in the Krylov–Bogolyubov averaging for nonlinear stochastic perturbations of linear PDEs with pure imaginary spectrum and show that if the involved effective equation is mixing, then the convergence is uniform in time.

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For any dynamical system T : X → X of a compact metric space X with g-almost product property and uniform separation property, under the assumptions that the periodic points are dense in X and the periodic measures are dense in the space of invariant measures, we distinguish various periodiclike recurrences and find that they all carry full topological entropy and so do their gap-sets. In particular, this implies that any two kind of periodic-like recurrences are essentially different. Moreover, we coordinate periodic-like recurrences with (ir)regularity and obtain lots of generalized multifractal analyses for all continuous observable functions. These results are suitable for all β-shifts (β > 1), topological mixing subshifts of finite type, topological mixing expanding maps or topological mixing hyperbolic diffeomorphisms, etc. Roughly speaking, we combine many different “eyes” (i.e., observable functions and periodic-like recurrences) to observe the dynamical complexity and obtain a Refined Dynamical Structure for Recurrence Theory and Multi-fractal Analysis.

The modified Camassa-Holm (also called FORQ) equation is one of numerous <i>cousins</i> of the Camassa-Holm equation possessing non-smoth solitons (<i>peakons</i>) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissipative^a) the Sobolev <i>H</i>¹ norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the <i>H</i>¹ norm invariant. In this Letter, it is shown that the conservative peakon equations of the modified Camassa-Holm can be given an appropriate Poisson structure relative to which the equations are Hamiltonian and, in fact, Liouville integrable. The latter is proved directly by exploiting the inverse spectral techniques, especially asymptotic analysis of solutions, developed elsewhere [3].

Let f be a holomorphic automorphism of a compact Kähler manifold. Assume that f admits a unique maximal dynamic degree d_p with only one eigenvalue of maximal modulus. Let μ be its equilibrium measure. In this paper, we prove that μ is exponentially mixing for all d.s.h. test functions.

We consider a measurable matrix-valued cocycle A: Z+ X Rd d, driven by a measurepreserving transformation T of a probability space (X,, ), with the integrability condition log+ A (1,) L1 (). We show that for -ae x X, if limn 1 n log A (n, x) v= 0 for all v Rd\{0}, then the trajectory {A (n, x) v} n= 0 is far away from 0 (ie lim supn A (n, x) v&gt; 0) and there is some nonzero v such that lim supn A (n, x) v v. This improves the classical multiplicative ergodic theorem of Oseledec. We here present an application to linear random processes to illustrate the importance.

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Consider a perturbed KdV equation: ut +uxxx −6uux = εf(u(·)), x ∈ T = R/Z, where the nonlinear perturbation defines analytic operators u(·) 􏰀→ f(u(·)) in sufficiently smooth Sobolev spaces. Assume that the equation has an ε-quasi- invariant measure μ and satisfies some additional mild assumptions. Let uε(t) be a solution. Then on time intervals of order ε−1, as ε → 0, its actions I(uε(t,·)) can be approximated by solutions of a certain well-posed averaged equation, provided that the initial datum is μ-typical.

Let S be a compact, orientable surface of hyperbolic type and let ψ be a mapping class of S. The present article shows that there exists a finite cover X of S and a lift ψ_X of ψ to X such that the spectral radius of ψ_X, viewed as an automorphism of H_1(X,C), is greater than one if and only if ψ has infinite order and is not a Dehn multitwist. Among the corollaries of this result is the fact that a compact, irreducible 3-manifold with toroidal or empty boundary and positive simplicial volume admits a finite cover for which the multivariable Alexander polynomial is not identically zero and has Mahler measure strictly greater than one; such a manifold also admits a finite cover whose integral first homology has nontrivial torsion. An independent proof of the main result of this paper was given by A. Hadari [Geom. Topol. 24 (2020), no. 4, 1717–1750; MR4173920]. In that paper, the surface was assumed to have at least one boundary component, but stronger control over the required finite covers was obtained.

We investigate Sarnak's M\"obius Disjointness Conjecture through asymptotically periodic functions. It is shown that Sarnak's conjecture for rigid dynamical systems is equivalent to the disjointness of M\"obius from asymptotically periodic functions. We give sufficient conditions and a partial answer to the later one. As an application, we show that Sarnak's conjecture holds for a class of rigid dynamical systems, which improves an earlier result of Kanigowski-Lema{\'{n}}czyk-Radziwi{\l}{\l}.

The classical Morse theory proceeds by considering sublevel sets f^{−1}(−∞,a] of a Morse function f:M→R, where M is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets f^{−1}(a) and give conditions under which the topology of f^{−1}(a) changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse functions, the topology of a regular level f^{−1}(a) always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold M. When f is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the base space. (Counter-)examples and applications to celestial mechanics are also discussed.

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We consider a damped/driven nonlinear Schr\"odinger equation in an $n$-cube $K^{n}\subset\mathbb{R}^n$, $n\in\mathbb{N}$, under Dirichlet boundary conditions \[ u_t-\nu\Delta u+i|u|^2u=\sqrt{\nu}\eta(t,x),\quad x\in K^{n},\quad u|_{\partial K^{n}}=0, \quad \nu>0, \] where $\eta(t,x)$ is a random force that is white in time and smooth in space. It is known that the Sobolev norms of solutions satisfy $ \| u(t)\|_m^2 \le C\nu^{-m}, $ uniformy in $t\ge0$ and $\nu>0$. In this work we prove that for small $\nu>0$ and any initial data, with large probability the Sobolev norms $\|u(t,\cdot)\|_m$ of the solutions with $m>2$ become large at least to the order of $\nu^{-\kappa_{n,m}}$ with $\kappa_{n,m}>0$, on time intervals of order $\mathcal{O}(\frac{1}{\nu})$.

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