Specification Property for Topological Spaces

We introduce and study here the notion of specification property termed as topological specification property for homeomorphisms on non-compact non-metrizable spaces. We prove that if a self homeomorphism on a totally bounded uniform space is mixing, topologically expansive and topologically shadowing then the map has topological specification property.     

Devaney’s and Li-Yorke’s Chaos in Uniform Spaces

A definition of chaos in the sense of Li-Yorke is given for an action of a group on a uniform space, and it is shown that if a continuous action of an Abelian group G on a second countable Baire Hausdorff uniform space X without isolated points is chaotic in the sense of Devaney, then it is also chaotic in the sense of Li-Yorke.     

Regions Where the Exponential Map at Regular Points of Sub-Riemannian Manifolds is a Local Diffeomorphism

For a k-step sub-Riemannian manifold which admits a bracket generating vector at a point, we describe a region near the point where the exponential map is a local diffeomorphism. This is proved by taking the Taylor series of the exponential map and calculating the first nonzero term, which has order , where n is the topological dimension and is the Hausdorff dimension of the metric space associated to the sub-Riemannian manifold.      

Specification and Shadowing Properties for Non-autonomous Systems

In this paper, various properties of non-autonomous dynamical systems having periodic shadowing property and local weak specification property are studied. The main theorem gives an interrelation among the shadowing property, periodic shadowing property and local weak specification property of an expansive non-autonomous system.

     

On Topological Equivalence of Linear Lows with Applications to Bilinear Control Systems

This paper classifies continuous linear flows using concepts and techniques from topological dynamics. Specifically, the concepts of equivalence and conjugacy are adapted to flows on vector bundles, and the Lyapunov decomposition is characterized using the induced flows on the Grassmann and the flag bundles. These results are then applied to bilinear control systems, for which their behavior in , on the projective space , and on the Grassmannians is characterized. This research was partially supported by Proyecto FONDECYT No. 1060981 and Proyecto FONDECYT de Incentivo a la Cooperación Internacional No. 7020439.     

Stability of Saul'yev's Methods with Nonuniform Grids

Stability of Saul'yev's methods for heat conduction with nonuniform grids is investigated. Though these methods are known to be unconditionally stable with uniform grids, the author shows that their stability with nonuniform grids depends on time step Δ τ, space intervals Δ X, and ratios of neighboring space intervals. The author also shows that physical reality is broken when Patankar's positive-coefficients rule is not satisfied, even if the methods are applied to uniform grids. This article presents a stability criterion for Saul'yev's methods with both nonuniform and uniform grids.     

Almost Additive Topological Pressure of Proper Systems

We introduce a natural extension of almost additive topological pressure for a proper map of locally compact metric spaces and present a corresponding variational principle. As an application, for a class of almost additive sequence, we describe a multi-fractal analysis of the u-dimension spectra of their generalized Birkhorff averages.     

Multifractal Spectrum for Barycentric Averages

Let \(\left (X,\nu \right ) \) and Y be a measured space and a C A T(0) space, respectively. If \(\mathcal {M}_{2}(Y)\) is the set of measures on Y with finite second moment then a map \(bar:\mathcal {M}_{2}(Y)\rightarrow Y\) can be defined. Also, for any x∈X and for a map \(\varphi :X\rightarrow Y\), a sequence \(\left \{\mathcal {E}_{N,\varphi }(x)\right \} \) of empirical measures on Y can be introduced. The sequence \(\left \{ bar\left (\mathcal {E}_{N,\varphi }(x)\right ) \right \} \) replaces in C A T(0) spaces the usual ergodic averages for real valuated maps. It converges in Y (to a map \(\overline {\varphi }\left (x\right )\)) almost surely for any x∈X (Austin J Topol Anal. 2011;3: 145–152). In this work, we shall consider the following multifractal decomposition in X:

$$K_{y,\varphi}=\left\{ x:\lim\limits_{N\rightarrow\infty}bar\left(\mathcal{E}_{N,\varphi}(x)\right) =y\right\} , $$

and we will obtain a variational formula for this multifractal spectrum.

     

A Note on the Kepler Problem

J. Moser proved that the flow arising in the Kepler problem and restricted to the manifold of the constant energy E < 0 is equivalent to the geodesic flow on a sphere. This was proved by means of some algebraic manipulations with the Hamilton function. In a similar way Yu. S. Osipov proved that this flow is equivalent to the geodesic flow on the Euclidean space for E = 0 and on the Lobachevskii space for E < 0. In this paper results of such kind are related to the approach to the Kepler problem suggested by Hamilton (this approach seems to be the simplest one). For the planar Kepler problem one first considers the picture arising on the hodograph plane, where the hodograph curves turn out to be circles or arcs of circles. For fixed E one obtains a net (2-parameter linear system) of circles which in the case of E < 0 can be obtained from the system of great circles on a sphere by a stereographic projection; related geometric construction exists also for other E. This leads in a geometrical way to Moser's result. Moser showed also that for E < 0 the trajectory space of the covering flow on the universal covering space (which is a three-dimensional sphere ³) is a two-dimensional sphere ²; the corresponding map ^{3 2} is the Hopf fibration. An additional remark made below is that under appropriate normalizations and modifications this is the map

When a Minimal Map Is Totally Transitive on a <Emphasis Type="Italic">G</Emphasis>-Space

In this paper, we introduce the notion of G-regular periodic decomposition (GRPD) for maps on G-spaces and investigate its relation with G-transitivity. It is shown that if a pseudoequivariant, G-transitive map on a G-space has a GRPD of some length n, then its nth iterate is not G-transitive. On the other hand, if a pseudoequivariant, G-transitive map on a G-space has a non-G-transitive nth iterate, then it admits a GRPD of length p for some prime p dividing n. Using the notion of GRPD, it is obtained that a pseudoequivariant, G-minimal map is totally G-transitive on a connected G-space.     

Examples of Integrable Sub-Riemannian Geodesic Flows

We exhibit examples of sub-Riemannian metrics with integrable geodesic flows and positive topological entropy. The underlying distribution is contact in 3 dimensions and is Goursat in the higher-dimensional case. We also give an example of Poisson ³-action on 6-dimensional manifold such that topological entropies for all nonzero vectors (Hamiltonians) are positive.     

KdV Hamiltonian as a Function of Actions

We prove that the non-linear part of the Hamiltonian of the KdV equation on the circle, written as a function of the actions, defines a continuous convex function on the ? ² space and derive for it lower and upper bounds in terms of some functions of the ? ²-norm. The proof is based on a new representation of the Hamiltonian in terms of the quasimomentum, obtained via the conformal mapping theory.     

Shadowable Points for Flows

A shadowable point for a flow is a point where the shadowing lemma holds for pseudo-orbits passing through it. We prove that this concept satisfies the following properties: the set of shadowable points is invariant and a G _δ set. A flow has the pseudo-orbit tracing property if and only if every point is shadowable. The chain recurrent and nonwandering sets coincide when every chain recurrent point is shadowable. The chain recurrent points which are shadowable are exactly those that can be are approximated by periodic points when the flow is expansive. These results extends those presented in Morales (Dyn Syst. 2016;31(3):347–356). We study the relations between shadowable points of a homeomorphism and the shadowable points of its suspension flow. We characterize the set of forward shadowable points for transitive flows and chain transitive flows. We prove that the geometric Lorenz attractor does not have shadowable points. We show that in the presence of shadowable points chain transitive flows are transitive and that transitivity is a necessary condition for chain recurrent flows with shadowable points whenever the phase space is connected. Finally, as an application, these results we give concise proofs of some well known theorems establishing that flows with POTP admitting some kind of recurrence are minimal.     

水电工程数值模型快速查错方法研究

针对土木工程领域的地质模型构建了基于六面体连续数值模型的快速查错方法。该方法首先进行六面体有限分割,得节点及单元索引信息,再将每个单元拆分为6个空间面元结构,分为公用面元与独享面元两类,利用单元的节点坐标、网格拓扑关系进行自检索,将可能存在错误的独享面元输出至AutoCAD中进行可视化显示。案例分析表明:该方法效率高,可使模型中的错误一次性显现,然后对照错误出现位置在模型中进行一次性修改,达到省时省力的目的。研究的方法是连续数值模拟前处理的有效补充,可避免因几何联结不当引起的计算结果失真。     

Highly Singular L-Q Problems: Solutions in Distribution Spaces

It is well known that singular problems may fail to have optimal solution in the class of ordinary (say, square-integrable) controls, even in the cases where the cost is bounded from below. In this paper, we suggest a method for overcoming this difficulty by defining an order r of singularity of the problem and extending both the input-trajectory map and the cost functional to an adequate subspace of the Sobolev space H__r-r. We show that the extended problem has a minimum if and only if the infimum of the original problem is finite. The extended problem can be transformed in a natural way into a regular L-Q problem with strictly smaller controllable space and (possibly) smaller control space. We use this transformation to describe the structure of relaxed optimal controls and the corresponding relaxed trajectories. We provide a method for computing optimal relaxed solutions from the solution of an adequate Riccati differential equation. We also show how the relaxed minimizers can be approximated by square-integrable functions.     

Genericity for Non-Wandering Surface Flows

Consider the set \(\chi ^{0}_{\text {nw}}\) of non-wandering continuous flows on a closed surface M. Then we show that such a flow can be approximated by a non-wandering flow v such that the complement M?Per(v) of the set of periodic points is the union of finitely many centers and finitely many homoclinic saddle connections. Using the approximation, the following are equivalent for a continuous non-wandering flow v on a closed connected surface M: (1) the non-wandering flow v is topologically stable in \(\chi ^{0}_{\text {nw}}\); (2) the orbit space M/v is homeomorphic to a closed interval; (3) the closed connected surface M is not homeomorphic to a torus but consists of periodic orbits and at most two centers. Moreover, we show that a closed connected surface has a topologically stable continuous non-wandering flow in \(\chi ^{0}_{\text {nw}}\) if and only if the surface is homeomorphic to either the sphere \(\mathbb {S}^{2}\), the projective plane \(\mathbb {P}^{2}\), or the Klein bottle \(\mathbb {K}^{2}\).     

C ∞ and L ∞ Observability of Single-Input C ∞-Systems

For single-input multi-output C -systems, we state conditions under which observability for every C -input implies observability for every almost everywhere continuous, bounded input (for every L -input in the control-affine case). A normal system is then defined as a system whose only bad inputs are smooth on some nonempty open set. When the state space is compact, normality turns out to be generic and enables us to extend some results of genericity of observability to nonsmooth inputs.     

Totally analytical closure of space filtered Navier–Stokes for arbitrary Reynolds number: Part I. Theory,resolutions

ABSTRACT

Rigorously space filtering the thermal, multispecies Navier–Stokes (NS) conservation principle partial differential equation (PDE) system embeds a priori undefined tensor and vector quadruples. Large eddy simulation (LES) computational fluid dynamics algorithm resolutions replace the tensor quadruple with a single tensor then secures closure through “physics-based” modeling, assuming the velocity field is turbulent, i.e., the Reynolds number (Re) is large. In complete distinction, a totally analytical closure is derived for the rigorously generated tensor/vector quadruples, achieved totally absent any modeling component or Re assumption. For Gaussian filter of uniform measure δ, derived analytical filtered Navier–Stokes (aFNS) theory PDE system state variable is significance scaled O(1; δ²; δ³) through classic fluid mechanics perturbation theory. That uniform measure δ filter penetrates domain boundaries requires O(1) resolved scale PDE system inclusion of boundary commutation error (BCE) integrals, (unfiltered) NS state variable extension in the sense of distributions, and domain enlargement to encompass all surfaces with Dirichlet boundary condition (DBC) specification. Theory-derived O(δ²) resolved–unresolved scale interaction PDE system, also the O(1) system, is rendered bounded domain, well posed through a priori identification of O(1; δ²) state variable nonhomogeneous DBCs. BCE and DBC resolution algorithm derivations use O(δ⁴) approximate deconvolution (AD) differential definition Galerkin weak forms. Theory analytically derived unresolved scale O(δ³) state variable annihilates discretization-induced O(h²) dispersion error at unresolved scale threshold δ, h the mesh measure. Net is an analytical theory closing rigorously space-filtered NS exhibiting potential for first principles prediction of viscous laminar–turbulent transition, separation, and relaminarization.     

Topological Entropy of Free Semigroup Actions and Skew-Product Transformations

A definition of topological entropy for a free semigroup action is suggested. Suppose that a free semigroup acts on a compact metric space by continuous self-maps. To this action, we assign a skew-product transformation whose fiber entropy is taken to be the entropy of the initial action. The main result is Theorem 1, a topological analogue of the Abramov–Rokhlin formula.     

Knots as Topological Invariants for Gradient-Like Diffeomorphisms of the Sphere S 3

We show how knots in appear in a natural way as complete invariants of topological conjugacy for the simplest gradient-like diffeomorphisms on 3-manifolds.