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.     

On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections do not Have Cusps

We consider the problem P _{c u r v e} of minimizing \(\int \limits _{0}^{L} \sqrt {\xi ^{2} + \kappa ^{2}(s)} \, \mathrm {d}s\) for a curve x in \(\mathbb {R}^{3}\) with fixed boundary points and directions. Here, the total length L≥0 is free, s denotes the arclength parameter, κ denotes the absolute curvature of x, and ξ>0 is constant. We lift problem P _{c u r v e} on \(\mathbb {R}^{3}\) to a sub-Riemannian problem P _{m e c} on SE(3)/({0}×SO(2)). Here, for admissible boundary conditions, the spatial projections of sub-Riemannian geodesics do not exhibit cusps and they solve problem P _{c u r v e} . We apply the Pontryagin Maximum Principle (PMP) and prove Liouville integrability of the Hamiltonian system. We derive explicit analytic formulas for such sub-Riemannian geodesics, relying on the co-adjoint orbit structure, an underlying Cartan connection, and the matrix representation of SE(3) arising in the Cartan-matrix. These formulas allow us to extract geometrical properties of the sub-Riemannian geodesics with cuspless projection, such as planarity conditions, explicit bounds on their torsion, and their symmetries. Furthermore, they allow us to parameterize all admissible boundary conditions reachable by geodesics with cuspless spatial projection. Such projections lay in the upper half space. We prove this for most cases, and the rest is checked numerically. Finally, we employ the formulas to numerically solve the boundary value problem, and visualize the set of admissible boundary conditions.     

On Symmetries in Time Optimal Control,Sub-Riemannian Geometries,and the <Emphasis Type="Italic">K</Emphasis>−<Emphasis Type="Italic">P</Emphasis> Problem

The goal of this paper is to describe a method to solve a class of time optimal control problems which are equivalent to finding the sub-Riemannian minimizing geodesics on a manifold M. In particular, we assume that the manifold M is acted upon by a group G which is a symmetry group for the dynamics. The action of G on M is proper but not necessarily free. As a consequence, the orbit space M/G is not necessarily a manifold but it presents the more general structure of a stratified space. The main ingredients of the method are a reduction of the problem to the orbit space M/G and an analysis of the reachable sets on this space. We give general results relating the stratified structure of the orbit space, and its decomposition into orbit types, with the optimal synthesis. We consider in more detail the case of the so-called K?P problem where the manifold M is itself a Lie group and the group G is determined by a Cartan decomposition of M. In this case, the geodesics can be explicitly calculated and are analytic. As an illustration, we apply our method and results to the complete optimal synthesis on S O(3).     

Adjoint Orbits of <Emphasis Type="Italic">sl</Emphasis>(2, ℝ) on Real Simple Lie Algebras and Controllability

This paper shows that for any Lie group G whose Lie algebra L is the split real form of a complex simple Lie algebra, and for any arbitrary root α, there exists a Cartan decomposition of L, related to α, which characterizes some controllability properties by using the adjoint orbits of sl(2, ?). For a class of invariant control systems evolving on G, it is proved that the necessary full rank condition for controllability is also sufficient.     

Turbulent convective heat transfer of methane at supercritical pressure in a helical coiled tube

The heat transfer of methane at supercritical pressure in a helically coiled tube was numerically investigated using the Reynolds Stress Model under constant wall temperature. The effects of mass flux (G), inlet pressure (P_in) and buoyancy force on the heat transfer behaviors were discussed in detail. Results show that the light fluid with higher temperature appears near the inner wall of the helically coiled tube. When the bulk temperature is less than or approach to the pseudocritical temperature (T _pc ), the combined effects of buoyancy force and centrifugal force make heavy fluid with lower temperature appear near the outer-right of the helically coiled tube. Beyond the T _pc , the heavy fluid with lower temperature moves from the outer-right region to the outer region owing to the centrifugal force. The buoyancy force caused by density variation, which can be characterized by Gr/Re² and Gr/Re^2.7, enhances the heat transfer coefficient (h) when the bulk temperature is less than or near the T _pc , and the h experiences oscillation due to the buoyancy force. The oscillation is reduced progressively with the increase of G. Moreover, h reaches its peak value near the T _pc . Higher G could improve the heat transfer performance in the whole temperature range. The peak value of h depends on P_in. A new correlation was proposed for methane at supercritical pressure convective heat transfer in the helical tube, which shows a good agreement with the present simulated results.     

The Stability of Eigenvalues for the p(x)-Laplacian Equations Involving Robin Boundary Conditions

In this paper, the stability of eigenvalues for the p(x)-Laplacian equations involving Robin boundary conditions has been studied in the framework of Γ-convergence. We prove that the uniform convergence of the exponents is enough to guarantee the convergence of the mth eigenvalues for Robin problems, which can be defined by inf sup sequences of Rayleigh quotient for the Luxemburg norms.     

Regional Gradient Stabilization of Semilinear Distributed Systems

This paper deals with regional stabilization of the gradient of semilinear distributed system evolving on a spatial domain Ω. It consists in studying the asymptotic behavior of gradient of such a system in a subregion ω of Ω. Then we characterize gradient stabilizing controls, and the one that stabilizes the gradient on ω and minimizes a given performance cost. The developed results are successfully illustrated by simulations.     

Harmonic Analysis of Finite Lamplighter Random Walks

Recently, several papers have been devoted to the analysis of lamplighter random walks, in particular, in the case where the underlying graph is the infinite path \( \mathbb{Z} \). In the present paper, we develop a spectral analysis for lamplighter random walks on finite graphs. In the general case, we use the C ₂-symmetry to reduce the spectral computations to a series of eigenvalue problems on the underlying graph. If the graph has a transitive isometry group G, we also describe the spectral analysis in terms of the representation theory of the wreath product C ₂?G. We apply our theory to the lamplighter random walks on the complete graph and on the discrete circle. These examples have already been studied by Häggström and Jonasson by probabilistic methods.     

General Expansiveness for Diffeomorphisms from the Robust and Generic Properties

Let f:M → M be a diffeomorphism on a closed smooth d(d ≥ 2)-dimensional manifold. For each \(n\in \mathbb N\), if f belongs to C ¹-interior of the set of the n-expansive diffeomorphisms, then f satisfies quasi-Anosov. For C ¹-generic f, if f is n-expansive then f satisfies both Axiom A and the no-cycle condition.     

Free Time Optimization of Second-Order Differential Inclusions with Endpoint Constraints

The present paper deals with the optimal control theory given by second-order differential inclusions (P _C ) with a non-fixed time interval and endpoint constraints. Our aim is to establish well-verifiable sufficient conditions of optimality for second-order differential inclusions. Thus, the sufficient conditions, including distinctive t ₁-attainability condition ones, are formulated by using the Euler-Lagrange and Hamiltonian type of inclusions. Here, the basic apparatus of locally adjoint mappings (L A M s) is suggested. Application of these results is illustrated by solving some linear control problem with second-order differential inclusions.     

Hierarchical Control for the One-dimensional Plate Equation with a Moving Boundary

In this paper, we investigate the controllability for the one-dimensional plate equation in intervals with a moving boundary. This equation models the vertical displacement of a point x at time t in a bar with uniform cross section. We assume the ends of the bar with small and uniform variations. More precisely, we have introduced functions α(t) and β(t) modeling the motion of these ends. We present the following results: the existence and uniqueness of Nash equilibrium, the approximate controllability with respect to the leader control, and the optimality system for the leader control.     

On High Dimensional Schrödinger Equation with Quasi-Periodic Potentials

In this paper, we consider the high dimensional Schrödinger equation \( -\frac {d^{2}y}{dt^{2}} + u(t)y= Ey, y\in \mathbb {R}^{n}, \) where u(t) is a real analytic quasi-periodic symmetric matrix, \(E= \text {diag}({\lambda _{1}^{2}}, \ldots , {\lambda _{n}^{2}})\) is a diagonal matrix with λ _j >0,j=1,…,n, being regarded as parameters, and prove that if the basic frequencies of u satisfy a Bruno-Rüssmann’s non-resonant condition, then for most of sufficiently large λ _j ,j=1,…,n, there exist n pairs of conjugate quasi-periodic solutions.     

Influence of gamma radiation on spectral photosensitivity of (Si<Subscript>2</Subscript>)<Subscript>1 − <Emphasis Type="Italic">x</Emphasis></Subscript> (ZnSe)<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript> solid solution

Increase of the photosensitivity of the pSi-n(Si₂)_{1 ? x} (ZnSe) _x (0 ≤ x ≤ 0.01) structure exposed to gamma radiation with photon energy E _ph ≥ 2.3 eV has been demonstrated. It is shown that irradiation with dose up to 10⁴ rad raises and radiation with dose up to 10⁵ rad reduces the forward current of the pSi-n(Si₂)_{1 ? x} (ZnSe) _x structure.     

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.

     

Attractor for the nonlinear Schr?dinger equation with a nonlocal nonlinear term

In this paper, we consider the long-time behavior of solutions of the dissipative 1D nonlinear Schrödinger (NLS) equation with nonlocal integral term and with periodic boundary conditions. We prove the existence of the global attractor \( \mathcal{A} \) for the nonlocal equation in the strong topology of H ¹(Ω). We also prove that the global attractor is regular, i.e., \( \mathcal{A} \subset {H^2}\left( \Omega \right) \), assuming that f(x) is of class C ². Furthermore, we estimate the number of the determining modes for this equation.     

Fabrication and investigation of ultravialet Au-Zn<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>Cd<Subscript>1–<Emphasis Type="Italic">x</Emphasis></Subscript>S-Mo-structures

For the first time fabricated and investigated the photovoltaic characteristics of Au-Zn _x Cd_1–x S-Mofilm structural injection photo detectors sensitive to narrow the ultraviolet region of the electromagnetic spectrum, based on polycrystalline Zn _x Cd_1–x S layers. It was found by adjusting the flow of ZnS and CdS coming to the surface Mo substrate can control the shape of the spectral sensitivity of the Au-Zn _x Cd_1–x S-Mo-film structural injection photo detectors. The results will allow to optimize the structure of photo detectors and solar cells based on polycrystalline thin film solar cells.     

Analysis of turbulence and surface growth models on the estimation of soot level in ethylene non-premixed flames

Soot prediction in a combustion system has become a subject of attention, as many factors influence its accuracy. An accurate temperature prediction will likely yield better soot predictions, since the inception, growth and destruction of the soot are affected by the temperature. This paper reported the study on the influences of turbulence closure and surface growth models on the prediction of soot levels in turbulent flames. The results demonstrated that a substantial distinction was observed in terms of temperature predictions derived using the k-ε and the Reynolds stress models, for the two ethylene flames studied here amongst the four types of surface growth rate model investigated, the assumption of the soot surface growth rate proportional to the particle number density, but independent on the surface area of soot particles, f(A_s) = ρN_s, yields in closest agreement with the radial data. Without any adjustment to the constants in the surface growth term, other approaches where the surface growth directly proportional to the surface area and square root of surface area, f(A_s) = A_s and f(A_s) = √A_s, result in an under-prediction of soot volume fraction. These results suggest that predictions of soot volume fraction are sensitive to the modelling of surface growth.     

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.     

Approximate Controllability of Fractional Semilinear Stochastic System of Order <Emphasis Type="Italic">α</Emphasis>∈ (1,2]

A family of dynamical control systems described by nonlinear fractional of order (1,2] stochastic differential equations in L _p spaces is considered. We discussed the approximate controllability of stochastic semilinear fractional control system of order α∈(1,2] under the assumption that the corresponding linear system is approximately controllable. A new set of sufficient conditions for approximate controllability of system are obtained by the theory of strongly continuous α-order cosine family, fixed point theorem, and stochastic analysis techniques. At the end, an example is given to illustrate the theory.     

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 \(\mathbb {K}^{2}\).