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.     

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.

     

First Variation of the Hausdorff Measure of Non-horizontal Submanifolds in Sub-Riemannian Stratified Lie Groups

We determine necessary conditions for a non-horizontal submanifold of a sub-Riemannian stratified Lie group to be of minimal measure. We calculate the first variation of the measure for a non-horizontal submanifold and find that the minimality condition implies the tensor equation H + σ = 0, where H is analogous to the mean curvature and σ is the mean torsion. We also discuss new examples of minimal non-horizontal submanifolds in the Heisenberg group, in particular surfaces in \(\mathbb {H}^{2}\).     

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).     

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 \(\mathbb {K}^{2}\).     

Second-Order Viability Problems with Perturbation in Hilbert Spaces

In this paper, we study the existence of viable solutions to the differential inclusion

$ \ddot{x}(t) \in f\left( {t,x(t),\dot{x}(t)} \right) + F\left( {x(t),\dot{x}(t)} \right), $

where f is a Carathéodory single-valued map and F is an upper semi-continuous multifunction with compact values contained in the Clarke subdifferential ? _c V of an uniformly regular function V.

     

Existence of Weak Solutions for Generalized Quasilinear Schrödinger Equations

This paper shows the existence of nontrivial weak solutions for the generalized quasilinear Schrödinger equations

$$ -div(g^{p}(u)|\nabla u|^{p-2}\nabla u)+g^{p-1}(u)g^{\prime}(u)|\nabla u|^{p}+ V(x)|u|^{p-2}u=h(u),\,\, x\in \mathbb{R}^{N}, $$

where N ≥ 3, \(g(s): \mathbb {R}\rightarrow \mathbb {R}^{+}\) is C ¹ nondecreasing function with respect to |s|, V is a positive potential bounded away from zero and h(u) is a nonlinear term of subcritical type. By introducing a variable replacement and using minimax methods, we show the existence of a nontrivial solution in \(C^{\alpha }_{loc}(\mathbb {R}^{N})\).

     

Geodesics in the Engel Group with a Sub-Lorentzian Metric

Let E be the Engel group and D be a rank 2 bracket generating left invariant distribution with a Lorentzian metric, which is a nondegenerate metric of index 1. In this paper, we first study some properties of horizontal curves on E. Second, we prove that time-like normal geodesics are locally maximizers in the Engel group and calculate the explicit expression of non-space-like geodesics.     

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.     

Small Time Asymptotics on the Diagonal for Hörmander’s Type Hypoelliptic Operators

We compute the small time asymptotics of the fundamental solution of Hörmander’s type hypoelliptic operators with drift, on the diagonal at a point x ₀. We show that the order of the asymptotics depends on the controllability of an associated control problem and of its approximating system. If the control problem of the approximating system is controllable at x ₀, then so is also the original control problem, and in this case we show that the fundamental solution blows up as \(\phantom {\dot {i}\!}t^{-\mathcal {N}/2}\), where \(\phantom {\dot {i}\!}\mathcal {N}\) is a number determined by the Lie algebra at x ₀ of the fields, that define the hypoelliptic operator.     

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.     

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.     

Gauss-Bonnet Theorem in Sub-Riemannian Heisenberg Space $\mathbb H^{1}$

We prove a version of the Gauss-Bonnet theorem in sub-Riemannian Heisenberg space \(\mathbb H^{1}\). The sub-Riemannian distance makes \(\mathbb H^{1}\) a metric space that consequently has a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transverse to the tangent space of S. If all points of S have this property, we prove a Gauss-Bonnet formula and, for compact surfaces (which are topologically a torus), we obtain \({\int }_{S}K=0\).     

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.     

Output-Based Stabilization of Timoshenko Beam with the Boundary Control and Input Distributed Delay

In this paper, we consider the output-feedback exponential stabilization of Timoshenko beam with the boundary control and input distributed delay. Suppose that the outputs of controllers are of the forms \(\alpha _{1}u_{1}(t)+\beta _{1}u_{1}(t-\tau )+{\int }_{-\tau }^{0}g_{1}(\eta )u_{1} (t+\eta )d\eta \) and \(\alpha _{2}u_{2}(t)+\beta _{2}u_{2}(t-\tau ) +{\int }_{-\tau }^{0}g_{2}(\eta )u_{2}(t+\eta )d\eta \) respectively, where u ₁(t) and u ₂(t) are the inputs of controllers. Using the tricks of the Luenberger observer and partial state predictor, we translate the system with delay into a system without delay. And then, we design the feedback controls to stabilize the system without delay. Finally, we prove that under the choice of such controls, the original system also is stabilized exponentially.     

Real-Formal Orbital Rigidity for Germs of Real Analytic Vector Fields on the Real Plane

For \(n \geqslant 2\), we consider \(\mathcal {V}^{\mathbb {R}}_{n}\) the class of germs of real analytic vector fields on \(\left (\mathbb {R}^{2}, \widehat {0}\right )\) with zero (n?1)-jet and nonzero n-jet. We prove, for generic germs of \(\mathcal {V}^{\mathbb {R}}_{n}\), that the real-formal orbital equivalence implies the real-analytic orbital equivalence, that is, the real-formal orbital rigidity takes place. This is the real analytic version of Voronin’s formal orbital rigidity theorem.     

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.     

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.     

Horizontal Holonomy for Affine Manifolds

In this paper, we consider a smooth connected finite-dimensional manifold M, an affine connection ? with holonomy group H ^? and Δ a smooth completely non integrable distribution. We define the Δ-horizontal holonomy group \({H^{\;\nabla }_{\Delta }}\) as the subgroup of H ^? obtained by ?-parallel transporting frames only along loops tangent to Δ. We first set elementary properties of \({H^{\;\nabla }_{\Delta }}\) and show how to study it using the rolling formalism Chitour and Kokkonen (2011). In particular, it is shown that \({H^{\;\nabla }_{\Delta }}\) is a Lie group. Moreover, we study an explicit example where M is a free step-two homogeneous Carnot group with m ≥ 2 generators, and ? is the Levi-Civita connection associated to a Riemannian metric on M, and show in this particular case that \({H^{\;\nabla }_{\Delta }}\) is compact and strictly included in H ^? as soon as m ≥ 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.