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.     

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.     

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.     

Natural heat accumulation in large enclosures of premises with an insulation passive solar heating system

This article provides the results of design studies intended to determine the specific amount of heat accumulated in the inner layers of vast enclosures used in premises with an insulation passive solar heating system (Q _acc) by the temperature-wave method. As shown by the calculation data, the value of Q _acc depends significantly on the ratio of the F _te surface area of the translucent enclosure of the considered premise to the F _fl surface area of its floor slabs and the amplitude of air motion in the heated premise. According to the calculated data, Q _acc, at the end of the second decade of March, is 169.0 kJ/m² at F _te/F _fl = 0.30–0.40 whereas the coefficient of natural heat accumulation in the inner layers of the given vast enclosure is ≈0.12.     

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.

     

Effect of partial premixing on the sooting structure of methane flames

An experimental investigation of the sooting structure of diluted methane–oxygen counterflow flames is reported for partial premixing in the following two nonpremixed flame configurations:

• Case 1:Nonpremixed flame on the oxidizer side of the stagnation plane,
• Case 2:Nonpremixed flame on the fuel side of the stagnation plane.

Effects of both fuel-side and oxidizer-side partial premixing for Cases 1 and 2 were investigated in a low-strain-rate (∼6–8 s⁻¹) counterflow flame. Computations using OPPDIF code were in excellent agreement with the measured concentrations of major species and [OH]. Distribution of measured soot volume fraction and particle sizes are presented along with measured distributions of C₂ hydrocarbon species. Soot loading can increase or decrease depending on (a) the level of partial premixing, (b) the side of partial premixing (fuel side or oxidizer side), and (c) the nonpremixed flame configuration. Of particular interest is the trend for fuel-side partial premixing of Case 1, where the peak soot loading, the peak soot particle diameter, and the thickness of the soot zone initially decrease and then increase with progressive partial premixing. The trends presented are discussed based on chemical, dilution, and flow-field effects of partial premixing on soot growth in counterflow flames. Unlike previous literature, which focused on soot inception, this work emphasizes the role of partial premixing on soot growth by taking into account the changes in the temperature–time history of soot particulates in addition to the previously reported “chemical” and “dilution” effects.     

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.     

Influence of composition and heat treatment in CdCl<Subscript>2</Subscript> solution on intrinsic point defects in CdTe films

CdTe films of different compositions were grown by the chemical molecular-beam deposition method. The activation energy and the nature of deep levels in relation with the composition of films based on the temperature dependence of electroconductivity are defined, and the influence on these levels of heat treatment process in CdCl₂ solution has been studied. Deep levels with the following activation energies are defined: E_v +0.31 eV; E_v + 0.42 ± 0.03 eV; E_c–0.44 ± 0.01 eV; E_c–0.28 eV; E_v + 0.24 ± 0.01 eV.     

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.     

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.

     

Effect of Marangoni number on thermocapillary convection and free-surface deformation in liquid bridges

Floating zone technique is a crucible-free process for growth of high quality single crystals. Unstable thermocapillary convection is a typical phenomenon during the process under microgravity. Therefore, it is very important to investigate the instability of thermocapillary convection in liquid bridges with deformable free-surface under microgravity. In this works, the Volume of Fluid(VOF) method is employed to track the free-surface movement. The results are presented as the behavior of flow structure and temperature distribution of the molten zone. The impact of Marangoni number(Ma) is also investigated on free-surface deformation as well as the instability of thermocapillary convection. The free-surface exhibits a noticeable axisymmetric(but it is non-centrosymmetric) and elliptical shape along the circumferential direction. This specific surface shape presents a typical narrow ‘neck-shaped' structure with convex at two ends of the zone and concave at the mid-plane along the axial direction. At both θ = 0° and θ = 90°, the deformation ratio ξ increases rapidly with Ma at first, and then increases slowly. Moreover, the hydrothermal wave number m and the instability of thermocapillary convection increase with Ma.     

Thin Semiconductor Films of Fullerene C<Subscript>70</Subscript> Nanoaggregates on the Surface of a Plane Glass Substrate

It was shown for the first time that, in the volume of an evaporating drop of a fullerene C₇₀ solution in toluene located on the surface of a glass substrate, the self-organization of C₇₀ molecules occurs and crystalline nC₇₀ nanoaggregates of a spherical shape are synthesized. Films of nC₇₀ nanoaggregates are deposited on a plane surface of a glass substrate. The physical mechanism of the self-organization of C₇₀ molecules and the formation of large nC₇₀ nanoaggregates are proposed, namely, Ostwald ripening, according to which relatively large nC₇₀ nanoaggregates grow through smaller ones. The possibility of using thin films as solar transducers is discussed.     

Effect of Rotational Speed on the Stability of Two Rotating Side-by-side Circular Cylinders at Low Reynolds Number

Flow around two rotating side-by-side circular cylinders of equal diameter D is numerically studied at the Reynolds number 40≤ Re ≤200 and various rotation rate θ_i. The incoming flow is assumed to be two-dimensional laminar flow. The governing equations are the incompressible Navier-Stokes equations and solved by the finite volume method(FVM). The ratio of the center-to-center spacing to the cylinder diameter is T/D=2. The objective of the present work is to investigate the effect of rotational speed and Reynolds number on the stability of the flow.The simulation results are compared with the experimental data and a good agreement is achieved. The stability of the flow is analyzed by using the energy gradient theory, which produces the energy gradient function K to identify the region where the flow is the most prone to be destabilized and the degree of the destabilization. Numerical results reveal that K is the most significant at the separated shear layers of the cylinder pair. With Re increases,the length of the wake is shorter and the vortex shedding generally exhibits a symmetrical distribution forθ_iθ_(crit), It is also shown that the unsteady vortex shedding can be suppressed by rotating the cylinders in the counter-rotating mode.     

Modification of aluminum alloys in a solar furnace

The processes of the modification of the D16 aluminum alloy at the focus of a solar furnace are studied experimentally. The surface heat treatment at the solar furnace focus was performed without melting the surface. The experiment showed that the grain size decreases from 10 to 3 microns with the exposure of the specimens at the focal spot for 35 seconds, and small regions with the grain size 0.3 micron appear. Grains of dimension 0.3 micron are observed with the exposure of the specimen at the solar furnace focus for 40 seconds, and the microhardness is H _v = 74.8 kg/mm². The repeat measurement of the microhardness after 40 days corresponds to H _v = 104.8 kg/mm², i.e., the microhardness after modification in the solar furnace and natural aging increases by 16%.     

Prediction of the theoretical and semi-empirical model of ambient temperature

It is well known that the ambient temperature is a sensitive parameter which has a great effect on biology, technology, geology and even on human behavior. A prediction is a statement about an uncertain event. It is often, but not always, based upon experience or knowledge. Although guaranteed accurate information about the future is in many cases impossible, prediction can be useful to assist in making plans about possible developments. As a result, temperature profiles can be developed which accurately represent the expected ambient temperature exposure that this environment experiences during measurement. The ambient temperature over time is modeled based on the previous T _min and T _max data and using a Lagrange interpolation. To observe the comprehensive variation of ambient temperature the profile must be determined numerically. The model proposed in this paper can provide an acceptable way to measure the change in ambient temperature.     

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 \(\mathbb {K}^{2}\).     

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.     

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 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).