Differential Independence of <Emphasis Type="Italic">Γ</Emphasis> and <Emphasis Type="Italic">ζ</Emphasis>

In a celebrated theorem H?lder proved that the Euler Γ-function is differential transcendental, i.e. Γ(z) is not a solution of any (non-trivial) algebraic ordinary differential equation with coefficients that are complex numbers; and we extend his methods to the Riemann ζ-function. Moreover, we conjecture that Γ and ζ are differential independent, i.e. Γ(z) is not a solution of any such algebraic differential equation—even allowing coefficients that are differential polynomials in ζ(z). However, we are able to demonstrate only the partial result that Γ(z) and ζ(sin 2πz) are differential independent.     

2<Emphasis Type="Italic">T</Emphasis>-periodic solution for<Emphasis Type="Italic">m</Emphasis> order neutral type differential equations with time delays

Periodic solution of m order linear neutral equations with constant coefficient and time delays was studied. Existence and uniqueness of 2 T-periodic solutions for the equation were discussed by using the method of Fourier series. Some new necessary and sufficient conditions of existence and uniqueness of 2T-periodic solutions for the equation are obtained. The main result is used widely. It contains results in some correlation paper for its special case, improves and extends the main results in them. Existence of periodic solution for the equation in larger number of particular case can be checked by using the result, but cannot be checked in another paper. In other words, the main result in this paper is most generalized for (1), the better result cannot be found by using the same method. Foundation item: the Natural Science Foundation of Yunnan Education Committee (990002) Biography: ZHANG Bao-sheng (1962-)     

Motion of a Graph by <Emphasis Type="Italic">R</Emphasis>-Curvature

We show the existence of weak solutions to the partial differential equation which describes the motion by R-curvature in R ^d , by the continuum limit of a class of infinite particle systems. We also show that weak solutions of the partial differential equation are viscosity solutions and give the uniqueness result on both weak and viscosity solutions.     

A new method for calculating hysteretic <Emphasis Type="Italic">K</Emphasis>(<Emphasis Type="Italic">S</Emphasis>) relationship

A new method for calculating the hysteretic relationship between hydraulic conductivity (K) and suction (S) is proposed. This method uses the experimental (K–S) data of the main wetting and drying branches and predicts satisfactorily the scanning drying and wetting curves. The proposed method is applicable to those porous media where the hysteretic Θ–S relationship complies with the independent domain concept.     

Inequalities relating to Lp-version of Petty＇s conjectured projection inequality

Petty＇s conjectured projection inequality is a famous open problem in the theory of convex bodies. In this paper, it is shown that an inequality relating to Lp-version of the Petty＇s conjectured projection inequality is developed by using the notions of the Lp-mixed volume and the Lp-dual mixed volume, the relation of the Lp-projection body and the geometric body Г-pK, the Bourgain-Milman inequality and the Lp-Bnsemann-Petty inequality. In addition, for each origin-symmetric convex body, by applying the Jensen inequality and the monotonicity of the geometric body Г-pK, the reverses of Lp-version of the Petty＇s conjectured projection inequality and the Lp-Petty projection inequality are given, respectively.     

Asymptotic properties of solutions of an equation of order <Emphasis Type="Italic">n</Emphasis> with almost constant coefficients and pulse action

We investigate the asymptotic behavior of solutions of linear differential equations with almost constant coefficients and pulse action at fixed times as t tends to infinity. We establish conditions for the times of pulse action under which there exist values of pulse action for which the solution of the considered Cauchy problem with initial conditions that coincide with the initial conditions for a certain (arbitrary but fixed) solution of the original equation without pulse action is bounded, unbounded, or tending to infinity. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 4, pp. 444–455, October–December, 2005.     

g-η-Monotone mapping and resolvent operator technique for solving generalized implicit variational-like inclusions

A new class of g-η-monotone mappings and a class of generalized implicit variational-like inclusions involving g-η-monotone mappings are introduced. The resolvent operator of g-η-monotone mappings is defined and its Lipschitz continuity is presented. An iterative algorithm for approximating the solutions of generalized implicit variational-like inclusions is suggested and analyzed. The convergence of iterative sequences generated by the algorithm is also proved.     

<Emphasis Type="Italic">J</Emphasis><Subscript>2</Subscript> invariant relative orbits via differential correction algorithm

This paper describes a practical method for finding the invariant orbits in J ₂ relative dynamics. Working with the Hamiltonian model of the relative motion including the J ₂ perturbation, the effective differential correction algorithm for finding periodic orbits in three-body problem is extended to formation flying of Earth’s orbiters. Rather than using orbital elements, the analysis is done directly in physical space, which makes a direct connection with physical requirements. The asymptotic behavior of the invariant orbit is indicated by its stable and unstable manifolds. The period of the relative orbits is proved numerically to be slightly different from the ascending node period of the leader satellite, and a preliminary explanation for this phenomenon is presented. Then the compatibility between J ₂ invariant orbit and desired relative geometry is considered, and the design procedure for the initial values of the compatible configuration is proposed. The influences of measure errors on the invariant orbit are also investigated by the Monte–Carlo simulation. The project supported by the Innovation Foundation of Beihang University for Ph.D. Graduates, and the National Natural Science Foundation of China (60535010).     

Measurement of velocity profiles in a rectangular microchannel with aspect ratio <Emphasis Type="Italic">α</Emphasis> = 0.35

In this work, we measured 14 horizontal velocity profiles along the vertical direction of a rectangular microchannel with aspect ratio α = h/w = 0.35 (h is the height of the channel and w is the width of the channel) using microPIV at Re = 1.8 and 3.6. The experimental velocity profiles are compared with the full 3D theoretical solution, and also with a Poiseuille parabolic profile. It is shown that the experimental velocity profiles in the horizontal and vertical planes are in agreement with the theoretical profiles, except for the planes close to the wall. The discrepancies between the experimental data and 3D theoretical results in the center vertical plane are less than 3.6%. But the deviations between experimental data and Poiseuille’s results approaches 5%. It indicates that 2D Poiseuille profile is no longer a perfect theoretical approximation since α = 0.35. The experiments also reveal that, very near the hydrophilic wall (z = 0.5–1 μm), the measured velocities are significantly larger than the theoretical velocity based on the no-slip assumption. A proper discussion on some physical effects influencing the near wall velocity measurement is given.     

An experimental—Numerical evaluation of the<Emphasis Type="Italic">T</Emphasis><Stack><Subscript>ε</Subscript><Superscript>*</Superscript></Stack> integral for a three-dimensional crack front

TheT _ε ^* integral was calculated on the surface of single edge notched, three-point bend (SE(B)) specimens using experimentally obtained displacements. Comparison was made withT _ε ^* calculated with the measured surface displacements andT _ε ^* calculated at several points through the thickness of a finite element (FE) model of the SE(B) specimen. Good comparison was found between the surfaceT _ε ^* calculated from displacements extracted from the FE model and the surfaceT _ε ^* calculated from experimentally obtained displacements. The computedT _ε ^* integral was also observed to decrease as the crack front was traversed from the surface to the mid-plane of the specimen. Mid-planeT _ε ^* values tend to be approximately 10% of the surface values.     

<Emphasis Type="Italic">In-situ</Emphasis> AFM Experiments with Discontinuous DIC Applied to Damage Identification in Biomaterials

Natural materials (e.g. nacre, bone, and spider silk) exhibit unique and outstanding mechanical properties. This performance is due to highly evolved hierarchical designs. Building a comprehensive understanding of the multi-scale mechanisms that enable this performance represents a critical step toward realizing strong and tough bio-inspired materials. This paper details a multi-scale experimental investigation into the toughening mechanisms in natural nacre. By applying extended digital image correlation and other image processing techniques, quantitative information is extracted from otherwise prodominantly qualitative experiments. In situ three point bending fracture tests are performed to identify and quantify the toughening mechanisms involved during the fracture of natural nacre across multiple length scales. At the macro and micro scales, fracture tests performed in situ with a macro lens and optical microscope enable observation of spreading of damage outward from the crack tip. This spreading is quantified using an iso-contour technique to assess material toughness. At the nanoscale, fracture tests are performed in situ an atomic force microscope to link the larger-scale damage spreading to sliding within the tablet-based microstructure. To quantify the magnitude of sliding and its distribution, images from the in situ AFM fracture tests are analyzed using new algorithms based on digital image correlation techniques which allow for discontinuous displacement fields. Ultimately, this comprehensive methodology provides a framework for broad experimental investigations into the failure mechanisms of bio- and bio-inspired materials.     

Variational Analysis of the Asymptotics of the <Emphasis Type="Italic">XY</Emphasis> Model

In this paper we consider the XY (N-dimensional possibly anisotropic) spin-type model and, by comparison with a Ginzburg–Landau-type functional, we perform a variational analysis in the limit when the number of particles diverges. In particular we show how the appearance of vortex-like singularities can be described by properly scaling the energy of the system through a Γ-convergence procedure. We also address the problem in the case of long-range interactions and solve it in two dimensions.     

Green’s function for T-stress of semi-infinite plane crack

Green’s function for the T-stress near a crack tip is addressed with an analytic function method for a semi-infinite crack lying in an elastical, isotropic, and infinite plate. The cracked plate is loaded by a single inclined concentrated force at an interior point. The complex potentials are obtained based on a superposition principle, which provide the solutions to the plane problems of elasticity. The regular parts of the potentials are extracted in an asymptotic analysis. Based on the regular parts, Gre...     

On the L_p intersection body

In this paper,by using the Brunn-Minkowski-Firey mixed volume theory and dual mixed volume theory,associated with L_p intersection body and dual mixed volume,some dual Brunn-Minkowski inequalities and their isolate forms are established for L_p intersection body about the normalized L_p radial addition and L_p radial linear combination.Some properties of operator Lp are given.     

Degradation of nylon-6/clay nanocomposites in NO<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>

Nylon-6 is an important engineering polymer that, in its fully spherulitic (bulk) form, has many applications in gears, rollers, and other long life cycle components. In 1993, Toyota commercialized a nylon-6/clay nanocomposite out of which it produced the timing belt cover for the Camry. Although these hybrid nanocomposites show significant improvements in their mechanical response characteristics, including yield strength and heat distortion temperature, little is known about the degradation of these properties due to environmental pollutants like NO _x . Nylon-6 fibers are severely degraded by interaction with NO _x and other pollutants, showing a strong synergy between applied load and environmental degradation. While the nanocomposites show a significant reduction in permeability of gases and water due to the incorporation of lamellar clay, their susceptibility to nondiffusional mechano-chemical degradation is unknown. The fracture toughness of these nylon-6/clay nanocomposites increases, not as a function of clay content, but as a function of the volume of nylon-6 polymer chains influenced by the clay lamellae surfaces. Both the clay and the constrained volume offer the nanocomposites some protection from the deleterious effects of NO _x . The time-to-failure at a given stress intensity factor as a function of clay content and constrained volume is discussed along with fracture toughness of the materials.     

Navier–Stokes Equations: Almost <Emphasis Type="Italic">L</Emphasis><Subscript>3,∞</Subscript>-Case

A sufficient condition of regularity for solutions to the Navier–Stokes equations is proved. It generalizes the so-called L _3,∞-case.     

Weakly R-KKM mappings——intersection theorems and minimax inequalities in topological spaces

In this paper, we introduce the concepts of weakly R-KKM mappings, R-convex and ,R-β-quasiconvex in general topological spaces without any convex structure. Relating to these, we obtain an extension to general topological spaces of Fan's matching theorem, namely that Lemma 1.2 in this paper. On this basis, two intersection theorems are proved in topological spaces. By using intersection theorems, some minimax inequalities of Ky Fan type are also proved in topological spaces. Our results generalize and improve the corresponding results in the literature.     

<Emphasis Type="Italic">In situ</Emphasis><Emphasis Type="Italic">μ</Emphasis> CT-scan Mechanical Tests: Fast 4D Mechanical Identification

A recently proposed “Projection-based Digital Volume Correlation” (P-DVC) method is extended in this work to a cone-beam lab-tomograph in which a mechanical test is performed. This consists of a crack propagation test in an elastic-brittle gypsum specimen. Kinematic analysis is performed based on a reduced finite element modeling for which the appropriate boundary conditions and crack propagation stage are determined from the radiographs. By considering only two projections per loading step, an integrated model-based analysis of the entire test provides a full space and time identification of the kinematics, including the crack position and the determination of two material parameters. This is achieved with a drastic reduction in the acquisition time compared to classical digital volume correlation analysis. In the examples presented, the acquisition time was reduced by a factor of 350.     

MODAL SYNTHESIS METHOD FOR NORM COMPUTATION OF H∞ DECENTRALIZED CONTROL SYSTEMS （ I ）

When using H∞ techniques to design decentralized controllers for large systems,the whole system is divided into subsystems, which are analysed using H∞ control theorybefore being recombined. An analogy was established with substructural analysis instructural mechanics, in which H∞ decentralized control theory corresponds to substructuralmodal synthesis theory so that the optimal H∞ norm of the whole system corresponds to thefundamental vibration frequency of the whole structure. Hence, modal synthesismethodology and the extended Wittrick-Williams algorithm were transplanted from structuralmechanics to compute the optimal H∞ norm of the control system. The orthogonality and theexpansion theorem of eigenfunctions of the subsystems H∞ control are presented in part(I) of the paper. The modal synthesis method for computation of the optimal H∞ norm ofdecentralized control systems and numerical examples are presented in part (Ⅱ).     

Assessment of <Emphasis Type="Italic">Σ−Y</Emphasis><Subscript>liq</Subscript> model predictions for air-assisted atomisation

In a series of recent works, R. Borghi and co-workers proposed a new Eulerian model of two-phase turbulent flows which introduced a transport equation for the average area of the liquid–gas interface. This work is devoted to the assessment of this model’s ability to predict the effects of liquid properties and injection regimes on the atomisation quality. Air-assisted atomisation, for which extensive experimental data are available, is chosen as a test case. It is shown that the model predictions are in good agreement with the observed trends for a wide range of variations of the liquid properties, such as density and surface tension, as well as the injection regimes, defined by the liquid and gas jet exit velocities.