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.     

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.     

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

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).     

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).     

<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).     

MAXIMAL ELEMENTS FOR GB-MAJORIZED MAPPINGS IN PRODUCT G-CONVEX SPACES AND APPLICATIONS (Ⅰ)

A new family of set-valued mappings from a topological space into generalized convex spaces was introduced and studied. By using the continuous partition of unity theorem and Brouwer fixed point theorem, several existence theorems of maximal elements for the family of set-valued mappings were proved under noncompact setting of product generalized convex spaces. These theorems improve, unify and generalize many important results in recent literature. Contributed by DING Xie-ping Foundation items: the National Natural Science Foundation of China (19871059); the Natural Science Foundation of Education Department of Sichuan Province of China ([2000]25) Biography: DING Xie-ping (1938∼)     

MAXIMAL ELEMENTS FOR G-B-MAJORIZED MAPPINGS IN PRODUCT G-CONVEX SPACES AND APPLICATIONS (Ⅱ)

By applying existence theorems of maximal elements for a family of G_B-majorized mappings in a product space of G-convex spaces, some coincidence theorem, Fan-Browder type fixed point theorem and some existence theorems of solutions for a system of minimax inequalities are proved under noncompact setting of G-convex spaces. These theorems improve and generalize many important known results in literature. Contributed by Duping Xie-ping Foundation items: the National Natural Science Foundation of China (19871059); the Natural Science Foundation of Education Department of Sichuan Province ([2000]25) Biography: Duping Xie-ping (1938∼), Professor (E-mail: dingxip@sichu.edu.cn)     

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.     

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.     

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...     

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.      

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.     

A class of nonlinear boundary value problems for the second-order<Emphasis Type="Italic">E</Emphasis><Subscript>2</Subscript> class elliptic systems in general form

A class of nonlinear boundary value problems (BVP) for the second-order E₂ class elliptic systems in general form is discussed. By introducing a kind of transformation, this kind of BVP is reduced to a class of generalized nonlinear Riemann-Hilbert BVP. And then some singular integral operators are introduced to establish the equivalent nonlinear singular integral equations. The solvability is proved under some suitable hypotheses by means of the properties of singular integral operators and the function theoretic methods. Foundation items: the National Natural Science Foundation of China (19671056); Shanghai Municipal Natural Scientific Foundation (99ZA14030, 01ZA14023); Jiangxi Provincial Natural Scientific Foundation (981102, 0211014) Biographies: LI Ming-zhong (1935−); XU Ding-hua (1967−)     

Effects of measurement techniques on the experimental characterization of crack tip opening angle-Δ<Emphasis Type="Italic">a</Emphasis> resistance curves

One of the most promising approaches to characterizing the stable tearing process for lower constraint configurations in metallic materials has been indicated to be the crack tip opening angle (CTOA) criterion. In this paper we examine the effect of measurement distance selection and the impact of measurement procedures/analysis on the experimental CTOA-Δa resistance curve behavior of a Ti-6Al-4V and an Al5083 alloy. A new, systematic procedure for carrying out the experimental, surface CTOA measurement process has been used, which has uncovered artifacts of measurement distance selection that have not been presented prior to this study. By studying the CTOA-Δa behavior using a rigidly defined and consistent method, it is apparent that the measurement distance behind the crack tip at which angles are measured has a direct impact on the shape of the early transitional behavior of the CTOA-Δa behavior as well as the critical, experimental CTOA values produced.     

Mechanical properties of cocoons constructed consecutively by a single silkworm caterpillar, <Emphasis Type="Italic">Bombyx mori</Emphasis>

Most animals have the ability to adapt, to some extends and in different ways, the variation or disturbance of environment. In our experiments, we forced a silkworm caterpillar to spin two, three or four thin cocoons by taking it out from the cocoon being constructed. The mechanical properties of these cocoons were studied by static tensile tests and dynamic mechanical thermal analysis. Though external disturbances may cause the decrease in the total weight of silk spun by the silkworm, a gradual enhancement was interestingly found in the mechanical properties of these thin cocoons. Scanning electron microscopy observations of the fractured specimens of the cocoons showed that there exist several different energy dissipation mechanisms occurred simultaneously at macro-, meso-, and micro-scales, yielding a superior capacity of cocoons to adsorb the energy of possible attacks from the outside and to protect efficiently its pupa against damage. Through evolution of millions of years, therefore, the silkworm Bombyx mori seems to have gained the ability to adapt external disturbances and to redesign a new cocoon with optimized protective function when its first cocoon has been damaged for some reasons.     

Hausdorff dimension of set generated by exceptional oscillations of a class of N-parameter Gaussian processes

A class of N-parameter Gaussian processes are introduced, which are more general than the N-parameter Wiener process. The definition of the set generated by exceptional oscillations of a class of these processes is given, and then the Hausdorff dimension of this set is defined. The Hausdorff dimensions of these processes are studied and an exact representative for them is given, which is similar to that for the two-parameter Wiener process by Zacharie （2001）. Moreover, the time set considered is a hyperrectangle which is more general than a hyper-scluare used by Zacharie （2001）. For this more general case, a Fernique-type inequality is established and then using this inequality and the Slepian lemma, a Levy＇s continuity modulus theorem is shown. Independence of increments is required for showing the representative of the Hausdorff dimension by Zacharie （2001）. This property is absent for the processes introduced here, so we have to find a different way.