Simultaneous Representation and Approximation Formulas and High-Order Sobolev Embedding Theorems on Stratified Groups

We give various integral representation formulas simultaneously for a function and its derivatives in terms of vector field gradients of the function of appropriately high order. When the function has compact support, simpler formulas can be derived. Many of the results proved here appear to be new even in the special case of classical Euclidean space. For instance, Theorem 2.2 below reduces to the following result in the usual Euclidean case: Let B be a ball in R^N with radius r(B), let m be a positive integer, and let f\in W^m,1(B). Then there is a polynomial P=_m(B,f) of degree m-1 such that for any integers i, j with 0 j < i m and a.e. x\in B, |\nabla^j(f-P)(x)| C\int_{B}\frac{|\nablaⁱf(y)|}{|x-y|^N-(i-j)} dy +Cr(B)^i-j-N\int_B|\nablaⁱf(y)| dy. Moreover, if 0 < i-j N, then for a.e. x \in B we have the more refined formula |\nabla^j(f-P)(x)| C\int_B\frac{|\nablaⁱf(y)|}{|x-y|^N-i+j} dy.     

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: Theoretical formulation and model validation

This paper is first of the two papers dealing with analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables – which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations – are presented. A multi-dimensional Galerkin expansion of the solution of nonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effects of quadratic/cubic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation.     

Mindlin second-gradient elastic properties from dilute two-phase Cauchy-elastic composites Part II: Higher-order constitutive properties and application cases

Starting from a Cauchy elastic composite with a dilute suspension of randomly distributed inclusions and characterized at first-order by a certain discrepancy tensor (see part I of the present article), it is shown that the equivalent second-gradient Mindlin elastic solid: (i) is positive definite only when the discrepancy tensor is negative defined; (ii) the non-local material symmetries are the same of the discrepancy tensor, and (iii) the non-local effective behaviour is affected by the shape of the RVE, which does not influence the first-order homogenized response. Furthermore, explicit derivations of non-local parameters from heterogeneous Cauchy elastic composites are obtained in the particular cases of: (a) circular cylindrical and spherical isotropic inclusions embedded in an isotropic matrix, (b) n-polygonal cylindrical voids in an isotropic matrix, and (c) circular cylindrical voids in an orthotropic matrix.     

Dynamic Response of Fractional-Order Viscoelastic High-Order Shear Beam Based on Nonlocal Strain Gradient Elasticity

The dynamic behavior of a viscoelastic high-order shear microbeam is studied based on a new constitutive model which incorporates size effects and viscoelasticity simultaneously.The size effects are modeled by the nonlocal gradient elasticity,while viscoelastic effects are modeled by fractional-order derivatives.The constitutive relation and the equations of motion are both differential equations with fractional-order derivatives.Based on the Laplace transform and inverse transform,the analytical solution of the dynamic response under a step load is obtained in terms of the Mittag-Leffler function.In order to verify the reliability of the analytical solution,a comparison with the numerical solution is also provided.Based on the numerical results,the effects of the nonlocal parameter,strain gradient parameter,fractional-order parameter,and viscosity coefficient on the dynamic response of the viscoelastic microbeam are discussed.It is found that the influences of the fractional order and the coefficient of viscosity on the dynamic response of the microbeam are very different,although both are related to the viscoelastic behavior.     

Activity coefficients for ternary systems: VI. The system HCl + MgCl2 + H2O at different temperatures; application of Pitzer's equations

Electromotive-force measurements have been made on HCl–MgCl₂–H₂O mixtures at 5, 15, 25, 35 and 45°C at eleven different ionic strengths from 0.1–5.0 mol-kg ^–1 . The results are interpreted in terms of the simple Harned's equations, as well as the more complicated Pitzer ion-component treatment of multicomponent electrolyte mixtures. Activity coefficients for HCl in the salt mixtures obey Harned's rule up to and including I=5.0. For the salt in the acid mixtures, Harned's rule holds true up to and including I=0.5. The contribution of higher-order electrostatic terms (E and E') in the Pitzer equations is important for accurate evaluations of unlike cation-cation interactions (_H,Mg), and cation-anion-cation interactions (_H,Mg,Cl). The values of^S_H,Mg and _H,Mg,Cl (determined with E and E'), _H,Mg and _H,Mg,Cl (determined without E and E'), as well as the trace activity coefficients of HCl, _tr ^A , in solutions of MgCl₂ (where ionic strength fraction of the salt,y _B = 1) at all the experimental temperatures and ionic strengths, are reported. Results of this study are compared with those for similar systems. At I=0.1 and 25°C, the results of the Brönsted-Guggenheim specific interaction theory are discussed briefly.     

Application of least square method to arbitrary-order problems with separated boundary conditions

In this paper, differential equations of arbitrary order with separated boundary conditions are converted into an optimal control problem. Then a convergent approximate solution is constructed such that the exact boundary conditions are satisfied. In fact it will be shown that for every ε>0$\epsilon >0$, there exists an approximate solution _vε$v_{\epsilon }$ of B-spline functions such that the corresponding least square error is less than ε>0$\epsilon >0$, and also _vε$v_{\epsilon }$ satisfies the exact boundary conditions. Some examples are given and the optimal errors are obtained for the sake of comparison.     

Higher-order diffraction images in photorefractive materials

We report some photorefractive higher-order diffraction phenomena of a liquid crystal polymer composite. The photorefractive composite consists of the polymer poly[N-vinylcarbazole] (PVK) doped with the chromophore 4,4′-n-entylcyanobiphenyl (5CB) and the sensitizer C₆₀. In two-wave coupling experiments, the signal beam was interfered by a mutually coherent reference beam within composite materials at a small incident angle. Results are given when the composite was placed behind the focal plane, in front of the focal plane, and in the focal plane. Higher-order diffraction images were obtained by amplification and reduction. A theory of the properties of higher-order diffraction images in a photorefractive liquid crystal polymer composite is developed. The theory is in good agreement with the results of experimental work. The experiments demonstrate the feasibility of optical image processing.