Some estimates for the oscillation of the deformation gradient

As a measure of deformation we can take the difference D - R, where D is the deformation gradient of the mapping and R is the deformation gradient of the mapping , which represents some proper rigid motion. In this article, the norm is estimated by means of the scalar measure e( ) of nonlinear strain. First, the estimates are given for a deformation W ^1,p() satisfying the condition . Then we deduce the estimate in the case that (x) is a bi-Lipschitzian deformation and .     

Exponential growth of the vorticity gradient for the Euler equation on the torus

We prove that there are solutions to the Euler equation on the torus with C^1,α$C^{1,\alpha }$ vorticity and smooth except at one point such that the vorticity gradient grows in L^∞$L^{\infty }$ at least exponentially as t→∞$t\to \infty$. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Šverák [6].     

Singular quasilinear equations with quadratic growth in the gradient without sign condition

Given a bounded domain Ω in RN, and a function a∈Lq(Ω) with q>N/2, we study the existence of a positive solution for the quasilinear problem

     

Approximation and regularization of Lipschitz functions: Convergence of the gradients

We examine the possible extensions to the Lipschitzian setting of the classical result on -convergence: first (approximation), if a sequence of functions of class from to converges uniformly to a function of class , then the gradient of is a limit of gradients of in the sense that ; second (regularization), the functions can be chosen to be of class and -converging to in the sense that . In other words, the space of functions is dense in the space of functions endowed with the pseudo-norm.

We first deepen the properties of Warga's counterexample (1981) for the extension of the approximation part to the Lipschitzian setting. This part cannot be extended, even if one restricts the approximation schemes to the classical convolution and the Lasry-Lions regularization. We thus make more precise various results in the literature on the convergence of subdifferentials.

We then show that the regularization part can be extended to the Lipschitzian setting, namely if is a locally Lipschitz function, we build a sequence of smooth functions such that

In other words, the space of functions is dense in the space of locally Lipschitz functions endowed with an appropriate Lipschitz pseudo-distance. Up to now, Rockafellar and Wets (1998) have shown that the convolution procedure permits us to have the equality , which cannot provide the exactness of our result.

As a consequence, we obtain a similar result on the regularization of epi-Lipschitz sets. With both functional and set parts, we improve previous results in the literature on the regularization of functions and sets.

     

Strength and deformation properties of fiberglass-plastic in the flat stressed state as a function of reinforcement structure

Conclusion Limiting strength values have been ascertained in the flat stressed state as a function of reinforcement structure. The change in each strength surface tensor component as a function of reinforcement intensity has been approximated by the piecewise-linear approximation method. A strength condition has been derived which can be used in optimization problems. The problem of the optimum reinforcement structure of a composite at various ratios of the stresses ₁₁, ₂₂, and ₁₂ has been examined. By using the strength condition, one can predict strength values for structures which appear in the class of materials in question with various reinforcement intensities. The procedure developed can be used in designing composite materials.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 848–859, September–October, 1978.     

Some specific unboundedness property in smoothness Morrey spaces. The non-existence of growth envelopes in the subcritical case

We study smoothness spaces of Morrey type on Rn and characterise in detail those situations when such spaces of type A_(p,q)~(s,r)(R~n) or A_(u,p,q)~s(R~n) are not embedded into L_(∞)(R~n).We can show that in the so-called sub-critical,proper Morrey case their growth envelope function is always infinite which is a much stronger assertion.The same applies for the Morrey spaces M_(u,p)(R~m) with p u.This is the first result in this direction and essentially contributes to a better understanding of the structure of the above spaces.     

Computable upper bounds for the constants in Poincaré‐type inequalities for fields of bounded deformation

We collect various Poincaré‐type inequalities valid for fields of bounded deformation and give explicit upper bounds for the constants being involved. Copyright © 2011 John Wiley & Sons, Ltd.     

Growth rate,growth curve and growth prediction of tumour in the competitive model

ABSTRACT

The growth of cancer is still the focus of many research works in the scientific community. So far, various models have been introduced to analyse the behaviour of cancers, including the mathematical growth models such as Logistic, Gompertz and Bertalany. Despite the advances in the analysis of the cancer behaviour, the lack of definitive treatment of cancer disease indicates the need for new perspectives which are supported by more biological background. Recently, a model has been proposed, in which, the tumour growth is interpreted as the outcome of the competition of healthy and cancer cells over the available oxygen, nutrients and space. We have modified this model in order to provide the necessary preparations for wider use of the model in growth rate, growth curve and growth prediction of tumours. Meanwhile, the model is performed on some experimental data to show its capabilities.     

Propagation of S-wave in a non-homogeneous anisotropic incompressible and initially stressed medium under influence of gravity field

In this paper, propagation of shear waves in a non-homogeneous anisotropic incompressible, gravity field and initially stressed medium is studied. Analytical analysis reveals that the velocity of propagation of the shear waves depends upon the direction of propagation, the anisotropy, gravity field, non-homogeneity of the medium, and the initial stress. The frequency equation that determines the velocity of the shear wave has been obtained. The dispersion equations have been obtained and investigated for different cases. A comparison is made with the results predicted by Abd-Alla et al. [22] in the absence of initial stress and gravity field. The results obtained are discussed and presented graphically.     

Energy scattering theory for the nonlinear Schrödinger equations with exponential growth in lower spatial dimensions

For one and two spatial dimensions, we show the existence of the scattering operators for the nonlinear Schrödinger equation with exponential nonlinearity in the whole energy spaces.     

Mathematical modelling of tongue deformation during swallow in patients with head and neck cancer

ABSTRACT

Cancer localized to the tongue is often characterized by increased stiffness in the affected region. This stiffness affects swallow in a manner that is difficult to quantify in patients. A biomechanical model was developed to simulate the spatio-temporal deformation of the tongue during the pharyngeal phase of swallow in patients with cancer of the tongue base. The model involves finite element analysis (FEA) of a three-dimensional (3D) model of the tongue reconstructed from magnetic resonance images (MRI). The tongue tissue is assumed to be hyper-elastic. In order to examine the effects of tissue change (increased stiffness) due to the presence of cancer localized to the tongue base, various sections of the 3D geometry are modified to exhibit different elastic properties. Three cases are considered, representing the normal tongue, a tongue with early-stage cancer, and tongue with late-stage cancer. Early- and late-stage cancers are differentiated by the degree of stiffness within the base of tongue tissue. Analysis of the model suggests that healthy tongue has a maximum deformation of 9.38 mm, whereas tongues having mild cancer and severe cancer have a maximum deformation of 8.65 and 6.17 mm, respectively. Biomechanical modelling is a useful tool to explain and estimate swallowing abnormalities associated with tongue cancer and post-treatment characteristics.     

On the homogenization of a diffusion–deformation problem in strongly heterogeneous media

We consider a porous fluid-saturated medium with periodic distribution of heterogeneities where the value of permeability decreases with the scale parameters. Homogenization of such double-porous material is performed using the method of periodic unfolding. The resulting homogenized macroscopic model is featured by the fading memory effect in the viscoelastic behaviour. This paper is based upon the work sponsored by the Ministry of Education of the Czech Republic under the research project MSM 49777513 03.     

Modelling the role of cell-cell adhesion in the growth and development of carcinomas

In this paper, a mathematical model is presented to describe the evolution of an avascular solid tumour in response to an externally-supplied nutrient. The growth of the tumour depends on the balance between expansive forces caused by cell proliferation and cell-cell adhesion forces which exist to maintain the tumour's compactness. Cell-cell adhesion is incorporated into the model using the Gibbs-Thomson relation which relates the change in nutrient concentration across the tumour boundary to the local curvature, this energy being used to preserve the cell-cell adhesion forces.

Our analysis focuses on the existence and uniqueness of steady, radially-symmetric solutions to the model, and also their stability to time-dependent and asymmetric perturbations. In particular, our analysis suggests that if the energy needed to preserve the bonds of adhesion is large then the radially-symmetric configuration is stable with respect to all asymmetric perturbations, and the tumour maintains a radially-symmetric structure—this corresponds to the growth of a benign tumour. As the energy needed to maintain the tumour's compactness diminishes so the number of modes to which the underlying radially-symmetric solution is unstable increases—this corresponds to the invasive growth of a carcinoma. The strength of the cell-cell bonds of adhesion may at some stage provide clinicians with a useful index of the invasive potential of a tumour.     

Spectral cycles and average velocity of clusters in discrete two-contours system with two nodes

The paper considers a discrete dynamical system containing two contours. There are $n$ cells and $m$ particles in each contour. At any time, the particles of each contour form a cluster. There are two common points of the contours. These common points are called nodes. The nodes divide the contours into two nonequal parts. The system belongs to the class of dynamical systems introduced by A. P. Buslaev. In these systems, the movement of particles (clusters) can be interpreted as the mass transfer on regular networks. A cyclic trajectory in the system state space is called a spectral cycle. The system state space is divided into sets such that any of this set contains the states of the spectral cycle and the nonrecurrent states from which the system results in the set of the spectral cycle states. We have found that, in the general case, what spectral cycle is realized and with what average velocity the clusters move depends on the initial state of the system. We have found the set of spectral cycles and the values of the clusters average velocities taking into account the delays due to that the clusters cannot pass through the same node simultaneously. In one of two considered versions of the system, the clusters move counterclockwise (one-directional movement). In the other version, one of the clusters moves clockwise and the other cluster moves counterclockwise (codirectional movement). It is turned out that the behavior of system is significantly different in these versions. In particular, in the case of the codirectional movement, the average clusters velocity does not depend on the initial state.     

Existence and Multiplicity for Elliptic Problems with Quadratic Growth in the Gradient

We show that a class of divergence-form elliptic problems with quadratic growth in the gradient and non-coercive zero order terms are solvable, under essentially optimal hypotheses on the coefficients in the equation. In addition, we prove that the solutions are in general not unique. The case where the zero order term has the opposite sign was already intensively studied and the uniqueness is the rule.     

The velocity and shear stress functions of the Falkner-Skan equation arising in boundary layer theory

New compactness results on the velocity functions and shear stress functions of the well-known Falkner-Skan equation are obtained. The methodology is to utilize the equivalence between the Falkner-Skan equation and a singular integral equation established recently by Lan and Yang.     

On the solutions and conservation laws of the model for tumor growth in the brain

We study the problem of tumor growth and its monitoring ranging from the simple model for the radially symmetric to the more complex case being the radially non-symmetric one. In each case, we take killing rate of the cancer cells dependent on the concentration of the cells. A number of invariant reductions whose further analysis leads to exact solutions are obtained. Conservation laws for the model are also studied.     

Picard values and the growth of meromorphic functions

The purpose of this paper is to investigate the growth of meromorphic functions concern- ing Picard values with a radially distributed value. This generalizes a classic result due to Hayman [Hayman, W. K.： Picard values of meromorphic functions and their derivatives. Ann. of Math., 70, 9-42 （1959）].     

裂纹尖端场弹塑性分析的加权残数法及塑性应力强度因子的计算

本文以幂强化材料,平面应变情形为例,系统地提出了裂纹尖端场弹塑性分析的加权残数法,并根据此法,得出了裂纹尖端场的解析式弹塑性近似解.在此基础上.对整个裂纹区域,构造了弹塑性解叠加非线性有限元计算塑性应力强度因子的方法,从而为裂纹尖端场和整个裂纹体的分析和计算,提供了一个方法.