Necessary and sufficient conditions for isotropic rank-one convex functions in dimension 2

We provide some new necessary and sufficient conditions for regular isotropic rank-one convex functions on M ₂ ⁺={2×2 matrices such that det M0}. It is well known that isotropic functions W (M) can be written as W (M)=G(₁, ₂) where _i are the singular values of M. One of these conditions allows us to understand better the gap between the rank-one convexity and the quasiconvexity.     

Necessary and sufficient stability conditions for invariant sets of nonlinear impulsive systems

Necessary and sufficient conditions for the uniform asymptotic stability of the invariant set of a nonlinear impulsive system are established __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 132–142, February 2008.     

Necessary and sufficient conditions for complete blow-up and extinction for one-dimensional quasilinear heat equations

We characterize the occurrence of complete or incomplete blow-up (and extinction) for a general quasilinear heat equation of the form in terms of the constitutive functions and f We assume that (u)>0 for u>0 and that f(u)0. For the positive sign + before f(u) in (HE), with f(u) superlinear as u, blow-up occurs in finite time: sup _x u(x, t) as tT<. For the negative sign, we consider the case of singular absorption: f(u) as u0. Then initially positive solutions vanish at some point in finite time (extinction), and a singularity in the equation occurs there. An important aspect of blow-up or extinction problems is the possibility of having a nontrivial extension of the solution for t>T, i.e., after the singularity occurs. If such continuation exists, we say that the blow-up (extinction) is incomplete; otherwise it is called complete. Our characterization is based on the qualitative behaviour of the family of travelling-wave solutions and a proper use of the Intersection-Comparison argument. The analysis applies to other nonlinear models, like the equations with gradient-dependent diffusivity.     

Necessary and sufficient conditions for the stability of flows of incompressible viscous fluids

Summary We perturb a steady flow of an incompressible viscous fluid and derive a necessary and sufficient condition for the marginal cases for monotonie energy stability and stability against small (infinitesimal) disturbances to coincide. Evaluation of this condition in two examples singles out, in terms of the parameters of the problem, the cases where necessary and sufficient conditions for stability coincide and thus the steady flow first becomes unstable, together with the class of perturbations responsible for the instability. The analysis is done within the range of strict solutions of each underlying problem; the precise regularity and existence classes are given in Sec. 0. The examples we treat are plane parallel shear flow with a non-symmetric profile in an infinite rotating layer and the effect of rotation on convection.     

Necessary and sufficient conditions for the absolute stability of discrete type lurie control system

ＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＯＮＤＩＴＩＯＮＳＦＯＲＴＨＥＡＢＳＯＬＵＴＥＳＴＡＢＩＬＩＴＹＯＦＤＩＳＣＲＥＴＥＴＹＰＥＬＵＲＩＥＣＯＮＴＲＯＬＳＹＳＴＥＭＺｈａｎｇＪｉｙｅ（张继业）（ＲｅｃｅｉｘｅｄＯｃｔ．５．１９９４Ｃｏｍｍｕ...     

Necessary and sufficient conditions for existence of stationary solutions of some nonlinear stochastic systems

At the state of statistical stationarity, the response of a nonlinear system under multiplicative random excitations can be either trivial or non-trivial, depending on the spectral levels of the excitations and the values of certain system parameters. Assuming that the random excitations are Gaussian white noises, the two types of response may be investigated by way of their stationary densities, which are obtainable for first order dynamical systems and for higher order dynamical systems belonging to the class of generalized stationary potential. Alternatively, the Lyapunov exponents can be computed for perturbation from either the trivial or non-trivial solution, since a negative sign for the greatest Lyapunov exponent provides both the necessary and sufficient conditions for the stability of sample functions with probability one. It is shown in two specific examples, that the boundary at which the greatest Lyapunov exponent changes its sign coincides with the boundary for regularity (or being normalizable) for the probability density in both the trivial and non-trivial solutions. Thus, the stability conditions in the strong sense of probability one and the weak sense in distribution are identical in these cases.     

Necessary and sufficient conditions of global nonlinear stability for rotating double-diffusive convection in a porous medium

The nonlinear stability of the conduction-diffusion solution of a fluid mixture heated and salted from below (and of a homogeneous fluid heated from below) and saturating a porous medium is studied with the Lyapunov direct method. Both Darcy and Brinkman models have been used. The porous medium is bounded by two horizontal parallel planes and is rotating about a vertical axis. Necessary and sufficient conditions of unconditional stability are proved (i.e., the critical linear and nonlinear stability Rayleigh numbers coincide). Our results generalize those given by Straughan [21] for a homogeneous fluid in the Darcy regime. In the case of a mixture two stabilizing effects act: that of the rotation and of the concentration of the solute. Received March 05, 2002 / Published online June 4, 2002 RID="a" ID="a" e-mail: lombardo@dmi.unict.it RID="b" ID="b" e-mail: mulone@dmi.unict.it Communicated by Brian Straugham, Durham     

Necessary conditions for the existence of positive solutions of second-order linear difference equations and sufficient conditions for the oscillation of solutions

We consider the difference equation

Necessary and sufficient conditions for description of the motion of objects in a viscous fluid under ultrasonic actions

Necessary and sufficient conditions are formulated for theories that describe the motion of solid particles and gas bubbles in viscous fluid under the action of acoustic harmonic waves. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 2, pp. 64–71, February, 2000.     

Necessary and sufficient criteria for absolute stability of the direct control system

ＮＥＣＥＳＳＡＲＹＡＮＤＳＵＦＦＩＣＩＥＮＴＣＲＩＴＥＲＩＡＦＯＲＡＢＳＯＬＵＴＥＳＴＡＢＩＬＩＴＹＯＦＴＨＥＤＩＲＥＣＴＣＯＮＴＲＯＬＳＹＳＴＥＭＺｈａｎｇＪｉ－ｙｅ（张继业）;ＳｈｕＺｈｏｎｇ－ｚｈｏｕ（舒仲周）（ＤｅｐａｒｔｍｅｎｔｏｆＥｎｇｉ...     

Necessary conditions for shear thickening in associating polymer networks

We study shear-thickening phenomena observed in thermoreversible gels of telechelic associating polymers on the basis of the transient network theory. We introduce nonlinear models for the elongational properties of the middle chain; the tension along middle chains contains a nonlinear term with an amplitude A that diverges if the chain is fully stretched. Models for the dissociation rate of end chains from network junctions are also introduced; the dissociation rate is coupled to the middle-chain tension by a coupling constant g. The balance between the nonlinearity strength and coupling intensity determines the necessary condition for thickening. A strong nonlinearity in the tension caused by a large A leads to thickening if the dissociation rate of the end chains is weakly coupled to the middle-chain tension. In contrast, if it is strongly coupled, the end chains easily dissociate upon a small nonlinear elongation of the middle chains, thus leading to thinning. We present theoretical phase diagrams showing the boundary of the thickening/thinning transition in the A–g plane, and we show the existence of a critical point for thickening in these diagrams.     

Necessary and sufficient condition for the correct formulation of boundary integral equations of harmonic functions

A necessary and sufficient condition for the correct formulation of boundary integral equations of harmonic functions is established in the present paper. A super-determined problem of harmonic functions is proposed for the first time. Then a necessary and sufficient condition for the existence of solution of the super-determined problem is proved. At the same time, it is a necessary and sufficient condition for the correct formulation of boundary integral equations with direct unknown quantities. A relation between boundary integral equations and variational principles is discovered for the first time. And a one-to-one correspondence between boundary integral equations with direct and indirect unknown quantities is indicated. The concept and route of the present paper can be applied to other boundary value problems possessing variational principles.     

Necessary conditions at the boundary for minimizers in finite elasticity

Dedicated to Bernard Coleman on the occasion of his 60^th birthday     

Stiffness formulations and necessary and sufficient conditions for exponential stability of prestressable structures

Prestressable structures composed of elastic tendons and rigid bars are investigated for their stiffness and stability properties. Two analytical formulations of the tangent stiffness matrix are analyzed, the first using tendon lengths and forces, and the second using tendon lengths squared and force densities. Several necessary and sufficient conditions for the exponential stability of prestressable configurations are proved using both formulations. Advantages of these conditions, primarily with respect to the traditional test which uses the matrix of the first order system, are discussed and illustrated via examples. Advantages of the second formulation of the tangent stiffness matrix in analytical manipulations and computations are also revealed.     

Methods for obtaining sufficient conditions for the stability of autonomous conservative systems

A computational method for obtaining sufficient conditions for the stability of the stationary solution of autonomous conservative systems is proposed in the paper. This method is adapted to linear autonomous gyroscopic systems with three degrees of freedom. It is based on the positive definiteness of a parametric quadratic form composed of the gyroscopic force matrices and the potential function. The control parameters for the stability of the zero solution of the gyroscopic system are the entries of the gyroscopic force matrix. The algorithm of the computational method includes estimating one gyroscopic force parameter in the equation constructed from a necessary stability condition.A special example is used to demonstrate the application of this algorithm. Comparison is performed with some well-known methods for obtaining sufficient conditions on the basis of an incomplete set of first integrals of motion. It is shown that the positive definiteness of the modified potential energy may result in stable as well as unstable motions.     

Necessary and sufficient fracture criteria for a composite with a brittle matrix. Part 1. Weak reinforcement

Fracture of a composite medium with a brittle matrix is studied. The brittle or plastic material of the reinforcing elements is highly deformable. For normalrupture macrocracks, necessary criteria of brittle strength and sufficient criteria of quasibrittle strength are proposed. Simple analytical dependences of the macrocrack length on the loading parameter, structural, rigidity, and strength parameters of the medium, and damage parameters of the material of the components are obtained. The critical loads for these criteria may differ substantially even if the reinforcement coefficient is small and the material of reinforcing elements is highly deformable. If the necessary criterion is satisfied, crack extension occurs and microcracks are formed in the bonds of the structure located ahead of the macrocrack tip. The number of damaged bonds depends on the macrocrack opening and characteristics of postcritical deformation of the damaged bonds.     

Necessary conditions for partial controllability of a rigid body — Gyrodyne system

Institute of Applied Mathematics and Mechanics, National Academy of Sciences of Ukraine, Donetsk. Translated from Prikladnaya Mekhanika, Vol. 30, No. 11, pp. 90–96, November, 1994.     

A simple derivation of necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials in plane strain

Necessary and sufficient conditions for the strong ellipticity of isotropic hyperelastic materials were first given by Knowles and Sternberg [3,4] by means of a lengthy calculation. Since then Aubert and Tahraoui [1] have shown the necessity of these conditions and simplified one of them using a different but still complicated method. The purpose of this note is to show how the conditions can be derived in a very simple way.