Discrete time semigroup transformations with random perturbations

Let (X, ) and (Y,C) be two measurable spaces withX being a linear space. A system is determined by two functionsf(X): X X and:X×YX, a (small) positive parameter and a homogeneous Markov chain {y _n } in (Y,C) which describes random perturbations. States of the system, say {x _n X, n=0, 1,}, are determined by the iteration relations:x _n+1 =f(x _n )+(x _n ,Y_n+1) forn0, wherex ₀ =x ₀ is given. Here we study the asymptotic behavior of the solutionx _n as 0 andn under various assumptions on the data. General results are applied to some problems in epidemics, genetics and demographics.Supported in part by NSF Grant DMS92-06677.Supported in part by NSF Grant DMS93-12255.     

Existence and regularity for solutions of the Korteweg-de Vries equation

The construction suggested by an inverse-scattering analysis establishes the existence of solutions u(x, t) of the Korteweg-de Vries equation subject to an initial condition u(x, 0)=U(x), where U has certain regularity and decay properties. It is assumed that UC³(), that U is piecewise of class C ⁴, and that U ^(j) decays at an algebraic rate for j4. The faster the decay of U ^(j) the smoother the solution will be for t0. If U and its first four derivatives decay faster than ¦x¦^–n for all n, then the solution will be infinitely differentiable for t0. For t>0, the decay rate of u(x, t) as x + increases with the decay rate of U; but the decay rate as x - depends on the regularity of U. A solution u ₁ of the Korteweg-de Vries equation such that u ₁(·, 0)C() may fail to remain in class C for all time if u ₁(x, 0) does not decay fast enough as ¦x¦.This research was performed in part as a Visiting Member of the Courant Institute of Mathematical Science.     

Variants of the Stokes Problem: the Case of Anisotropic Potentials

We investigate the smoothness properties of local solutions of the nonlinear Stokes problem$\begin{eqnarray*}-\diverg \{T(\eps(v))\} + \nabla \pi &=& g \msp \mbox{on $\Omega$,}\\\diverg v&\equiv & 0 \msp \mbox{on $\Omega$,}\end{eqnarray*}$where v: ⁿ is the velocity field, $\pi$: $ denotes the pressure function, and g: ⁿ represents a system of volume forces, denoting an open subset of ⁿ . The tensor T is assumed to be the gradient of some potential f acting on symmetric matrices. Our main hypothesis imposed on f is the existence of exponents 1 < p q < \infty such that\lambda (1+|\eps|^{2})^{\frac{p-2}{2}} |\sigma|^{2} \leq D^{2}f(\eps)(\sigma ,\sigma) \leq \Lambda (1+|\eps|^{2})^{\frac{q-2}{2}} |\sigma|^{2}holds with suitable constants , > 0, i.e. the potential f is of anisotropic power growth. Under natural assumptions on p and q we prove that velocity fields from the space W ¹ _{p, loc} (; ⁿ ) are of class C ^1, on an open subset of with full measure. If n = 2, then the set of interior singularities is empty.Dedicated to O. A. Ladyzhenskaya on the occasion of her 80th birthday     

The deviation of heat transfer calculation for laminar free convection of gas due to ignoring the variable thermophysical properties

According to the similarity transformation method described in [1–3], the ordinary differential governing equations of laminar free convection of gases with Boussinesq's approximation are set up. The equations are corresponding to the caseT _w /T 1 and _{c p} 0 as discussed in [2, 3]. The deviations of the predicted local heat transfer coefficients and local Nusselt numbers due to ignoring the variable thermophysical properties are calculated out theT _w /T T for several gases, and the necessity of the treatment for considering variable thermophysical properties is pointed out.

---

Die Abweichungen der Berechnungsergebnisse für die laminare freie Konvektionsströmung von Gasen infolge Vernachlässigung der Veränderlichkeit ihrer Stoffwerte

Zusammenfassung Entsprechend der in [1–3] beschriebenen Methode der Ähnlichkeitstransformation werden die gewöhnlichen Differential — Grundgleichungen der laminaren freien Konvektion von Gasen unter Verwendung der Boussinesq — Approximation aufgestellt. Diese Gleichungen entsprechen, wie in [2, 3] beschrieben, dem FallT _w /T 1 undn _{c p} 0. Die Abweichungen der berechneten örtlichen Nusselt-Zahlen infolge Vernachlässigung der Veränderlichkeit der thermophysikalischen Stoffwerte werden in Abhängigkeit vonT _w /T für mehrere Gase ermittelt, wobei sich herausstellt, daß die Veränderlichkeit der Stoffwerte bei der Berechnung zu berücksichtigen ist.

     

Some remarks on differential equations of quadratic type

In this paper we study differential equations of the formx(t) + x(t)=f(x(t)), x(0)=x ₀ C HereC is a closed, bounded convex subset of a Banach spaceX,f(C) C, and it is often assumed thatf(x) is a quadratic map. We study the differential equation by using the general theory of nonexpansive maps and nonexpansive, non-linear semigroups, and we obtain sharp results in a number of cases of interest. We give a formula for the Lipschitz constant off: C C, and we derive a precise explicit formula for the Lipschitz constant whenf is quadratic,C is the unit simplex inR ⁿ, and thel ₁ norm is used. We give a new proof of a theorem about nonexpansive semigroups; and we show that if the Lipschitz constant off: CC is less than or equal to one, then lim_tf(x(t))–x(t)=0 and, if {x(t):t 0} is precompact, then lim_tx(t) exists. Iff¦C=L¦C, whereL is a bounded linear operator, we apply the nonlinear theory to prove that (under mild further conditions on C) lim_t f(x(t))–x(t)=0 and that lim_t x(t) exists if {x(t):t 0} is precompact. However, forn 3 we give examples of quadratic mapsf of the unit simplex ofR ⁿ into itself such that lim_t x(t) fails to exist for mostx ₀ C andx(t) may be periodic. Our theorems answer several questions recently raised by J. Herod in connection with so-called model Boltzmann equations.     

Gap Estimates for Double-Well Schrödinger Operators and Quantum Stabilization of Anharmonic Crystals

The spectrum of the Schrödinger operator of a one-dimensional quantum anharmonic oscillator of mass m is studied. This spectrum consists of simple (nondegenerate) eigenvalues E _n , $$n\in {\mathbb N}_0$$ such that n ^– E _n + as n + with a certain > 1. The gap parameter =min _n (E _n – E _n-1) is in the center of the study. It is proven that this parameter is a continuous function of m; its small mass and large mass asymptotics are found. The influence of the dependence of on m on the stability of systems of interacting quantum anharmonic oscillators is briefly discussed.     

A delta-function method for the n-th approximation of relaxation or retardation time spectrum from dynamic data

A function series g(x; n, m) is presented that converges in the limiting case n and m = constant to the delta-function located at x = = 1. For every finite n, there exists 2n+1(–nmn) approximations of the delta-function _(n)(x–x _n,m ). x _n,m is the argument where the function reaches its maximum. A formula for the calculation is given.The delta-function approximation is the starting point for the approximative determination of the logarithmic density function of the relaxation or retardation time spectrum. The n-th approximation of density functions based on components of the complex modulus (G*) or the complex compliance (J*) is given. It represents an easy differential operator of order n.This approach generalizes the results obtained by Schwarzl and Staverman, and Tschoegl. The symmetry properties of the approximations are explained by the symmetry properties of the function g(x; n, m). Therefore, the separate equations for each approximation given by Tschoegl can be subsumed in a single equation for G and G, and in another for J and J.     

Boundary layer effect on thermal NO decomposition behind incident shock waves

The thermal decomposition of nitric oxide (diluted in Argon) has been measured behind incident shock waves by means of IR diode laser absorption spectroscopy. In two independent runs the diode laser was tuned to the=0 =1² _3/2 R(18.5)-rotational vibrational transition and the=1 =2² _3/2 R(20.5)-rotational vibrational transition of nitric oxide, respectively. These two transitions originating from the vibrational ground state (=0) and the first excited vibrational state (=1) were selected in order to probe the homogeneity along the absorption path. The measured NO decomposition could satisfactorily be described by a chemical reaction mechanism after taking into account boundary layer corrections according to the theory of Mirels. The study forms a further proof of Mirels' theory including his prediction of the laminar-turbulent transition. It also shows, that the inhomogeneities from the boundary layer do not affect the IR linear absorption markedly.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

One dimensional lattice gases with rapidly decreasing interaction

We consider the pressure and the correlation functions of a one dimensional lattice gas in which the mutual interaction decreases as r exp-n ^t, (r, t>0), when the interparticle distance n. We prove that such a system cannot show phase transitions of order k1 in the sense that the pressure and the correlation functions are infinitely differentiable with respect to any relevant parameter (such as the temperature or the chemical potential).     

On K-BKZ and other viscoelastic models as continuum generalizations of the classical spring-dashpot models

An alternate constitutive formulation for visco-elastic materials, with particular emphasis on macromolecular viscoelastic fluids, is presented by generalizing Maxwell's idealized separation of elastic and relaxation mechanisms. The notion ofrelative rate of change of elastic stress is identified, abstracted, and formulated with the help of the established theory of finitely elastic isotropic materials. This given a local rate-type constitutive relation for an elastic mechanism in a simple material.For the simplest class of viscoelastic polymer melts, the notion of rate of change of elastic stress and its damped accumulation is identified and formulated. Under conditions of moderate strain rates, this scheme implies the reliable K-BKZ model for a class of polymer melts. An obvious extension generalizes the remaining classical spring-dashpot models. I Set of second-order tensors.A I is identified with a 3 × 3 matrix in a Cartesian co-ordinate system - I ^sym Set of symmetric second order tensors - Q Orthogonal tensor, i.e.Q ^T=Q ^–1. - Symbol for the value of the functional H:X I ^sym, whereX is the set of piecewise continuous and differentiable strain historiesF _to : [t ₀,t] I Other functionals, unless otherwise specified, should be interpreted in a similar manner.     

On the Problem of Dissipative Perturbations of Nonexpansive Mappings

ＩｎｔｒｏｄｕｃｔｉｏｎａｎｄＰｒｅｌｉｍｉｎａｒｉｅｓＴｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒ,ｗｅａｓｓｕｍｅｔｈａｔＨｉｓａｒｅａｌＨｉｌｂｅｒｔｓｐａｃｅ ,〈· ,·〉ｉｓｔｈｅｉｎｎｅｒｐｒｏｄｕｃｔｏｎＨ ,ＰｉｓａｃｏｎｅｉｎＨ .ＢｙｖｉｒｔｕｅｏｆｔｈｅｃｏｍｅＰ ,ａｎｏｒｄｅｒ“≤”ｉｓｉｎｄｕｃｅｄｉｎＨ ,ｉ.ｅ .,ｆｏｒａｎｙｇｉｖｅｎｘ,ｙ∈Ｈ ,ｘ≤ｙｉｆａｎｄｏｎｌｙｉｆｙ -ｘ∈Ｐ .Ａｍｕｌｔｉ＿ｖａｌｕｅｄｍａｐｐｉｎｇＡ :Ｄ(Ａ) Ｈ → 2 Ｈｉｓｓａｉｄｔｏｂｅａｃｃｒｅｔｉｖ…     

The Rellich-Kondrachov theorem for unbounded domains

Summary The full Kondrachov compactness theorem for Sobolev imbeddings of the type W ₀ ^m,p (G) W ₀ ^j,r (G) on bounded domains G in R ⁿ is extended to a large class of unbounded domains with reasonable n- 1 dimensional boundaries. A Poincaré inequality is obtained for such domains and a compactness theorem for traces of functions in W ₀ ^m,p (G) on lower dimensional hyperplanes is also proved.     

K+ uptake by root systems grown in soil under salinity: II. Sensitivity analysis

A numerical algorithm for the solution of multicomponent transport of Ca²⁺, Mg²⁺, Na⁺, K⁺, Cl^– in soil and their uptake by plant roots has been developed. The model emphasizes adsorptiondesorption due to cation exchange mechanism, dissolution-precipitation of CaCO₃, and pH changes at the root surface controlled by the anion-cation influx balance. A fully implicit finite difference scheme is used for numerical implementation. Sensitivity analysis was conducted to evaluate the effect of each parameter on nutrient uptake. Each parameter (independent of all others) was varied between 0.25 to 4 times its speculated average level. Predicted K⁺ uptake was found to be more sensitive to changes of root radius and the parameter indicating maximal influx of K⁺. Effective diffusion coefficient and soil moisture are less influential. The influence of C_aCO₃ dissolution and different kinds of boundary conditions were also considered.Nomenclature A, B, E matrices of coefficients for finite difference scheme - b _i coefficients of equation for H⁺ concentration - C ^I concentration of theI-component in water - C ₀ ^I initial concentration of theI-component - _I ^r0 concentration of theI-component at the inner side of the root surface - C _I ^r1 concentration of theI-component at the external boundary - C _Na ^cr critical concentration for Na⁺ influx into root - CEC cation exchange capacity - D ^*I effective diffusion coefficient of theI-component in soil - F ^I concentration of theI-component on the exchange complex - G vector of coefficients in finite difference scheme - h Hill's cooperativity index for K⁺ influx - h ⁰ value ofh whenC ^Na=0 - J ^I uptake of theI-component by a root length unit - J _I ^r0 influx at the root surface of theI-component - J _max maximal influx of K⁺ - J _max ⁰ value ofJ ^max whenC ^Na=0 - K _a apparent Michaelis-Menten coefficient for K⁺ influx - K _a ⁰ value ofK _a whenC ^Na=0 - K _i selectivity coefficient of the exchange reaction - P _m ^I permeability of root surface for theI-component - P_CO2 partial pressure of CO₂ - r radial distance - r ₀ root radius - r ₁ half the distance between adjacent root (external boundary) - R ^I retardation factor of the-component in mass balance equation for theI-component - t time - t ₀ initial time - T simulation time - v ₀ water radial velocity at the root surface - x _i coordinate of nodes of finite difference mesh Greek coefficient of linear change of K⁺ influx - coefficient of linear change of Na⁺ influx - _s mass density of soil solid phase - soil porosity - volumetric content of liquid in soil - _i coefficients in formulae for parameters of K⁺ influx - parameter of perturbation in finite difference scheme - gg ^I activity coefficient of theI-component - accuracy of iteration convergence - time step for finite difference scheme - steps of finite difference mesh Special Symbols [...] activity symbol     

Nonlinear motion equations for a Non-Newtonian incompressible fluid in an orthogonal coordinate system

Summary Starting with an assumed relationship between the stress tensor, the non-Newtonian viscosity, and the strain rate tensor, the nonlinear equations of motion are developed for use in any orthogonal coordinate system. The resulting equations are written in terms of the scalar velocities, the non-Newtonian viscosity, the metric coefficients, and their derivatives.The non-Newtonian viscosity is assumed to be a scalar function of the strain rate tensor, and so depends upon the invariants of the strain rate tensor. For convenience, the necessary invariants are written out in complete form for use in any orthogonal coordinate system, in terms of the scalar velocities, the metric coefficients, and their derivatives.Using the resulting motion equations and a model of this type of viscosity, theOstwald-de Waele model, an example of time dependent flow is solved using a continuous time-discrete space method programmed on an analog computer. e _ij strain rate tensor - body force density, dynes/cm³ - F ₁,F ₂,F ₃ components of body force density, dynes/cm³ - g acceleration of gravity - H function of time - h ₁,h ₂,h ₃ metric coefficients - I ₁,I ₂,I ₃ invariants - m constant - P pressure, dynes/cm² - r radius, cm - t time, sec - velocity vector, cm/sec - v ₁,v ₂,v ₃ velocities in thex ₁,x ₂ andx ₃ directions, respectively, cm/sec - v _n (t) velocity of thenth node, cm/sec - x ₁,x ₂,x ₃ coordinate directions - z coordinate, cm - unit tensor - _ij Kronecker delta - _ij 2e _ij - nabla - _ijk alternating unit tensor - non-Newtonian viscosity, dynes/cm² - ₀, ₁ constant viscosities, dynes sec/cm², dynes sec ^m /cm² - angle, radians - v ₀,v ₁ constant kinematic viscosities, cm²/sec, cm² sec ^m-2 - density, g/cm³ - _ij stress tensor - fluid dilation With 3 figures     

Lack of Critical Phase Points and Exponentially Faint Illumination

The Stationary Phase Principle (SPP) states that in the computation of oscillatory integrals, the contributions of non-stationary points of the phase are smaller than any power n of 1/k, for k. Unfortunately, SPP says nothing about the possible growth in the constants in the estimates with respect to the powers n. A quantitative estimate of oscillatory integrals with amplitude and phase in the Gevrey classes of functions shows that these contributions are asymptotically negligible, like exp(–ak^b), a,b > 0. An example in Optics is given.     

The site model theory and the standard linear solid

Summary The site model theory (SMT) is shown to lead to the same deformation behaviour as that displayed by the standard linear solid (SLS), group I, for all loading conditions. If a second deformation mechanism (inter-molecular slip) is introduced the result is the same as that obtained with the standard linear solid, group II, and models the behaviour of a polymer melt near to the solidification temperature.

---

Zusammenfassung Es wird gezeigt, daß ein einfaches Platzwechsel-Modell (site model theory) bei allen Belastungsbedingungen das gleiche Deformationsverhalten voraussagt wie der lineare Drei-Parameter-Festkörper (standard linear solid, group I). Wenn ein weiterer Deformationsmechanismus (zwischenmolekulare Gleitung) eingeführt wird, entspricht das Verhalten dagegen demjenigen einer linearen Drei-Parameter-Flüssigkeit (standard linear solid, group II), welche das Verhalten einer Polymerschmelze in der Nähe der Schmelztemperatur beschreibt.

---

a = ₁₂ ⁰ + ₂₁ ⁰ , see eq. [1] - b =N ₁ ⁰ ₁₂ ⁰ (V ₁₂ +V ₂₁), see eq. [1] - c = 2N _s ⁰ V _s see eq. [6] - k Boltzmann constant - t time - E,E ₁,E ₂ spring constants, see figures 1 and 3 - E _u unrelaxed modulus - N ₁ ⁰ site 1 equilibrium population in the unstressed state - N _s number of units available for slip - N(t) decrease in site 1 population - N _s (t) net number of slip jumps in the stressaided direction - T temperature (K) - V _i,j activation volume for jumps in directioni j - V _s activation volume for the slip process - strain - strain rate - incremental change in strain per unit change in site population - µ,µ ₁,µ ₂ dashpot constants, see figures 1 and 3 - applied stress - ₀ initial applied stress, (stress relaxation) =(t) (creep) - incremental change in stress per unit change in site population - ⁰ jump rate for slip in the unstressed state - _i,j ⁰ jump rate in the directioni j in the unstressed state With 3 figures and 3 tables     

Ishikawa Iterative Process in Uniformly Smooth Banach Spaces

ＬｅｔＥｂｅａｕｎｉｆｏｒｍｌｙｓｍｏｏｔｈＢａｎａｃｈｓｐａｃｅ ,ＫｂｅａｎｏｎｅｍｐｔｙｃｌｏｓｅｄｃｏｎｖｅｘｓｕｂｓｅｔｏｆＥ ,ａｎｄｓｕｐｐｏｓｅＴ :Ｋ→ＫｉｓａｃｏｎｔｉｎｕｏｕｓΦ＿ｓｔｒｏｎｇｌｙｐｓｅｕｄｏｃｏｎｔｒａｃｔｉｖｅｏｐｅｒａｔｏｒ.ＤｅｎｏｔｅｔｈｅｄｕａｌｓｐａｃｅｏｆＥｂｙＥ .ＷｅｄｅｎｏｔｅｂｙＪｔｈｅｄｕａｌｉｔｙｍａｐｆｒｏｍＥｔｏ 2 Ｅ ｄｅｆｉｎｅｄｂｙＪ(ｘ) =ｆ∈Ｅ :〈ｘ ,ｆ〉=‖ｘ‖2 =‖ｆ‖2 . ( 1 )Ｉｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｉｆＥｉｓｕ…     

Ordinary differential equations (ODEs) on inertial manifolds for reaction-diffusion systems in a singularly perturbed domain with several thin channels

Dynamics of solutions to a reaction-diffusion system in a domain of specific shape is investigated under the homogeneous Neumann boundary conditions. It is assumed that the domain hasN large regionsD _i ,i=1,...,N, and thin channelsQ _i,j () connectingD _i andD _j , which approach a line segment as 0 in some sense. In such a domain the firstN eigenvalues of – with the Neumann boundary conditions tend to zero as 0, while the (N + 1)-th eigenvalue is bounded away from zero. By virtue of this gap of the eigenvalues, an inertial manifold which is invariant and attracts every solution exponentially can be constructed under a certain condition. Moreover, the ODE describing the dynamics on the inertial manifold can be given in quite an explicit form through the analysis of the limit of the manifold as 0.     

Orientational anisotropy for Rouse eigenmodes during creep and recovery process

The Rouse model is a well established model for nonentangled polymer chains and its dynamic behavior under step strain has been fully analyzed in the literature. However, to the knowledge of the authors, no analysis has been made for the orientational anisotropy for the Rouse eigenmodes during the creep and creep recovery processes. For completeness of the analysis of the Rouse model, this anisotropy is calculated from the Rouse equation of motion. The calculation is simple and straightforward, but the result is intriguing in a sense that respective Rouse eigenmodes do not exhibit the single Voigt-type retardation. Instead, each Rouse eigenmode has a distribution in the retardation time. This behavior, reflecting the interplay among the Rouse eigenmodes of different orders under the constant stress condition, is quite different from the behavior under rate-controlled flow (where each eigenmode exhibits retardation/relaxation associated with a single characteristic time).List of abbreviations and symbols a Average segment size at equilibrium - A_p(t) Normalized orientational anisotropy for the p-th Rouse eigenmode defined by Eq. (14) - p-th Fourier component of the Brownian force (=x, y) - F_B(n,t) Brownian force acting on n-th segment at time t - G(t) Relaxation modulus - J(t) Creep compliance - J_R(t) Recoverable creep compliance - k_B Boltzmann constant - N Segment number per Rouse chain - Q_j(t) Orientational anisotropy of chain sections defined by Eq. (21) - r(n,t) Position of n-th segment of the chain at time t - S(n,t) Shear orientation function (S(n,t)=a^–2<u_x(n,t)u_y(n,t)>) - T Absolute temperature - u(n,t) Tangential vector of n-th segment at time t (u = r/n) - V(r(n,t)) Flow velocity of the frictional medium at the position r(n,t) - X_p(t), Y_p(t), and Z_p(t) x-, y-, and z-components of the amplitudes of p-th Rouse eigenmode at time t - Strain rate being uniform throughout the system - Segmental friction coefficient - ₀ Zero-shear viscosity - _p Numerical coefficients determined from Eq. (25) - Gaussian spring constant ( = 3k_BT/a²) - Number of Rouse chains per unit volume - (t) Shear stress of the system at time t - _steady Shear stress in the steadily flowing state - _R Longest viscoelastic relaxation time of the Rouse chain     

Drag of a sphere in an unsteady flow of viscous fluid at small reynolds numbers

An unsteady flow of viscous incompressible fluid past a sphere is investigated. The values of the inertial and unsteady terms in the Navier-Stokes equations are characterized by translational (R) and vibrational (R_k) Reynolds numbers, which are assumed small. The solution is constructed in the form of an expansion with respect to max(R, R _k ^1/2 ) by the method of matched asymptotic expansions. A correction to the Stokes force, correct to o[max(R, R _k ^1/2 )], is calculated. It is shown that the result depends strongly on the ratio R/R _k ^1/2 and goes over into the well-known equations for the cases R 0, R_k 0.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 11–16, January–February, 1988.