Singular perturbations of piecewise monotonic maps of the interval

Let t: [0, 1] [0, 1] be a piecewise monotonic, C₂, and expanding map. In computing an orbit { ⁱ (x ₀)} _i=0 , we model the roundoff error at each iteration by a singular perturbation; i.e.,X _n+1=(X _n )+W , whereW is a random variable taking on discrete values in an interval (-ε, ). The main result proves that this process admits an absolutely continuous invariant measure which approaches the absolutely continuous measure invariant under the deterministic map t as the precision of computation 0.     

Simulation of carbohydrate-protein interactions: Computer-aided design of a second generation GM1 mimic

The oligosaccharide of ganglioside GM1 [Gal1-3GalNAc1-4(NeuAc2-3)Gal1-4Glc1-1Cer] is the cellular target of two bacterial enterotoxins: the cholera toxin (CT) and the heat-labile toxin of E.coli (LT). We recently reported that the pseudosaccharide 2[Gal1-3GalNAc1-4(NeuAc2-3)DCCHD] is a high-affinity ligand for CT, and thus a functional mimic of GM1 (Bernardi, A., Checchia, A., Brocca, P., Sonnino, S. and Zuccotto, F., J. Am. Chem. Soc., 121 (1999) 2032–2036). In this paper we describe the design of a second-generation mimic, formally obtained from 2 by inverting the configuration of a single stereocenter, thus transforming a N-acetyl galactosamine into a N-acetyl glucosamine. The design process involved modeling of the free ligand and its LT complex, followed by qualitative and quantitative comparison with the corresponding structures of 2. The protocol employed relied on both conformational search and molecular dynamics methodologies to account for the flexibility of both the ligand and the protein receptor. The conformational search of the LT:inhibitor complex showed that, compared to 2, the new compound can insert one more hydroxy group within the protein binding site. Molecular dynamics simulations showed that, in turn, this may trigger a series of rearrangements and reorientations of side chains and crystallographic water molecules in the toxin, leading to new H-bond contacts which may result in enhanced affinity of the new inhibitor. FEP calculations were performed by mutating the structure of 2 in solution and in the protein complex, and the prediction was made that the second-generation mimic should be a stronger binder than its parent compound.     

Uniformly valid feedback expansions for optimal control of singularly perturbed dynamic systems

This paper shows how to construct a feedback control law for a class of singularly perturbed autonomous optimization problems. The control law is expressed as a single power series in the small parameter representing the ratio of the two effective time scales of the problem. The present approach avoids the need of expansion matching. The method is applied to a constant-speed interception problem. Comparison of numerical results with the exact solution shows an excellent agreement.Dedicated to G. Leitmann     

Solution of εY″+YY′−Y=0 by a nonasymptotic method

The solution of y(x)+y(x)y(x)–y(x)=0, with the boundary conditionsy(0)=,y(1)=, is obtained by a nonasymptotic method. It is shown that the nature of the inner solution for both left-hand and right-hand boundary layers depends on the roots of a transcendental equation. From sketches of this function, the location of the roots can be found. For the left-hand boundary layer, depending on the relative size and signs of and , 13 cases exist for the possible solution of the transcendental equation. Of these cases, only five correspond to acceptable solutions. Similar remarks apply to the right-hand boundary layer solutions. Numerical experience with the method is also reported to confirm the theoretical analysis.The author expresses his thanks to Dr. J. Greenstadt for his suggestions on sketching the roots of (C).     

Time evolution of infinitely many particles: An existence theorem

In this paper we deal with systems of infinitely many particles in ³, given by a two-body, short-range potential and an external potential, depending on the position of the particles. We show the existence of dynamics for a set of initial configurations, which has measure one with respect to the Gibbs measure induced by a suitable family of Hamiltonians.     

Complete controllability of perturbed linear control systems,revisited

LetL ¹(I, ^n,n )×M(I, ^n,m ) be the space of all pairs (A, B), whereA andB are measurable functions from a compact intervalI to ^n,n and ^n,m , respectively, andA is Lebesgue integrable. Also, let this space be endowed with the topology of theL ¹-norm with respect toA and the topology of convergence in measure with respect toB. Then, the set of all pairs (A, B), for which the corresponding linear control system

The Schrödinger equation with a moving point interaction in three dimensions

In the case of a single point interaction we improve, by using different methods, the existence theorem for the unitary evolution generated by a Schrödinger operator with moving point interactions obtained by Dell'Antonio, Figari and Teta.

     

Nonaxisymmetric vibrations of ellipsoidal shells under nonstationary distributed loads

Problems of forced nonaxisymmetric vibrations of ellipsoidal shells under nonstationary loads are formulated. A numerical algorithm to solve them is developed. The solutions obtained are analyzed Translated from Prikladnaya Mekhanika, Vol. 44, No. 9, pp. 73–84, September 2008.     

Nonstationary indentation of a blunt rigid body into an elastic layer: An axisymmetric problem

The nonstationary indentation of a blunt rigid body into an elastic layer is studied. The general formulation of the problem includes different boundary conditions in the contact area and on the free surface of the layer. The simplified nonmixed problem that arises at the early stage of interaction and allows obtaining approximate results for later times is solved exactly. The solution obtained is compared with that for the plane case __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 7, pp. 35–46, July 2008.