Stability and Bifurcation Analysis of a Nonlinear DDE Model for Drilling

We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate patterns of synchrony in networks with small numbers of cells and in lattices (and periodic arrays) of cells that cannot readily be explained by conventional symmetry considerations. We also show that different types of dynamics can coexist robustly in single solutions of systems of coupled identical cells. The examples include a three-cell system exhibiting equilibria, periodic, and quasiperiodic states in different cells; periodic $2n\times 2n$ arrays of cells that generate $2^n$ different patterns of synchrony from one symmetry-generated solution; and systems exhibiting multirhythms (periodic solutions with rationally related periods in different cells). Our theoretical results include the observation that reduced equations on a center manifold of a skew product system inherit a skew product form.     

Continuous dependence results for inhomogeneous ill-posed problems in Banach space

We apply semigroup theory and other operator-theoretic methods to prove Hölder-continuous dependence on modeling for the inhomogeneous ill-posed Cauchy problem in Banach space. The inhomogeneous ill-posed Cauchy problem is given by , u(0)=χ, 0?t<T; where −A is the infinitesimal generator of a holomorphic semigroup on a Banach space X, χ∈X, and . For a suitable function f, the approximate problem is given by , v(0)=χ. Under certain stabilizing conditions, we prove that for a related norm, where and M are computable constants independent of β, 0<β<1, and ω(t) is a harmonic function. These results extend earlier work of Ames and Hughes on the homogeneous ill-posed problem.     

Extension of the 2-p-opt and 1-shift algorithms to the heterogeneous probabilistic traveling salesman problem

The probabilistic traveling salesman problem is a well known problem that is quite challenging to solve. It involves finding the tour with the lowest expected cost for customers that will require a visit with a given probability. There are several proposed algorithms for the homogeneous version of the problem, where all customers have identical probability of being realized. From the literature, the most successful approaches involve local search procedures, with the most famous being the 2-p-opt and 1-shift procedures proposed by Bertsimas [D.J. Bertsimas, L. Howell, Further results on the probabilistic traveling salesman problem, European Journal of Operational Research 65 (1) (1993) 68–95]. Recently, however, evidence has emerged that indicates the equations offered for these procedures are not correct, and even when corrected, the translation to the heterogeneous version of the problem is not simple. In this paper we extend the analysis and correction to the heterogeneous case. We derive new expressions for computing the cost of 2-p-opt and 1-shift local search moves, and we show that the neighborhood of a solution may be explored in O(n²) time, the same as for the homogeneous case, instead of O(n³) as first reported in the literature.     

GMRES and the minimal polynomial

We present a qualitative model for the convergence behaviour of the Generalised Minimal Residual (GMRES) method for solving nonsingular systems of linear equationsAx =b in finite and infinite dimensional spaces. One application of our methods is the solution of discretised infinite dimensional problems, such as integral equations, where the constants in the asymptotic bounds are independent of the mesh size.Our model provides simple, general bounds that explain the convergence of GMRES as follows: If the eigenvalues ofA consist of a single cluster plus outliers then the convergence factor is bounded by the cluster radius, while the asymptotic error constant reflects the non-normality ofA and the distance of the outliers from the cluster. If the eigenvalues ofA consist of several close clusters, then GMRES treats the clusters as a single big cluster, and the convergence factor is the radius of this big cluster. We exhibit matrices for which these bounds are tight.Our bounds also lead to a simpler proof of existing r-superlinear convergence results in Hilbert space.This research was partially supported by National Science Foundation grants DMS-9122745, DMS-9423705, CCR-9102853, CCR-9400921, DMS-9321938, DMS-9020915, and DMS-9403224.     

On the elastic-viscoplastic proportional combined twist and stretch of a circular bar of rate-sensitive material

The elastic-plastic behaviour of a solid circular bar made of a homogeneous, isotropic and nonhardening material but of a rate-dependent type, subjected to different proportional deformation programmes of twist and stretch, is analysed. In this investigation the material is assumed to yield according to the von Mises criterion and then to follow a rate-dependent post-yield constitutive law of the Perzyna type. Four radial paths in the angte-of-twist and axial-stretch plane were investigated. For each deformation path the numerical solution of the governing system of quasi-linear partial differential equations gave the corresponding loading path and the radial variation of the stress field at selected times. This study complements that of S. A. Meguid and J.D. Campbell (1979) which considered mainly nonproportional deformation paths. In this work residual stress profiles were also calculated, based upon instantaneous unloading after various amounts of deformation. Comparison with predictions of a rateindependent theory for a limiting state, where elastic strains are negligible, is made. Over a considerable range, the loading paths beyond the initial yield were nearly straight, at the elastic slope. The load trajectory then bends over and approaches a point on the limit-state locus. The results showed that the approximate procedure of increasing the flow stress by an empirically determined factor corresponding to a mean plastic strain-rate gave a good estimate for the limit-state stresses.     

Electron excitation of the Schumann-Runge continuum, longest band, and second band electronic states in O2

We report measurements of differential and integral cross sections for electron excitation of the Schumann-Runge continuum, longest band, and second band electronic states in molecular oxygen. The energy range of the present study is 15-200 eV, with the angular range of the differential cross section (DCS) measurements from 2 to 130°. A generalized oscillator strength analysis is then employed in order to derive integral cross sections (ICSs) from the corresponding DCSs, and these ICSs are compared with relevant energy and oscillator strength scaled Born cross section (BEf-scaling [Y.-K. Kim, J. Chem. Phys. 126, 064305 (2007)]) results determined as a part of this investigation. Interestingly, while the present Schumann-Runge continuum and second band ICSs were in reasonable agreement with the respective BEf-scaling results, agreement for the longest band was poor below 100 eV with a possible reason for this apparently anomalous behavior being canvassed here. Finally, where possible all present data are compared with the results from earlier measurements and calculations with the level of agreement found being very good in some cases and marginal in others.     

Effects of peripheral and axial substitutions on electronic transitions of tin naphthalocyanines

Tin naphthalocyanine molecules display strong absorption in the infrared region (IR), making them ideal as components of organic photodiodes and solar cells. We use density functional theory and time-dependent density functional theory (TD-DFT) at the B3LYP level to study the influence of axial and peripheral ligands on the absorption wavelength of tin naphthalocyanines. We find that TD-DFT is successful at reproducing the experimental absorption spectra of free base naphthalocyanine and tin naphthalocyanine molecules and can be used as a reliable tool to predict absorption spectra of substituted naphthalocyanines. Functional groups attached axially to tin (-F, -Cl, -Br, -I) and peripherally to the inner ring (-F, -Cl, -Br, -Ph, -OH, -COCH(3), -O(CH(2))(3)CH(3)) of the tin naphthalocyanine molecule tune the excitation wavelength in the near-infrared region between 770 and 940 nm. While substituents to the outer naphthalocyanine ring (-Cl, -Br) affect the intensity of the absorption peaks in the NIR region, they do not influence their absorption wavelength. Asymmetric substitution of naphthalocyanine pendant arms can be employed to decrease intensity of the absorption peaks in the visible region with respect to the intensity of the peaks in the NIR.