Finite activation energy reactions Part 1: Reactants with spatially varying surface temperature

Summary The effect of a finite activation energy of reaction on the stability of reactants undergoing steady zero-order exothermic reactions is investigated. The reactants are exposed to a surface temperature which decreases slowly away from a global line maximum ofO(1). For both slab and cylinder geometries, it is found that large but finite activation energies can counteract the destabilising effect of the globalO(1) maximum in surface temperature.     

The Two-Dimensional Finite Bin Packing Problem. Part II: New lower and upper bounds

This paper is the second of a two part series and describes new lower and upper bounds for a more general version of the Two-Dimensional Finite Bin Packing Problem (2BP) than the one considered in Part I (see Boschetti and Mingozzi 2002). With each item is associated an input parameter specifying if it has a fixed orientation or it can be rotated by . This problem contains as special cases the oriented and non-oriented 2BP. The new lower bound is based on the one described in Part I for the oriented 2BP. The computational results on the test problems derived from the literature show the effectiveness of the new proposed lower and upper bounds.     

A note on thetheoretical existence of invariants: Part II

We propose a framework in which to study invariants of difference equations with the ultimate goal of establishing the (theoretical) existence of effective invariants. The paper is divided into two parts.Part I began with several examples which illustrated some of the difficulties to be overcome and ended with a discussion of the connection between invariants and decompositions of the phase space.In PartII, which is more topological in nature, we now pursue the theme of invariant decompositions     

Forced lattice vibrations: Part II

This is the second part of a two-part series on forced lattice vibrations in which a semi-infinite lattice of one-dimensional particles {x_n}_n≧1, is driven from one end by a particle x₀. This particle undergoes a given, periodically perturbed, uniform motion x0(t) = 2at + h(yt) where a and γ are constants and h(·) has period 2π. Results and notation from Part I are used freely and without further comment. Here the authors prove that sufficiently ample families of traveling-wave solutions of the doubly infinite system exist in the cases γ > γ₁ and γ₁ > γ > γ₂ for general restoring forces F. In the case with Toda forces, F(x) = e^x, the authors prove that sufficiently ample families of traveling-wave solutions exist for all k, γ_k > γ > γ_k+1. By a general result proved in Part I, this implies that there exist time-periodic solutions of the driven system (i) with k-phase wave asymptotics in n of the type with k = 0 or 1 for general F and k arbitrary for F(x) = e^x (when k = 0, take γ₀ = ∞ and X₀ ≡ 0).     

Variable Programming: A Generalized Minimax Problem. Part II: Algorithms

In this part of the two-part series of papers, algorithms for solving some variable programming (VP) problems proposed in Part I are investigated. It is demonstrated that the non-differentiability and the discontinuity of the maximum objective function, as well as the summation objective function in the VP problems constitute difficulty in finding their solutions. Based on the principle of statistical mechanics, we derive smooth functions to approximate these non-smooth objective functions with specific activated feasible sets. By transforming the minimax problem and the corresponding variable programming problems into their smooth versions we can solve the resulting problems by some efficient algorithms for smooth functions. Relevant theoretical underpinnings about the smoothing techniques are established. The algorithms, in which the minimization of the smooth functions is carried out by the standard quasi-Newton method with BFGS formula, are tested on some standard minimax and variable programming problems. The numerical results show that the smoothing techniques yield accurate optimal solutions and that the algorithms proposed are feasible and efficient.This work was supported by the RGC grant CUHK 152/96H of the Hong Kong Research Grant Council.     

On well-covered triangulations: Part II

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part I, Discrete Appl. Math., 132, 2004, 97-108] that there are no 5-connected planar well-covered triangulations. It is the aim of the present paper to completely determine the 4-connected well-covered triangulations containing two adjacent vertices of degree 4. In a subsequent paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part III (submitted for publication)], we show that every 4-connected well-covered triangulation contains two adjacent vertices of degree 4 and hence complete the task of characterizing all 4-connected well-covered planar triangulations. There turn out to be only four such graphs. This stands in stark contrast to the fact that there are infinitely many 3-connected well-covered planar triangulations.     

Generalized Bosbach states: Part II

We continue the investigation of generalized Bosbach states that we began in Part I, restricting our research to the commutative case and treating further aspects related to these states. Part II is concerned with similarity convergences, continuity of states and the construction of the s-completion of a commutative residuated lattice, where s is a generalized Bosbach state.     

A Chebyshev Quadrature Rule for One Sided Finite Part Integrals

This paper is concerned with a Chebyshev quadrature rule for approximating one sided finite part integrals with smooth density functions. Our quadrature rule is based on the Chebyshev interpolation polynomial with the zeros of the Chebyshev polynomial T_N+1(τ)−T_N−1(t). We analyze the stability and the convergence for the quadrature rule with a differentiable function. Also we show that the quadrature rule has an exponential convergence when the density function is analytic.     

Equivalential algebras. Part II: Independence

Received November 27, 1996; accepted in final form February 26, 1997.     

Finite dimensional convex structures II: the invariants

Various relations between the dimension and the classical invariants of a topological convex structure have been obtained, leading to an equivalence between Helly's and Carathéodory's theorem, and to the closedness of the hull of compact sets in finite-dimensional convexities. It is also shown that the Radon number of an n-dimensional binary convexity is in most cases equal to the Radon number of the n-cube, and a natural condition is presented under which the invariants are equal to dimension plus one.     

Computing finite commutative semigroups: Part II

New precedence results are obtained for semigroups with zero, which describe the first non-zero row in detail and provide effective comparisons with subsequent rows. These results are used to further sharpen existing algorithms for the computation of finite commutative semigroups, confirm the number of distinct commutative semigroups of order 9, and determine the number of distinct commutative semigroups of order 10.     

A boundary element methodology for viscoelastic analysis: Part II without cells

In this study Kelvin and Boltzmann viscoelastic models are implemented in a two-dimensional boundary element atmosphere. This general methodology is based on differential constitutive relations for viscoelasticity, avoiding the use of relaxation functions. In this part of the study, important algebraic operations are introduced into the formulation allowing analysing viscoelastic problems without using internal cells. This improvement is very important to model infinite and semi-infinite regions. The formulation is verified comparing the numerical results with analytical solutions. An extension of the formulation to consider soil–structure interaction is presented in order to demonstrate the vast applicability of the technique.     

Radial sums of ridge functions: A characterization

We extend and sharpen the characterization of radial sums of ridge functions of two variables given in an earlier paper. More precisely, a bound on the degree of certain polynomials is given which depends on the angles between the ridges; not only is this bound best possible but also a converse result holds. Furthermore, variants of this characterization which hold for higher dimensional analogues are also given. These results are then applied to certain parallel am models in computed tomography to obtain more precise and otherwise improved characterizations of algorithms which commute with rigid motions.     

A Degenerate Fourth-Order Parabolic Equation Modeling Bose-Einstein Condensation Part II: Finite-Time Blow-Up

A degenerate fourth-order parabolic equation modeling condensation phenomena related to Bose-Einstein particles is analyzed. The model can be motivated from the spatially homogeneous and isotropic Boltzmann-Nordheim equation by a formal Taylor expansion of the collision integral. It maintains some of the main features of the kinetic model, namely mass and energy conservation and condensation at zero energy. The existence of local-in-time weak solutions satisfying a certain entropy inequality is proven. The main result asserts that if a weighted L ¹ norm of the initial data is sufficiently large and the initial data satisfies some integrability conditions, the solution blows up with respect to the L ^∞ norm in finite time. Furthermore, the set of all such blow-up enforcing initial functions is shown to be dense in the set of all admissible initial data. The proofs are based on approximation arguments and interpolation inequalities in weighted Sobolev spaces. By exploiting the entropy inequality, a nonlinear integral inequality is proved which implies the finite-time blow-up property.     

A revisit of spinning disk models. Part II: linear transverse vibrations

In this paper, two recently derived linear models for the transverse vibrations of a spinning plate are considered. The disk is modelled as a pure plate with no membrane effects. Furthermore, the effect of the rotary inertia of the plate is taken into account. The first model is based on the assumption of linear (Kirchhoff) strains. The second model is based on the assumption of non-linear (von Karman) strains. The merits of both models are considered and their predictions are compared with those of the traditional linear model.     

Smoothing minimally supported frequency wavelets: Part II

The main purpose of this paper is to give a procedure to “mollify” the low-pass filters of a large number ofMinimally Supported Frequency (MSF) wavelets so that the smoother functions obtained in this way are also low-pass filters for an MRA. Hence, we are able to approximate (in the L ² -norm) MSF wavelets by wavelets with any desired degree of smoothness on the Fourier transform side. Although the MSF wavelets we consider are bandlimited, this may not be true for their smooth approximations. This phenomena is related to the invariant cycles under the transformation x ↦2x (mod2π). We also give a characterization of all low-pass filters for MSF wavelets. Throughout the paper new and interesting examples of wavelets are described.