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This paper outlines an account of numbers based on the numerical equivalence schema (NES), which consists of all sentences of the form ' # x. F x = n if and only if ∃ n x   F x ', where # is the number-of operator and ∃ n is defined in standard Russellian fashion. In the first part of the paper, I point out some analogies between the NES and the T-schema for truth. In light of these analogies, I formulate a minimalist account of numbers, based on the NES, which strongly parallels the minimalist (deflationary) account of truth. One may be tempted to develop the minimalist account in a fictionalist direction, according to which arithmetic is useful but untrue, if taken at face value. In the second part, I argue that this suggestion is not as attractive as it may first appear. The NES suffers from a similar problem to the T-schema: it is deductively weak and does not enable the derivation of any non-trivial generalizations. In the third part of the paper, I explore some strategies to deal with the generalization problem, again drawing inspiration from the literature on truth. In closing this paper, I briefly compare the minimalist to some other accounts of numbers. [ABSTRACT FROM AUTHOR]

In this paper, we prove the following differential analog of the Noether normalization lemma: for every d-dimensional differential algebraic variety over differentially closed field of zero characteristic there exists a surjective map on to the d-dimensional affine space. Equivalently, for every integral differential algebra A over differential field of zero characteristic there exist differentially independent b1, . . . , bd such that A is differentially algebraic over subalgebra B differentially generated by b1, . . . , bd, and whenever p ⊂ B is a prime differential ideal, there exists a prime differential ideal q ⊂ A such that p = B ∩ q. We also prove the analogous theorem for differential algebraic varieties over the ring of formal power series over an algebraically closed differential field and present some applications to differential equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with an ordinary non-linear differential equation that occurs in the theory of thermal explosion and, for the spherically symmetric case, in the theory of stellar structure. For plane and axial symmetry, closed form solutions are well known, but the spherically symmetric case can so far only be obtained numerically. By examining the problem in the phase plane, Enig (1967, Critical parameters in the Poisson-Boltzmann equation of steady-state thermal explosion theory. Combust. Flame, 10, 197–199enig1967 was able to obtain an equation from which the critical parameters of the equations can be determined. The equation is of Abel type and there is some difficulty in determining an integrating factor. We note that the partial differential equation for the integrating factor is more difficult to solve then the original equation. A general method is presented that allows the solution to be found in parametric form. Methods of solution for the spherically symmetric case have been presented by various authors. It is shown that these may all be reduced to the equation of Enig. The spherically symmetric case has been examined in some detail near its singular point. In a brilliant but largely ignored paper, it was Jules Enig who first discovered the existence of a vortex in the phase plane. It is hoped that my contribution has solved some of the mysteries of the vortex. [ABSTRACT FROM PUBLISHER]

In the paper cited in the heading, we treated the problem of stabilizing a flexible cable with two rigid loads, which was described by two kinds of hyperbolic equations. To show the asymptotic stability of the closed-loop system with a controller derived there, we used the LaSalle's invariance principle. However, in that paper, we need to supplement the proof of Theorem 5.1 and to revise the proof of Theorem 5.2. Throughout this note, we use the same notation as in the paper cited in the heading. [ABSTRACT FROM PUBLISHER]

In this paper we analyse a two‐dimensional in‐diffusion problem for a model for the dissociative (Frank–Turnbull) diffusion mechanism which incorporates charge effects; the paper is the second of a pair, the first of which deals with a corresponding one‐dimensional problem. The equations are analysed using both asymptotic and numerical methods. Some of the impurity profiles and contours are observed to have a highly unusual character, an example being contours which have a ‘fan’ shape. The formulation and results are also potentially relevant to many other reaction–diffusion processes. [ABSTRACT FROM PUBLISHER]

We consider a Landau-Lifshitz-Gilber equation perturbed by a multiplicative space-dependent noise for a ferromagnet filling a bounded three-dimensional domain. We show the existence of weak martingale solutions taking values in a sphere . The regularity of weak solutions is also discussed. Our research is in response to the paper by Kohn et al. “Magnetic elements at finite temperature and large deviation theory.” Journal of Nonlinear Science 15 (2005): 223–53, which calls for the study of stochastic equations with spatially varying magnetization. [ABSTRACT FROM AUTHOR]

In this paper, we consider a three-component system of 1D long wave–short wave interaction equations. The system has two-parameter family of solitary wave solutions. We prove orbital stability of the solitary wave solutions using variational methods. [ABSTRACT FROM PUBLISHER]

In this paper, we study the decay properties of the solutions to Timoshenko beam with a weak non-linear dissipation. [ABSTRACT FROM PUBLISHER]

Donaldson introduced an interesting geometric structure (the Donaldson metric) on the space of volume forms for any compact Riemannian manifold, which has nonpositive sectional curvature formally. The geodesic equation and its perturbed equations are fully nonlinear elliptic equations. These equations are also equivalent to two free boundary problems of the Laplacian equation, and it also has a relationship with many interesting problems, such as Nahm's equation. In this paper, we solve these equations and demonstrate the geometric structure of the space of volume forms; in particular, we show that the space of volume forms with the Donaldson metric is a metric space with non-positive curvature in Alexanderov sense. [ABSTRACT FROM PUBLISHER]

This paper is concerned with rational Szegő quadrature formulas to approximate integrals of the form by a formula such as , where the weights λk are positive and the nodes zk are carefully chosen on the complex unit circle. It will be shown that, for a given set of poles, the quadrature formulas can be chosen to be exact in certain subspaces of rational functions of dimension 2n. Also, the problem where one node (Radau) or two nodes (Lobatto) are prefixed will be analysed and the corresponding rational Szegő–Radau and rational Szegő–Lobatto quadrature formulas will be characterized. [ABSTRACT FROM PUBLISHER]

This is a foundational paper on flows of G2-structures. We use local coordinates to describe the four torsion forms of a G2-structure and derive the evolution equations for a general flow of a G2-structure ϕ on a 7-manifold M. Specifically, we compute the evolution of the metric g, the dual 4-form ψ and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernández–Gray. [ABSTRACT FROM PUBLISHER]

The application of the direct Lyapunov method to the stability analysis of neutral systems with time-varying delays usually encounters a restrictive assumption on the function in the right side of the differential equation. This function is supposed to satisfy the Lipschitz condition with respect to the delayed state derivative with a constant less than 1. In the present paper, we extend the input–output approach to consider the stability of neutral type systems with uncertain time-varying delays and norm-bounded uncertainties. The assumption on the Lipschitzian constant can then be avoided. Sufficient stability criteria are derived in the frequency domain and the time domain, where the descriptor discretized Lyapunov–Krasovskii functional is applied. As a by-product, new necessary conditions for neutral-delay-independent/retarded-delay-dependent stability criteria are obtained. The method can be easily extended to L2-gain analysis and can be applied to design problems. [ABSTRACT FROM PUBLISHER]

In this paper, we study the complex version of the Kharitonov theorem without the constant-degree assumption. It is proved that, for a complex-interval polynomial with degree drop, robust Hurwitz stability is equivalent to the Hurwitz stability of the eight Kharitonov polynomials. Furthermore, it is also shown that for a robustly stable complex-interval polynomial, degree drop cannot exceed one and degree drop can only occur at one of the corners of the rectangular region in the complex plane corresponding to the leading coefficient of the uncertain polynomial. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate pattern formation in a coupled system of reaction–diffusion equations in two spatial dimensions. These equations arise as a model of isothermal chemical autocatalysis with termination in which the orders of autocatalysis and termination, m and n, respectively, are such that 1 < n < m. We build on the preliminary work by Leach & Wei (2003, Physica D, 180, 185–209) for this coupled system in one spatial dimension, by presenting rigorous stability analysis and detailed numerical simulations for the coupled system in two spatial dimensions. We demonstrate that spotty patterns are observed over a wide parameter range. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the equation [ABSTRACT FROM PUBLISHER]

The upper free surface z = u(x, y) of a static fluid with gravity acting in the z direction, occupying a volume V, satisfies the Laplace–Young equation. The fluid wets the vertical boundaries of V so that the usual capillary contact conditions hold. This paper considers wedgeshaped volumes V with corner angle 2α, that belong to the intermediate corner angle case of π/2−γ < α < π/2, where γ is the contact angle and determines explicitly a regular power series expansion for the height u(r, θ) of the fluid near the corner, r = 0, to all orders in r. Miersemann (1988, Pacific J. Math., 134, 99–311), shows that it is possible to have logarithmic terms for a general corner expansion of the Laplace–Young equation, with appropriate boundary conditions. However, we suggest that the usual practical cases do not possess any singular terms near the corner, and we analytically and explicitly produce a non-singular series to any order in r, and propose that near the corner the far-field effects are lost through any ‘interior or inner flat’ region in exponentially small terms. We give computational solutions for these regular (energy minimizing) cases based on a numerical finite volume method on an unstructured mesh, which fully support our assertions and our analytical series results, including the (minor) influence of the far field on local corner behaviour. [ABSTRACT FROM PUBLISHER]

In this paper stabilization of infinite‐dimensional undampedsecond‐order systems is considered in the case where the inputand output operators are collocated. The systems have aninfinite number of poles and zeros on the imaginary axis. In thecase where only position feedback is available, a parallelcompensator is effective. The stabilizer is constructed by aP‐controller for the augmented system which consists of thecontrolled system and a parallel compensator. The asymptoticstability of the augmented system is proved by LaSalle'sinvariance principle under compactness of the resolvent. [ABSTRACT FROM PUBLISHER]

In a recent paper Savin, Smith and Allen (Proc. R. Soc. A 455 (1999)) performed a nonlinear critical layer analysis based upon the long wavelength instability of an inflectional sublayer flow. This limit made it possible for the authors to consider the evolution of general disturbances to the base flow for a number of possible transition paths. The present analysis follows one of those paths and derives, in closed form, the amplitude evolution equation for a viscous non‐equilibrium critical layer. This equation is then solved numerically in the inviscid limit for a representative initial condition. [ABSTRACT FROM PUBLISHER]

This paper analyses the existence of invariant manifolds of periodic orbits for a specific piecewise linear three‐dimensional system with two zones, whose linear parts share a pair of imaginary eigenvalues. This degenerate situation is obtained from the lack of controllability. The analysis proceeds by its reduction to a periodic one‐dimensional equation for which some results of the Ambrosetti–Prodi type are given. [ABSTRACT FROM PUBLISHER]

The aim of this paper is to give explicit limit expressions, for diffusion equations involving a small parameter ε, describing both nonperiodic homogenization and reduction of dimension. In other words, we give the limit behaviour, when ε tends to zero, of the diffusion equation in a thin domain, with thickness of order ε, when the coefficients of the equation also depend on ε and may present rapid, nonperiodic oscillations, provided they satisfy a suitable compensated compactness condition. We consider two kinds of reduction of dimension: the case of thin plates (3D → 2D) and the case of thin cylinders (3D → 1D). In particular, we give the limit diffusion equation for laminated plates. This is completely explicit and requires no special assumption, except stratification. In the case of thin cylinders, the formulae are less explicit, but we also indicate some simple applications. [ABSTRACT FROM PUBLISHER]

The common core of A-level Mathematics includes the study of numerical methods for solution of equations. One particular syllabus requires each student to study an example of failure in the case of the Newton-Raphson method. The construction of such failures is a contrivance; yet this attempt to do so in this paper opens up a fascinating case that enlivens other aspects of the Mathematics' syllabus at that level. [ABSTRACT FROM AUTHOR]

In a parametric model the maximum likelihood estimator of a parameter of interest ψ may be viewed as the solution to the equation l′p(ψ) = 0, where lp denotes the profile loglikelihood function. It is well known that the estimating function l′p(ψ) is not unbiased and that this bias can, in some cases, lead to poor estimates of ψ. An alternative approach is to use the modified profile likelihood function, or an approximation to the modified profile likelihood function, which yields an estimating function that is approximately unbiased. In many cases, the maximum likelihood estimating functions are unbiased under more general assumptions than those used to construct the likelihood function, for example under first‐ or second‐moment conditions. Although the likelihood function itself may provide valid estimates under moment conditions alone, the modified profile likelihood requires a full parametric model. In this paper, modifications to l′p(ψ) are presented that yield an approximately unbiased estimating function under more general conditions. [ABSTRACT FROM PUBLISHER]

In an earlier paper on a malignant cell invasion model (Marchant et al., SIAM J. Appl. Math, 60, 2000) we introduced a novel form of discontinuous travelling wave solution. These solutions could be studied easily by combining behaviour within a phase plane with the Rankine–Hugoniot shock conditions, which describe properties (such as the ratio of the jump discontinuities to the speed of propagation) that solutions may possess. These results were new for several reasons. The shock conditions relate to hyperbolic equations (which the model is) but were applied in a travelling wave ordinary differential equation phase plane using techniques that usually apply to parabolic reaction–diffusion systems. In addition the solutions possess singular behaviour near several points in the phase plane but in spite of this there exists a robust and stable family of physically interesting solutions. [ABSTRACT FROM PUBLISHER]

In this paper we address an initial-boundary-value problem for a generalized Fisher equation. In particular, we use the method of matched asymptotic expansions to develop a rational approach for determining the propagation speed for the large-t (time) travelling wave structures which evolve in the initial-boundary-value problem. This approach resolves apparent paradoxes which arise in the much used linearized approximation (in the cases R(u) ≤ U and R(u) ≤ u, where R(u) is the associated reaction function) and is readily adaptable to systems of Fisher-Kolmogorov type and to problems in higher spatial dimensions. [ABSTRACT FROM AUTHOR]