We prove the compactness of the Whittaker sublocus of the moduli space of Riemann surfaces (complex algebraic curves). This is the subset of points representing hyperelliptic curves that satisfy Whittaker's conjecture on the uniformization of hyperelliptic curves via the monodromy of Fuchsian differential equations. In the last part of the paper we devote our attention to the statement made by R. A. Rankin more than 40 years ago, to the effect that the conjecture ‘has not been proved for any algebraic equation containing irremovable arbitrary constants’. We combine our compactness result with other facts about Teichmüller theory to show that, in the most natural interpretations of this statement we can think of, this result is, in fact, impossible.

Let An and Gn (respectively, A′n and G′n) be the weighted arithmetic and geometric means of x1, …, xn (respectively, 1 – x1, …, 1 – xn). We present sharp upper and lower bounds for the differences and . And we determine the best possible constants r and s such that

holds for all xi ∈ [a, b] (i = 1, …, n; 0 < a < b < 1). Our theorems extend and sharpen results of Fan, Cartwright and Field, McGregor and the author.

The solution of a problem in classical dynamics can be described, as Hamilton showed in 1834, by its Principal Function. Considering for simplicity a conservative problem with one degree of freedom, let the co-ordinate at the instant t be q, and let the Lagrangean function be L. Let Q be the value of the co-ordinate at a previous instant T. Let the quantity , after the integration has been performed, be expressed in terms of (q, Q, t — T), and let the function thus obtained, which is Hamilton's Principal Function, be denoted by W.

The paper is concerned with a Three-dimensional theory of non-linear magnetosonic waves in a turbulent plasma. A perturbation method is used that allows us to obtain a transport equation, like Burgers equation, but with a variable coefficient.

A modified form of the centroid method used in factor analysis is described. Various large sample results are obtained, including a test of significance of the residuals. The method is compared with the corresponding form of maximum likelihood estimation and its efficiency is investigated. A numerical illustration is given of some of the foregoing theory.

α-Truxillic acid (1, R = CO2H) (Criegee and Höver 1960) underwent the Arndt-Eistert reaction to yield l,3-diphenylcyclobutane-2,4-diacetic acid (1, R = CH2·CO2H), cyclisation of which with hydrogen fluoride gave cis-trans-cis-5,6,6a,6b, 11,12,12a, 12b-octahydro-5,11-dioxodibenzo[a,g]biphenylene (11). Attempts to aromatise the compound by bromination with N-bromosuccinimide followed by treatment with pyridine or potassium acetate were unsuccessful. The ketone was therefore reduced with potassium borohydride to the corresponding diol (11, CO replaced by CHOH) in the hope that dehydration of the product followed by mild dehydrogenation would yield dibenzo[a,g]biphenylene, but the only product isolated was naphthalene.

Given a regular two-dimensional local ring $(R,\mathfrak{m}_R)$ and a complete $\mathfrak{m}_{R}$-primary ideal $I\subset R$, we describe the (Zariski) factorization of the complete $\mathfrak{m}_{R}$-primary ideals $J\subset I$ of codimension $1$ in terms of the factorization of $I$.

Let X ⊂ ℂn be an equidimensional complex algebraic set and let f: X → ℂ be a polynomial function. For each c ∈ ℂ, we define the global Brasselet number of f at c, a global counterpart of the Brasselet number defined by the authors in a previous work, and the Brasselet number at infinity of f at c. Then we establish several formulas relating these numbers to the topology of X and the critical points of f.

The classical notions of monotonicity and convexity can be characterized via the nonnegativity of the first and the second derivative, respectively. These notions can be extended applying Chebyshev systems. The aim of this note is to characterize generalized monotonicity in terms of differential inequalities, yielding analogous results to the classical derivative tests. Applications in the fields of convexity and differential inequalities are also discussed.

In this paper we determine the structure of some new types of ordered inverse semigroup in which the ordering need not be the natural ordering. In particular, we generalise some results of McFadden and O'Carroll on F-inverse semigroups.

We consider the behaviour at x = ±∞ of solutions to reaction-diffusion equations modelling laminar flames in a premixed reactive gas. We show that if the initial data have limits at ±∞, then the solutions satisfy ODEs at ±∞ for all positive time. We then analyse the qualitative behaviour of solutions to the ODEs. Our applications include extensions of previous results on questions of flame propagation versus extinction, and a new decay result: if the initial temperature is above ignition temperature at one end of the domani and if the initial concentration vanishes at the other, then we show that the concentration decays^to zero uniformly as the time variable goes to infinity.

The main goal of this paper is to obtain the lower bounds

for solutions uν of nonlinear Schrödinger equations with small viscosity ν. We also discuss the application of the above estimate to the theory of turbulence. Namely, we are interested in time-averaged lower bounds, which are important in establishing upper bounds for the turbulent space-scale.

The following conjecture of Race will be proved: if τ is a formally J-symmetric quasi-differential expression on a real interval I, such that for some λ = ℂ all solutions of τy = τy belong to L2(I), then λ belongs to the regularity field of the minimal operator To generated by τ in L2(l).

Given a complete m-primary ideal J in a local regular two-dimensional ring (R,m), we describe every adjacent complete ideal above J as the integral closure of some ideal (f, g) for suitable f, g associated to J. We also provide a geometrical procedure that gives its base points, thus determining its equisingularity class. We decompose the set IJ of these adjacent ideals in terms of the Rees valuations of J. As a consequence, we obtain a geometrical characterization of the finiteness of IJ.

We study the weak stability index of an immersion ϕ: M → Sn+1 (1) ⊂ Rn+2 of an n-dimensional compact Riemannian manifold. We prove that the weak stability index of a compact hypersurface M with constant scalar curvature in Sn+1 (1), which is not totally umbilical, is greater than or equal to n + 2 if the mean curvature H1 and H3 are constant, and that the equality holds if and only if M is . As an application, we show that the weak stability index of an n-dimensional compact hypersurface with constant scalar curvature in Sn+1 (1), which is neither totally umbilical nor a Clifford hypersurface, is greater than or equal to 2n + 4 if the mean curvature H1 and H3 are constant.

In feedback stabilization studies, a control scheme is designed so that the ‘state’ of the system decays with a designated rate as t → ∞. Thus, any linear functional of the state also decays with at least the same decay rate. We study a class of linear parabolic systems, and show the existence of a feedback control scheme with a specific property such that some non-trivial linear functionals decay faster than the state. In constructing the control scheme, we also solve the new problem of an arbitrary allocation of the eigenvalues of some coefficient matrix which is subject to constraint, and give the necessary and sufficient condition in terms of controllability and observabililty assumptions. This is an essential extension of the celebrated 1967 theory of pole allocation by W. M. Wonham.

The even and odd Hilbert transformations, H+ and H−, are known to be bounded from the power-weighted space ℒμp to itself for −1 < μ 1 and 0 < μ < 2 respectively. We show that they fail to be surjective on ℒ0,p and ℒ1, p respectively, and we characterise the spaces H+(ℒ0, p), H- (ℒ1, p) and find inverses for H+, H− on them.

The purpose of this paper is to study the stability of some unilateral free-discontinuity problems in two-dimensional domains, with the density of the volume part having p-growth, with 1 < p < ∞, under perturbations of the discontinuity sets in the Hausdorff metric.

We prove monotonicity properties of positive weak solutions to – Δpu = f(u) in half-spaces for the case in which (2N + 2)/(N + 2) < p < 2 when zero Dirichlet boundary conditions are imposed. The model nonlinearity is given by f(s) := sq − λsm with q > m ≥ p − 1 and λ > 0.

Polynomial vector fields which admit a prescribed Darboux integrating factor are quite well understood when the geometry of the underlying curve is non-degenerate. In the general setting, morphisms of the affine plane may remove degeneracies of the curve, and thus allow more structural insight. In the present paper we establish some properties of integrating factors subjected to morphisms, and we discuss in detail one particular class of morphisms related to finite reflection groups. The results indicate that degeneracies for the underlying curve generally impose additional restrictions on vector fields admitting a given integrating factor.