Nonoscillation of Elliptic Integrals Related to Cubic Polynomials with Symmetry of Order Three

We study zeros of elliptic integrals I(h)=_HhR(x,y)dxdy, whereH(x,y) is a real cubic polynomial with a symmetry of order three,and R(x,y) is a real polynomial of degree at most n. It turnsout that the vector space A_n formed by such integrals is a Chebishevsystem: the number of zeros of each elliptic integral I(h)A_nis less than the dimension of the vector space A_n. 1991 MathematicsSubject Classification 34C10.     

On Extensions of Myers' Theorem

Let M be a compact Riemannian manifold, and let h be a smoothfunction on M. Let p^h(x) = inf||–₁(Ric_x(,)–2Hess(h_x(,)).Here Ric_x denotes the Ricci curvature at x and Hess(h) is theHessian of h. Then M has finite fundamental group if ^h–p^h<0. Here ^h =:+2L_h is the Bismut-Witten Laplacian. This leadsto a quick proof of recent results on extension of Myers' theoremto manifolds with mostly positive curvature. There is also asimilar result for noncompact manifolds.     

On the Solution of Some Elliptic Difference Equations

Iterative methods for the solution of some nonlinear ellipticdifference systems, approximating the first boundary value problemare considered. If h > 0 is the network step in the spaceof variables x = (x₁, x₂,..., x_p) and 2m is the order of theoriginal boundary value problem, then the iterative methodsproposed give solution of accuracy with the expenditure ofO(|In | h^–(p+m–)) and O(|In | |In h| h^–p)arithmetic operations in the case of a general region and arectangular parallelepiped respectively. In the case p = 2 theestimate O(|In | h^{–[2+ (m/2)]}) is obtained if the regionis made up of rectangles with sides parallel to the co-ordinateaxes.     

On the Sloan iteration applied to integral equations of the first kind

When the piecewise constant collocation method is used to solvean integral equation of the first kind with logarithmic kernel,the convergence rate is O(h) in the L₂ norm. In this note weshow that O(h³) or O(h⁵) convergence in any Sobolev norm (andthus, for example, in L) may be obtained by a simple cheap postprocessingof the original collocation solution. The construction of thepostprocessor is based on writing the first kind equation asa second kind equation, and applying the Sloan iteration tothe latter equation. The theoretical convergence rates are verifiedin a numerical example.     

The structure of the normalisers of the congruence subgroups of the Hecke group G5

Let = 2cos (/5) and let []. Denote the normaliser ofG₀() of the Hecke group G₅ in PSL₂() by N(G₀()). Then N(G₀())= G₀(/h), where h is the largest divisor of 4 such that h² divides. Further, N(G₀())/G₀() is either 1 (if h = 1), ₂ x ₂ (if h= 2) or ₄ x ₄ (if h = 4).     

Existence of unique solutions for perturbed autonomous differential equations

We consider the existence of unique absolutely continuous solutionsfor x' = p(t)f(x) + p(t)h(t), t 0, x(0) = 0, where p, f, andh are positive almost everywhere, but none of them needs becontinuous or monotone. Moreover, p and f can be unbounded aroundzero. Our uniqueness results are not based on assumptions onthe differences f(x) – f(y), as it is usual in most uniquenessresults, and they are new even when p, f, and h are continuous.     

Linear maps preserving essential spectral functions and closeness of operator ranges

Let H be an infinite-dimensional complex Hilbert space and (H)the algebra of all bounded linear operators on H. Some characterizationsare obtained of linear maps on (H) preserving essential spectralfunctions such as the intersection of left essential spectrumand right essential spectrum of operators, preserving Fredholmoperators, and preserving semi-Fredholm operators, and preservingcloseness of operator ranges. A result of Mbekhta, Rodman andemrl (Integral Equations Operator Theory 55 (2006) 93-109) ongeneralized invertibility preserving linear maps is improved.     

Random Walk Conditioned to Stay Positive

A random walk that is certain to visit (0, ) has associatedwith it, via a suitable h-transform, a Markov chain called ‘randomwalk conditioned to stay positive’, which is defined properlybelow. In continuous time, if the random walk is replaced byBrownian motion then the analogous associated process is Bessel-3.Let (x) = log log x. The main result obtained in this paper,which is stated formally in Theorem 1, is that, when the randomwalk has zero mean and finite variance, the total time for whichthe random walk conditioned to stay positive is below x ultimatelylies between Lx²/(x) and Ux²(x), for suitable (non-random) positiveL and finite U, as x goes to infinity. For Bessel-3, the bestL and U are identified.     

Arithmetical Results on Certain Multivariate Power Series

Power series with non-zero convergence radius R(f) are considered, and the arithmeticalnature (that is, irrationality, or even transcendence) of thecorresponding multivariate series is studied if x₁, ..., x_m and the sequence (f(n)) satisfy appropriatearithmetical conditions. It follows that such arithmetical resultscan be written down easily if linear independence results onthe function F(x), defined in |x| < R(f) by the originalone-dimensional power series, and possibly its derivatives atthe points x_µ are known. Some typical applications areexplicitly stated. 2000 Mathematics Subject Classification 11J72.     

Lorentz Spaces Of Vector-Valued Measures

Given a non-atomic, finite and complete measure space (,,µ)and a Banach space X, the modulus of continuity for a vectormeasure F is defined as the function _F(t) = sup_µ(E)t |F|(E)and the space V^p,q(X) of vector measures such that t^–1/p'_F(t) L^q((0,µ()],dt/t) is introduced. It is shown thatV^p,q(X) contains isometrically L^p,q(X) and that L^p,q(X) = V^p,q(X)if and only if X has the Radon–Nikodym property. It isalso proved that V^p,q(X) coincides with the space of cone absolutelysumming operators from L^p',q^' into X and the duality V^p,q(X^*)=(Lp^',q^'(X))^*where 1/p+1/p^'= 1/q+1/q^' = 1. Finally, V^p,q(X) is identifiedwith the interpolation space obtained by the real method (V¹(X),V(X))_1/p^',q. Spaces where the variation of F is replaced bythe semivariation are also considered.     

The Harmonious Chromatic Number of Bounded Degree Graphs

A harmonious colouring of a simple graph G is a proper vertexcolouring such that each pair of colours appears together onat most one edge. The harmonious chromatic number h(G) is theleast number of colours in such a colouring. Let d be a fixed positive integer, and >0. We show that thereis a natural number M such that if G is any graph with mM edgesand maximum degree at most d, then the harmonious chromaticnumber h(G) satisfies      

The Derivation Problem for Group Algebras of Connected Locally Compact Groups

The derivation problem for a locally compact group G is to decidewhether for each derivation D from L¹(G) into L¹(G) there isa bounded measure µM(G) with D(a) = aµ–µa(a L¹(G)). In this paper we obtain an affirmative answer forthe case of connected groups. To explain the contents of thispaper we give an equivalent formulation of the problem. Supposethat the group G acts as a group of homeomorphisms of the locallycompact space X. Related to this there is an action of G onM(X). A bounded crossed homomorphism from G to M(X) is a map with bounded range and satisfying (gh) = g(h)+(g) (g, h G).The problem for bounded crossed homomorphisms is to decide iffor each such there is an element µ of M(X) with (g)= gµ– µ (g G). The derivation problem isequivalent to this bounded crossed homomorphism problem forthe special case X = G where G acts on X by conjugation (togetherwith some mild continuity hypotheses about the map :GM(X) whichare often automatically satisfied). The bounded crossed homomorphismproblem always has a positive solution if G is amenable anda closely related calculation shows that in solving the boundedcrossed homomorphism problem we need only solve it for functions which are zero on H where H is a given amenable subgroup ofG. It can happen that this condition of being zero on H forces to be zero even when H is a comparatively small subgroup ofG. If h is an element of G such that ‘hⁿx ’ asn for all x X then for any two measures µ and , forlarge values of n, µ and hⁿ have little overlap so ||µ+ hⁿ|| ||µ|| + ||||. Thus if H is the subgroup generatedby h, for any g G .     

Strengthening Hadwiger's Conjecture

We consider the following strengthening of Hadwiger's Conjecture. Let G be any graph of chromatic number k, S any subset of V(G)which takes all k colours in each proper k-colouring of G. Thenthere are k pairwise adjacent connected subgraphs of G, eachof whose vertex sets has non-trivial intersection with S. We show that the truth of this conjecture for all graphs ofchromatic number k implies the truth of Hadwiger's Conjecturefor all graphs of chromatic number k + 1. We also show thatits truth implies the following statement (which is at firstsight even stronger). For any graph G of chromatic number k and any subset S of V(G),define (S; G) to be the least number of colours that can appearon S in any proper k-colouring of G, and h(S; G) to be the largestnumber of pairwise adjacent connected subgraphs of G each havingnon-trivial intersection with S. Then (S; G) h(S; G). We define the number w(S; G) to be the largest cardinality ofa subset T of S such that, however T is partitioned into pairs(possibly with one spare element), there are vertex-disjointpaths linking the elements in each pair, none passing throughthe spare element if it exists. We show that (S; G) (|S| +w(S; G))/2 for any graph G and subset S of V(G). Finally, we show that for any graph G, (S; G) h(S; G) whenever(S; G) 3. 1991 Mathematics Subject Classification 05C15.     

Approximate Monotonicity: Theory and Applications

The ideas of value distribution for measurable functions fromR to R are applied to functions which are approximately monotonicon sets of positive measure. (For definitions see 1.) A functionp(x) is introduced, describing the local relative value distributionin the neighbourhood of a point x, and it is shown that almosteverywhere p(x) = 0 or wherever p(x) exists, implying approximate differentiability, with thefunction approximately oscillatory elsewhere. These resultsare applied to the analysis of angular boundary behaviour forHerglotz functions, where they have implications for the spectralanalysis of differential and other operators.     

Global errors of numerical ODE solvers and Lyapunov's theory of stability

The error made by a numerical method in approximating the solutionof the initial value problem (t) = f (t,x), x (0) = x₀, t 0, x (t) R^d varies with the time of integration.The increase of the global error ||(t; h)– x (t)||, where (t, h) is an approximationderived by a numerical method with time step h, with time tdetermines the feasibility of approximating the solution accuratelyfor increasing t. However, the best available theoretical boundsinvolve the Lipshitz constant and are exponential in t for someproblems where the actual increase of global error is only linearin time. Using techniques from Lyapunov's theory of stability, we provethat the increase of global errors is linear in time for trajectoriesof dynamical systems which fall into a hyperbolic and attractingcycle or into a hyperbolic and attracting torus, with the flowon the torus being quasi-periodic. The increase is linear fornon-linear problems when certain stability properties of thesolution can be verified. The error analysis uses a conditioningfunction E(t) associated with the exact solution, which capturesthe propagation and accumulation of global errors. Received 4 January 1999. Accepted 15 December 1999.     

K-Theory for Algebras of Operators on Banach Spaces

It is proved that, for each pair (m, n) of non-negative integers,there is a Banach space X for which K₀(B(X))Z^m and K₁(B(X))Zⁿ.The K-groups of all closed ideals of operators contained inthe ideal of strictly singular operators are computed, and someresults about the existence of splittings of certain short exactsequences are derived.     

On the Centred Hausdorff Measure

Let v be a measure on a separable metric space. For t, q R,the centred Hausdorff measures µ^h with the gauge functionh(x, r) = r^t(vB(x, r))^q is studied. The dimension defined bythese measures plays an important role in the study of multifractals.It is shown that if v is a doubling measure, then µ^h isequivalent to the usual spherical measure, and thus they definethe same dimension. Moreover, it is shown that this is trueeven without the doubling condition, if q 1 and t 0 or ifq 0. An example in R² is also given to show the surprisingfact that the above assertion is not necessarily true if 0 <q < 1. Another interesting question, which has been askedseveral times about the centred Hausdorff measure, is whetherit is Borel regular. A positive answer is given, using the aboveequivalence for all gauge functions mentioned above.     

Convergent and spurious solutions of nonlinear elliptic equations

In this paper we investigate finite element approximations ofnonlinear elliptic equations in three dimensions. By applyingand extending the results of Lopez-Marcos and Sanz-Serna, weprove that the finite element approximation on a mesh of sizeh, has a solution U_k which converges to an exact solution ofthe differential equation as h0. This solution is unique withina suitably defined stability ball Bh. For the particular nonlinearequation u + (u + u^p) we show that the size of B_h depends uponh only if p > 5 when it tends to zero as h 0. In this casewe prove the existence of spurious solutions V_h of the Galerkinapproximation which become unbounded in the maximum norm ash0. The stability ball B_h then acts to separate the convergentand the spurious solutions. We present the results of some numericalexperiments to substantiate our claims.     

Normal families and omitted functions II

Let k 2 be an integer and let be a family of functions meromorphicon a domain D in , all of whose poles are multiple and whosezeros all have multiplicity at least k + 1. Let h be a functionmeromorphic on D, h 0, . Suppose that for each f , f^(k)(z) h(z) for z D. Then is a normal family on D.