A global Riesz decomposition theorem on trees without positive potentials

We study the potential theory of trees with nearest-neighbortransition probability that yields a recurrent random walk andshow that, although such trees have no positive potentials,many of the standard results of potential theory can be transferredto this setting. We accomplish this by defining a non-negativefunction H, harmonic outside the root e and vanishing only ate, and a substitute notion of potential which we call H-potential.We define the flux of a superharmonic function outside a finiteset of vertices, give some simple formulas for calculating theflux and derive a global Riesz decomposition theorem for superharmonicfunctions with a harmonic minorant outside a finite set. Wediscuss the connection of the H-potentials with other notionsof potentials for recurrent Markov chains in the literature.     

Strategy and Complexity of The Game of Squares

We introduce a game called Squares where the single player ispresented with a pattern of black and white squares and hasto reduce the pattern to white by making as few moves as possible.We present a method for solving the game, and show that thefollowing problem is NP-complete. PROBLEM 1 (Squares-Solvability). Given a pattern X and kN, canX be solved in k or less moves? We demonstrate a reduction to this problem from Not-All-Equal-3SAT.We also present another NP-complete problem that Squares-Solvabilitycan be reduced to. 1991 Mathematics Subject Classification 68Q25.     

On the proportion of permutations of order a multiple of the degree

We study permutations of a set of size n for which the orderis a multiple of n. We prove that, for large n, most such elementslie in one of two families. The first family consists of thosepermutations with a single very large cycle of order dividingn and includes the n-cycles, and the second consists of permutationsfor which the cycles of length dividing n have total lengthsignificantly less than n. This work was inspired by the algorithmicproblem of fast recognition of large symmetric groups actingprimitively on subsets.     

On the mixed Cauchy problem with data on singular conics

We consider a problem of mixed Cauchy type for certain holomorphicpartial differential operators with the principal part Q_2p(D)essentially being the (complex) Laplace operator to a power,^p. We provide inital data on a singular conic divisor givenby P = 0, where P is a homogeneous polynomial of degree 2p.We show that this problem is uniquely solvable if the polynomialP is elliptic, in a certain sense, with respect to the principalpart Q_2p(D).     

The spine of a Fourier-Stieltjes algebra

We define the spine A *(G) of the Fourier–Stieltjes algebraB (G) of a locally compact group G. This algebra encodes informationabout much of the fine structure of B (G), particularly informationabout certain homomorphisms and idempotents. We show that A *(G) is graded over a certain semi-lattice, thatof non-quotient locally precompact topologies on G. We computethe spine's spectrum G*, which admits a semi-group structure.We discuss homomorphisms from A *(G) to B (H) where H is anotherlocally compact group; and we show that A *(H) contains theimage of every completely bounded homomorphism from the Fourieralgebra A (H) of any amenable group G. We also show that A *(G)contains all of the idempotents in B (G). Finally, we computeexamples for vector groups, abelian lattices, minimally almostperiodic groups and the (ax + b)-group; and we explore the complexityof A *(G) for the discrete rational numbers and free groups.     

Discretization of the Time Variable in Evolution Equations

We prove convergence of a discretization method of evolutionequations in Banach spaces, based on substitution for the strongderivative with respect to t of the corresponding incrementalratio. We examine both the homogeneous and the non-homogeneous equationsand consider in detail the particular case in which the evolutionoperator A(t) has the form B + K(t) with B independent of tand K(t) B(X).     

A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives ofarbitrary order, posed on the domain {t > 0, 0 < x <L}. We show that the solution can be expressed as an integralin the complex k-plane. This integral is defined in terms ofan x-transform of the initial condition and a t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the global relation, which is an algebraicrelation defined in the complex k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite intervalcannot be expressed in terms of an infinite series.     

Optimal Regularity and Fredholm Properties of Abstract Parabolic Operators in Lp Spaces on the Real Line

We study the operator Lu(t):= u'(t) – A(t) u(t) on L^p(R; X) for sectorial operators A(t), with t R, on a Banachspace X that are asymptotically hyperbolic, satisfy the Acquistapace–Terreniconditions, and have the property of maximal L^p-regularity.We establish optimal regularity on the time interval R showingthat L is closed on its minimal domain. We further give conditionsfor ensuring that L is a semi-Fredholm operator. The Fredholmproperty is shown to persist under A(t)-bounded perturbations,provided they are compact or have small A(t)-bounds. We applyour results to parabolic systems and to generalized Ornstein–Uhlenbeckoperators. 2000 Mathematics Subject Classification 35K20, 35K90,47A53.     

Zero-Sets of Quaternionic and Octonionic Analytic Functions with Central Coefficients

We prove that the zero set of any quaternionic (or octonionic)analytic function f with central (that is, real) coefficientsis the disjoint union of codimension two spheres in R⁴ or R⁸(respectively) and certain purely real points. In particular,for polynomials with real coefficients, the complete root-setis geometrically characterisable from the lay-out of the rootsin the complex plane. The root-set becomes the union of a finitenumber of codimension 2 Euclidean spheres together with a finitenumber of real points. We also find the preimages f^–1for any quaternion (or octonion) A. We demonstrate that this surprising phenomenon of complete spheresbeing part of the solution set is very markedly a special ‘real’phenomenon. For example, the quaternionic or octonionic Nthroots of any non-real quaternion (respectively octonion) turnout to be precisely N distinct points. All this allows us todo some interesting topology for self-maps of spheres.     

A Fast Method for the Solution of Fredholm Integral Equations

Methods described to date for the solution of linear Fredholmintegral equations have a computing time requirement of O(N³),where N is the number of expansion functions or discretizationpoints used. We describe here a Tchebychev expansion method,based on the FFT, which reduces this time to O(N² ln N), andreport some comparative timings obtained with it. We give alsoboth a priori and a posteriori error estimates which are cheapto compute, and which appear more reliable than those used previously.     

Small Deviations of Riemann-Liouville Processes In lq Spaces With Respect To Fractal Measures

We investigate Riemann–Liouville processes R_H, with H> 0, and fractional Brownian motions B_H, for 0 < H <1, and study their small deviation properties in the spacesL_q([0, 1], µ). Of special interest here are thin (fractal)measures µ, that is, those that are singular with respectto the Lebesgue measure. We describe the behavior of small deviationprobabilities by numerical quantities of µ, called mixedentropy numbers, characterizing size and regularity of the underlyingmeasure. For the particularly interesting case of self-similarmeasures, the asymptotic behavior of the mixed entropy is evaluatedexplicitly. We also provide two-sided estimates for this quantityin the case of random measures generated by subordinators. While the upper asymptotic bound for the small deviation probabilityis proved by purely probabilistic methods, the lower bound isverified by analytic tools concerning entropy and Kolmogorovnumbers of Riemann–Liouville operators. 2000 MathematicsSubject Classification 60G15 (primary), 47B06, 47G10, 28A80(secondary).     

Geometrical Spines of Lens Manifolds

We introduce the concept of ‘geometrical spine’for 3-manifolds with natural metrics, in particular, for lensmanifolds. We show that any spine of L_p,q that is close enoughto its geometrical spine contains at least E(p,q) – 3vertices, which is exactly the conjectured value for the complexityc(L_p,q). As a byproduct, we find the minimal rotation distance(in the Sleator–Tarjan–Thurston sense) between atriangulation of a regular p-gon and its image under rotation.     

Pseudo-Abelian integrals along Darboux cycles

We study polynomial perturbations of integrable, non-Hamiltoniansystem with first integral of Darboux-type with positive exponents.We assume that the unperturbed system admits a period annulus.The linear part of the Poincaré return map is given bypseudo-Abelian integrals. In this paper we investigate analyticproperties of these integrals. We prove that iterated variationsof these integrals vanish identically. Using this relation weprove that the number of zeros of these integrals is locallyuniformly bounded under generic hypothesis. This is a genericanalog of the Varchenko-Khovanskii theorem for pseudo-Abelianintegrals. Finally, under some arithmetic properties of exponents,the pseudo-Abelian integrals are a sum over exponents a_j ofpolynomials in log h with meromorphic functions of h^1/a_j ascoefficients.     

K0 and the dimension filtration for p-torsion Iwasawa modules

Let G be a compact p-adic analytic group. We study K-theoreticquestions related to the representation theory of the completedgroup algebra kG of G with coefficients in a finite field kof characteristic p. We show that if M is a finitely generatedkG-module with canonical dimension smaller than the dimensionof the centralizer, as a p-adic analytic group, of any p-regularelement of G, then the Euler characteristic of M is trivial.Writing _i for the abelian category consisting of all finitelygenerated kG-modules of dimension at most i, we provide an upperbound for the rank of the natural map from the Grothendieckgroup of _i to that of _d, where d denotes the dimension of G.We show that this upper bound is attained in some special cases,but is not attained in general.     

An explicit solution for an instantaneous two-phase Stefan problem with nonlinear thermal coefficients

We consider a nonlinear heat conduction problem for a semi-infinitematerial x > 0, with phase-change temperature T₁, an initialtemperature T₂ (> T₁) and a heat flux of the type q (t) =q₀/t imposed on the fixed face x = 0. We assume that the volumetricheat capacity and the thermal conductivity are particular nonlinearfunctions of the temperature in both solid and liquid phases. We determine necessary and/or sufficient conditions on the parametersof the problem in order to obtain the existence of an explicitsolution for an instantaneous nonlinear twophase Stefan problem(solidification process).     

Use of extrapolation for improving the order of convergence of eigenelement approximations

We consider the approximation of the eigenelements of a compactintegral operator defined on C[0, 1] with a smooth kernel. Weuse the iterated collocation method based on r Gauss pointsand piecewise polynomials of degree r – 1 on each subintervalof a nonuniform partition of [0, 1]. We obtain asymptotic expansionsfor the arithmetic means of m eigenvalues and also for the associatedspectral projections. Using Richardson extrapolation, we showthat the order of convergence O(h^2r) in the iterated collocationmethod can be improved to O(h^2r+2). Similar results hold forthe Nyström method and for the iterated Galerkin method.We illustrate the improvement in the order of convergence bynumerical experiments.     

Bounded Solutions of Some Second Order Nonlinear Differential Equations

We prove Landesman–Lazer type existence conditions forthe solutions bounded on the real line, together with theirfirst derivatives, for some second order nonlinear differentialequations of the form x'+cx'+f(t, x)=0. The proofs use upperand lower solutions techniques together with some limiting arguments.     

Optimal error estimates for the hp-version interior penalty discontinuous Galerkin finite element method

We consider the hp-version interior penalty discontinuous Galerkinfinite-element method (hp-DGFEM) for second-order linear reaction–diffusionequations. To the best of our knowledge, the sharpest knownerror bounds for the hp-DGFEM are due to Rivière et al.(1999,Comput. Geosci., 3, 337–360) and Houston et al.(2002,SIAM J. Numer. Anal., 99, 2133–2163). These are optimalwith respect to the meshsize h but suboptimal with respect tothe polynomial degree p by half an order of p. We present improvederror bounds in the energy norm, by introducing a new functionspace framework. More specifically, assuming that the solutionsbelong element-wise to an augmented Sobolev space, we deducefully hp-optimal error bounds.