A Petrov-Galerkin method with quadrature for elliptic boundary value problems

We propose and analyse a fully discrete Petrov–Galerkinmethod with quadrature, for solving second-order, variable coefficient,elliptic boundary value problems on rectangular domains. Inour scheme, the trial space consists of C² splines of degreer 3, the test space consists of C⁰ splines of degree r –2, and we use composite (r – 1)-point Gauss quadrature.We show existence and uniqueness of the approximate solutionand establish optimal order error bounds in H², H¹ and L² norms.     

Triangle group representations and constructions of regular maps

A regular map of type {m,n} is a 2-cell embedding of a graphin an orientable surface, with the property that for any twodirected edges e and e' there exists an orientation-preservingautomorphism of the embedding that takes e onto e', and in whichthe face length and the vertex valence are m and n, respectively.Such maps are known to be in a one-to-one correspondence withtorsion-free normal subgroups of the triangle groups T(2,m,n).We first show that some of the known existence results aboutregular maps follow from residual finiteness of triangle groups.With the help of representations of triangle groups in speciallinear groups over algebraic extensions of Z we then constructivelydescribe homomorphisms from T(2,m,n)=y,z|y^m=zⁿ=(yz)²=1 intofinite groups of order at most c^r where c=c(m,n), such thatno non-identity word of length at most r in x,y is mapped ontothe identity. As an application, for any hyperbolic pair {m,n}and any r we construct a finite regular map of type {m,n} ofsize at most C^r, such that every non-contractible closed curveon the supporting surface of the map intersects the embeddedgraph in more than r points. We also show that this result isthe best possible up to determining C=C(m,n). For r>m thegraphs of the above regular maps are arc-transitive, of valencen, and of girth m; moreover, if each prime divisor of m is largerthan 2n then these graphs are non-Cayley. 2000 Mathematics SubjectClassification: 05C10, 05C25, 20F99, 20H25.     

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.     

Generating Examples of Shock Propagation by Three Principles

In a medium characterized by a scalar speed C(x), a shock arrivesat the point x, after time T(x), with its magnitude decreasedby A(x). Symmetric C, T, and A in two dimensions can be convertedto cylindrically symmetric results in three dimensions by applyinga dimension-increasing principle: "Let C(x, y), T(x, y), andA(x, y) be even functions of y. They can be extended into threedimensions by using the formulas C(x, y)C(x, r), T(x, y)T(x,r), and A(x,y)A(x,r) [r^–1 cos(x, r)]^?, where r = (x²+2²)^?and is an auxiliary function." When C(x) is a function of asingle variable, the auxiliary function is given by cos(x,y) = T_y(x, y). In two dimensions, there is a conformal mappingprinciple: "Under the conformal mapping x+iy = f(x^*+iy^*), thefunctions T(x, y) and A(x, y) go into functions associated witha medium having speed C^*,y^*) = C(Re[f), Im[f]/f¹(x^*+iy."Thereis also an unchanged wavefronts principle: "If g is a smoothfunction with g(0) = 0 and g'(0)>0 then T^*(x) = g(T(x) andA^*(x) = A(x)[g'(x)/g'^1/2 are associated with a medium havingspeed C^*(x) = C(x)/g'(T(x))." in two dimensions, alternatingthe application of the last two principles generates a sequenceof media with their associated T(x, y) and A(x, y). Some ofthese can be extended into three dimensions by applying thefirst principle.     

Systems of cubic forms

We discuss the existence of rational and p-adic zeros of systemsof cubic forms. In particular, we prove that for p2 any systemof r cubic forms over Q_p in more than 125r³+705r²+210r variablesadmits a non-trivial p-adic zero, and that any system of r rationalcubic forms in more than O(r⁴ m⁶+r⁶ m⁵) variables admits a rationallinear space of zeros of dimension at least m.     

Graphs with many r-cliques have large complete r-partite subgraphs

Let r2 and c>0. Every graph on n vertices with at least cn^rcliques on r vertices contains a complete r-partite subgraphwith r–1 parts of size c^rlog n and one part of size greaterthan n^1–cr–1. This result implies a quantitativeform of the Erdös–Stone theorem.     

UNIVERSAL SPACES OF TWO-CELL COMPLEXES AND THEIR EXPONENT BOUNDS

Let P²ⁿ⁺¹ be a two-cell complex which is formed by attachinga (2n + 1)-cell to a 2m-sphere by a suspension map. We constructa universal space U for P²ⁿ⁺¹ in the category of homotopy associative,homotopy commutative H-spaces. By universal, we mean that Uis homotopy associative, homotopy commutative and has the propertythat any map f: P²ⁿ⁺¹ Y to a homotopy associative, homotopycommutative H-space Y extends to a uniquely determined H-map: U Y. We then prove upper and lowerbounds of the H-homotopy exponent of U. In the case of a modp^r, Moore space U is the homotopy fibre S²ⁿ⁺¹{p^r} of the p^r-powermap on S²ⁿ⁺¹, and we reproduce Neisendorfer's result that S²ⁿ⁺¹{p^r}is homotopy associative, homotopy commutative and that the p^r-powermap on S²ⁿ⁺¹{p^r} is null homotopic.     

On Sharp Sobolev Embedding and The Logarithmic Sobolev Inequality

The purpose of this note is to give a short proof of the Grosslogarithmic Sobolev inequality using the asymptotics of thesharp L² Sobolev constant and the product structure of Euclideanspace. Let FL^r(Rⁿ) for some positive r with ||F||_r=1. 1991 MathematicsSubject Classification 58G35.     

Bifurcations of Periodic Points of Holomorphic Maps From C2 into C2

Let F:Cⁿ Cⁿ be a holomorphic map, F^k be the kth iterate ofF, and p Cⁿ be a periodic point of F of period k. That is,F^k(p) = p, but for any positive integer j with j < k, F^j(p) p. If p is hyperbolic, namely if DF^k(p) has no eigenvalue ofmodulus 1, then it is well known that the dynamical behaviourof F is stable near the periodic orbit = {p, F(p),..., F^k–1(p)}.But if is not hyperbolic, the dynamical behaviour of F near may be very complicated and unstable. In this case, a veryinteresting bifurcational phenomenon may occur even though may be the only periodic orbit in some neighbourhood of : forgiven M N\{1}, there may exist a C^r-arc {F_t: t [0,1]} (wherer N or r = ) in the space H(Cⁿ) of holomorphic maps from Cⁿinto Cⁿ, such that F₀ = F and, for t (0,1], F_t has an Mk-periodicorbit _t with as t 0. Theperiod thus increases by a factor M under a C^r-small perturbation!If such an F_t does exist, then , as well as p, is said to beM-tupling bifurcational. This definition is independent of r. For the above F, there may exist a C^r-arc in H(Cⁿ), with t [0,1], such that and, for t (0,1], has two distinct k-periodic orbits _t,1 and _t,2 with d(_t,i, ) 0 as t 0 for i = 1,2. If such an does exist, then , as well as p, is said to be 1-tupling bifurcational. In recent decades, there have been many papers and remarkableresults which deal with period doubling bifurcations of periodicorbits of parametrized maps. L. Block and D. Hart pointed outthat period M-tupling bifurcations cannot occur for M >2 in the 1-dimensional case. There are examples showing thatfor any M N, period M-tupling bifurcations can occur in higher-dimensionalcases. An M-tupling bifurcational periodic orbit as defined here actsas a critical orbit which leads to period M-tupling bifurcationsin some parametrized maps. The main result of this paper isthe following. Theorem. Let k N and M N, and let F: C² C² be a holomorphicmap with k-periodic point p. Then p is M-tupling bifurcationalif and only if DF^k(p) has a non-zero periodic point of periodM. 1991 Mathematics Subject Classification: 32H50, 58F14.     

The Decomposition of Lie Powers

Let G be a group, F a field of prime characteristic p and Va finite-dimensional FG-module. Let L(V) denote the free Liealgebra on V regarded as an FG-submodule of the free associativealgebra (or tensor algebra) T(V). For each positive integerr, let L^r (V) and T^r (V) be the rth homogeneous components ofL(V) and T(V), respectively. Here L^r (V) is called the rth Liepower of V. Our main result is that there are submodules B₁,B₂, ... of L(V) such that, for all r, B_r is a direct summandof T^r(V) and, whenever m 0 and k is not divisible by p, themodule is the direct sum of , . Thus every Lie power is a direct sum of Lie powers of p-powerdegree. The approach builds on an analysis of T^r (V) as a bimodulefor G and the Solomon descent algebra. 2000 Mathematics SubjectClassification 17B01 (primary), 20C07, 20C20 (secondary).     

On Closed Sets with Convex Projections

A shadow of a subset A of Rⁿ is the image of A under a projectiononto a hyperplane. Let C be a closed nonconvex set in Rⁿ suchthat the closures of all its shadows are convex. If, moreover,there are n independent directions such that the closures ofthe shadows of C in those directions are proper subsets of therespective hyperplanes then it is shown that C contains a copyof R^n–2. Also for every closed convex set B ‘minimalimitations’ C of B are constructed, that is, closed subsetsC of B that have the same shadows as B and that are minimalwith respect to dimension.     

Higher string topology on general spaces

In this paper, I give a generalized analogue of the string topologyresults of Chas and Sullivan, and of Cohen and Jones. For afinite simplicial complex X and k 1, I construct a spectrumMaps(S^k, X)^S(X), which is obtained by taking a generalizationof the Spivak bundle on X (which however is not a stable spherebundle unless X is a Poincaré space), pulling back toMaps(S^k, X) and quotienting out the section at infinity. I showthat the corresponding chain complex is naturally homotopy equivalentto an algebra over the (k + 1)-dimensional unframed little diskoperad C_{k + 1}. I also prove a conjecture of Kontsevich, whichstates that the Quillen cohomology of a based C_k-algebra (inthe category of chain complexes) is equivalent to a shift ofits Hochschild cohomology, as well as prove that the operadC_*C_k is Koszul-dual to itself up to a shift in the derived category.This gives one a natural notion of (derived) Koszul dual C_*C_k-algebras.I show that the cochain complex of X and the chain complex of^k X are Koszul dual to each other as C_*C_k-algebras, and thatthe chain complex of Maps(S^k, X)^S(X) is naturally equivalentto their (equivalent) Hochschild cohomology in the categoryof C_* C_k-algebras. 2000 Mathematics Subject Classification 55P48(primary), 16E40, 55N45, 18D50 (secondary).     

Multistep approximation of linear sectorial evolution equations

This paper concerns the linear multistep approximation of alinear sectorial evolution equation u_t = Au on a complex Banachspace X. Given a strictly A()-stable q-step method of orderp whose stability region includes a sectorial region containingthe spectrum of the operator A, the corresponding evolutionsemigroup for the method is Cⁿ(hA), n 0, defined on X^q, whereC(z) L (C^q) denotes the one-step map associated with the method.It is shown that for appropriately chosen V, Y: C C^q, basedon the principal right and left eigenvectors of C(z), Cⁿ(hA)approximates the semigroup V(hA)e^nhAY^H(hA) with optimal orderp.     

Smooth Lipschitz Retractions of Starlike Bodies onto their Boundaries in Infinite-Dimensional Banach Spaces

Let X be an infinite-dimensional Banach space, and let A bea C^p Lipschitz bounded starlike body (for instance the unitball of a smooth norm). We prove that:

1. the boundary A is C^p Lipschitzcontractible;
2. there is a C^p Lipschitz retraction from A ontoA;
3. there is a C^p Lipschitz map T: A A with no approximatefixedpoints.

     

Ruled Special Lagrangian 3-Folds In C3

This is the fifth in a series of papers constructing explicitexamples of special Lagrangian submanifolds in C^m. A submanifoldof C^m is ruled if it is fibred by a family of real straightlines in C^m. This paper studies ruled special Lagrangian 3-foldsin C³, giving both general theory and families of examples.Our results are related to previous work of Harvey and Lawson,Borisenko, and Bryant. Special Lagrangian cones in C³ are automaticallyruled, and each ruled special Lagrangian 3-fold is asymptoticto a unique special Lagrangian cone. We study the family ofruled special Lagrangian 3-folds N asymptotic to a fixed specialLagrangian cone N0. We find that this depends on solving a linearequation, so that the family of such N has the structure ofa vector space. We also show that the intersection of N0 withthe unit sphere S⁵ in C³ is a Riemann surface, and constructa ruled special Lagrangian 3-fold N asymptotic to N0 for eachholomorphic vector field w on . As corollaries of this we writedown two large families of explicit special Lagrangian 3-foldsin C³ depending on a holomorphic function on C, which includemany new examples of singularities of special Lagrangian 3-folds.We also show that each special Lagrangian T2-cone N0 can beextended to a 2-parameter family of ruled special Lagrangian3-folds asymptotic to N0, and diffeomorphic to T²xR. 2000 Mathematical Subject Classification: 53C38, 53D12.     

Jacobi Fields Along Harmonic 2-Spheres in CP2 are Integrable

The paper shows that any Jacobi field along a harmonic map fromthe 2-sphere to the complex projective plane is integrable (thatis, is tangent to a smooth variation through harmonic maps).This provides one of the few known answers to the problem ofintegrability, which was raised in different contexts of geometryand analysis. It implies that the Jacobi fields form the tangentbundle to each component of the manifold of harmonic maps fromS² to CP² thus giving the nullity of any such harmonic map;it also has a bearing on the behaviour of weakly harmonic E-minimizingmaps from a 3-manifold to CP² near a singularity and the structureof the singular set of such maps from any manifold to CP².     

Hypersurfaces in a Unit Sphere Sn+1(1) with Constant Scalar Curvature

The paper considers n-dimensional hypersurfaces with constantscalar curvature of a unit sphere S^n–1(1). The hypersurfaceS^k(c₁)xS^n–k(c₂) in a unit sphere Sⁿ⁺¹(1) is characterized,and it is shown that there exist many compact hypersurfaceswith constant scalar curvature in a unit sphere Sⁿ⁺¹(1) whichare not congruent to each other in it. In particular, it isproved that if M is an n-dimensional (n > 3) complete locallyconformally flat hypersurface with constant scalar curvaturen(n–1)r in a unit sphere Sⁿ⁺¹(1), then r > 1–2/n,and (1) when r (n–2)/(n–1), if then M is isometric to S¹xS^n–1(c),where S is the squared norm of the second fundamental form ofM; (2) there are no complete hypersurfaces in Sⁿ⁺¹(1) with constantscalar curvature n(n–1)r and with two distinct principalcurvatures, one of which is simple, such that r = (n–2)/(n–1)and      

Isometries of the Space of Convex Bodies in Euclidean Space

The isometries of the space of convex bodies of E^d with respectto the Hausdorff-metric are precisely the mappings of the formC i(C) + D where i is a rigid motion of E^d and D a fixed convexbody.     

A Functional Relation for Accessory Parameters for Genus 2 Algebraic Curves with an Order 4 Automorphism

Let C be a genus 2 algebraic curve defined by an equation ofthe form y² = x(x² – 1)(x – a)(x – 1/a). Asis well known, the five accessory parameters for such an equationcan all be expressed in terms of a and the accessory parameter b corresponding to a. The main result of the paper is thatif a' = 1 – a², which in general yields a non-isomorphiccurve C', then b'a'(a'² – 1) = – – ba(a²– 1). This is proven by it being shown how the uniformizing functionfrom the unit disk to C' can be explicitly described in termsof the uniformizing function for C.     

Lie Powers of Modules for Groups of Prime Order

Let L(V) be the free Lie algebra on a finite-dimensional vectorspace V over a field K, with homogeneous components Lⁿ(V) forn 1. If G is a group and V is a KG-module, the action of Gextends naturally to L(V), and the Lⁿ(V) become finite-dimensionalKG-modules, called the Lie powers of V. In the decompositionproblem, the aim is to identify the isomorphism types of indecomposableKG-modules, with their multiplicities, in unrefinable directdecompositions of the Lie powers. This paper is concerned withthe case where G has prime order p, and K has characteristicp. As is well known, there are p indecomposables, denoted hereby J₁,...,J_p, where J_r has dimension r. A theory is developedwhich provides information about the overall module structureof LV) and gives a recursive method for finding the multiplicitiesof J₁,...,J_p in the Lie powers Lⁿ(V). For example, the theoryyields decompositions of L(V) as a direct sum of modules isomorphiceither to J₁ or to an infinite sum of the form J_r J_{p-1} J_{p-1} ... with r 2. Closed formulae are obtained for the multiplicitiesof J₁,..., J_p in Lⁿ(J_p and Lⁿ(J_{p-1). For r < p-1, the indecomposableswhich occur with non-zero multiplicity in Lⁿ(J_r) are identifiedfor all sufficiently large n. 2000 Mathematical Subject Classification:17B01, 20C20.