Initial Motion of the Free Boundary for a Non-linear Diffusion Equation

The free boundary between v > 0 and v = 0 for the porousmedium equation v₁= |v|² +nv²v can remain stationary for somepositive waiting-time and then start moving. It is of interestto know the way in which any part of the boundary first moves.It is already known that if the waiting time is given by purelylocal considerations then the boundary speed is continuous,i.e. the initial speed is zero. For cases where the boundary moves before the time found bysimply considering v(x, 0) close to the boundary, a perturbationanalysis indicates that it starts moving with positive speed.Two-dimensional problems show the possible formation of verticesin the boundary. At these points the normal velocity jumps fromzero to some positive value.     

Conditions for the Solvability of Systems of Two and Three Additive Forms Over p-Adic Fields

This paper is concerned with non-trivial solvability in p-adicintegers of systems of two and three additive forms. Assumingthat the congruence equation ax^k + by^k + cz^k d (modp) has asolution with xyz 0(modp) we have proved that any system oftwo additive forms of odd degree k with at least 6k + 1 variables,and any system of three additive forms of odd degree k withat least 14k + 1 variables always has non-trivial p-adic solutions,provided p does not divide k. The assumption of the solubilityof the congruence equation above is guaranteed for example ifp > k⁴. In the particular case of degree k = 5 we have proved the followingresults. Any system of two additive forms with at least n variablesalways has non-trivial p-adic solutions provided n 31 and p> 101 or n 36 and p > 11. Furthermore any system of threeadditive forms with at least n variables always has non-trivialp-adic solutions provided n 61 and p > 101 or n 71 andp > 11. 2000 Mathematics Subject Classification 11D72, 11D79.     

OPTIMAL INTEGRABILITY CONDITION FOR THE LOG-SOBOLEV INEQUALITY

Let M denote a connected complete Riemannian manifold (possiblywith a convex boundary), the Riemannian distance function froma fixed point and V C² (M) such that dµ_V e^V d xis a probability measure. For any K 0, we prove that K/2 isthe infimum over all > 0 such that Ric_M – Hess_V –Kand imply the log-Sobolevinequality for the Dirichlet form µ_V(| f |²).     

Existence Theorems for the Linear, Space-inhomogeneous Transport Equation

This paper studies existence problems in L¹ for the linear,space-inhomogeneous Boltzmann equation with periodic or (perfectly)absorbing boundary conditions under realistic assumptions onthe cross-sections. By an iteration technique, solutions arefirst constructed to an integral equation variant of the transportequation in the case of bounded impact parameters and an L¹type of cross-sections. They are then used to study the existenceof solutions of a measure form of the transport equation inthe case of unbounded impact parameters. These solutions conservemass. Estimates of their higher moments are also given. In particularthe results hold for inverse kth-power forces with 3 < k 5.     

The Mesa Problem: Diffusion Patterns for ut = {nabla} {middle dot} (um{nabla}u) as m -> +{infty}

In this paper we consider the limit m+ of solutions of the porous-mediumequation u_t = · (u^mu) (xR^N), with N > 1. We conjecturethat, for initial data with a unique maximum, the evolutionis characterized by the onset of a ‘mesa’ region,in which the solution is nearly spatially independent, surroundedby a region in which u is nearly equal to its initial value.The transition between these regions occurs near a surface whichis identified with the free boundary in a certain Stefan problemwhich can be studied using variational inequalities. Moreover,singular-perturbation theory can be used to describe the structureof the transition region.     

Construction of p-1 Irreducible Modules with Fundamental Highest Weight for the Symplectic Group in Characteristic p

Let K be a field and let V be a vector space of dimension 2mover K. Let V denote the exterior algebra of V and ^kV its kthexterior power for 0k2m. Let f be a non-degenerate alternatingbilinear form defined on VxV. The symplectic group Sp_2m(K) isthe group of all isometries of f and it acts as a group of vectorspace automorphisms on ^kV. In the case that K is algebraicallyclosed and 1km, it is known that ^kV contains a composition factorcorresponding to the fundamental weight _k of a root system oftype C_m. We shall refer to the irreducible module for Sp_2m(K)given by this composition factor as a fundamental module.     

On Lp boundedness of wave operators for 4-dimensional Schrodinger operators with threshold singularities

Let H=–+V(x) be a Schrödinger operator on L²(R⁴),H₀=–. Assume that |V(x)|+| V(x)|C x ^– for some>8. Let be the wave operators. It is known that W_± extend to bounded operators in L^p(R⁴)for all 1p, if 0 is neither an eigenvalue nor a resonance ofH. We show that if 0 is an eigenvalue, but not a resonance ofH, then the W_± are still bounded in L^p(R⁴) for all psuch that 4/3<p<4.     

Convergence of a finite-element approximation of surfactant spreading on a thin film in the presence of van der Waals forces

We consider a fully practical finite-element approximationof the following system of nonlinear degenerate parabolic equations: (u)/(t) + . (u² [(v)]) - (1)/(3) .(u³ w)= 0, w = - c u - u^-+ a u^-3 , (v)/(t) + . (u v [(v)]) - v - .(u² v w) = 0. The above models a surfactant-driven thin-film flow in the presenceof both attractive, a>0, and repulsive, >0 with >3,van der Waals forces; where u is the height of the film, v isthe concentration of the insoluble surfactant monolayer and(v):=1-v is the typical surface tension. Here 0 and c>0 arethe inverses of the surface Peclet number and the modified capillarynumber. In addition to showing stability bounds for our approximation,we prove convergence, and hence existence of a solution to thisnonlinear degenerate parabolic system, (i) in one space dimensionwhen >0; and, moreover, (ii) in two space dimensions if inaddition 7. Furthermore, iterative schemes for solving the resultingnonlinear discrete system are discussed. Finally, some numericalexperiments are presented.     

On the Small Squaring and Commutativity

Let G be a group and let k > 2 be an integer, such that (k²– 3)(k – 1) < |G|/15 if G is finite. Supposethat the condition |A²| k(k + 1)/2 + (k – 3)/2 is satisfiedby every it-element subset A G. Then G is abelian. The proofuses the structure of quasi-invariant sets.     

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.     

Long-Time Behavior of Solutions of the Fast Diffusion Equations with Critical Absorption Terms

This paper is devoted to the long-time behavior of solutionsto the Cauchy problem of the porous medium equation u_t = (u^m)– u^p in Rⁿ x (0,) with (1 – 2/n)₊ < m < 1and the critical exponent p = m + 2/n. For the strictly positiveinitial data u(x,0) = O(1 + |x|)^–k with n + mn(2 –n + nm)/(2[2 – m + mn(1 – m)]) k < 2/(1 –m), we prove that the solution of the above Cauchy problem convergesto a fundamental solution of u_t = (u^m) with an additional logarithmicanomalous decay exponent in time as t .     

Markov and Bernstein type Inequalities in Lp for Classes of Polynomials with Constraints

The Markov-type inequality is proved for all real algebraic polynomials f of degree atmost n having at most k, with 0 k n, zeros (counting multiplicities)in the open unit disk of the complex plane, and for all p >0, where c(p) = c^{p + 1}(l + p^–2) with some absolute constantc > 0. This inequality has been conjectured since 1983 whenthe L case of the above result was proved. It improves and generalizesmany earlier results. Up to the multiplicative constant c(p)>0 the above inequality is sharp. A sharp Bernstein-type analoguefor real trigonometric polynomials is also established, whichis interesting on its own, and plays a key role in the proofof the Markov-type inequality.     

A Unicity Theorem for Entire Functions

Let k be a non-negative integer. Suppose that f and g are nonconstantentire functions and that a and b (b a(^k) are small functionsrelated to f and g such that (a,f) + (a, g) > 1. Iff^(k) –b and g^k – b assume the same zeros with the same multiplicities,then f g unless (f – a^(k))(g^(k) – a^(k)) = (b –a(K))2. The problem is related to C. C. Yang's question. A correspondingresult was proved for the case where a 0, b 1, k 1 and theorder of f and g is finite.     

Spurious solutions of numerical methods for initial value problems

It is well known that some numerical methods for initial valueproblems admit spurious limit sets. Here the existence and behaviourof spurious solutions of Runge-Kutta, linear multistep and predictor-correctormethods are studied in the limit as the step-size h0. In particular,it is shown that for ordinary differential equations definedby globally Lipschitz vector fields, spurious fixed points andperiod 2 solutions cannot exist for h arbitrarily small, whilstfor locally Lipschitz vector fields, spurious solutions mayexist for h arbitrarily small, but must become unbounded ash0. The existence of spurious solutions is also studied forvector fields merely assumed to be continuous, and an exampleis given, showing that in this case spurious solutions may remainbounded as h0. It is shown that if spurious fixed points orperiod 2 solutions of continuous problems exist for h arbitrarilysmall, then as h0 spurious solutions either converge to steadysolutions of the underlying differential equation or divergeto infinity. A necessary condition for the bifurcation spurioussolutions from h=0 is derived. To prove the above results forimplicit Runge-Kutta methods, an additional assumption on theiteration scheme used to solve the nonlinear equations definingthe method is needed; an example of a Runge-Kutta method whichgenerates a bounded spurious solution for a smooth problem withh arbitrarily small is given, showing that such an assumptionis necessary. It is also shown that an Adams-Bashforth/Adams-Moultonpredictor-corrector method in PC^m implementation can generatespurious fixed point solutions for any m.     

Symplectic embeddings of 4-dimensional ellipsoids

We show how to reduce the problem of symplectically embeddingone 4-dimensional rational ellipsoid into another to a problemof embedding disjoint unions of balls into P². For example,the problem of embedding the ellipsoid E(1, k) into a ball Bis equivalent to that of embedding k disjoint equal balls intoP², and so can be solved by the work of Gromov, McDuff–Polterovich,and Biran. (Here k is the ratio of the area of the major axisto that of the minor axis.) As a consequence we show that theball may be fully filled by the ellipsoid E(1, k) for k = 1,4 and all k 9, thus answering a question raised by Hofer. Received March 31, 2008.     

Une Conjecture de Lebesgue

Let A > 0 be an integer. The equation x⁵–y⁵ = Az⁵ wasfirst studied by Dirichlet and Lebesgue. Lebesgue conjecturedin 1843 that if A has no prime divisors of the form 10k+1, theequation has no solutions except the visible ones. Partial resultswere obtained by Lebesgue and by Terjanian in 1987. The purposeof the paper is to prove Lebesgue's conjecture. The main toolused is the method known as the elliptic Chabauty method.     

A multigrid-type method for thin plate spline interpolation on a circle

We consider the problem of thin plate spline interpolation ton equally spaced points on a circle, where the number of datapoints is sufficiently large for work of O(n³ to be unacceptable.We develop an iterative multigrid-type method, each iterationcomprising ngrid stages, and n being an integer multiple of2^ngrid–1. We let the first grid, V₁ be the full set ofdata points, V say, and each subsequent (coarser) grid, V_k,k=2, 3,...,ngrid, contain exactly half of the data points ofthe preceding (finer) grid, these data points being equallyspaced. At each stage of the iteration, we correct our current approximationto the thin plate spline interpolant by an estimate of the interpolantto the current residuals on V_k, where the correction is constructedfrom Lagrange functions of interpolation on small local subsetsof p data points in V_k. When the coarsest grid is reached, however,then the interpolation problem is solved exactly on its q=n/2^ngrid–1points. The iterative process continues until the maximum residualdoes not exceed a specified tolerance. Each iteration has the effect of premultiplying the vector ofresiduals by an n x n matrix R, and thus convergence will dependupon the spectral radius, (R), of this matrix. We investigatethe dependence of the spectral radius on the values of n, p,and q. In all the cases we have considered, we find (R) <<1, and thus rapid convergence is assured.     

The gradient iteration for approximation in reproducing kernel Hilbert spaces

Consider the bounded linear operator, L: F Z, where Z R^N andF are Hilbert spaces defined on a common field X. L is madeup of a series of N bounded linear evaluation functionals, L_i:F R. By the Riesz representation theorem, there exist functionsk(x_i, ^·) F : L_if = f, k(x_i, ^·)_F. The functions,k(x_i, ^·), are known as reproducing kernels and F is areproducing kernel Hilbert space (RKHS). This is a natural frameworkfor approximating functions given a discrete set of observations.In this paper the computational aspects of characterizing suchapproximations are described and a gradient method presentedfor iterative solution. Such iterative solutions are desirablewhen N is large and the matrix computations involved in thebasic solution become infeasible. This is also exactly the casewhere the problem becomes ill-conditioned. An iterative approachto Tikhonov regularization is therefore also introduced. Unlikeiterative solutions for the more general Hilbert space setting,the proofs presented make use of the spectral representationof the kernel.     

On Finite and InfiniteCk-A-Determinacy

Let f: (Rⁿ,0) (R^p,0) be a C map-germ. We define f to be finitely,or -, A-determined, if there exists an integer m such that allgerms g with j^mg(0) = j^mf(0), or if all germs g with the sameinfinite Taylor series as f, respectively, are A-equivalentto f. For any integer k, 0 k < , we can consider A' sC^kcounterpart (consisting of C^k diffeomorphisms) A^(k), and wecan define the notion of finite, or -,A^(k)-determinacy in asimilar manner. Consider the following conditions for a C germf: (a_k) f is -A^(k)-determined, (b_k) f is finitely A^(k)-determined,(t) , (g) there exists a representative f : U R^p defined on some neighbourhood U of 0 in Rⁿ such thatthe multigerm of f is stable at every finite set , and (g') every f' with j f'(0)=j f(0) satisfiescondition (g). We also define a technical condition which willimply condition (g) above. This condition is a collection ofp+1 Lojasiewicz inequalities which express that the multigermof f is stable at any finite set of points outside 0 and onlybecomes unstable at a finite rate when we approach 0. We willdenote this condition by (e). With this notation we prove thefollowing. For any C map germ f:(Rⁿ,0) (R^p,0) the conditions(e), (t), (g') and (a) are equivalent conditions. Moreover,each of these conditions is equivalent to any of (a_k) (p+1 k < , (b_k) (p+1 k < ). 1991 Mathematics Subject Classification:58C27.     

A Criterion for Finite Topological Determinacy of Map-Germs

Let f, g: (Rⁿ, 0) (R^p, 0) be two C map-germs. Then f and gare C⁰-equivalent if there exist homeomorphism-germs h and lof (Rⁿ, 0) and (R^p, 0) respectively such that g = l f h^–1.Let k be a positive integer. A germ f is k-C⁰-determined ifevery germ g with j^k g(0) = j^k f(0) is C⁰-equivalent to f. Moreover,we say that f is finitely topologically determined if f is k-C⁰-determinedfor some finite k. We prove a theorem giving a sufficient conditionfor a germ to be finitely topologically determined. We explainthis condition below. Let N and P be two C manifolds. Consider the jet bundle J^k(N,P) with fiber J^k(n, p). Let z in J^k(n, p) and let f be suchthat z = j^kf(0). Define Whether (f) < k depends only on z, not on f. We can thereforedefine the set Let W^k(N, P) be the subbundle of J^k(N, P) with fiber W^k(n, p).Mather has constructed a finite Whitney (b)-regular stratificationS^k(n, p) of J^k(n, p) – W^k(n, p) such that all strata aresemialgebraic and K-invariant, having the property that if S^k(N,P) denotes the corresponding stratification of J^k(N, P) –W^k(N, P) and f C(N, P) is a C map such that j^kf is multitransverseto S^k(N, P), j^kf(N) W^k(N, P) = and N is compact (or f is proper),then f is topologically stable. For a map-germ f: (Rⁿ, 0) (R^p, 0), we define a certain ojasiewiczinequality. The inequality implies that there exists a representativef: U R^p such that j^kf(U – 0) W^k (Rⁿ, R^p = and suchthat j^kf is multitransverse to S^k (Rⁿ, R^p) at any finite setof points S U – 0. Moreover, the inequality controlsthe rate j^kf becomes non-transverse as we approach 0. We showthat if f satisfies this inequality, then f is finitely topologicallydetermined. 1991 Mathematics Subject Classification: 58C27.