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.     

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 .     

Small Sets which meet all the n-Term Arithmetic Progressions in the Interval [1, n2]

For any positive integers n and k, let f(n, k) denote the smallestsize of a subset of the integer interval I =[l, n] which meetsall the k-term arithmetic progressions contained in I. We showthat n+(1/2)n^1/2–2 < f(n²,n) , where p is the largest prime n, and for any real number x,[x] is the least integer x.     

Shapiro's Cyclic Sum

Shapiro's cyclic sum is defined by , If K is the cone in Rⁿ of points withnon-negative coordinates, it is shown that the minimum of Ein K is a fixed point of T², where T is the non-linear operatordefined by (Tx)_i = x_n–i+1/(x_n–i+2 + x_n–i+3)²for i = 1,2,...,n. It is conjectured that Tx = S^kx, where Sis the shift operator in Rⁿ, and a proof is given under someadditional hypotheses. One of the consequences is a simple proofthat at the minimum point, a_i(x) = a_n–i+1–k(x) fori = 1,2,...,n.     

Geometric Approximation Of The Fibre Of The Freudenthal Suspension

Let Rat_k(CPⁿ) denote the space of based holomorphic maps ofdegree k from the Riemannian sphere S² to the complex projectivespace CPⁿ. The basepoint condition we assume is that f()=[1,..., 1]. Such holomorphic maps are given by rational functions: Rat_k(CPⁿ) ={(p₀(z), ..., p_n(z)):each p_i(z) is a monic, degree-kpolynomial and such that there are no roots common to all p_i(z)}.(1.1) The study of the topology of Rat_k(CPⁿ) originated in [10]. Later,the stable homotopy type of Rat_k(CPⁿ) was described in [3] interms of configuration spaces and Artin's braid groups. LetW(S²ⁿ) denote the homotopy theoretic fibre of the Freudenthalsuspension E:S²ⁿ S²ⁿ⁺¹. Then we have the following sequenceof fibrations: ²S²ⁿ⁺¹ W(S²ⁿ)S²ⁿ S²ⁿ⁺¹. A theorem in [10] tellsus that the inclusion Rat_k(CPⁿ) ²_kCP^{n 2}S²ⁿ⁺¹ is a homotopy equivalenceup to dimension k(2n–1). Thus if we form the direct limitRat(CPⁿ)= lim_k Rat_k(CPⁿ), we have, in particular, that Rat(CPⁿ)is homotopy equivalent to ²S²ⁿ⁺¹. If we take the results of [3] and [10] into account, we naturallyencounter the following problem: how to construct spaces X_k(CPⁿ),which are natural generalizations of Rat_k(CPⁿ), so that X(CPⁿ)approximates W(S²ⁿ). Moreover, we study the stable homotopytype of X_k(CPⁿ). The purpose of this paper is to give an answer to this problem.The results are stated after the following definition. 1991Mathematics Subject Classification 55P35.     

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.     

A p-Adic Quasi-Quadratic Time Point Counting Algorithm

In this article, we give an algorithm for the computation ofthe number of rational points on the Jacobian variety of a genericordinary hyperelliptic curve defined over a finite field of cardinality q with time complexity O(n^2+o(1))and space complexity O(n²), where n = log(q). In the lattercomplexity estimate the genus and the characteristic are assumedas fixed. Our algorithm forms a generalization of both, theAGM algorithm of J.-F. Mestre and the canonical lifting methodof T. Satoh. We canonically lift a certain arithmetic invariantof the Jacobian of the hyperelliptic curve in terms of thetaconstants. The theta null values are computed with respect toa semi-canonical theta structure of level 2p where >0 isan integer and . Theresults of this paper suggest a global positive answer to thequestion whether there exists a quasi-quadratic time algorithmfor the computation of the number of rational points on a genericordinary abelian variety defined over a finite field.     

Economical Covers with Geometric Applications

A cover of a hypergraph is a collection of edges whose unioncontains all vertices. Let H = (V, E) be a k-uniform, D-regularhypergraph on n vertices, in which no two vertices are containedin more than o(D/e^2k log D) edges as D tends to infinity. Ourresults include the fact that if k = o(log D), then there isa cover of (1 + o(1))n/k edges, extending the known result thatthis holds for fixed k. On the other hand, if k 4 log D thenthere are k-uniform, D-regular hypergraphs on n vertices inwhich no two vertices are contained in more than one edge, andyet the smallest cover has at least (nk) log (k log D)) edges.Several extensions and variants are also obtained, as well asthe following geometric application. The minimum number of linesrequired to separate n random points in the unit square is,almost surely, (n^2/3 / (log n)^1/3). 2000 Mathematical SubjectClassification: 05C65, 05D15, 60D05.     

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.     

ON THE INTEGRAL GALOIS MODULE STRUCTURE OF CYCLIC EXTENSIONS OF p-ADIC FIELDS

We strengthen results of Miyata on the integral Galois modulestructure of totally ramified cyclic Kummer extensions K ofdegree pⁿ of a p-adic field k. Let c₁(K/k) be the first ramificationnumber of K/k, and let c(K/k) be the least non-negative residueof c₁(K/k) modulo pⁿ. Suppose that K is of the form k() with^pn k and val _K(–1)>0, (val _K(–1), p)= 1. Thenthe valuation ring of K is free over its associated order ifc(K/k) divides p^m–1 for some m with 1mn; the converseholds if n= 2; and is a Hopf order (or a Gorenstein order)if and only if c(K/k) = pⁿ–1.     

Classification Des Ideaux a Droite De A1(k)

Les études récentes sur les idéaux àdroite de A₁(k), la première algèbre de Weyl surun corps algébriquement clos et de caractéristiquenulle k, nous montrent que : pour tout idéal I 0 àdroite de A₁(k), il existe x Q = frac(A₁(k)), et V V telsque : I = xD(R, V) o V est l'ensemble des sous-espaces primairementdécomposables de k[t] = R, et D(R, V), l'idéalà droite {d A₁(k/d(R V}. Dans cet article nous montreronsprincipalement que: pour tout 0 I idéal à droitede A₁(k, !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),où X_n est la courbe d'algèbre des fonctions régulières: O(X_n = k+tⁿ⁺¹k[t]. La forme des idéaux décriteci-dessus permet de voir dans une hypothèse de Letzteret Makar-Limanov, pour deux courbes algébriques affinesX et X' on a : D(XD(X') co dim D(X = co dim D(X'). Recent studies on right ideals of the first Weyl algebra A₁(k)over an algebraic closed field k with characteristic zero showthat: for each right ideal I 0 of A₁(k), there exist x Q =fracA₁(k)) and a primary decomposable sub-space V of k[t] suchthat I=xD(R,V), where D(R,V) : = {d A₁(k)/d(R) V} is a rightideal of A₁(k). In this paper, we show that for all right idealsI 0 of A₁(k), !n N, (x, ) Q^* x Aut_k(A₁(k)) : I = x(D(R, O(X_n))),where X_n denotes the affine algebraic curve with ring of regularfunctions O(X_n=k+tⁿ⁺¹k[t]. With ideals as described above, onecan easily see, under a hypothesis given by Letzter and Makar-Limanov,that for two affine algebraic curves X and X', D(X)D(X') codim D(X) = co dim D(X'). 2000 Mathematics Subject Classification16S32.     

On the Number of Singularities in Generic Deformations of Map Germs

Let f:Cⁿ, 0C^p, 0 be a K-finite map germ, and let i=(i₁, ...,i_k) be a Boardman symbol such that ⁱ has codimension n in thecorresponding jet space J^k(n, p). When its iterated successorshave codimension larger than n, the paper gives a list of situationsin which the number of ⁱ points that appear in a generic deformationof f can be computed algebraically by means of Jacobian idealsof f. This list can be summarised in the following way: f musthave rank n–i₁ and, in addition, in the case p=6, f mustbe a singularity of type ^i₁,i₂.     

BENFORD'S LAW FOR THE 3x + 1 FUNCTION

Benford's law (to base B) for an infinite sequence {x_k : k 1} of positive quantities x_k is the assertion that {log_B x_k: k 1} is uniformly distributed (mod 1). The 3x + 1 functionT(n) is given by T(n) = (3n + 1)/2 if n is odd, and T(n) = n/2if n is even. This paper studies the initial iterates x_k = T^(k)(x₀)for 1 k N of the 3x + 1 function, where N is fixed. It showsthat for most initial values x₀, such sequences approximatelysatisfy Benford's law, in the sense that the discrepancy ofthe finite sequence {log_B x_k : 1 k N} is small.     

On the number of homotopy types of fibres of a definable map

In this paper we prove a single exponential upper bound on thenumber of possible homotopy types of the fibres of a Pfaffianmap in terms of the format of its graph. In particular, we showthat if a semi-algebraic set SR^m+n, where R is a real closedfield, is defined by a Boolean formula with s polynomials ofdegree less than d, and : R^m+nRⁿ is the projection on a subspace,then the number of different homotopy types of fibres of doesnot exceed s^2(m+1)n(2^m nd)^O(nm). As applications of our mainresults we prove single exponential bounds on the number ofhomotopy types of semi-algebraic sets defined by fewnomials,and by polynomials with bounded additive complexity. We alsoprove single exponential upper bounds on the radii of ballsguaranteeing local contractibility for semi-algebraic sets definedby polynomials with integer coefficients.     

Turning Euler's Factoring Method into a Factoring Algorithm

An algorithm is presented which, given a positive integer n,will either factor n or prove it to be prime. The algorithmtakes O(n^1/3+) steps.     

The Fast Numerical Solution of Very Large Elliptic Difference Schemes

Let L_kv_k = g_k be a system of difference equations discretizingan elliptic boundary value problem. Assume the system to be"very large", that means that the number of unknowns exceedsthe capacity of storage. We present a method for solving theproblem with much less storage requirement. For two-dimensionalproblems the size of the needed storage decreases from O(h^–2)to (or even O(h^–5/4)). The computational work increasesonly by a factor about six. The technique can be generalizedto nonlinear problems. The algorithm is also useful for computerswith a small number of parallel processors.     

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      

The Consistency of Holt's Conjectures on Cohomological Dimension of Locally Finite Groups

Let G be a locally finite group of cardinality _n where n isa natural number. Let (G) be the set of primes p for which Ghas an element of order p. In [5], Holt conjectures that ifk is a finite field with char k (G) then (1) G has cohomological dimension n+1 over k; (2) Hⁿ⁺¹(G, kG) has cardinality 2^_n; (3) Hⁱ(G, kG) = 0 for 0 i n.