On Artin's Conjecture, I: Systems of Diagonal Forms

As a special case of a well-known conjecture of Artin, it isexpected that a system of R additive forms of degree k, say [formula] with integer coefficients a_ij, has a non-trivial solution inQ_p for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non-trivialif not all the x_i are 0. To date, this has been verified onlywhen R=1, by Davenport and Lewis [4], and for odd k when R=2,by Davenport and Lewis [7]. For larger values of R, and in particularwhen k is even, more severe conditions on N are required toassure the existence of p-adic solutions of (1) for all primesp. In another important contribution, Davenport and Lewis [6]showed that the conditions [formula] are sufficient. There have been a number of refinements of theseresults. Schmidt [13] obtained N>>R²k³ log k, and Low,Pitman and Wolff [10] improved the work of Davenport and Lewisby showing the weaker constraints [formula] to be sufficient for p-adic solubility of (1). A noticeable feature of these results is that for even k, onealways encounters a factor k³ log k, in spite of the expectedk² in (2). In this paper we show that one can reach the expectedorder of magnitude k². 1991 Mathematics Subject Classification11D72, 11D79.     

Supplementary Note on 'Rational Invariants of Certain Orthogonal and Unitary Groups'

Given a field k and a finite group G acting on the rationalfunction field k(X₁, ..., X_n) as a group of k-automorphisms,an important Noether's problem asks whether the invariant subfield [forumal] is purely transcendental over k. 1991 Mathematics Subject Classification12F20, 20G40.     

The Strength of a Fabric

Consider fabrics in which every warp (and weft) strand passesover and under at least k weft (warp) strands. There is a 4kby 4k fabric of this kind that does not hang together. We provethat any n by n fabric of this kind, where n < 4k, does hangtogether. Moreover, we introduce a measure of how well a fabrichangs together, by defining the strength of a fabric, and twotheorems are proved that extend the preceding results.     

Stability in the Sense of Bounded Average Power

An R^m-valued sequence (x_k): = (x_k : k = 1, 2, ...), e.g. generatedrecursively by x_k = f_k (x_k–_k, U_k), is called ‘averagepth power bounded’ if (1/K) is bounded uniformly in K= 1, 2,.... (The case p = 2 may correspond to ‘power’in the physical sense.) This is a notion of stability. Givenestimates of the form: f_k (x, u) < a x + ¶ _k conditionsare obtained on the coefficient sequence (a_k) and the inputestimates e_k:=¶_k (u_k) which ensure this form of stabilityfor the output (x_k). In particular, a condition (utilized inan application to adaptive control) is obtained which imposes(i) a bound b on (a_k) and a ‘sparsity measure’ m(K) on #{kK: a_k>} as K ( >1) (ii) average pth power boundednesson (e_k), and (iii) a growth condition on (e_k) related to b andm (•). This condition is sharp.     

On Automorphisms of Polynomial Rings

Implicit in the work of Lane [1] is the result that if k isan algebraically closed field then every k-automorphism leavesa non-zero proper ideal of k[x, y] invariant. The purpose ofthis note is to show that this result does not hold for k[x,y, z].     

On a Question of Herman, Baker and Rippon Concerning Siegel Disks

Consider the family of exponential maps E_k(z) = exp(z)+k. Thispaper shows that any unbounded Siegel disk U of E_k containsthe singular value k on its boundary. By a result of Herman,this implies that k U if the rotation number is diophantine.2000 Mathematics Subject Classification 37F10, 30D05.     

Determination of a Convex Body from Minkowski Sums of its Projections

For a convex body K in R^d and 1 K d – 1, let P_K (K)be the Minkowski sum (average) of all orthogonal projectionsof K onto k-dimensional subspaces of R^d. It is Known that theoperator P_k is injective if kd/2, k=3 for all d, and if k =2, d 14. It is shown that P_2k (K) determines a convex body K among allcentrally symmetric convex bodies and P_2k+1(K) determines aconvex body K among all bodies of constant width. Correspondingstability results are also given. Furthermore, it is shown thatany convex body K is determined by the two sets P_k (K) and P_k'(K) if 1 < k < k'. Concerning the range of P_k , 1 k d–2, it is shown that its closure (in the Hausdorff-metric)does not contain any polytopes other than singletons.     

Conversion of Polynomials between Different Polynomial Bases

The purpose of this paper is to derive a recursive scheme forthe evaluation of the coefficients in the expansion , in terms of the coefficients in the expansion , where both q_k(x) and Q_k(x) are polynomials in xof degree k, and where both q_k(x) and Q_k{x} satisfy recursionformulae of the type satisfied by orthogonal polynomials. Thesets {Q_k(x)} and {q_k(x)} need not be orthogonal polynomials,though they usually are in the applications. An applicationis made to the evaluation of integrals with oscillatory andsingular integrands.     

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).     

Least Squares Smoothing of Univariate Data to achieve Piecewise Monotonicity

If measurements of a univariate function include uncorrelatederrors, then it is usual for the first-order divided differencesof the measurements to show far more sign changes than the correspondingdifferences of the underlying function. Therefore we addressthe problem of making the least sum of squares change to thedata so that the piecewise linear interpolant to the smootheddata is composed of at most k monotonic sections, k being aprescribed positive integer. The positions of the joins of thesections are integer variables whose optimal values are determinedautomatically, which is a combinatorial problem that can haveO(n^k) local minima, where n is the number of data. Fortunatelywe find that a dynamic programming procedure can calculate theglobal minimum of the sum of squares in at most O(n² + kn logn) computer operations. Further, the complexity reduces to onlyO(n) when k = 1 or k = 2, this result being well known in themonotonic case (k = 1). Algorithms that achieve these efficienciesare described. They perform well in practice, but a discussionof complexity suggests that there is still room for improvementwhen k 3.     

Repulsive Fixpoints of Analytic Functions with Applications to Complex Dynamics

Let G be a family of functions analytic in a domain D in thecomplex plane. It is proved that G is a normal family, providedthat for each fG, there exists k = k(f) > 1 such that thekth iterate f^k has no repulsive fixpoint in D. A new proof ofa result of Bergweiler and Terglane concerning the dynamicsof entire functions is also given.     

The BP<N> Cohomology of Elementary Abelian Groups

In this paper we define three elements of a certain generalisedcohomology ring BPm, n* BV_k. Here m, n and k are non-negativeintegers with k+mn+1, there is a fixed prime p not exhibitedin the notation, and V_k is an elementary Abelian p-group ofrank k. We show that these elements are equal; this is striking,because the three definitions are very different. The significanceof our equation is not yet entirely clear, but it makes contactwith other work in the literature in a number of fascinatingways.     

On the anticipation phenomenon for implicit linear discrete-time systems on finite intervals

Implicit linear discrete-time systems are systems describedby the difference equation Ex_k+1 = Fx_k + Gu_k. It is well knownthat, for such systems, the current value of a trajectory (x_k)may be dependent on future values of an input sequence (u_k).This property, which is called the anticipation phenomenon,has previously been studied by the authors for k running throughthe whole set of nonnegative integers. In this paper, we studythis phenomenon for systems defined on finite intervals. Inparticular, we give a quantitative characterization of anticipationphenomenon by introducing the so-called anticipation index ofa system.     

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.     

Anneaux de definition des dg-algebres propres et lisses

RÈSUMÈ: Soit k = colim_i k_i une colimite filtrante d’anneaux commutatifs.On montre que la théorie homotopique des dg-algèbrespropres et lisses sur k est la colimite des théorieshomotopiques des dg-algèbres propres et lisses sur lesk_i. Nous en déduisons en particulier que toute dg-algèbrepropre et lisse est définissable sur une -algèbrecommutative de type fini. Let k = colim_i k_i be a filtered colimit of commutative rings.We show that the homotopy theory of smooth and proper dg-algebrasover k is the colimit of the homotopy theories of smooth andproper dg-algebras over the k_i. We deduce, in particular, thatevery smooth and proper dg-algebra can be defined over a commutative-algebra of finite type.     

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.     

Asymptotics for the Heat Content of a Planar Region with a Fractal Polygonal Boundary

Let k 3 be an integer. For 0<s<1, let D_s R² be the setthat is constructed iteratively as follows. Take a regular openk-gon with sides of unit length, attach regular open k-gonswith sides of length s to the middles of the edges, and so on.At each stage of the iteration the k-gons that are added area factor s smaller than the previous generation and are attachedto the outer edges of the family grown so far. The set D_s isdefined to be the interior of the closure of the union of allthe k-gons. It is easy to see that there must exist some s_k> 0 such that no k-gons overlap if and only if 0 < s s_k. We derive an explicit formula for s_k. The set D_s is open, bounded, connected and has a fractal polygonalboundary. Let denote the heat content of D_s at time t when D_s initially has temperature 0and D_s is kept at temperature 1. We derive the complete short-timeexpansion of up to terms that are exponentially small in 1/t. It turns out that there arethree regimes, corresponding to 0<s<1/(k–1), s=1/(k–1),and 1/(k–1)<s s_k. For s 1/(k–1) the expansionhas the form where p_s is a log (1/s²)-periodic function, d_s=log (k–1)/log(1/s) is a similarity dimension, A_s and B are constants relatedto the edges and vertices, respectively, of D_s, and r_s is anerror exponent. For s=1/(k–1), the t^1/2-term carries anadditional log t. 1991 Mathematics Subject Classification: 11D25,11G05, 14G05.     

Perturbation Solutions of the Whittaker-Hill Equation

When the Helmholtz equation ²V+k²V = 0 is separated in the generalparaboloidal co-ordinate system, the three ordinary differentialequations obtained each take, after a suitable change of variable,the form of the Whittaker-Hill equation. For the case k²<0,a considerable amount is known about the periodic solutionsof this equation. For k²>0, however, very little is so farknown. In this paper solutions of the Whittaker-Hill equationfor small positive k² are derived. These are the first explicitsolutions to be obtained for the case k²>0, and they couldbe employed to solve the Dirichlet or Neumann problem for ageneral paraboloid when k² is small. Three limiting cases arenoted, involving the reduction of the solutions to Mathieu functionsand the reduction of the co-ordinate system to the rotation-paraboloidalsystem.     

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.     

On Waring's Problem in Number Fields

Let K be an algebraic number field of degree n over the rationals,and denote by J_k the subring of K generated by the kth powersof the integers of K. Then G_K(k) is defined to be the smallests1 such that, for all totally positive integers vJ_k of sufficientlylarge norm, the Diophantine equation (1.1) is soluble in totally non-negative integers _i of K satisfying N(_i)<<N(v)^1/k (1is). (1.2) In (1.2) and throughout this paper, all implicit constants areassumed to depend only on K, k, and s. The notation G_K(k) generalizesthe familiar symbol G(k) used in Waring's problem, since wehave G_Q(k) = G(k). By extending the Hardy–Littlewood circle method to numberfields, Siegel [8, 9] initiated a line of research (see [1–4,11]) which generalized existing methods for treating G(k). Thistypically led to upper bounds for G_K(k) of approximate strengthnB(k), where B(k) was the best contemporary upper bound forG(k). For example, Eda [2] gave an extension of Vinogradov'sproof (see [13] or [15]) that G(k)(2+o(1))k log k. The presentpaper will eliminate the need for lengthy generalizations assuch, by introducing a new and considerably shorter approachto the problem. Our main result is the following theorem.