Languages of k-interval exchange transformations

This paper gives a complete characterization of those sequencesof subword complexity (k – 1)n + 1that are natural codings of orbits of k-interval exchange transformations,thereby answering an old question of Rauzy.     

The Milnor number of a function on a space curve germ

Given a finite function germ f:(X, 0) (, 0) on a reduced spacecurve singularity (X, 0), we show that µ(f) = µ(X,0) + deg(f) – 1. Here, µ(f) and µ(X, 0) denotethe Milnor numbers of the function and the curve, respectively,and deg(f) is the degree of f. We use this formula to obtainseveral consequences related to the topological triviality andWhitney equisingularity of families of curves and families offunctions on curves.     

On time regularity and related conditions for power-bounded operators

Let T be a bounded linear operator in a complex Banach space.Our main result gives various characterizations of the condition:T is power-bounded and an estimate ||(I – T)Tⁿ || cn^–1/2 holds for all positive integers n. In particular, this conditionholds if and only if T = β S + (1 – β)I, forsome β (0, 1) and some power-bounded operator S; or ifand only if T is power-bounded and the discrete semigroup (Tⁿ)is dominated by the continuous semigroup (e^{– t(I –T)})_{t 0} in a natural sense. As a consequence of our main results,for 1/2 < 1 we characterize the condition that T is power-boundedand ||(I – T)Tⁿ || c n^– for all n, in terms ofestimates on the semigroup e^{–t(I – T)}.     

The Global Mckay-Ruan Correspondence Via Motivic Integration

The purpose of this paper is to show how the methods of motivicintegration of Kontsevich, Denef–Loeser (Invent. Math.135 (1999) 201–232 and Compositio Math. 131 (2002) 267–290)and Looijenga (Astérisque 276 (2002) 267–297) canbe adapted to prove the McKay–Ruan correspondence, a generalizationof the McKay–Reid correspondence to orbifolds that arenot necessarily global quotients. 2000 Mathematics Subject Classification14A20, 14E15, 14F43.     

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      

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.     

Pad?-approximant Bounds and Approximate Solutions for Kirkwood-Riseman Integral Equations

Two-point Pad? approximants are used to calculate tight upperand lower bounds on the quantity <?, f> associated withKirkwood-Riseman integral equations (1+yL)?=f, which arise inthe diffusion theory of flexible macromolecules. The self-adjointoperator L is an integral operator on –1 x 1, with weaklysingular kernel |x – x'|^–?, and the two specificcases (i) f = 1, (ii) f = x² are studied. In case (i) directbounds on <?, 1> are obtained; this quantity is inverselyproportional to the translational diffusion constant. In case(ii) bounds on <?, 1 > are found by a new technique involvingcombinations of bounds for the three cases f = 1, f = x² andf = bx²?b^–1. Various types of Pade and related approximantsare compared, using the information <f, Lⁿf>, n = –2,–1, 0, 1, 2, 3 and (an upper bound on L) for severalvalues of the positive parameter y. Pad?-approximant-generating trial vectors are investigated anda convergence theorem is established. The vector consistingof an optimum linear combination of L^–1f, f and Lf isfound to be an accurate approximation to a numerical solutionin case (ii), for all values of y and x. Specific analyticalexpressions are derived for the approximate solutions.     

Amplitude Bounds on Periodic and Aperiodic Oscillations: Second-order Systems

The global behaviour of the control systems described by thepair of differential equations x{dot} = –f(x)±g(y)+p(t), y{dot} = –f(x)±g(y)+p(t) has been investigated. Here f(x), g(y), h(x) and k(y) are polynomialsof odd degree with leading coefficients positive and p(t) andq(t) are bounded functions of time. Sufficient conditions havebeen found under which the trajectories of the above systemmay eventually be confined in a subset of (x, y, t)-space, thusgiving bounds on the amplitude of periodic as well as aperiodicoscillations. Further bounds on the amplitude of oscillationshave been investigated by finding regions in (x,y,t)-space fromwhich all trajectories eventually leave and into which no trajectoriesenter. Thus sufficient conditions have been derived for theexistence of an annulus in which oscillatory behaviour may beconfined.     

Bounded-Norm Matrix-Inverse Mappings

This paper introduces the concept of bounded-norm matrix-inversemappings, i.e. mappings µ : R^mxnR^nxm such that, for allnonzero mxn matrices A, the matrix µ(A) is a generalizedinverse of A and ||µ(A)||> k/s(A), where K < 0 isa constant and s(A) is the nonzero singular value of A havingsmallest absolute value. It is shown how the definition of suchmappings is motivated by the need to ensure finite terminationof the inner-iterations of generalized elimination methods forthe solution of nonlinearly constrained optimization problems.The main result of the paper is that the mapping defined byµ(A) = A^b is a bounded-norm matrix-inverse mapping, providedthat the basic inverse A^b is calculated using Gaussian eliminationwith complete pivoting. The concept of bounded-norm matrix-inversemappings is then extended to that of boundednorm least-squaresmatrix-inverse mappings. It is proved that the mapping definedby µ(A) = A^ß is a bounded-norm least-squaresmatrix-inverse mapping, provided that the basic least-squaresinverse A^ß is calculated using the QR decompositionwith column pivoting.     

Linear Systems and Quotients of Projective Space

A complete characterization of the categorical quotients of(P¹)ⁿ by the diagonal action of SL(2, C) with respect to anypolarization is given by M. Polito, in ‘SL(2, C)-quotientsde (P¹)ⁿ’, C. R. Acad. Sci. Paris Sér. I 321 (1995)1577–1582. In this paper, these categorical quotientsare obtained by certain linear systems on P^n–3, dependingon the given polarization. 2000 Mathematics Subject Classification14L24, 14L30     

Hemisystems on the Hermitian Surface

The natural geometric setting of quadrics commuting with a Hermitiansurface of PG(3,q²), q odd, is adopted and a hemisystem on theHermitian surface H(3,q²) admitting the group P^–(4,q)is constructed, yielding a partial quadrangle PQ((q–1)/2,q²,(q–1)²/2) and a strongly regular graph srg((q³+1)(q+1)/2,(q²+1)(q–1)/2,(q–3)/2,(q–1)²/2).For q>3, no partial quadrangle or strongly regular graphwith these parameters was previously known, whereas when q=3,this is the Gewirtz graph. Thas conjectured that there are nohemisystems on H(3,q²) for q>3, so these are counterexamplesto his conjecture. Furthermore, a hemisystem on H(3,25) admitting3.A₇.2 is constructed. Finally, special sets (after Shult) andovoids on H(3,q²) are investigated.     

Tameness and complexity of finite group schemes

Using a representation-theoretic interpretation of support varietiesdue to Friedlander and Pevtsova (Amer. J. Math. 127 (2005) 379–420;Erratum Amer. J. Math. 128 (2006) 1067–1068, we show thatthe complexity of tame blocks of finite group schemes is boundedby 2. In this context, our result salvages a theorem by Rickard(Bull. London Math. Soc. 22 (1990) 540–546), the proofof which is flawed.     

The Direct Limits Of The Banach-mazur Compacta

Let 1 p . For each n-dimensional Banach space E = (E, || ·||), we define a norm || · ||_p on E x R as follows: [formula] It is shown that the correspondence (E, || · ||) (Ex R, || · ||_p) defines a topological embedding of oneBanach–Mazur compactum, BM(n), into another, BM(n 1),and hence we obtain a tower of Banach–Mazur compacta:BM(1) BM(2) BM(3) ···. Let BM_p be thedirect limit of this tower. We prove that BM_p is homeomorphicto Q = dir lim Qⁿ, where Q = [0, 1] is the Hilbert cube. 1991Mathematics Subject Classification 46B04, 46B20, 52A21, 57N20,54H15.     

On Periodic Expansions of Pisot Numbers and Salem Numbers

Let ß > 1 be a real number, and let T_ßbe the associated ß-transformation of the unit interval[0,1) given by T_ß = ß (mod 1). We writeQ for the set of rational numbers, Q (ß) for the smallestsub-field of the reals containing ß, and Per (ß)for the set of (eventually) periodic points for T_ß,i.e. for the set of points whose orbits under T_ß,are finite. In this note we prove the following results: (1) If Q [0,1) Per (ß), then ß is eithera Pisot- or a Salem-number. (2) If ß is a Pisot-number, then Per (ß)= Q(ß) [0,1). The last section contains explicit formulae for the cardinalitiesof the sets {T^k_ß: k 0}, Q [0, 1), if ßsatisfies an equation ß² = nß + 1 with n 1. Present address: Mathematics Institute, University of Warwick,Coventry CV4 7AL.     

The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings

Motivated in part by the first author's work [23] on the Weyl-Berryconjecture for the vibrations of ‘fractal drums’(that is, ‘drums with fractal boundary’), M. L.Lapidus and C. Pomerance [31] have studied a direct spectralproblem for the vibrations of ‘fractal strings’(that is, one-dimensional ‘fractal drums’) and establishedin the process some unexpected connections with the Riemannzeta-function = (s) in the ‘critical interval’0 < s < 1. In this paper we show, in particular, thatthe converse of their theorem (suitably interpreted as a naturalinverse spectral problem for fractal strings, with boundaryof Minkowski fractal dimension D (0,1)) is not true in the‘midfractal’ case when D = , but that it is true for all other D in the criticalinterval (0,1) if and only if the Riemann hypothesis is true.We thus obtain a new characterization of the Riemann hypothesisby means of an inverse spectral problem. (Actually, we provethe following stronger result: for a given D (0,1), the aboveinverse spectral problem is equivalent to the ‘partialRiemann hypothesis’ for D, according to which = (s)does not have any zero on the vertical line Re s = D.) Therefore,in some very precise sense, our work shows that the question(à la Marc Kac) "Can one hear the shape of a fractalstring?" – now interpreted as a suitable converse (namely,the above inverse problem) – is intimately connected withthe existence of zeros of = (s) in the critical strip 0 <Res < 1, and hence to the Riemann hypothesis.     

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.     

The Kirwan Map for Singular Symplectic Quotients

Let M be a Hamiltonian K-space with proper moment map µ.The symplectic quotient X = µ^–1(0)/K is a singularstratified space with a symplectic structure on the strata.In this paper we generalise the Kirwan map, which maps the Kequivariant cohomology of µ^–1(0) to the middle perversityintersection cohomology of X, to this symplectic setting. The key technical results which allow us to do this are Meinrenken'sand Sjamaar's partial desingularisation of singular symplecticquotients and a decomposition theorem, proved in Section 2 ofthis paper, exhibiting the intersection cohomology of a ‘symplecticblowup’ of the singular quotient X along a maximal depthstratum as a direct sum of terms including the intersectioncohomology of X.     

Low-Dimensional Representations of Special Linear Groups in Cross Characteristics

The low-dimensional projective irreducible representations incross characteristics of the projective special linear groupPSL_n(q) are investigated. If n 3 and (n, q) (3,2), (3,4), (4,2), (4,3), all such representationsof the first degree (which is (qⁿ – q)/(q – 1) – with = 0 or 1) and the second degree (which is (qⁿ –1)/(q – 1) come from Weil representations. We show thatthe gap between the second and the third degree is roughly q^2n-4.1991 Mathematics Subject Classification: 20C20, 20C33.     

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).     

Stability Regions of Hill's Equation

The stability of the solutions of Hill's equation+[B—µø(t)]x= 0 is studied through the use of a fundamental matrix of solutionsevaluated after one period. This fundamental matrix is expandedin a power series in µ, but its determinant is computedhere by using series expansion rather than the Jacobi-Liouvilleformula. Using this procedure, it is shown that successive approximationsof boundaries of stability regions in the (B, µ) planecan be obtained simply, in particular without expansion of Bin power series of µ and Ø(t) in a Fourier series.A simple expression is obtained for the first approximation.Two examples are treated, with stability curves drawn up tothe second approximation.