Christoffel Words and Markoff Triples: An Algebraic Approach

We introduce a family of modules, called Markoff modules, generated by a cluster-mutation-like iterative process. We show that these modules are combinatorially similar to Christoffel words. Furthermore, we construct a bijective map between the set of Markoff module triples and the set of proper Markoff triples. This allows us to interpret the uniqueness conjecture for Markoff numbers within an algebraic framework.     

A simple proof of the Markoff conjecture for prime powers

We give a simple and independent proof of the result of Jack Button and Paul Schmutz that the Markoff conjecture on the uniqueness of the Markoff triples (a, b, c) where a ≤ b ≤ c holds whenever c is a prime power. We also indicate some further directions for investigation.      

On Certain Exponential Sums and the Distribution of Diffie-Hellman Triples

Let g be a primitive root modulo a prime p. It is proved thatthe triples (g^x, g^y, g^xy), x, y = 1, ..., p–1, are uniformlydistributed modulo p in the sense of H. Weyl. This result isbased on the following upper bound for double exponential sums.Let >0 be fixed. Then uniformly for any integers a, b, c with gcd(a, b, c, p) = 1.Incomplete sums are estimated as well. The question is motivated by the assumption, often made in cryptography,that the triples (g^x, g^y, g^xy) cannot be distinguished fromtotally random triples in feasible computation time. The resultsimply that this is in any case true for a constant fractionof the most significant bits, and for a constant fraction ofthe least significant bits.     

The Uniqueness of the Prime Markoff Numbers

Given the Diophantine equation a²+b²+c²=3abc, a solution tripleof natural numbers (a, b, c) can be arranged in ascending orderso that abc. Then, given the largest element c, one can askwhether this uniquely determines the triple. This is referredto as the Markoff conjecture. The paper proves that, if c isprime, then there is indeed only one triple that solves theequation with c as the largest element. The proof uses onlystandard algebraic number theory, but it was prompted by geometricconsiderations.     

A Comparison Theorem for a Second Order Linear Quasi-Differential Equation

We are concerned with the non-oscillatory behaviour of a secondorder linear quasi-differential equation defined a.e. over ahalf-line. More particularly given that (py')' + qy = 0 is non-oscillatorywe show that by writing q in a form which is amenable to furtheranalysis we can generalise Sturm's comparison theorem.     

Markoff数性质的研究

1913年,Frobenius对Markoff方程a~2+b~2+c~2=3abc提了一个著名猜想:若abc是Markoff方程的正整数解,则a,b的值由最大的数c唯一确定.此猜想仍未得到解决.本文证明了:任给定正整数s_i,t_i,w,u,v=1,2),若(a_i,b_i,c)是Markoff方程的两组不同的正整数解,且a_ib_ic(i=1,2),则gcd(s_1a_1+s_2a_2+t_1b_1+t_2b_2+w,uc+v)≤K(uc+v)~(13/14),其中K是仅与s_i,t_i,w,u,v(i=1,2)有关的正数.     

The character of the Weil representation

Let V be a symplectic vector space over a finite or local field.We compute the character of the Weil representation of the metaplecticgroup Mp(V). The final formulas are overtly free of choices(for example, they do not involve the usual choice of a Lagrangiansubspace of V). Along the way, in results similar to those ofMaktouf, we relate the character to the Weil index of a certainquadratic form, which may be understood as a Maslov index. Thisrelation also expresses the character as the pullback of a certainsimple function from Mp(V V).     

Zariski Pairs of Index 19 and Mordell-Weil Groups of K3 Surfaces

We find Zariski pairs of sextics with simple singularities havingmaximal total Milnornumber. We also relate them to the existenceof distinct Mordell–Weil groups of extremal elliptic K3surfaces with a fixed set of semistable singular fibres. 1991Mathematics Subject Classification: 14F45, 14F27, 14F28.     

On Markoff's Inequality

For compact subsets of the real line the Markoff factors increase at least as fast as n ² . In this paper it is shown that there are sets of measure zero with Markoff factors of order n ² . We shall also show that this cannot happen for compact sets of logarithmic capacity zero. In connection with Markoff's inequality for polynomials of two variables we show a set E

COHOMOLOGICAL DIMENSION OF MACKEY FUNCTORS FOR INFINITE GROUPS

We consider the cohomology of Mackey functors for infinite groupsand define the Mackey-cohomological dimension cdG of a groupG. We will relate this dimension to other cohomological dimensionssuch as the Bredon-cohomological dimension cdG and the relativecohomological dimension -cdG. In particular, we show that forvirtually torsion free groups the Mackey-cohomological dimensionis equal to both -cdG and the virtual cohomological dimension.     

L1 Properties of Second order Elliptic Operators

We present a connected account of various spectral and regularityproperties of the semigroup associated with a fairly generalsymmetric second order elliptic operator on a Riemannian manifold.Our main goal is to relate the L² theory to the less well understoodL¹ theory, and hence to the approach via the theory of stochasticdifferential equations.     

Formal Markoff maps are positive

This note defines a family of Laurent polynomials indexed in which generalize the Markoff numbers and relate to the character variety of the one-cusped torus. We describe which monomials appear in each polynomial and prove all the coefficients are positive integers. We also conjecture a generalization of that positivity result.      

L-splines--a manifestation of optimal control

We show connections between L-splines and optimal control theory,leading to the conclusion that L-splines are manifestationsof an optimal behaviour.     

Sliding mode control of uncertain systems: a singular perturbation approach

In this paper we consider a nonlinear control system affectedby deterministic uncertainty and described by a system of ordinarydifferential equations. The uncertainty is modelled by a multivaluedmap whose t-measurable and x-Lipschitz selections representthe possible system dynamics of the uncertain system. We proposea dynamical feedback control design, based on the singular perturbationtheory, which allows all the possible system trajectories correspondingto the system dynamics to have the same prescribed behaviour.Specifically, given a manifold K of the state space, definedas the zeros of a smooth map, the proposed control steers andthen holds, during finite or infinite time intervals, any possiblesystem trajectory to any prescribed neighbourhood of K. A resultensuring the exact attainability of K is also provided. Someexamples illustrating the obtained results are presented.     

Holomorphic Triples on Elliptic Curves

Here we study the α-stability for holomorphic triples over curves of genus g = 1. We provide necessary and sufficient conditions for the moduli space of α-stable triples to be non-empy and, in these cases, we show that it is smooth and irreducible. Received: January 10, 2007.     

Twisted equivariant K-theory with complex coefficients

Using a global version of the equivariant Chern character, wedescribe the complexified twisted equivariant K-theory of aspace with a compact Lie group action in terms of fixed-pointdata. We apply this to the case of a compact group acting onitself by conjugation and relate the result to the Verlindealgebra and to the Kac numerator at q=1. Verlinde's formulais also discussed in this context. Received February 28, 2007.     

On the Topology of Fibrations with Section and Free Loop Spaces

We relate the brace products of a fibration with section tothe differentials in its Serre spectral sequence. In the particularcase of free loop fibrations, we establish a link between thesedifferentials and Browder operations in the fiber. Applicationsand several calculations (for the particular cases of spheresand wedges of spheres) are given. 2000 Mathematical SubjectClassification: 55P35, 55Q15, 55R05, 55R20.     

From Klein to Painleve Via Fourier, Laplace and Jimbo

We describe a method for constructing explicit algebraic solutionsto the sixth Painlevé equation, generalising that ofDubrovin and Mazzocco. There are basically two steps. Firstwe explain how to construct finite braid group orbits of triplesof elements of SL₂(C) out of triples of generators of three-dimensionalcomplex reflection groups. (This involves the Fourier–Laplacetransform for certain irregular connections.) Then we adapta result of Jimbo to produce the Painlevé VI solutions.(In particular, this solves a Riemann–Hilbert problemexplicitly.) Each step is illustrated using the complex reflection groupassociated to Klein's simple group of order 168. This leadsto a new algebraic solution with seven branches. We also provethat, unlike the algebraic solutions of Dubrovin and Mazzoccoand Hitchin, this solution is not equivalent to any solutioncoming from a finite subgroup of SL₂(C). The results of this paper also yield a simple proof of a recenttheorem of Inaba, Iwasaki and Saito on the action of Okamoto'saffine D₄ symmetry group as well as the correct connection formulaefor generic Painlevé VI equations. 2000 Mathematics SubjectClassification 34M55, 34M40, 20F55.     

Conformal Conjugacies in Baker Domains

Let F be a meromorphic function in the complex plane. We investigatethe behaviour of the iterates of F in a Baker domain B. In particular,we describe the dynamics of the orbits with the help of conformalconjugacies; that is, we determine a function which is univalentin a large simply connected subdomain of B such that (F(z))=T((z))holds throughout B. Here T is either a parabolic or hyperbolicMöbius transformation mapping either a half plane or Conto itself. This functional equation is always solvable ina Baker domain if F has only finitely many poles. Moreover,there is an example of a function with infinitely many poleswhere one cannot find an appropriate conformal conjugacy inan invariant Baker domain.