Hauteurs normalisées des sous-variétés de produits de modules de Drinfeld

After establishing some important results on the usual height of projective varieties in positive characteristic, we construct a normalized height for subvarieties of products of Drinfeld modules and investigate its properties. In case the Drinfeld modules are pairwise isogeneous, we obtain in particular that the normalized height vanishes exactly on torsion varieties, that is on translates of sub-T-modules by torsion points.     

Non-linear higher-order boundary value problems describing thin viscous flows near edges

Two boundary value problems for non-linear higher-order ordinary differential equations are analyzed, which have been recently proposed in the modeling of steady and quasi-steady thin viscous flows over a bounded solid substrate. The first problem concerns steady states and consists of a third-order ODE for the height of the liquid; the ODE contains an unknown parameter, the flux, and the boundary conditions relate, near the edges of the substrate, the height and its second derivative to the flux itself. For this problem, (non-)existence and non-uniqueness results are proved depending on the behavior, as the flux approaches zero, of the “height-function” (the function which relates the height to the flux near the edge out of which the liquid flows). The second problem concerns quasi-steady states and consists of a fourth-order ODE for the (suitably scaled) height of the liquid; non-linear boundary conditions relate the height to the flux near the edges of the substrate. For this problem, the existence of a solution is proved for a suitable class of height-functions.     

Nathanson heights in finite vector spaces

Let p be a prime, and let Zp denote the field of integers modulo p. The Nathanson height of a point is the sum of the least nonnegative integer representatives of its coordinates. The Nathanson height of a subspace is the least Nathanson height of any of its nonzero points. In this paper, we resolve a quantitative conjecture of Nathanson [M.B. Nathanson, Heights on the finite projective line, Int. J. Number Theory, in press], showing that on subspaces of of codimension one, the Nathanson height function can only take values about . We show this by proving a similar result for the coheight on subsets of Zp, where the coheight of A⊆Zp is the minimum number of times A must be added to itself so that the sum contains 0. We conjecture that the Nathanson height function has a similar constraint on its range regardless of the codimension, and produce some evidence that supports this conjecture.     

Heights for line bundles on arithmetic varieties

Let X be an arithmetic variety and L be an element of the Néron-Severi group of its generic fiber X _K . Then there are only finitely many line bundles on X, generically belonging to L, such that the degrees of on the irreducible components of the special fibers of X and the height of are bounded. The concept of a height used here is recalled. Several elementary properties of this height are proven. Received: 9 March 1996     

Restructuring ordered binary trees

We consider the problem of restructuring an ordered binary tree T, preserving the in-order sequence of its nodes, so as to reduce its height to some target value h. Such a restructuring necessarily involves the downward displacement of some of the nodes of T. Our results, focusing both on the maximum displacement over all nodes and on the maximum displacement over leaves only, provide (i) an explicit tradeoff between the worst-case displacement and the height restriction (including a family of trees that exhibit the worst-case displacements) and (ii) efficient algorithms to achieve height-restricted restructuring while minimizing the maximum node displacement.     

On the number of points of bounded height¶on arithmetic projective spaces

Let K be a number field and its ring of integers. Let be a Hermitian vector bundle over . In the first part of this paper we estimate the number of points of bounded height in (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d>1 in of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of ℙ ^N _K blown up at a linear subspace of codimension two. Received: 20 February 1998 / Revised version: 9 November 1998     

On metric heights

Metric heights are modified height functions on the non-zero algebraic numbers Q which can be used to define a metric on certain cosets of . They have been defined with a view to eventually applying geometric methods to the study of . In this paper we discuss the construction of metric heights in general. More specifically, we study in some detail the metric height obtained from the na"ve height of an algebraic number (the maximum modulus of the coefficients of its minimal polynomial). In particular, we compute this metric height for some classes of surds. This revised version was published online in August 2006 with corrections to the Cover Date.     

Holding a regular pyramid by a circle

Among other things, we prove the following (1) If a regular pyramid whose base is a regular polygon of circum-radius 1 has height at least 1, then the pyramid can be held by a circle, while for every ${0 < \varepsilon < 1}$ , there is a regular pyramid with height ${1 - \varepsilon}$ and base polygon of circum-radius 1 that cannot be held by any circle. (2) A regular pyramid of height h whose base is an equilateral triangle of circum-radius 1 can be held by a circle if and only if h > 0.479 . . . (which complements a theorem by Tanoue). (3) A regular pyramid with square base of unit circum-radius can be held by a circle if and only if its height is greater than 0.828 . . ..     

Existence of homogeneous ideals fitting into long Bourbaki sequences

For any finitely generated torsion-free graded module over a polynomial ring, there exists a homogeneous ideal fitting into an exact sequence similar to a Bourbaki sequence even though its height is not restricted to two.

     

On p-Adic Heights in Families of Elliptic Curves

The non-degeneracy of the canonical p-adic height pairing definedby Perrin-Riou and Schneider on an elliptic curve over a numberfield with good, ordinary reduction is still unknown. Following the work done for the real-valued pairing, the behaviourof the p-adic height is analysed as a point varies on a sectionof a family of elliptic curves, and so new information is obtainedabout this pairing. In particular, the variation is p-adicallycontinuous and the non-degeneracy of a set of sections can bechecked simultaneously for almost all elements of the family.The paper contains some conjectures about the valuation of thep-adic regulator and its consequences for the growth of theMordell–Weil group in cyclotomic Z_p-extensions.     

Numerical simulation for a long-range dispersion of a pollutant using Chernobyl data

A Lagrangian particle model has been developed and applied to a long-range atmospheric dispersion. The developed numerical model has been tested by comparing its predictions with the ¹³⁷Cs air concentrations recorded over European areas during the Chernobyl accident. Sensitivity studies were performed to investigate the numerical accuracy according to a variation of the parameters such as the mixing height and diffusion coefficient in the model. From a comparative study, the calculated concentration distributions were more sensitive to a variation of the mixing height than to the changes of the diffusion coefficient values. Also, the calculated concentrations agreed with the time series of the measured ones at some sampling points.     

Light and low 5-stars in normal plane maps with minimum degree 5

It is known that there are normal plane maps (NPMs) with minimum degree δ = 5 such that the minimum degree-sum w(S₅) of 5-stars at 5-vertices is arbitrarily large. The height of a 5-star is the maximum degree of its vertices. Given an NPM with δ = 5, by h(S₅) we denote the minimum height of a 5-stars at 5-vertices in it.     

On mean distance and girth

We bound the mean distance in a connected graph which is not a tree in terms of its order n and its girth g. On one hand, we show that the mean distance is at most if g is even and at most if g is odd. On the other hand, we prove that the mean distance is at least unless G is an odd cycle. This resolves two conjectures of AutoGraphiX.     

Quasi sure analysis of local times of anticipating smooth semimartingales

Let be the anticipating smooth semimartingale and be its generalized local time. In this paper, we give some estimates about the quasi sure property of Xt and its quadratic variation process t〈X〉. We also study the fractional smoothness of and prove that the quadratic variation process of can be constructed as the quasi sure limit of the form , where is a sequence of subdivisions of [a,b], , i=0,1,…,n².     

Quantized symplectic oscillator algebras of rank one

A quantized symplectic oscillator algebra of rank 1 is a PBW deformation of the smash product of the quantum plane with . We study its representation theory, and in particular, its category .     

The compactified Picard scheme of the compactified Jacobian

Let C be an integral projective curve in any characteristic. Given an invertible sheaf L on C of degree 1, form the corresponding Abel map , which maps C into its compactified Jacobian, and form its pullback map , which carries the connected component of 0 in the Picard scheme back to the Jacobian. If C has, at worst, double points, then is known to be an isomorphism. We prove that always extends to a map between the natural compactifications, , and that the extended map is an isomorphism if C has, at worst, ordinary nodes and cusps.     

Qualitative uncertainty principles for groups with finite dimensional irreducible representations

Let G be a locally compact group of type I and its dual space. Roughly speaking, qualitative uncertainty principles state that the concentration of a nonzero integrable function on G and of its operator-valued Fourier transform on is limited. Such principles have been established for locally compact abelian groups and for compact groups. In this paper we prove generalizations to the considerably larger class of groups with finite dimensional irreducible representations.     

Hecke group algebras as quotients of affine Hecke algebras at level 0

The Hecke group algebra of a finite Coxeter group , as introduced by the first and last authors, is obtained from by gluing appropriately its 0-Hecke algebra and its group algebra. In this paper, we give an equivalent alternative construction in the case when is the finite Weyl group associated to an affine Weyl group W. Namely, we prove that, for q not a root of unity of small order, is the natural quotient of the affine Hecke algebra H(W)(q) through its level 0 representation.The proof relies on the following core combinatorial result: at level 0 the 0-Hecke algebra H(W)(0) acts transitively on . Equivalently, in type A, a word written on a circle can be both sorted and antisorted by elementary bubble sort operators. We further show that the level 0 representation is a calibrated principal series representation M(t) for a suitable choice of character t, so that the quotient factors (non-trivially) through the principal central specialization. This explains in particular the similarities between the representation theory of the 0-Hecke algebra and that of the affine Hecke algebra H(W)(q) at this specialization.     

Canonical height functions for affine plane automorphisms

Let be a polynomial automorphism of dynamical degree δ≥2 over a number field K. We construct height functions defined on that transform well relative to f, which we call canonical height functions for f. These functions satisfy the Northcott finiteness property, and a -valued point on is f-periodic if and only if its height is zero. As an application, we give an estimate on the number of points with bounded height in an infinite f-orbit.     

An upper bound on the “dimension of interval orders”

The main results of this paper are two distinct characterizations of interval orders and an upper bound on the dimension of an interval order as a function of its height. In particular, interval orders of height 1 have dimension of at most 2.