Coupling leads to chaos

The main goal of this paper is to prove analytically the existence of strange attractors in a family of vector fields consisting of two Brusselators linearly coupled by diffusion. We will show that such a family contains a generic unfolding of a 4-dimensional nilpotent singularity of codimension 4. On the other hand, we will prove that in any generic unfolding Xμ of an n-dimensional nilpotent singularity of codimension n there are bifurcation curves of (n−1)-dimensional nilpotent singularities of codimension n−1 which are in turn generically unfolded by Xμ. Arguments conclude recalling that any generic unfolding of the 3-dimensional nilpotent singularity of codimension 3 exhibits strange attractors.     

Tactical decompositions and orbits of projective groups

We consider decompositions of the incidence structure of points and lines of PG(n, q) (n?3) with equally many point and line classes. Such a decomposition, if line-tactical, must also be point-tactical. (This holds more generally in any 2-design.) We conjecture that such a tactical decomposition with more than one class has either a singleton point class, or just two point classes, one of which is a hyperplane. Using the previously mentioned result, we reduce the conjecture to the case n=3, and prove it when q²+q+1 is prime and for very small values of q. The truth of the conjecture would imply that an irreducible collineation group of PG(n, q) (n?3) with equally many point and line orbits is line-transitive (and hence known).     

Thom polynomials, symmetries and incidences of singularities

As an application of the generalized Pontryagin-Thom construction [RSz] here we introduce a new method to compute cohomological obstructions of removing singularities — i.e. Thom polynomials [T]. With the aid of this method we compute some sample results, such as the Thom polynomials associated to all stable singularities of codimension ≤8 between equal dimensional manifolds, and some other Thom polynomials associated to singularities of maps N ⁿ ?P ^n+k for k>0. We also give an application by reproving a weak form of the multiple point formulas of Herbert and Ronga ([H], [Ro2]). As a byproduct of the theory we define the incidence class of singularities, which — the author believes — may turn to be an interesting, useful and simple tool to study incidences of singularities. Oblatum 4-II-1999 & 19-VII-2000?Published online: 30 October 2000     

On the Existence of Absolutely Simple Abelian Varieties of a Given Dimension over an Arbitrary Field

We prove that for every field k and every positive integer n there exists an absolutely simple n-dimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we let S(k, n) denote the fraction of the isogeny classes of n-dimensional abelian varieties over k that consist of absolutely simple ordinary abelian varieties. Then for every n we have S(F_q, n)→1 as q→∞ over the prime powers.     

Degenerate orbits of adjoint representation of orthogonal and unitary groups regarded as algebraic submanifolds

We suggest a method for describing some types of degenerate orbits of orthogonal and unitary groups in the corresponding Lie algebras as level surfaces of a special collection of polynomial functions. This method allows one to describe orbits of the types SO(2n)/SO(2k)×SO(2) ^n?k , SO(2n+1)/SO(2k+1)×SO(2) ^n?k , and (S)U(n)/(S)(U(2k)×U(2) ^n?k ) in so(2n), so(2n+1), and (s)u(n), respectively. In addition, we show that the orbits of minimal dimensions of the groups under consideration can be described in the corresponding algebras as intersections of quadries. In particular, this approach is used for describing the orbit CP ^n?1?u(n).     

Topological change in mean convex mean curvature flow

Consider the mean curvature flow of an (n+1)-dimensional compact, mean convex region in Euclidean space (or, if n<7, in a Riemannian manifold). We prove that elements of the mth homotopy group of the complementary region can die only if there is a shrinking S ^k ×R ^n?k singularity for some k≤m. We also prove that for each m with 1≤m≤n, there is a nonempty open set of compact, mean convex regions K in R ⁿ⁺¹ with smooth boundary ?K for which the resulting mean curvature flow has a shrinking S ^m ×R ^n?m singularity.     

On the geometry of conjugacy classes in classical groups

Summary We study closures of conjugacy classes in the Lie algebras of the orthogonal and symplectic groups and determine which ones are normal varieties. Furthermore we give a complete classification of the minimal singularities which arise in this context, i.e. the singularities which occur in the open classes in the boundary of a given conjugacy class. In contrast to the results for the general linear group ([KP1], [KP2]) there are classes with non normal closure; they are branched in a class of codimension two and give rise to normal minimal singularities. The methods used are (classical) invariant theory and algebraic geometry. Supported in part by the SFB Theoretische Mathematik, University of Bonn, and by the University of Hamburg     

The vanishing of a higher codimension analogue of Hochster’s theta invariant

We study H. Dao’s invariant ${\eta_c^R}$ of pairs of modules defined over a complete intersection ring R of codimension c having an isolated singularity. Our main result is that ${\eta_c^R}$ vanishes for all pairs of modules when R is a graded complete intersection ring of codimension c > 1 having an isolated singularity. A consequence of this result is that all pairs of modules over such a ring are c-Tor-rigid.     

A characterization of projective subspaces of codimension two as quasi-symmetric designs with good blocks

Consider an incidence structure whose points are the points of a PG_n(n+2,q) and whose block are the subspaces of codimension two, where n?2. Since every two subspaces of codimension two intersect in a subspace of codimension three or codimension four, it is easily seen that this incidence structure is a quasi-symmetric design. The aim of this paper is to prove a characterization of such designs (that are constructed using projective geometries) among the class of all the quasi-symmetric designs with correct parameters and with every block a good block. The paper also improves an earlier result for the special case of n=2 and obtains a Dembowski-Wagner-type result for the class of all such quasi-symmetric designs.     

Finiteness for the k-factor model and chirality varieties

This paper deals with two families of algebraic varieties arising from applications. First, the k-factor model in statistics, consisting of n×n covariance matrices of n observed Gaussian random variables that are pairwise independent given k hidden Gaussian variables. Second, chirality varieties inspired by applications in chemistry. A point in such a chirality variety records chirality measurements of all k-subsets among an n-set of ligands. Both classes of varieties are given by a parameterisation, while for applications having polynomial equations would be desirable. For instance, such equations could be used to test whether a given point lies in the variety. We prove that in a precise sense, which is different for the two classes of varieties, these equations are finitely characterisable when k is fixed and n grows.     

Slow entropy for some smooth flows on surfaces

We study slow entropy in some classes of smooth mixing flows on surfaces. The flows we study can be represented as special flows over irrational rotations and under roof functions which are C² everywhere except one point (singularity). If the singularity is logarithmic asymmetric (Arnol’d flows), we show that in the scale a_n(t) = n(log n)^t slow entropy equals 1 (the speed of orbit growth is n log n) for a.e. irrational α. If the singularity is of power type (x^?γ, γ ∈ (0, 1)) (Kochergin flows), we show that in the scale a_n(t) = n^t slow entropy equals 1 + γ for a.e. α.We show moreover that for local rank one flows, slow entropy equals 0 in the n(log n)^t scale and is at most 1 for scale n^t. As a consequence we get that a.e. Arnol’d and a.e Kochergin flow is never of local rank one.     

Wonderful varieties of rank two

LetG be a complex connected reductive group. Well known wonderfulG-varieties are those of rank zero, namely the generalized flag varietiesG/P, those of rank one, classified in [A], and certain complete symmetric varieties described in [DP] such as the famous space of complete conics. Recently, there is a renewed interest in wonderful varieties of rank two since they were shown to hold a keystone position in the theory of spherical varieties, see [L], [BP], and [K]. The purpose of this paper is to give a classification of wonderful varieties of rank two. These are nonsingular completeG-varieties containing four orbits, a dense orbit and two orbits of codimension one whose closuresD ₁ andD ₂ intersect transversally in the fourth orbit which is of codimension two. We have gathered our results in tables, including isotropy groups, explicit basis of Picard groups, and several combinatorial data in relation with the theory of spherical varieties.     

A splitting criterion for an isolated singularity at x = 0 in a family of even hypersurfaces

Assume given a family of even local analytic hypersurfaces, whose central fiber has an isolated singularity at x =?0 which is not an ordinary double point. We prove that if the family is sufficiently general, for instance if the general fiber is smooth and the general singular fiber has only ordinary double points, then the singularity at x = 0 “splits in codimension one”, i.e., the local discriminant divisor has an irreducible component, over which a general fiber has more than one singularity specializing to the original one. As a corollary, we deduce the result by Grushevsky and Salvati Manni (Singularities of the theta divisor at points of order two, IMRN, 2007, Proposition 8) that on a principally polarized abelian variety (A, Θ) with dim(A) = g ≥ 4, a singularity of even multiplicity on Θ, isolated or not, at a point of order two and not an ordinary double point, must be a limit of two distinct ordinary double points {x, ?x} on nearby theta divisors.     

On the metric type of measurable functions and convergence in distribution

In the present paper, sequences of real measurable functions defined on a measure space ([0, 1], µ), where µ is the Lebesgue measure, are studied. It is proved that for every sequence f_n that converges to f in distribution, there exists a sequence of automorphisms S_n of ([0, 1], µ) such that f_n(S_n(t)) converges to f(t) in measure. Connection with some known results is also discussed.     

Commuting involutions with fixed point set of constant codimension

Special generators of the unoriented cobordism ringMO _* are constructed to determine the groupsJ _n,κ ^r ofn-dimensional cobordism classes inMO _n containing a representativeM ⁿ admitting a (Z ₂) ^k -action with fixed point set of constant codimension. This work is supported by HNSF     

Multi-layer canard cycles and translated power functions

The paper deals with two-dimensional slow-fast systems and more specifically with multi-layer canard cycles. These are canard cycles passing through n layers of fast orbits, with n?2. The canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n. We prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions.     

Steinitz classes of tamely ramified Galois extensions of algebraic number fields

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers Ok of k, which, together with the degree [K:k] of the extension determines the Ok-module structure of OK. We call Rt(k,G) the set of classes which are Steinitz classes of a tamely ramified G-extension of k. We will say that those classes are realizable for the group G; it is conjectured that the set of realizable classes is always a group. We define A^′-groups inductively, starting with abelian groups and then considering semidirect products of A^′-groups with abelian groups of relatively prime order and direct products of two A^′-groups. Our main result is that the conjecture about realizable Steinitz classes for tame extensions is true for A^′-groups of odd order; this covers many cases not previously known. Further we use the same techniques to determine Rt(k,Dn) for any odd integer n. In contrast with many other papers on the subject, we systematically use class field theory (instead of Kummer theory and cyclotomic descent).     

A notion of robustness and stability of manifolds

Starting from the notion of thickness of Parks we define a notion of robustness for arbitrary subsets of Rk and we investigate its relationship with the notion of positive reach of Federer. We prove that if a set M is robust, then its boundary ∂M is of positive reach and conversely (under very mild restrictions) if ∂M is of positive reach, then M is robust. We then prove that a closed non-empty robust set in Rk (different from Rk) is a codimension zero submanifold of class C¹ with boundary. As a partial converse we show that any compact codimension zero submanifold with boundary of class C² is robust. Using the notion of robustness we prove a kind of stability theorem for codimension zero compact submanifolds with boundary: two such submanifolds, whose boundaries are close enough (in the sense of Hausdorff distance), are diffeomorphic.     

Constructing polynomials of minimal growth

In [2] a cyclic diagonal operator on the space of functions analytic on the unit disk with eigenvalues (λ _n ) is shown to admit spectral synthesis if and only if for each j there is a sequence of polynomials (p _n ) such that lim _n→∞ p _n (λ _k ) = δ _j,k and lim sup _n→∞ sup _k>j |p _n (λ _k )|^1/k ≤ 1. The author also shows, through contradiction, that certain classes of cyclic diagonal operators are synthetic. It is the intent of this paper to use the aforementioned equivalence to constructively produce examples of synthetic diagonal operators. In particular, this paper gives two different constructions for sequences of polynomials that satisfy the required properties for certain sequences to be the eigenvalues of a synthetic operator. Along the way we compare this to other results in the literature connecting polynomial behavior ([4] and [9]) and analytic continuation of Dirichlet series ([1]) to the spectral synthesis of diagonal operators.