Spin structures and spectra of ℤ<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>-manifolds

We give necessary and sufficient conditions for the existence of pin^± and spin structures on Riemannian manifolds with holonomy group ₂^k. For any n4 (resp. n6) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M₁, M₂, non homeomorphic to each other, that are Laplace isospectral on functions and on p-forms for any p and such that M₁ admits a pin^± (resp. spin) structure whereas M₂ does not.Mathematics Subject Classification (2000):58J53, 57R15, 20H15Partially supported by Conicet and grants from SecytUNC, Foncyt and AgCba.     

Heteroclinic cycles emanating from local bifurcations

LetK be a quadratic imaginary number field. Letf denote a rational integer which is coprime to 6. We letK (f ²) (resp.K(f)) denote the ray class field ofK of conductorf ² (resp.f). We write Γ for the Galois group ofK(f ²)/K(f) and denote the unique maximal order ofK(f)Γ by . We show that the Kummer order is a free, rank one -module. An explicit generator is given by singular values of certain modular functions.     

Totally complex submanifolds inH P m (1)

Let (resp.K) be the second fundamental form (resp. the sectional curvature) of a compact submanifoldM of the quaternion projective spaceH P ^m (1). We determine all compact totally complex submanifolds of complex dimensionn inH P ^m (1) satisfying either ||² n orK 1/8.Supported by the JSPS postdoctoral fellowship.     

Aspects of Number Theory in Terms of Potential Theory

We develop the Potential Theory on the set N^* of all natural numbers 2 associated with the kernel V (resp. V ^*) given by

Approximation durch Polynomfunktionen auf universellen Algebren

LetA be a topological universal algebra,F(A)=A ^A the full function algebra endowed with the producttopology andP(A) the algebra of polynomial functions onA, i. e. the subalgebra ofF(A) generated by the constant mappings and the identity.A has the local approximation property ifP(A) is dense inF(A)., TopologicalT ₂-groupsG (resp.T ₂-ringsR) of order >2 have the local approximation property if an only if all proper normal subgroups (resp. ideals) are dense inG (resp.R) andG is non-abelian (R is not a zeroring). In particular all subringsR of satisfying R ( resp. denoting the field of rational resp. real numbers) yield examples of rings having this property thus showing that the local approximation property is a far-reaching generalization of the well-known interpolation property.

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Meinem verehten Lehrer, Herrn Professor Dr. E. Hlawka, zum 65. Geburtstag gewidmet     

Singuläre definition der äquivarianten Bredon homologie

Let G be a discrete group,o(G) the orbit category of G and M:o(G)a a covariant (contravariant) functor to abelian groups. We define a singular equivariant homology theory H_*(X;M) (resp. H^*(X;M)) which satisfies a dimension axiom, in the sense of Bredon (Lecture notes 34). It turns out, that all fundamental properties of these theories directly follow by naturality from the analogous theorems in the classical non equivariant case.     

Root Configurations for Hyperbolic Polynomials of Degree 3, 4, and 5

A real polynomial of one real variable is (strictly) hyperbolic if it has only real (and distinct) roots. There are 10 (resp. 116) possible non-degenerate configurations between the roots of a strictly hyperbolic polynomial of degree 4 (resp. 5) and of its derivatives (i.e., configurations without equalities between roots). The standard Rolle theorem allows 12 (resp. 286) such configurations. The result is based on the study of the hyperbolicity domain of the family P(x,a)=x ⁿ+a ₁ x ^n-1+...+a _n for n=4,5 (i.e., of the set of values of aⁿ for which the polynomial is hyperbolic) and its stratification defined by the discriminant sets Res(P ⁽ⁱ⁾,P ^(j))=0, 0 i < jn-1.     

Processus des misanthropes

Summary We consider a system of identical interacting particles moving on the lattice ^d . The rate at which a particle at the site x jumps to the site y is p(y–x)b((x), (y)) where p is an irreducible probability on ^d and b((x), (y)) is an increasing (resp. decreasing) function of the number (x) (resp. (y)) of particles at site x (resp. y). We study the convergence of the system to equilibrium and describe the invariant measures.     

Paradoxical subsets of hyperbolic and spherical discs

Given a disc D of radius r in H ² (resp. S ²) with <r (resp. <r), we determine the pairs (m,n) for which there is an (m,n)-paradoxical subset of D but not an (m–1, n)-paradoxical subset of D or an (m, n–1)-paradoxical subset of D.     

Some Landau's type inequalities for infinitesimal generators

Summary Lett T(t) be a strongly continuous contraction semigroup on a complex Banach space and letA be its infinitesimal generator. We prove that, forx D(A ³), the following inequalities hold true: Ax³ 243/8 x²A ³ x, A ² x 24 xA ³ x². Ift T(t) is a contraction group (resp. cosine function) we get the analogous but better inequalities with constants 9/8 and 3 (resp. 81/40 and 72/25) instead of 243/8 and 24. We consider also uniformly bounded semigroups, groups and cosine functions.     

The Banach-Mazur distance between symmetric spaces

We show that the Banach-Mazur distance betweenN-dimensional symmetric spacesE andF satisfies , wherec is a numerical constant. IfE is a symmetric space, then max (M ⁽²⁾(E),M ₍₂₎(E)), whereM ⁽²⁾(E) (resp.M ₍₂₎(E)) denotes the 2-convexity (resp. the 2-concavity) constant ofE. We also give an example of a spaceF with an 1-unconditional basis and enough symmetries that satisfiesd(F, l ₂ ^dimF )=M ⁽²⁾(F)M ₍₂₎(F). Partially supported by NSF Grant MCS-8201044.     

Orbites hétérocliniques au voisinage d'une bifurcation de Poincaré dégénérée pour les champs de vecteurs de ℝ2

We consider a ²-ordinary differential equation, where the fixed point (0, 0) presents a degenerate Poincaré-bifurcation of resonancek(=2k) and dominanced(=k–1). We prove the existence of a 2-dimensional linear manifoldV in the parameter space. , on which the perturbed dominant differential system (SD) possesses heteroclinic orbits between fixed points. The numerical continuation of the local stable or unstable manifolds of the saddle fixed points shows that for any neighborhood, in , of a point ofV corresponding to a saddle heteroclinic orbit, there exists only one stable (resp. unstable) periodic orbit close to the stable — in the Andronov sense [1]-(resp. unstable) heteroclinic orbit. Applications are given fork=4 andk=6.     

Asymptotic stability at infinity of planar vector fields

Let >0 andX be aC ¹ vector field on the plane such that: (i) for allq², Det(DX(q))>0; and (ii) for allp², with p, Trace(D(X(p))<0. IfX has a singularity and ² Trace(DX)dxdy is less than 0 (resp. greater or equal than 0), then the point at infinity of the Riemann sphere ²{} is a repellor (resp. an attractor) ofX.     

C r -finite elements of Powell-Sabin type on the three direction mesh

Let be the triangulation generated by a uniform three direction mesh of the plane. Let ₆ be the Powell-Sabin subtriangulation obtained by subdividing each triangleT by connecting each vertex to the midpoint of the opposite side.Given a smooth functionu, we construct a piecewise polynomial function C ^r (²) of degreen=2r (resp. 2r+1) forr odd (resp. even) in each triangle of ₆, interpolating derivatives ofu up to orderr at the vertices of.     

Construction of ɛ-nets

LetS be a set ofn points in the plane and let be a real number, 0<<1. We give a deterministic algorithm, which in timeO(n ^–2 log(1/)+ ^–8) (resp.O(n ^–2 log(1/)+ ^–10) constructs an-netNS of sizeO((1/) (log(1/))²) for intersections ofS with double wedges (resp. triangles); this means that any double wedge (resp. triangle) containing more thatn points ofS contains a point ofN. This givesO(n logn) deterministic preprocessing for the simplex range-counting algorithm of Haussler and Welzl [HW] (in the plane).We also prove that given a setL ofn lines in the plane, we can cut the plane intoO( ^–2) triangles in such a way that no triangle is intersected by more thann lines ofL. We give a deterministic algorithm for this with running timeO(n ^–2 log(1/)). This has numerous applications in various computational geometry problems.     

The Hopf algebra Rep U_{q} \widehat{\frak g \frak l}_\infty

We define the Hopf algebra structure on the Grothendieck group of finite-dimensional polynomial representations of in the limitN→∞. The resulting Hopf algebra Rep is a tensor product of its Hopf subalgebras Rep_a ,a ∈ ℂ^×/q^2ℤ. Whenq is generic (resp.,q ² is a primitive root of unity of orderl), we construct an isomorphism between the Hopf algebra Rep _a and the algebra of regular functions on the prounipotent proalgebraic group (resp., ). Whenq is a root of unity, this isomorphism identifies the Hopf subalgebra of Rep _a spanned by the modules obtained by pullback with respect to the Frobenius homomorphism with the algebra generated by the coefficients of the determinant of an element of considered as anl×l matrix over the Taylor series. This gives us an explicit formula for the Frobenius pullbacks of the fundamental representations. In addition, we construct a natural action of the Hall algebra associated to the infinite linear quiver (resp., the cyclic quiver withl vertices) on Rep _a and describe the span of tensor products of evaluation representations taken at fixed points as a module over this Hall algebra.     

Generalized pseudoinverses of matrix valued functions

We define the pseudoinverse (resp. a generalized pseudoinverse) of a matrix-valued functionF to be the functionF ^x such that, for each in the domain ofF, F ^x () is the inverse (resp. a generalized inverse) of the matrixF(). We derive a state space formula for a generalized pseudoinverse of a rational matrix function without a pole or zero at infinity. This derivation makes use of the theorem characterizing the factorization of a nonregular rational matrix functionW in terms of the decomposition of the state space of a realization ofW. We also give a formula for a generalized pseudoinverse of an arbitrary rational matrix function in the form of a centered realization. We indicate some applications of generalized pseudoinverses of matrix valued functions.     

On Generic Well-posedness of Restricted Chebyshev Center Problems in Banach Spaces

Let B （resp. K, BC,KC） denote the set of all nonempty bounded （resp. compact, bounded convex, compact convex） closed subsets of the Banach space X, endowed with the Hausdorff metric, and let G be a nonempty relatively weakly compact closed subset of X. Let B° stand for the set of all F ∈B such that the problem （F, G） is well-posed. We proved that, if X is strictly convex and Kadec, the set KC ∩ B° is a dense Gδ-subset of KC ／ G. Furthermore, if X is a uniformly convex Banach space, we will prove more, namely that the set B ／B° （resp. K ／ B°, BC ／B°, KC ／ B°） is a-porous in B （resp. K,BC, KC）. Moreover, we prove that for most （in the sense of the Baire category） closed bounded subsets G of X, the set K ／ B° is dense and uncountable in K.     

Clean triangulations

A polyhedron on a surface is called a clean triangulation if each face is a triangle and each triangle is a face. LetS _p (resp.N _p ) be the closed orientable (resp. nonorlentable) surface of genusp. If (S) is the smallest possible number of triangles in a clean triangulation ofS, the results are: (N ₁)=20, (S ₁)=24, lim(S _p )p ^–1=4, lim(N _p )p ^–1=2 forp.     

Kurventheorie im konform isotropen RaumC 3 (1)

LetM be a 3-dimensional manifold with metric tensorg. (M,g) is called a conformally isotropce space C ₃ ⁽¹⁾ if there exists a chart (M, ) of M, for which (i)(M)=³, (ii) the components ofg with respect to areg ₁₁=g ₂₂=q,g ₁₂=g ₁₃=g ₂₃=g ₃₃=0q(x ¹,x²,x³) > 0, . In this note, first we consider some metric properties ofC ₃ ⁽¹⁾ . Further it is shown that there exists a unique linear connection inC ₃ ⁽¹⁾ , the so-called standard connection. Finally we develop the fundamentals of the theory of curves inC ₃ ⁽¹⁾ up to Frenet's formula, and give a geometric interpretation of the conformally isotropic curvature resp. torsion of a curvec inC ₃ ⁽¹⁾ . It is shown, that , where resp. are the isotropic curvature resp. isotropic torsion of the curvec in the threedimensional isotropic spaceI ₃ ⁽¹⁾ when using a special coordinate system ofC ₃ ⁽¹⁾ as the standard coordinate system ofI ₃ ⁽¹⁾ .

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Herrn Prof. Dr. Oswald Giering zum 60. Geburtstag gewidmet