Sheared algebra maps and operation bialgebras for mod 2 homology and cohomology
David
J.
Pengelley;
Frank
Williams
1453-1492
Abstract: The mod 2 Steenrod algebra $\mathcal{A}$ and Dyer-Lashof algebra $\mathcal{R}$ have both striking similarities and differences arising from their common origins in ``lower-indexed'' algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra $\mathcal{K}$, whose module actions are equivalent to, but quite different from, those of $\mathcal{A}$ and $\mathcal{R}$. The exact relationships emerge as ``sheared algebra bijections'', which also illuminate the role of the cohomology of $\mathcal{K}$. As a bialgebra, $\mathcal{K}^{*}$ has a particularly attractive and potentially useful structure, providing a bridge between those of $\mathcal{A^{*}}$ and $\mathcal{R^{*}}$, and suggesting possible applications to the Miller spectral sequence and the $\mathcal{A}$ structure of Dickson algebras.
Rational homotopy theory for non-simply connected spaces
Antonio
Gómez-Tato;
Stephen
Halperin;
Daniel
Tanré
1493-1525
Abstract: We construct an algebraic rational homotopy theory for all connected CW spaces (with arbitrary fundamental group) whose universal cover is rationally of finite type. This construction extends the classical theory in the simply connected case and has two basic properties: (1) it induces a natural equivalence of the corresponding homotopy category to the homotopy category of spaces whose universal cover is rational and of finite type and (2) in the algebraic category, homotopy equivalences are isomorphisms. This algebraisation introduces a new homotopy invariant: a rational vector bundle with a distinguished class of linear connections.
On $BC$ type basic hypergeometric orthogonal polynomials
Jasper
V.
Stokman
1527-1579
Abstract: The five parameter family of Koornwinder's multivariable analogues of the Askey-Wilson polynomials is studied with four parameters generically complex. The Koornwinder polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partly discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain is given for which the partly discrete orthogonality measure is positive. The orthogonality relations and norm evaluations for multivariable $q$-Racah polynomials and multivariable big and little $q$-Jacobi polynomials are proved by taking suitable limits in the orthogonality relations for the Koornwinder polynomials. In particular new proofs of several well-known $q$-analogues of the Selberg integral are obtained.
Lorentzian affine hyperspheres with constant affine sectional curvature
Marcus
Kriele;
Luc
Vrancken
1581-1599
Abstract: We study affine hyperspheres $M$ with constant sectional curvature (with respect to the affine metric $h$). A conjecture by M. Magid and P. Ryan states that every such affine hypersphere with nonzero Pick invariant is affinely equivalent to either \begin{displaymath}(x_{1}^{2} \pm x_{2}^{2})(x_{3}^{2}\pm x_{4}^{2})\dots (x_{2m-1}^{2}\pm x_{2m}^{2}) = 1\end{displaymath} or \begin{displaymath}(x_{1}^{2} \pm x_{2}^{2})(x_{3}^{2}\pm x_{4}^{2})\dots (x_{2m-1}^{2}\pm x_{2m}^{2})x_{2m+1} = 1\end{displaymath} where the dimension $n$ satisfies $n=2m-1$ or $n=2m$. Up to now, this conjecture was proved if $M$ is positive definite or if $M$ is a $3$-dimensional Lorentz space. In this paper, we give an affirmative answer to this conjecture for arbitrary dimensional Lorentzian affine hyperspheres.
Sur les intégrales premières dans la classe de Nilsson d'équations différentielles holomorphes
Frédéric
Touzet
1601-1622
Abstract: We study two classes of holomorphic differential equations. The first one is constitued by elements admitting solutions defined in an algebraic way (the so called Liouville class) and the second of elements admitting solutions defined in an analytic way (the Nilsson class). We build up links between these two classes using special properties of the holonomy and its results on the monodromy.
On Chow motives of 3-folds
Pedro
Luis del
Angel;
Stefan
Müller-Stach
1623 - 1633
Abstract: Let $k$ be a field of characteristic zero. For every smooth, projective $k$-variety $Y$ of dimension $n$ which admits a connected, proper morphism $f: Y \to S$ of relative dimension one, we construct idempotent correspondences (projectors) $\pi _{ij}(Y) \in CH^{n}(Y \times Y,\mathbb{Q})$generalizing a construction of Murre. If $n=3$ and the transcendental cohomology group $H^{2}_{\text{tr}}(Y)$ has the property that $H^{2}_{\text{tr}}(Y,\mathbb{C})=f^{*}H^{2}_{\text{tr}}(S,\mathbb{C})+ {\text{Im... ...(S,\mathbb{C}) \otimes H^{1}(Y,\mathbb{C}) \to H^{2}_{\text{tr}}(Y,\mathbb{C}))$, then we can construct a projector $\pi _{2}(Y)$ which lifts the second Künneth component of the diagonal of $Y$. Using this we prove that many smooth projective 3-folds $X$ over $k$ admit a Chow-Künneth decomposition $\Delta =p_{0}+...+p_{6}$ of the diagonal in $CH^{3}(X \times X,{\mathbb{Q}})$.
A theorem on smoothness- Bass-Quillen, Chow groups and intersection multiplicity of Serre
S.
P.
Dutta
1635-1645
Abstract: We describe here an inherent connection of smoothness among the Bass-Quillen conjecture, the Chow-group problem and Serre's Theorem on Intersection Multiplicity. Extension of a theorem of Lindel on smoothness plays a key role in our proof of the Serre-multiplicity theorem in the geometric (resp. unramified) case. We reduce the complete case of the theorem to the above case by using Artin's Approximation. We do not need the concept of ``complete Tor''. Similar proofs are sketched for Quillen's theorem on Chow groups and its extension due to Gillet and Levine.
A bracket power characterization of analytic spread one ideals
L.
J.
Ratliff Jr.;
D.
E.
Rush Jr.
1647-1674
Abstract: The main theorem characterizes, in terms of bracket powers, analytic spread one ideals in local rings. Specifically, let $b_{1},\dots ,b_{g},x$ be regular nonunits in a local (Noetherian) ring $(R,M)$ and assume that $I$ $\subseteq$ $(xR)_{a}$, the integral closure of $xR$, where $I$ $=$ $(b_{1},\dots ,b_{g},x)R$. Then the main result shows that for all but finitely many units $u_{1},\dots ,u_{g}$ in $R$ that are non-congruent modulo $M$ and for all large integers $n$ and $k$ it holds that $I^{jn}$ $=$ $I^{[j]n}$ for $j$ $=$ $1,\dots ,k$ and $j$ not divisible by $char(R/M)$, where $I^{[j]}$ is the $j$-th bracket power $((b_{1}+u_{1}x)^{j}, \dots ,(b_{g}+u_{g}x)^{j},x^{j})R$ of $I$ $=$ $(b_{1}+u_{1}x, \dots ,b_{g}+u_{g}x,x)R$. And, conversely, if there exist positive integers $g$, $n$, and $k$ $\ge$ ${\binom{{n+g} }{{g}}}$ such that $I$ has a basis $\beta _{1},\dots ,\beta _{g} ,x$ such that $I^{kn}$ $=$ $({\beta _{1}}^{k},\dots ,{\beta _{g}}^{k},x^{k})^{n}R$, then $I$ has analytic spread one.
Sharp bounds on Castelnuovo-Mumford regularity
Chikashi
Miyazaki
1675-1686
Abstract: The Castelnuovo-Mumford regularity is one of the most important invariants in studying the minimal free resolution of the defining ideals of the projective varieties. There are some bounds on the Castelnuovo-Mumford regularity of the projective variety in terms of the other basic measures such as dimension, codimension and degree. In this paper we consider an upper bound on the regularity $\operatorname{reg}(X)$ of a nondegenerate projective variety $X$, $\operatorname{reg}(X)\le\lceil(\deg (X) - 1)/\operatorname{codim}(X)\rceil +k \cdot \dim (X)$, provided $X$ is $k$-Buchsbaum for $k \ge 1$, and investigate the projective variety with its Castelnuovo-Mumford regularity having such an upper bound.
Prehomogeneous vector spaces and ergodic theory II
Dave
Witte;
Akihiko
Yukie;
Roger
Zierau
1687-1708
Abstract: We apply M. Ratner's theorem on closures of unipotent orbits to the study of three families of prehomogeneous vector spaces. As a result, we prove analogues of the Oppenheim Conjecture for simultaneous approximation by values of certain alternating bilinear forms in an even number of variables and certain alternating trilinear forms in six and seven variables.
Shintani functions on $GL(2,\mathbf{R})$
Miki
Hirano
1709-1721
Abstract: In this paper, we give a formulation and an explicit formula for Shintani function on $GL(2,{\mathbf{R}})$, which has been studied by Murase and Sugano in the theory of automorphic $L$-functions. In particular, we obtain the multiplicity of this function.
Complexes galoisiens
Yves
Ladegaillerie
1723-1741
Abstract: We construct special $n$-complexes categories which are the object of a Galois theory. Their topological supports are pseudo-manifolds which are branched coverings of spheres. They are a generalization in every dimension of hypercartes. Every category of Galois $n$-complexes is equivalent to a 2-complexes one. Reducing operations are introduced in dimensions two and three. It leads to a notion of irreducible complex which is used in three-dimensions for a simplified generation of 3-manifolds. Résumé. On construit des catégories de $n$-complexes simpliciaux particuliers qui sont l'objet d'une théorie galoisienne. Topologiquement, ce sont des pseudo-variétés revêtements ramifiés de sphères. En particulier, ce sont des généralisations des hypercartes aux dimension supérieures. Tout catégorie de $n$-complexes galoisiens est équivalente à une catégorie de 2-complexes. Des opérations de réduction sont introduites en dimensions deux et trois. Elles mènent à une notion de complexe irréductible qui est utilisée en dimension trois pour obtenir une génération simplifiée des 3-variétés.
A continuous circle of pseudo-arcs filling up the annulus
Janusz
R.
Prajs
1743-1757
Abstract: We prove an early announcement by Knaster on a decomposition of the plane. Then we establish an announcement by Anderson saying that the plane annulus admits a continuous decomposition into pseudo-arcs such that the quotient space is a simple closed curve. This provides a new plane curve, ``a selectible circle of pseudo-arcs", and answers some questions of Lewis.
An approach to symmetrization via polarization
Friedemann
Brock;
Alexander
Yu.
Solynin
1759-1796
Abstract: We prove that the Steiner symmetrization of a function can be approximated in $L^p ({\mathbb R}^n )$ by a sequence of very simple rearrangements which are called polarizations. This result is exploited to develop elementary proofs of many inequalities, including the isoperimetric inequality in Euclidean space. In this way we also obtain new symmetry results for solutions of some variational problems. Furthermore we compare the solutions of two boundary value problems, one of them having a "polarized" geometry and we show some pointwise inequalities between the solutions. This leads to new proofs of well-known functional inequalities which compare the solutions of two elliptic or parabolic problems, one of them having a "Steiner-symmetrized" geometry. The method also allows us to investigate the case of equality in the inequalities. Roughly speaking we prove that the equality sign is valid only if the original problem has the symmetrized geometry.
How parabolic free boundaries approximate hyperbolic fronts
Brian
H.
Gilding;
Roberto
Natalini;
Alberto
Tesei
1797-1824
Abstract: A rather complete study of the existence and qualitative behaviour of the boundaries of the support of solutions of the Cauchy problem for nonlinear first-order and second-order scalar conservation laws is presented. Among other properties, it is shown that, under appropriate assumptions, parabolic interfaces converge to hyperbolic ones in the vanishing viscosity limit.
Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems
Julián
López-Gómez
1825 - 1858
A Banach space block finitely universal for monotone bases
E.
Odell;
Th.
Schlumprecht
1859-1888
Abstract: A reflexive Banach space $X$ with a basis $(e_{i})$ is constructed having the property that every monotone basis is block finitely representable in each block basis of $X$.
Local product structure for Equilibrium States
Renaud
Leplaideur
1889-1912
Abstract: The usual way to study the local structure of Equilibrium State of an Axiom-A diffeomorphism or flow is to use the symbolic dynamic and to push results on the manifold. A new geometrical method is given. It consists in proving that Equilibrium States for Hölder-continuous functions are related to other Equilibrium States of some special sub-systems satisfying a sort of expansiveness. Using different kinds of extensions the local product structure of Gibbs-measure is proven.
Periodic Groups Covered by Transitive Subgroups of Finitary Permutations or by Irreducible Subgroups of Finitary Transformations
Felix
Leinen;
Orazio
Puglisi
1913-1934
Abstract: Let $\mathfrak{X}$ be either the class of all transitive groups of finitary permutations, or the class of all periodic irreducible finitary linear groups. We show that almost primitive $\mathfrak{X}$-groups are countably recognizable, while totally imprimitive $\mathfrak{X}$-groups are in general not countably recognizable. In addition we derive a structure theorem for groups all of whose countable subsets are contained in totally imprimitive $\mathfrak{X}$-subgroups. It turns out that totally imprimitive $p$-groups in the class $\mathfrak{X}$ are countably recognizable.
Simple and semisimple Lie algebras and codimension growth
Antonio
Giambruno;
Amitai
Regev;
Michail
V.
Zaicev
1935-1946
Abstract: We study the exponential growth of the codimensions $c_ n^{L}(B)$ of a finite dimensional Lie algebra $B$ over a field of characteristic zero. In the case when $B$ is semisimple we show that $\lim _{n\to \infty } \sqrt[{n}]{c_ n^{L}(B)}$ exists and, when $F$ is algebraically closed, is equal to the dimension of the largest simple summand of $B$. As a result we characterize central-simplicity: $B$ is central simple if and only if $\dim B = \lim _{n\to \infty } \sqrt[{n}]{c_ n^{L} (B)}$.
Correction to ``Harish-Chandra's Plancherel theorem for $\mathfrak{p}-adic groups''
Allan
J.
Silberger
1947-1949
A correction to ``Equivariant cohomology and lower bounds for chromatic numbers''
Igor
Kriz
1951-1952