The residual spectrum of $U(2,2)$
Takyua
Kon-No
1285 - 1358
Covers of algebraic varieties III. The discriminant of a cover of degree 4 and the trigonal construction
G.
Casnati
1359-1378
Abstract: For each Gorenstein cover $\varrho \colon X\to Y$ of degree $4$ we define a scheme $\Delta (X)$ and a generically finite map $\Delta (\varrho )\colon \Delta (X)\to Y$ of degree $3$ called the discriminant of $\varrho$. Using this construction we deal with smooth degree $4$ covers $\varrho \colon X\to {{\mathbb P}^{n}_{\mathbb{C}}}$ with $n\ge 5$. Moreover we also generalize the trigonal construction of S. Recillas.
Geometric families of constant reductions and the Skolem property
Barry
Green
1379-1393
Abstract: Let $F|K$ be a function field in one variable and $\mathcal V$ be a family of independent valuations of the constant field $K.$ Given $v\in \mathcal V ,$ a valuation prolongation $\mathrm v$ to $F$ is called a constant reduction if the residue fields $F\mathrm v |Kv$ again form a function field of one variable. Suppose $t\in F$ is a non-constant function, and for each $v\in \mathcal V$ let $V_{t}$ be the set of all prolongations of the Gauß valuation $v_{t}$ on $K(t)$ to $F.$ The union of the sets $V_{t}$ over all $v\in \mathcal V$ is denoted by ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}}}_{t}.$ The aim of this paper is to study families of constant reductions ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}}}$ of $F$ prolonging the valuations of $\mathcal V$ and the criterion for them to be principal, that is to be sets of the type ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}}}_{t}.$ The main result we prove is that if either $\mathcal V$ is finite and each $v\in \mathcal V$ has rational rank one and residue field algebraic over a finite field, or if $\mathcal V$ is any set of non-archimedean valuations of a global field $K$ satisfying the strong approximation property, then each geometric family of constant reductions ${{\mathchoice {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\textstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}} {{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}{{\hbox {{\mathsurround =0pt{\setbox 0=\hbox {${\scriptscriptstyle {V}}$}\setbox 1=\hbox {\hbox to.1pt{}\copy 0}\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1\kern -\wd 0\copy 1}}}}}} }$ prolonging $\mathcal V$ is principal. We also relate this result to the Skolem property for the existence of $\mathcal V$-integral points on varieties over $K,$ and Rumely's existence theorem. As an application we give a birational characterization of arithmetic surfaces $\mathcal X /S$ in terms of the generic points of the closed fibre. The characterization we give implies the existence of finite morphisms to $\mathbb P ^{1}_{S}.$
Extreme points of the distance function on convex surfaces
Tudor
Zamfirescu
1395-1406
Abstract: We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus' conjecture that ``almost all" points admit a single farthest point on the surface.
The behavior of the heat operator on weighted Sobolev spaces
G.
N.
Hile;
C.
P.
Mawata
1407-1428
Abstract: Denoting by ${\mathcal{H}}$ the heat operator in $R^{n+1}$, we investigate its properties as a bounded operator from one weighted Sobolev space to another. Our main result gives conditions on the weights under which ${\mathcal{H}}$ is an injection, a surjection, or an isomorphism. We also describe the range and kernel of ${\mathcal{H}}$ in all the cases. Our results are analogous to those obtained by R. C. McOwen for the Laplace operator in $R^{n}$.
Fox calculus, symplectic forms, and moduli spaces
Valentino
Zocca
1429-1466
Abstract: An ``open pre-symplectic form'' on surfaces with boundary and glueing formulae are provided to symplectically integrate the symplectic form on the deformation space of representations of the fundamental group of a Riemann surface into a reductive Lie group $G$.
Berezin's quantization on flag manifolds and spherical modules
Alexander
V.
Karabegov
1467-1479
Abstract: We show that the theory of spherical Harish-Chandra modules naturally gives rise to Berezin's symbol quantization on generalized flag manifolds. It provides constructions of symbol algebras and of their representations for covariant and contravariant symbols, and also for symbols which so far have no explicit definition. For all these symbol algebras we give a general proof of the correspondence principle.
Limit theorems for random transformations and processes in random environments
Yuri
Kifer
1481-1518
Abstract: I derive general relativized central limit theorems and laws of iterated logarithm for random transformations both via certain mixing assumptions and via the martingale differences approach. The results are applied to Markov chains in random environments, random subshifts of finite type, and random expanding in average transformations where I show that the conditions of the general theorems are satisfied and so the corresponding (fiberwise) central limit theorems and laws of iterated logarithm hold true in these cases. I consider also a continuous time version of such limit theorems for random suspensions which are continuous time random dynamical systems.
The trace of jet space $J^k\Lambda^\omega$ to an arbitrary closed subset of $\mathbb{R}^n$
Yuri
Brudnyi;
Pavel
Shvartsman
1519-1553
Abstract: The classical Whitney extension theorem describes the trace $J^k|_X$ of the space of $k$-jets generated by functions from $C^k(\mathbb R^n)$ to an arbitrary closed subset $X\subset\mathbb R^n$. It establishes existence of a bounded linear extension operator as well. In this paper we investigate a similar problem for the space $C^k\Lambda^\omega(\mathbb R^n)$ of functions whose higher derivatives satisfy the Zygmund condition with majorant $\omega$. The main result states that the vector function $\vec f=(f_\alpha \colon X\to\mathbb R)_{|\alpha |\le k}$ belongs to the corresponding trace space if the trace $\vec f|_Y$ to every subset $Y\subset X$ of cardinality $3\cdot 2^\ell$, where $\ell=(\begin{smallmatrix}n+k-1 k+1\end{smallmatrix})$, can be extended to a function $f_Y\in C^k\Lambda^\omega(\mathbb R^n)$ and $\sup _Y|f_Y|_{C^k\Lambda^\omega}<\infty$. The number $3\cdot 2^l$ generally speaking cannot be reduced. The Whitney theorem can be reformulated in this way as well, but with a two-pointed subset $Y\subset X$. The approach is based on the theory of local polynomial approximations and a result on Lipschitz selections of multivalued mappings.
Linearization, Dold-Puppe stabilization, and Mac Lane's $Q$-construction
Brenda
Johnson;
Randy
McCarthy
1555-1593
Abstract: In this paper we study linear functors, i.e., functors of chain complexes of modules which preserve direct sums up to quasi-isomorphism, in order to lay the foundation for a further study of the Goodwillie calculus in this setting. We compare the methods of Dold and Puppe, Mac Lane, and Goodwillie for producing linear approximations to functors, and establish conditions under which these methods are equivalent. In addition, we classify linear functors in terms of modules over an explicit differential graded algebra. Several classical results involving Dold-Puppe stabilization and Mac Lane's $Q$-construction are extended or given new proofs.
Commuting Toeplitz operators with pluriharmonic symbols
Dechao
Zheng
1595-1618
Abstract: By making use of $\mathcal M$-harmonic function theory, we characterize commuting Toeplitz operators with bounded pluriharmonic symbols on the Bergman space of the unit ball or on the Hardy space of the unit sphere in $n$-dimensional complex space.
$n$-unisolvent sets and flat incidence structures
Burkard
Polster
1619-1641
Abstract: For the past forty years or so topological incidence geometers and mathematicians interested in interpolation have been studying very similar objects. Nevertheless no communication between these two groups of mathematicians seems to have taken place during that time. The main goal of this paper is to draw attention to this fact and to demonstrate that by combining results from both areas it is possible to gain many new insights about the fundamentals of both areas. In particular, we establish the existence of nested orthogonal arrays of strength $n$, for short nested $n$-OAs, that are natural generalizations of flat affine planes and flat Laguerre planes. These incidence structures have point sets that are ``flat'' topological spaces like the Möbius strip, the cylinder, and strips of the form $I \times \mathbb{R}$, where $I$ is an interval of $\mathbb{R}$. Their circles (or lines) are subsets of the point sets homeomorphic to the circle in the first two cases and homeomorphic to $I$ in the last case. Our orthogonal arrays of strength $n$ arise from $n$-unisolvent sets of half-periodic functions, $n$-unisolvent sets of periodic functions, and $n$-unisolvent sets of functions $I\to \mathbb{R}$, respectively. Associated with every point $p$ of a nested $n$-OA, $n>1$, is a nested $(n-1)$-OA-the derived $(n-1)$-OA at the point $p$. We discover that, in our examples that arise from $n$-unisolvent sets of $n-1$ times differentiable functions that solve the Hermite interpolation problem, deriving in our geometrical sense coincides with deriving in the analytical sense.
Geometric properties of the double-point divisor
Bo
Ilic
1643-1661
Abstract: The locus of double points obtained by projecting a variety $X^{n} \subset \mathbf P^N$ to a hypersurface in $\mathbf{P}^{n+1}$ moves in a linear system which is shown to be ample if and only if $X$ is not an isomorphic projection of a Roth variety. Such Roth varieties are shown to exist, and some of their geometric properties are determined.
Comultiplications on free groups and wedges of circles
Martin
Arkowitz;
Mauricio
Gutierrez
1663-1680
Abstract: By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group $F$. We consider individual comultiplications on $F$ and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of $F$. For a comultiplication $m$ of $F$ we define a subset $\Delta _{m} \subseteq F$ of quasi-diagonal elements which is basic to our investigation of associativity. The subset $\Delta _{m}$ can be determined algorithmically and contains the set of diagonal elements $D_{m}$. We show that $D_{m}$ is a basis for the largest subgroup $A_{m}$ of $F$ on which $m$ is associative and that $A_{m}$ is a free factor of $F$. We also give necessary and sufficient conditions for a comultiplication $m$ on $F$ to be a coloop in terms of the Fox derivatives of $m$ with respect to a basis of $F$. In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group $\operatorname{Aut} F$ on the set of comultiplications of $F$. We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.
Algebraic invariant curves for the Liénard equation
Henryk
Zoladek
1681-1701
Abstract: Odani has shown that if $\deg g\leq \deg f$ then after deleting some trivial cases the polynomial system $\dot {x}=y,\,\,\dot {y}=-f(x)y-g(x)$ does not have any algebraic invariant curve. Here we almost completely solve the problem of algebraic invariant curves and algebraic limit cycles of this system for all values of $\deg f$ and $\deg g$. We give also a simple presentation of Yablonsky's example of a quartic limit cycle in a quadratic system.
A new degree bound for vector invariants of symmetric groups
P.
Fleischmann
1703-1712
Abstract: Let $R$ be a commutative ring, $V$ a finitely generated free $R$-module and $G\le GL_R(V)$ a finite group acting naturally on the graded symmetric algebra $A=S(V)$. Let $\beta(V,G)$ denote the minimal number $m$, such that the ring $A^G$ of invariants can be generated by finitely many elements of degree at most $m$. For $G=\Sigma _n$ and $V(n,k)$, the $k$-fold direct sum of the natural permutation module, one knows that $\beta(V(n,k),\Sigma _n) \le n$, provided that $n!$ is invertible in $R$. This was used by E. Noether to prove $\beta(V,G) \le |G|$ if $|G|! \in R^*$. In this paper we prove $\beta(V(n,k),\Sigma _n) \le max\{n,k(n-1)\}$ for arbitrary commutative rings $R$ and show equality for $n=p^s$ a prime power and $R = \mathbb Z$ or any ring with $n\cdot 1_R=0$. Our results imply \begin{equation*}\beta(V,G)\le max\{|G|, \operatorname{rank}(V)(|G|-1)\}\end{equation*} for any ring with $|G| \in R^*$.