Vessiot structure for manifolds of $(p,q)$-hyperbolic type: Darboux integrability and symmetry
Peter
J.
Vassiliou
1705-1739
Abstract: It is well known that if a scalar second order hyperbolic partial differential equation in two independent variables is Darboux integrable, then its local Cauchy problem may be solved by ordinary differential equations. In addition, such an equation has infinitely many non-trivial conservation laws. Moreover, Darboux integrable equations have properties in common with infinite dimensional completely integrable systems. In this paper we employ a geometric object intrinsically associated with any hyperbolic partial differential equation, its hyperbolic structure, to study the Darboux integrability of the class $E$ of semilinear second order hyperbolic partial differential equations in one dependent and two independent variables. It is shown that the problem of classifying the Darboux integrable equations in $E$ contains, as a subproblem, that of classifying the manifolds of $(p,q)$-hyperbolic type of rank 4 and dimension $2k+3$, $k\ge2$; $p=2,q\ge 2$. In turn, it is shown that the problem of classifying these manifolds in the two (lowest) cases $(p,q)=(2,2),(2,3)$ contains, as a subproblem, the classification problem for Lie groups. This generalizes classical results of E. Vessiot. The main result is that if an equation in $E$ is (2,2)- or (2,3)-Darboux integrable on the $k$-jets, $k\ge 2$, then its intrinsic hyperbolic structure admits a Lie group of symmetries of dimension $2k-1$ or $2k-2$, respectively. It follows that part of the moduli space for the Darboux integrable equations in $E$ is determined by isomorphism classes of Lie groups. The Lie group in question is the group of automorphisms of the characteristic systems of the given equation which leaves invariant the foliation induced by the characteristic (or, Riemann) invariants of the equation, the tangential characteristic symmetries. The isomorphism class of the tangential characteristic symmetries is a contact invariant of the corresponding Darboux integrable partial differential equation.
A finiteness theorem for harmonic maps into Hilbert Grassmannians
Rodrigo
P.
Gomez
1741-1753
Abstract: In this article we demonstrate that every harmonic map from a closed Riemannian manifold into a Hilbert Grassmannian has image contained within a finite-dimensional Grassmannian.
Pfaffian systems with derived length one. The class of flag systems
María
A.
Cañadas-Pinedo;
Ceferino
Ruiz
1755-1766
Abstract: The incidence relations between a Pfaffian system and the characteristic system of its first derived system lead to obtain a characterization of Pfaffian systems with derived length one. Also, for flag systems, several properties are studied. In particular, an intrinsic proof of a result which determines the class of a system and of all the derived systems is given.
A new affine invariant for polytopes and Schneider's projection problem
Erwin
Lutwak;
Deane
Yang;
Gaoyong
Zhang
1767-1779
Abstract: New affine invariant functionals for convex polytopes are introduced. Some sharp affine isoperimetric inequalities are established for the new functionals. These new inequalities lead to fairly strong volume estimates for projection bodies. Two of the new affine isoperimetric inequalities are extensions of Ball's reverse isoperimetric inequalities.
On the number of ${L}_{\infty\omega_1}$-equivalent non-isomorphic models
Saharon
Shelah;
Pauli
Väisänen
1781-1817
Abstract: We prove that if $\operatorname{ZF}$ is consistent then $\operatorname{ZFC} + \operatorname{GCH}$ is consistent with the following statement: There is for every $k < \omega$ a model of cardinality $\aleph_1$ which is $L_{\infty{\omega_{1}}}$-equivalent to exactly $k$non-isomorphic models of cardinality $\aleph_1$. In order to get this result we introduce ladder systems and colourings different from the ``standard'' counterparts, and prove the following purely combinatorial result: For each prime number $p$ and positive integer $m$ it is consistent with $\operatorname{ZFC} + \operatorname{GHC}$ that there is a ``good'' ladder system having exactly $p^m$ pairwise nonequivalent colourings.
A universal continuum of weight $\aleph$
Alan
Dow;
Klaas
Pieter
Hart
1819-1838
Abstract: We prove that every continuum of weight $\aleph_1$ is a continuous image of the Cech-Stone-remainder $R^*$ of the real line. It follows that under $\mathsf{CH}$ the remainder of the half line $[0,\infty)$ is universal among the continua of weight $\mathfrak{c}$-- universal in the `mapping onto' sense. We complement this result by showing that 1) under $\mathsf{MA}$ every continuum of weight less than $\mathfrak{c}$ is a continuous image of $R^*$, 2) in the Cohen model the long segment of length $\omega_2+1$ is not a continuous image of $R^*$, and 3) $\mathsf{PFA}$ implies that $I_u$ is not a continuous image of $R^*$, whenever $u$ is a $\mathfrak{c}$-saturated ultrafilter. We also show that a universal continuum can be gotten from a $\mathfrak{c}$-saturated ultrafilter on $\omega$, and that it is consistent that there is no universal continuum of weight $\mathfrak{c}$.
Semi-dualizing complexes and their Auslander categories
Lars
Winther
Christensen
1839-1883
Abstract: Let $R$ be a commutative Noetherian ring. We study $R$-modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count among them such, potentially, diverse objects as dualizing complexes for $R$ on one side, and on the other, the ring itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes ``lying between'' these extremes is incentive.
Geometric interpretation of tight closure and test ideals
Nobuo
Hara
1885-1906
Abstract: We study tight closure and test ideals in rings of characteristic $p \gg 0$ using resolution of singularities. The notions of $F$-rational and $F$-regular rings are defined via tight closure, and they are known to correspond with rational and log terminal singularities, respectively. In this paper, we reformulate this correspondence by means of the notion of the test ideal, and generalize it to wider classes of singularities. The test ideal is the annihilator of the tight closure relations and plays a crucial role in the tight closure theory. It is proved that, in a normal $\mathbb Q$-Gorenstein ring of characteristic $p \gg 0$, the test ideal is equal to so-called the multiplier ideal, which is an important ideal in algebraic geometry. This is proved in more general form, and to do this we study the behavior of the test ideal and the tight closure of the zero submodule in certain local cohomology modules under cyclic covering. We reinterpret the results also for graded rings.
A brief proof of a maximal rank theorem for generic double points in projective space
Karen
A.
Chandler
1907-1920
Abstract: We give a simple proof of the following theorem of J. Alexander and A. Hirschowitz: Given a general set of points in projective space, the homogeneous ideal of polynomials that are singular at these points has the expected dimension in each degree of 4 and higher, except in 3 cases.
Arithmetic discriminants and morphisms of curves
Xiangjun
Song;
Thomas
J.
Tucker
1921-1936
Abstract: This paper deals with upper bounds on arithmetic discriminants of algebraic points on curves over number fields. It is shown, via a result of Zhang, that the arithmetic discriminants of algebraic points that are not pull-backs of rational points on the projective line are smaller than the arithmetic discriminants of families of linearly equivalent algebraic points. It is also shown that bounds on the arithmetic discriminant yield information about how the fields of definition $k(P)$ and $k(f(P))$ differ when $P$ is an algebraic point on a curve $C$ and $k(P) \not= k(f(P))$, with at most finitely many exceptions, whenever the degrees of $P$ and $f$ are sufficiently small, relative to the difference between the genera $g(C)$ and $g(C')$. The paper concludes with a detailed analysis of the arithmetic discriminants of quadratic points on bi-elliptic curves of genus 2.
Essential cohomology and extraspecial $p$-groups
Pham
Anh
Minh
1937-1957
Abstract: Let $p$ be an odd prime number and let $G$ be an extraspecial $p$-group. The purpose of the paper is to show that $G$ has no non-zero essential mod-$p$ cohomology (and in fact that $H^{*}(G,\mathbb{F}_{p})$ is Cohen-Macaulay) if and only if $\vert G\vert=27$ and $exp(G)=3$.
Spaces of rational loops on a real projective space
Jacob
Mostovoy
1959-1970
Abstract: We show that the loop spaces on real projective spaces are topologically approximated by the spaces of rational maps $\mathbf{RP}^{1}\rightarrow \mathbf{RP}^{n}$. As a byproduct of our constructions we obtain an interpretation of the Kronecker characteristic (degree) of an ornament via particle spaces.
The limits of refinable functions
Gilbert
Strang;
Ding-Xuan
Zhou
1971-1984
Abstract: A function $\phi$ is refinable ( $\phi \in S$) if it is in the closed span of $\{\phi (2x-k)\}$. This set $S$ is not closed in $L_{2}(\mathbb{R})$, and we characterize its closure. A necessary and sufficient condition for a function to be refinable is presented without any information on the refinement mask. The Fourier transform of every $f\in \overline{S} \setminus S$ vanishes on a set of positive measure. As an example, we show that all functions with Fourier transform supported in $[-{\frac{4}{3}}\pi , {\frac{4}{3}}\pi ]$ are the limits of refinable functions. The relation between a refinable function and its mask is studied, and nonuniqueness is proved. For inhomogeneous refinement equations we determine when a solution is refinable. This result is used to investigate refinable components of multiple refinable functions. Finally, we investigate fully refinable functions for which all translates (by any real number) are refinable.
Ahiezer-Kac type Fredholm determinant asymptotics for convolution operators with rational symbols
Sergio
Albeverio;
Konstantin
A.
Makarov
1985-1993
Abstract: Fredholm determinant asymptotics of convolution operators on large finite intervals with rational symbols having real zeros are studied. The explicit asymptotic formulae obtained can be considered as a direct extension of the Ahiezer-Kac formula to symbols with real zeros.
Vector $A_2$ weights and a Hardy-Littlewood maximal function
Michael
Christ;
Michael
Goldberg
1995-2002
Abstract: An analogue of the Hardy-Littlewood maximal function is introduced, for functions taking values in finite-dimensional Hilbert spaces. It is shown to be $L^2$ bounded with respect to weights in the class $A_2$ of Treil, thereby extending a theorem of Muckenhoupt from the scalar to the vector case.
Vanishing of the third simplicial cohomology group of $l^1(\mathbf{Z}_+)$
Frédéric
Gourdeau;
Michael
C.
White
2003-2017
Abstract: We show that $\mathcal{I}$ is the non-unital Banach algebra $l^1(\mathbf{N})$, and then prove that
Unconditional structures of weakly null sequences
S.
A.
Argyros;
I.
Gasparis
2019-2058
Abstract: The following dichotomy is established for a normalized weakly null sequence in a Banach space: Either every subsequence admits a convex block subsequence equivalent to the unit vector basis of $c_0$, or there exists a subsequence which is boundedly convexly complete.
Irreducible partitions and the construction of quasi-measures
D.
J.
Grubb
2059-2072
Abstract: A quasi-measure is a non-subadditive measure defined on only open or closed subsets of a compact Hausdorf space. We investigate the nature of irreducible partitions as defined by Aarnes and use the results to construct quasi-measures when $g(X)=1$. The cohomology ring is an important tool for this investigation.
A product formula for spherical representations of a group of automorphisms of a homogeneous tree, II
Donald
I.
Cartwright;
Gabriella
Kuhn
2073-2090
Abstract: Let $G=\text{Aut}(T)$ be the group of automorphisms of a homogeneous tree $T$and let $\pi$ be the tensor product of two spherical irreducible unitary representations of $G$. We complete the explicit decomposition of $\pi$commenced in part I of this paper, by describing the discrete series representations of $G$ which appear as subrepresentations of $\pi$.
Varieties of uniserial representations IV. Kinship to geometric quotients
Klaus
Bongartz;
Birge
Huisgen-Zimmermann
2091-2113
Abstract: Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field, and ${\mathbb{S} }$ a finite sequence of simple left $\Lambda$-modules. Quasiprojective subvarieties of Grassmannians, distinguished by accessible affine open covers, were introduced by the authors for use in classifying the uniserial representations of $\Lambda$ having sequence ${\mathbb{S} }$ of consecutive composition factors. Our principal objectives here are threefold: One is to prove these varieties to be `good approximations'--in a sense to be made precise--to geometric quotients of the (very large) classical affine varieties $\operatorname{Mod-Uni} ({\mathbb{S} })$ parametrizing the pertinent uniserial representations, modulo the usual conjugation action of the general linear group. We show that, to some extent, this fills the information gap left open by the frequent non-existence of such quotients. A second goal is that of facilitating the transfer of information among the `host' varieties into which the considered quasi-projective, respectively affine, uniserial varieties are embedded. For that purpose, a general correspondence is established, between Grassmannian varieties of submodules of a projective module $P$ on one hand, and classical varieties of factor modules of $P$ on the other. Our findings are applied towards the third objective, concerning the existence of geometric quotients. The main results are then exploited in a representation-theoretic context: Among other consequences, they yield a geometric characterization of the algebras of finite uniserial type which supplements existing descriptions, but is cleaner and more readily checkable.
Noncrossed products over $k_{\mathfrak{p}}(t)$
Eric
S.
Brussel
2115-2129
Abstract: Noncrossed product division algebras are constructed over rational function fields $k(t)$ over number fields $k$ by lifting from arithmetic completions $k(t)_{\mathfrak{p}}$. The existence of noncrossed products over $\mathfrak{p}$-adic rational function fields $k_{\mathfrak{p}}(t)$ is proved as a corollary.