Ramanujan's class invariants, Kronecker's limit formula, and modular equations
Bruce
C.
Berndt;
Heng Huat
Chan;
Liang-Cheng
Zhang
2125-2173
Abstract: In his notebooks, Ramanujan gave the values of over 100 class invariants which he had calculated. Many had been previously calculated by Heinrich Weber, but approximately half of them had not been heretofore determined. G. N. Watson wrote several papers devoted to the calculation of class invariants, but his methods were not entirely rigorous. Up until the past few years, eighteen of Ramanujan's class invariants remained to be verified. Five were verified by the authors in a recent paper. For the remaining class invariants, in each case, the associated imaginary quadratic field has class number 8, and moreover there are two classes per genus. The authors devised three methods to calculate these thirteen class invariants. The first depends upon Kronecker's limit formula, the second employs modular equations, and the third uses class field theory to make Watson's ``empirical method''rigorous.
A fixed point index for generalized inward mappings of condensing type
Kunquan
Lan;
Jeffrey
Webb
2175-2186
Abstract: A fixed point index is defined for mappings defined on a cone $K$ which do not necessarily take their values in $K$ but satisfy a weak type of boundary condition called generalized inward. This class strictly includes the well-known weakly inward class. New results for existence of multiple fixed points are established.
Resultants and the algebraicity of the join pairing on Chow varieties
Judith
Plümer
2187-2209
Abstract: The Chow/Van der Waerden approach to algebraic cycles via resultants is used to give a purely algebraic proof for the algebraicity of the complex suspension. The algebraicity of the join pairing on Chow varieties then follows. The approach implies a more algebraic proof of Lawson's complex suspension theorem in characteristic 0. The continuity of the action of the linear isometries operad on the group completion of the stable Chow variety is a consequence.
Proximity inequalities and bounds for the degree of invariant curves by foliations of \mathbb{P}_{\mathbb{C}}^2$
Antonio
Campillo;
Manuel
M.
Carnicer
2211-2228
Abstract: In this paper we prove that if $C$ is a reduced curve which is invariant by a foliation $\mathcal F$ in the complex projective plane then one has $\partial ^{\underline {\circ }} C\leq \partial^{\underline {\circ }} \mathcal F+2+a$ where $a$ is an integer obtained from a concrete problem of imposing singularities to projective plane curves. If $\mathcal F$ is nondicritical or if $C$ has only nodes as singularities, then one gets $a=0$ and we recover known bounds. We also prove proximity formulae for foliations and we use these formulae to give relations between local invariants of the curve and the foliation.
$L^2$-homology over traced *-algebras
William
L.
Paschke
2229-2251
Abstract: Given a unital complex *-algebra $A$, a tracial positive linear functional $\tau$ on $A$ that factors through a *-representation of $A$ on Hilbert space, and an $A$-module $M$ possessing a resolution by finitely generated projective $A$-modules, we construct homology spaces $H_k(A,\tau ,M)$ for $k = 0, 1, \ldots$. Each is a Hilbert space equipped with a *-representation of $A$, independent (up to unitary equivalence) of the given resolution of $M$. A short exact sequence of $A$-modules gives rise to a long weakly exact sequence of homology spaces. There is a Künneth formula for tensor products. The von Neumann dimension which is defined for $A$-invariant subspaces of $L^2(A,\tau )^n$ gives well-behaved Betti numbers and an Euler characteristic for $M$ with respect to $A$ and $\tau$.
Second variation of superminimal surfaces into self-dual Einstein four-manifolds
Sebastián
Montiel;
Francisco
Urbano
2253-2269
Abstract: The index of a compact orientable superminimal surface of a self-dual Einstein four-manifold $M$ with positive scalar curvature is computed in terms of its genus and area. Also a lower bound of its nullity is obtained. Applications to the cases $M=\mathbb {S}^4$ and $M=\mathbb {C}\mathbb {P}^2$ are given, characterizing the standard Veronese immersions and their twistor deformations as those with lowest index.
Discrete tomography: Determination of finite sets by X-rays
R.
J.
Gardner;
Peter
Gritzmann
2271-2295
Abstract: We study the determination of finite subsets of the integer lattice ${\Bbb Z}^n$, $n\ge 2$, by X-rays. In this context, an X-ray of a set in a direction $u$ gives the number of points in the set on each line parallel to $u$. For practical reasons, only X-rays in lattice directions, that is, directions parallel to a nonzero vector in the lattice, are permitted. By combining methods from algebraic number theory and convexity, we prove that there are four prescribed lattice directions such that convex subsets of ${\Bbb Z}^n$ (i.e., finite subsets $F$ with $F={\Bbb Z}^n\cap {\mathrm {conv}}\,F$) are determined, among all such sets, by their X-rays in these directions. We also show that three X-rays do not suffice for this purpose. This answers a question of Larry Shepp, and yields a stability result related to Hammer's X-ray problem. We further show that any set of seven prescribed mutually nonparallel lattice directions in ${\Bbb Z}^2$ have the property that convex subsets of ${\Bbb Z}^2$ are determined, among all such sets, by their X-rays in these directions. We also consider the use of orthogonal projections in the interactive technique of successive determination, in which the information from previous projections can be used in deciding the direction for the next projection. We obtain results for finite subsets of the integer lattice and also for arbitrary finite subsets of Euclidean space which are the best possible with respect to the numbers of projections used.
Doodle groups
Mikhail
Khovanov
2297-2315
Abstract: A doodle is a finite number of closed curves without triple intersections on an oriented surface. There is a ``fundamental'' group, naturally associated with a doodle. In this paper we study these groups, in particular, we show that fundamental groups of some doodles are automatic and give examples of doodles whose fundamental groups have non-trivial center.
Algebras associated to elliptic curves
Darin
R.
Stephenson
2317-2340
Abstract: This paper completes the classification of Artin-Schelter regular algebras of global dimension three. For algebras generated by elements of degree one this has been achieved by Artin, Schelter, Tate and Van den Bergh. We are therefore concerned with algebras which are not generated in degree one. We show that there exist some exceptional algebras, each of which has geometric data consisting of an elliptic curve together with an automorphism, just as in the case where the algebras are assumed to be generated in degree one. In particular, we study the elliptic algebras $A(+)$, $A(-)$, and $A({\mathbf {a}})$, where ${\mathbf {a}}\in \mathbb {P}^{2}$, which were first defined in an earlier paper. We omit a set $S\subset \mathbb {P}^2$ consisting of 11 specified points where the algebras $A({\mathbf {a}})$ become too degenerate to be regular. Theorem. Let $A$ represent $A(+)$, $A(-)$ or $A({\mathbf {a}})$, where ${\mathbf {a}} \in \mathbb {P}^2\setminus S$. Then $A$ is an Artin-Schelter regular algebra of global dimension three. Moreover, $A$ is a Noetherian domain with the same Hilbert series as the (appropriately graded) commutative polynomial ring in three variables. This, combined with our earlier results, completes the classification.
Unramified cohomology and Witt groups of anisotropic Pfister quadrics
R.
Sujatha
2341-2358
Abstract: The unramified Witt group of an anisotropic conic over a field $k$, with $char~k \neq 2$, defined by the form $\langle 1,-a,-b\rangle$ is known to be a quotient of the Witt group $W(k)$ of $k$ and isomorphic to $W( {k})/\langle 1,-a,-b,ab \rangle W( {k})$. We compute the unramified cohomology group $H^{3}_{nr}{k({C})}$, where $C$ is the three dimensional anisotropic quadric defined by the quadratic form $\langle 1,-a,-b,ab,-c\rangle$ over $k$. We use these computations to study the unramified Witt group of $C$.
A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics
Mark
Andrea A.
de Cataldo
2359-2370
Abstract: We prove that there are only finitely many families of codimension two nonsingular subvarieties of quadrics $\mathcal {Q}^{n}$ which are not of general type, for $n=5$ and $n\geq 7$. We prove a similar statement also for the case of higher codimension.
On singly-periodic minimal surfaces with planar ends
Joaquín
Pérez
2371-2389
Abstract: The spaces of nondegenerate properly embedded minimal surfaces in quotients of ${\mathbf R}^3$ by nontrivial translations or by screw motions with nontrivial rotational part, fixed finite topology and planar type ends, are endowed with natural structures of finite dimensional real analytic manifolds. This nondegeneracy is defined in terms of Jacobi functions. Riemann's minimal examples are characterized as the only nondegenerate surfaces with genus one in their corresponding spaces. We also give natural immersions of these spaces into certain complex Euclidean spaces which turn out to be Lagrangian immersions with respect to the standard symplectic structures.
The stretch of a foliation and geometric superrigidity
Raul
Quiroga-Barranco
2391-2426
Abstract: We consider compact smooth foliated manifolds with leaves isometrically covered by a fixed symmetric space of noncompact type. Such objects can be considered as compact models for the geometry of the symmetric space. Based on this we formulate and solve a geometric superrigidity problem for foliations that seeks the existence of suitable isometric totally geodesic immersions. To achieve this we consider the heat flow equation along the leaves of a foliation, a Bochner formula on foliations and a geometric invariant for foliations with leafwise Riemannian metrics called the stretch. We obtain as applications a metric rigidity theorem for foliations and a rigidity type result for Riemannian manifolds whose geometry is only partially symmetric.
On roots of random polynomials
Ildar
Ibragimov;
Ofer
Zeitouni
2427-2441
Abstract: We study the distribution of the complex roots of random polynomials of degree $n$ with i.i.d. coefficients. Using techniques related to Rice's treatment of the real roots question, we derive, under appropriate moment and regularity conditions, an exact formula for the average density of this distribution, which yields appropriate limit average densities. Further, using a different technique, we prove limit distribution results for coefficients in the domain of attraction of the stable law.
Some uniqueness and exact multiplicity results for a predator-prey model
Yihong
Du;
Yuan
Lou
2443-2475
Abstract: In this paper, we consider positive solutions of a predator-prey model with diffusion and under homogeneous Dirichlet boundary conditions. It turns out that a certain parameter $m$ in this model plays a very important role. A good understanding of the existence, stability and number of positive solutions is gained when $m$ is large. In particular, we obtain various results on the exact number of positive solutions. Our results for large $m$ reveal interesting contrast with that for the well-studied case $m=0$, i.e., the classical Lotka-Volterra predator-prey model.
Degenerations of K3 surfaces in projective space
Francisco
Javier
Gallego;
B.
P.
Purnaprajna
2477-2492
Abstract: The purpose of this article is to study a certain kind of numerical K3 surfaces, the so-called K3 carpets. These are double structures on rational normal scrolls with trivial dualizing sheaf and irregularity $0$. As is deduced from our study, K3 carpets can be obtained as degenerations of smooth K3 surfaces. We also study the Hilbert scheme near the locus parametrizing K3 carpets, characterizing those K3 carpets whose corresponding Hilbert point is smooth. Contrary to the case of canonical ribbons, not all K3 carpets are smooth points of the Hilbert scheme.
The Floer homotopy type of height functions on complex Grassmann manifolds
David
E.
Hurtubise
2493-2505
Abstract: A family of Floer functions on the infinite dimensional complex Grassmann manifold is defined by taking direct limits of height functions on adjoint orbits of unitary groups. The Floer cohomology of a generic function in the family is computed using the Schubert calculus. The Floer homotopy type of this function is computed and the Floer cohomology which was computed algebraically is recovered from the Floer homotopy type. Certain non-generic elements of this family of Floer functions were shown to be related to the symplectic action functional on the universal cover of the loop space of a finite dimensional complex Grassmann manifold in the author's preprint The Floer homotopy type of complex Grassmann manifolds.
Differential operators on Stanley-Reisner rings
J.
R.
Tripp
2507-2523
Abstract: Let $k$ be an algebraically closed field of characteristic zero, and let $R=k[x_{1},\dots ,x_{n}]$ be a polynomial ring. Suppose that $I$ is an ideal in $R$ that may be generated by monomials. We investigate the ring of differential operators $\mathcal {D}(R/I)$ on the ring $R/I$, and $\mathcal {I}_{R}(I)$, the idealiser of $I$ in $R$. We show that $\mathcal {D}(R/I)$ and $\mathcal {I}_{R}(I)$ are always right Noetherian rings. If $I$ is a square-free monomial ideal then we also identify all the two-sided ideals of $\mathcal {I}_{R}(I)$. To each simplicial complex $\Delta$ on $V=\{v_{1},\dots ,v_{n}\}$ there is a corresponding square-free monomial ideal $I_{\Delta }$, and the Stanley-Reisner ring associated to $\Delta$ is defined to be $k[\Delta ]=R/I_{\Delta }$. We find necessary and sufficient conditions on $\Delta$ for $\mathcal {D}(k[\Delta ])$ to be left Noetherian.
Herz-Schur multipliers and weakly almost periodic functions on locally compact groups
Guangwu
Xu
2525-2536
Abstract: For a locally compact group $G$ and $1<p<\infty$, let $A_{p}(G)$ be the Herz-Figà-Talamanca algebra and $B_{p}(G)$ the Herz-Schur multipliers of $G$, and $MA_{p}(G)$ the multipliers of $A_{p}(G)$. Let $W(G)$ be the algebra of continuous weakly almost periodic functions on $G$. In this paper, we show that (1), if $G$ is a noncompact nilpotent group or a noncompact [IN]-group, then $W(G)/B_{p}(G)^{-}$ contains a linear isometric copy of $l^{\infty }({\mathbb {N}})$; (2), for a noncommutative free group $F, B_{p}(F)$ is a proper subset of ${MA_{p}(F)\cap {W(F)}}$.