On maximally elliptic singularities
Stephen Shing Toung
Yau
269-329
Abstract: Let p be the unique singularity of a normal two-dimensional Stein space V. Let m be the maximal ideal in $ _V{\mathcal{O}_p}$, the local ring of germs of holomorphic functions at p. We first define the maximal ideal cycle which serves to identify the maximal ideal. We give an ``upper'' estimate for maximal ideal cycle in terms of the canonical divisor which is computable via the topological information, i.e., the weighted dual graph of the singularity. Let $M \to V$ be a resolution of V. It is known that $h\, = \,\dim \,{H^1}(M,\,\mathcal{O})$ is independent of resolution. Rational singularities in the sense of M. Artin are equivalent to $ h\, = \,0$. Minimally elliptic singularity in the sense of Laufer is equivalent to saying that $h\, = \,1$ and $ _V{\mathcal{O}_p}$ is Gorenstein. In this paper we develop a theory for a general class of weakly elliptic singularities which satisfy a maximality condition. Maximally elliptic singularities may have h arbitrarily large. Also minimally elliptic singlarities are maximally elliptic singularities. We prove that maximally elliptic singularities are Gorenstein singularities. We are able to identify the maximal ideal. Therefore, the important invariants of the singularities (such as multiplicity) are extracted from the topological information. For weakly elliptic singularities we introduce a new concept called ``elliptic sequence". This elliptic sequence is defined purely topologically, i.e., it can be computed explicitly via the intersection matrix. We prove that --K, where K is the canonical divisor, is equal to the summation of the elliptic sequence. Moreover, the analytic data $\dim \,{H^1}(M,\,\mathcal{O})$ is bounded by the topological data, the length of elliptic sequence. We also prove that $ h\, = \,2$ and $_V{\mathcal{O}_p}$ Gorenstein implies that the singularity is weakly elliptic.
The dependence of the generalized Radon transform on defining measures
Eric Todd
Quinto
331-346
Abstract: Guillemin proved that the generalized Radon transform R and its dual ${R^t}$ are Fourier integral operators and that $ {R^t}R$ is an elliptic pseudodifferential operator. In this paper we investigate the dependence of the Radon transform on the defining measures. In the general case we calculate the symbol of $ {R^t}R$ as a pseudodifferential operator in terms of the measures and give a necessary condition on the defining measures for $ {R^t}R$ to be invertible by a differential operator. Then we examine the Radon transform on points and hyperplanes in ${\textbf{R}^n}$ with general measures and we calculate the symbol of ${R^t}R$ in terms of the defining measures. Finally, if ${R^t}R$ is a translation invariant operator on ${\textbf{R}^n}$ then we prove that ${R^t}R$ is invertible and that our condition is equivalent to $ {({R^t}R)^{ - 1}}$ being a differential operator.
The lattice of $l$-group varieties
J. E.
Smith
347-357
Abstract: For any type of abstract algebra, a variety is an equationally defined class of such algebras. Recently, attempts have been made to study varieties of lattice-ordered groups (l-groups). Martinez has shown that the set L of all l-group varieties forms a lattice under set inclusion with a compatible associative multiplication. Certain varieties ${\mathcal{S}_p}$ (p prime) have been proved by Scrimger to be minimal nonabelian varieties in L. In the present paper, it is shown that these varieties can be used to produce varieties minimal with respect to properly containing various other varieties in L. Also discussed are the relations among the ${\mathcal{S}_n}\,(n\, \in \,N)$, and it is established that all infinite collections of the ${\mathcal{S}_n}$ have the same least upper bound in L. Martinez has also classified l-groups using torsion classes, a generalization of the idea of varieties. It is proved here that L is not a sublattice of T, the lattice of torsion classes.
$p$-adic gamma functions and Dwork cohomology
Maurizio
Boyarsky
359-369
Abstract: The relations of Gross and Koblitz between gauss sums and the p-adic gamma function is reexamined from the point of view of Dwork's formulation of p-adic cohomology. Some higher dimensional generalizations are proposed.
Decomposition of nonnegative group-monotone matrices
S. K.
Jain;
Edward K.
Kwak;
V. K.
Goel
371-385
Abstract: A decomposition of nonnegative matrices with nonnegative group inverses has been obtained. This decomposition provides a new approach to the solution of problems relating to nonnegative matrices with nonnegative group inverses. As a consequence, a number of results are derived. Our results, among other things, answer a question of Berman, extend the theorems of Berman and Plemmons, DeMarr and Flor.
A spectral theorem for $J$-nonnegative operators
Bernard N.
Harvey
387-396
Abstract: A J-space is a Hilbert space with the usual inner product denoted $ [x,y]$ and an indefinite inner product defined by $(x,y)\, =\, [Jx,y]$ where J is a bounded selfadjoint operator whose square is the identity. We define a J-adjoint ${T^ + }$ of an operator T with respect to the indefinite inner product in the same way as the regular adjoint $T^{\ast}$ is defined with respect to $[x,y]$. We say T is J-selfadjoint if $T = {T^ + }$. An operator-valued function is called a J-spectral function with critical point zero if it is defined for all $t \ne 0$, is bounded, J-selfadjoint and has the properties of a resolution of the identity on its domain. It has been proved by M. G. Krein and Ju. P. Smul'jan that bounded Jselfadjoint operators A with $(Ax,x) \geqslant 0$ for all x can be represented as a strongly convergent improper integral of t with respect to a J-spectral function with critical point zero plus a nilpotent of index 2. Further, the product of the nilpotent with the J-spectral function on intervals not containing zero is zero. The present paper extends this theory to the unbounded case. We show that unbounded J-selfadjoint operators with $(Ax,x) \geqslant 0$ are a direct sum of an operator of the above mentioned type and the inverse of a bounded operator of the same type whose nilpotent part is zero.
A construction of uncountably many weak von Neumann transformations
Karl
David
397-410
Abstract: We define weak von Neumann transformations and discuss some of their properties, using several examples of countable classes of these transformations. Then we construct an uncountable class by the cutting-and-stacking method. We show that each member of this class is ergodic and has zero entropy.
The free boundary for elastic-plastic torsion problems
Avner
Friedman;
Gianni A.
Pozzi
411-425
Abstract: Consider the variational inequality: Find $ u\, \in\, K$ such that $\int_Q {\nabla u \cdot\, \nabla\, (\upsilon\, -\, u)\, \geqslant\, \mu\, \int_Q\, {(\upsilon\, -\, u)\,(\mu\, >\, 0)} }$ for any $\upsilon\, \in\, K$, where $K\, =\, \{ w\, \in\, H_0^1(Q);\,\left\vert {\nabla\, w} \right\vert\, \leqslant\, 1\}$ and Q is a 2-dimensional simply connected domain in $ {R^2}$ with piecewise $ {C^3}$ boundary. The solution u represents the stress function in a torsion problem of an elastic-plastic bar with cross section Q. The sets $E\, =\, \{ x\, \in\, Q;\,\left\vert {\nabla\, u(x)} \right\vert\, <\, 1\} $, $P\, =\, \{ x\, \in\, Q;\,\left\vert {\nabla\, u(x)} \right\vert\, =\, 1\}$ are the elastic and plastic sets respectively. The purpose of this paper is to study the free boundary $\partial E\, \cap\, Q$; more specifically, an estimate is derived on the number of points of local maximum of the free boundary.
A lower central series for split Hopf algebras with involution
Bruce W.
Jordan
427-454
Abstract: A lower central series is defined for split Hopf algebras with involution over a field k. Various structure theorems for coalgebras and Hopf algebras are established.
A noncommutative generalization and $q$-analog of the Lagrange inversion formula
Ira
Gessel
455-482
Abstract: The Lagrange inversion formula is generalized to formal power series in noncommutative variables. A q-analog is obtained by applying a linear operator to the noncommutative formula before substituting commuting variables.
On the oscillatory behavior of singular Sturm-Liouville expansions
J. K.
Shaw
483-505
Abstract: A singular Sturm-Liouville operator $Ly\, =\, - (Py')'\, +\, Qy$, defined on an interval $[0,b^{\ast})$ of regular points, but singular at $b^{\ast}$, is considered. Examples are the Airy equation on $[0,\infty )$ and the Legendre equation on $ [0,1)$. A mode of oscillation of the successive iterates $f(t)$, $(Lf)(t)$, $ ({L^2}f)(t),\, \ldots$ of a smooth function f is assumed, and the resulting influence on f is studied. The nature of the mode is that for a fixed integer $N\, \geqslant\, 0$, each iterate $({L^k}f)(t)$ shall have on $(0,b^{\ast})$ exactly N sign changes which are stable, in a certain sense, as k varies. There is quoted from the literature the main characterization of such functions f which additionally satisfy strong homogeneous endpoint conditions at 0 and $ b^{\ast}$. An extended characterization is obtained by weakening the conditions of f at 0 and $b^{\ast}$. The homogeneous endpoint conditions are replaced by a summability condition on the values, or limits of values, of f at 0 and $ b^{\ast}$.
Continuously translating vector-valued measures
U. B.
Tewari;
M.
Dutta
507-519
Abstract: Let G be a locally compact group and A an arbitrary Banach space. $ {L^p}(G,A)$ will denote the space of p-integrable A-valued functions on G. $M(G,A)$ will denote the space of regular A-valued Borel measures of bounded variation on G. In this paper, we characterise the relatively compact subsets of $ {L^p}(G,A)$. Using this result, we prove that if $\mu\, \in\, M(G,A)$, such that either $x\, \to\, {\mu _x}$ or $x{ \to _x}\mu$ is continuous, then $\mu\, \in\, {L^1}(G,A)$.
On the cohomology of real Grassmanians
Howard L.
Hiller
521-533
Abstract: Let ${G_k}({\textbf{R}^{n + k}})$ denote the grassman manifold of k-planes in real $(n\, +\, k)$-space and $ {w_1}\, \in\, {H^1}({G_k}({\textbf{R}^{n + k}});\,{\textbf{Z}_2})$ the first Stiefel-Whitney class of the universal bundle. Using Schubert calculus techniques and the cohomology of flag manifolds we estimate the height of $ {w_1}$ in the cohomology ring. We then apply this to improve earlier lower bounds on the Lusternik-Schnirelmann category of real grassmanians.
Correction to: ``On the free boundary of a quasivariational inequality arising in a problem of quality control'' [Trans. Amer. Math. Soc. {\bf 246} (1978), 95--110; MR 80f:93086c]
Avner
Friedman
535-537