On $(K\sb\ast ({\bf Z}/p\sp{2}{\bf Z})$ and related homology groups
Leonard
Evens;
Eric M.
Friedlander
1-46
Abstract: It is shown that, for $p \geqslant 5$, $\displaystyle R = {\mathbf{Z}} / {p^2}{\mathbf{Z}},\,{K_3}(R) = {\mathbf{Z}} / {p^2}{\mathbf{Z}} + {\mathbf{Z}} / ({p^2} - 1){\mathbf{Z}}$ and $ {K_4}(R) = 0$. Similar calculations are made for $R$ the ring of dual numbers over ${\mathbf{Z}} / p{\mathbf{Z}}$. The calculation reduces to finding homology groups of $\operatorname{Sl} (R)$. A key tool is the spectral sequence of the group extension of $ \operatorname{Sl} (n,\,{p^2})$ over $\operatorname{Sl} (n,\,p)$. The terms of this spectral sequence depend in turn on the homology of $ \operatorname{Gl} (n,\,p)$ with coefficients various multilinear modules. Calculation of the differentials uses the Charlap-Vasquez description of ${d^2}$.
A hypersurface defect relation for a class of meromorphic maps
Aldo
Biancofiore
47-60
Abstract: Let ${D_1}, \ldots ,{D_q}$ be hypersurfaces of degree $ p$ in ${{\mathbf{P}}_n}$ with normal crossings. We prove for a certain class of meromorphic maps $ f:{{\mathbf{C}}^m} \to {{\mathbf{P}}_n}$ a defect relation ${\delta _f}\left( {{D_1}} \right) + \cdots + {\delta _f}({D_q}) \leqslant (n + 1)/p$ conjectured by Ph. Griffiths and B. Shiffman.
Uncountable admissibles. I. Forcing
Sy D.
Friedman
61-73
Abstract: Assume $V = L$. Let $\kappa$ be a regular cardinal and for $X \subseteq \kappa$ let $\alpha (X)$ denote the least ordinal $\alpha$ such that ${L_\alpha }[X]$ is admissible. In this paper we characterize those ordinals of the form $\alpha (X)$ using forcing and fine structure of $ L$ techniques. This generalizes a theorem of Sacks which deals with the case $\kappa = \omega $.
A family of links and the Conway calculus
Cole A.
Giller
75-109
Abstract: In 1969, J. H. Conway gave efficient methods of calculating abelian invariants of classical knots and links. The present paper includes a detailed exposition (with new proofs) of these methods and extensions in several directions. The main application given here is as follows. A link $ L$ of two unknotted components in ${S^3}$ has the distinct lifting property for $p$ if the lifts of each component to the $ p$-fold cover of $ {S^3}$ branched along the other are distinct. The $p$-fold covers of these lifts are homeomorphic, and so $L$ gives an example of two distinct knots with the same $p$-fold cover. The above machinery is then used to construct an infinite family of links, each with the distinct lifting property for all $p \geqslant 2$.
Duality between logics and equivalence relations
Daniele
Mundici
111-129
Abstract: Assuming $ \omega$ is the only measurable cardinal, we prove: (i) Let $\sim$ be an equivalence relation such that $ \sim \, = \,{ \equiv _L}$ for some logic $L \leqslant {L^{\ast}}$ satisfying Robinson's consistency theorem (with ${L^{\ast}}$ arbitrary); then there exists a strongest logic $ {L^ + } \leqslant {L^{\ast}}$ such that $\sim \, = \,{ \equiv _{{L^ + }}}$; in addition, $ {L^ + }$ is countably compact if $ \sim \, \ne \, \cong$. (ii) Let $\dot \sim$ be an equivalence relation such that $ \sim \, = \,{ \equiv _{{L^0}}}$ for some logic ${L^0}$ satisfying Robinson's consistency theorem and whose sentences of any type $\tau$ are (up to equivalence) equinumerous with some cardinal $ {\kappa _\tau }$; then $ {L^0}$ is the unique logic $ L$ such that $\sim \, = \,{ \equiv _L}$; furthermore, $ {L^0}$ is compact and obeys Craig's interpolation theorem. We finally give an algebraic characterization of those equivalence relations $\sim$ which are equal to ${ \equiv _L}$ for some compact logic $L$ obeying Craig's interpolation theorem and whose sentences are equinumerous with some cardinal.
Boolean algebras with no rigid or homogeneous factors
Petr
Štěpánek
131-147
Abstract: A simple construction of Boolean algebras with no rigid or homogeneous factors is described. It is shown that for every uncountable cardinal $\kappa$ there are $ {2^\kappa }$ isomorphism types of Boolean algebras of power $\kappa$ with no rigid or homogeneous factors. A similar result is obtained for complete Boolean algebras for certain regular cardinals. It is shown that every Boolean algebra can be completely embedded in a complete Boolean algebra with no rigid or homogeneous factors in such a way that the automorphism group of the smaller algebra is a subgroup of the automorphism group of the larger algebra. It turns out that the cardinalities of antichains in both algebras are the same. It is also shown that every $ \kappa$-distributive complete Boolean algebra can be completely embedded in a $ \kappa$-distributive complete Boolean algebra with no rigid or homogeneous factors.
Calibers of compact spaces
S.
Argyros;
A.
Tsarpalias
149-162
Abstract: Let $X$ be a compact Hausdorff space and $ \kappa$ its Souslin number.$^{2}$ We prove that if $\alpha$ is a cardinal such that either $ \alpha$ and $ \operatorname{cf} (\alpha )$ are greater than $\kappa$ and strongly $\kappa$-inaccessible or else $\alpha$ is regular and greater than $ \kappa$, then $ X$ has $ (\alpha ,\,\sqrt[{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\kappa } }]{\alpha })$ caliber. Restricting our interest to the category of compact spaces $X$ with $ S(X) = {\omega ^ + }$ (i.e. $X$ satisfy the countable chain condition), the above statement takes, under G.C.H., the following form. For any compact space $X$ with $ S(X) = {\omega ^ + }$, we have that (a) if $\alpha$ is a cardinal and $\operatorname{cf} (\alpha )$ does not have the form ${\beta ^ + }$ with $\operatorname{cf} (\beta ) = \omega$, then $ \alpha$ is caliber for the space $X$. (b) If $\varepsilon = {\beta ^ + }$ and $ \operatorname{cf} (\beta ) = \omega$ then $ (\alpha ,\,\beta )$ is caliber for $X$. A related example shows that the result of (b) is in a sense the best possible.
A Poisson summation formula for integrals over quadratic surfaces
Robert S.
Strichartz
163-173
Abstract: Let $S(t)$ denote Lebesgue measure on the sphere of radius $t > 0$ in $ {{\mathbf{R}}^n}$, and $\displaystyle {S_k}(t) = {\left( {\frac{\partial } {{\partial t}}\quad \frac{1} {t}} \right)^k}S(t).$ Let $P{\sum _k} = {S_k}(0) + 2\sum _{m = 1}^\infty {S_k}(m)$. Theorem. If $n$ is odd and $j$ and $k$ are nonnegative integers with $j + k = (n - 1) / 2$, then the Fourier transform of $P{\sum _j}$ is ${(2\pi )^{j - k}}P{\sum _k}$. There is an analogous, although slightly different, identity involving integrals over hyperboloids in odd dimensions. These results were inspired by recent work of M. Vergne.
A note on complete intersections
S. M.
Bhatwadekar
175-181
Abstract: Let $R$ be a regular local ring and let $ R[T]$ be a polynomial algebra in one variable over $R$. In this paper the author proves that every maximal ideal of $R[T]$ is complete intersection in each of the following cases: (1) $R$ is a local ring of an affine algebra over an infinite perfect field, (2) $R$ is a power series ring over a field.
Degeneracy theorems for holomorphic mappings between algebraic varieties
Robert
Molzon
183-192
Abstract: Degeneracy theorems are proved for holomorphic mappings from affine algebraic manifolds to projective algebraic manifolds of equal dimensions. A mapping is degenerate if it satisfies a growth estimate and omits a set of $\kappa$-plane sections of positive capacity; the capacity being defined in terms of a singular integral. The capacity is a more delicate method of measuring the size of a set of $\kappa$-plane sections than Hausdorff measure and arises naturally by considering the singular integrals in the First Main Theorem of Nevanlinna.
On nonseparable Banach spaces
Spiros A.
Argyros
193-216
Abstract: Combining combinatorial methods from set theory with the functional structure of certain Banach spaces we get some results on the isomorphic structure of nonseparable Banach spaces. The conclusions of the paper, in conjunction with already known results, give complete answers to problems of the theory of Banach spaces. An interesting point here is that some questions of Banach spaces theory are independent of Z.F.C. So, for example, the answer to a conjecture of Pełczynski that states that the isomorphic embeddability of ${L^1}{\{ - 1,\,1\} ^\alpha }$ into ${X^{\ast}}$ implies, for any infinite cardinal $ \alpha$, the isomorphic embedding of $ l_\alpha ^1$ into $ X$, gets the following form: if $\alpha = \omega $, has been proved from Pełczynski; if $\alpha > {\omega ^ + }$, the proof is given in this paper; if $ \alpha = {\omega ^ + }$, in $ {\text{Z}}{\text{.F}}{\text{.C}}{\text{.}} + {\text{C}}{\text{.H}}{\text{.}}$, an example discovered by Haydon gives a negative answer; if $\alpha = {\omega ^ + }$, in $ {\text{Z}}{\text{.F}}{\text{.C}}{\text{.}} + \urcorner {\text{C}}{\text{.H}}{\text{.}} + {\text{M}}{\text{.A}}{\text{.}}$, is also proved in this paper.
Closures of conjugacy classes in classical real linear Lie groups. II
Dragomir Ž.
Djoković
217-252
Abstract: By a classical group we mean one of the groups $G{L_n}(R)$, $G{L_n}(C)$, $G{L_n}(H)$, $U(p,\,q)$, ${O_n}(C)$, $O(p,\,q)$, $ S{O^{\ast}}(2n)$, $S{p_{2n}}(C)$, $ S{p_{2n}}(R)$, or $ Sp(p,\,q)$. Let $ G$ be a classical group and $L$ its Lie algebra. For each $x \in L$ we determine the closure of the orbit $G \cdot x$ (for the adjoint action of $ G$ on $L$). The problem is first reduced to the case when $x$ is nilpotent. By using the exponential map we also determine the closures of conjugacy classes of $ G$.
A study of the local components of the Hecke algebra mod $l$
Naomi
Jochnowitz
253-267
Abstract: We use information about modular forms $\bmod l$ to study the local structure of the Hecke ring. In particular, we find nontrivial lower bounds for the dimensions of the Zariski tangent spaces of the local components of the Hecke ring $\bmod l$. These results suggest that the local components of the Hecke ring $\bmod l$ are more complex than originally expected. We also investigate the inverse limits of the Hecke rings of weight $k\bmod l$ as $k$ varies within a fixed congruence class $\bmod l - 1$. As an immediate corollary to some of the above results, we show that when $ k$ is sufficiently large, an arbitrary prime $l$ must divide the index of the classical Hecke ring $ {{\mathbf{T}}_k}$ in the ring of integers of ${{\mathbf{T}}_k} \otimes {\mathbf{Q}}$.
Congruences between systems of eigenvalues of modular forms
Naomi
Jochnowitz
269-285
Abstract: We modify and generalize proofs of Tate and Serre in order to show that there are only a finite number of systems of eigenvalues for the Hecke operators with respect to ${\Gamma _0}(N)\bmod l$. We also summarize results for ${\Gamma _1}(N)$. Using these results, we show that an arbitrary prime divides the discriminant of the classical Hecke ring to a power which grows linearly with $k$. In this way, we find a lower bound for the discriminant of the Hecke ring. After limiting ourselves to cusp forms, we also find an upper bound. Lastly we use the constructive nature of Tate and Serre's result to describe the structure and dimensions of the generalized eigenspaces for the Hecke operators $\bmod l$.
Structure theory for a class of grade four Gorenstein ideals
Andrew
Kustin;
Matthew
Miller
287-307
Abstract: An ideal $ I$ in a commutative noetherian ring $R$ is a Gorenstein ideal of $\operatorname{grade} g$ if ${\operatorname{pd} _R}(R / I) = \operatorname{grade} \,I = g$ and the canonical module $ \operatorname{Ext} _R^g(R / I,\,R)$ is cyclic. Serre showed that if $g = 2$ then $I$ is a complete intersection, and Buchsbaum and Eisenbud proved a structure theorem for the case $g = 3$. We present generic resolutions for a class of Gorenstein ideals of $\operatorname{grade} 4$, and we illustrate the structure of the resolution with various specializations. Among these examples there are Gorenstein ideals of $ \operatorname{grade} \,4$ in $ k[[x,\,y,\,z,\,v]]$ that are $n$-generated for any odd integer $n \geqslant 7$. We construct other examples from almost complete intersections of $ \operatorname{grade} \,3$ and their canonical modules. In the generic case the ideals are shown to be normal primes. Finally, we conclude by giving an explicit associative algebra structure for the resolutions. It is this algebra structure that we use to classify the different Gorenstein ideals of $\operatorname{grade} \,4$, and which may be the key to a complete structure theorem.
A network of congruences on an inverse semigroup
Mario
Petrich;
Norman R.
Reilly
309-325
Abstract: A congruence $ \rho$ on an inverse semigroup $S$ is determined uniquely by its kernel and its trace. Denoting by $ {\rho ^{\min }}$ and ${\rho _{\min }}$ the least congruence on $ S$ having the same kernel and the same trace as $\rho$, respectively, and denoting by $\omega$ the universal congruence on $ S$, we consider the sequence $\omega$, $ {\omega ^{\min }}$, ${\omega _{\min }}$, ${({\omega ^{\min }})_{\min }}$, ${({\omega _{\min }})^{\min }} \ldots$. These congruences, together with the intersections of corresponding pairs, form a sublattice of the lattice of all congruences on $S$. We study the properties of these congruences and establish several properties of the quasivarieties of inverse semigroups induced by them.
Banach bundles of continuous functions and an integral representation theorem
Anthony Karel
Seda
327-332
Abstract: A construction is given of a Banach bundle $p:A \to X$ whose fibres are spaces of continuous functions which vanish at infinity. A Riesz type integral representation theorem is established which describes all functional on $A$.
The integrability tensor for bundle-like foliations
Richard H.
Escobales
333-339
Abstract: A certain function is introduced which is useful in the study of a bundle-like foliation on a Riemannian manifold. Under the assumption that the leaves are totally geodesic, the Laplacian of this function is computed along a leaf. From this computation a sufficient condition is provided for the ambient manifold to be locally isometric to a product.
On the radial limits of analytic and meromorphic functions
J. S.
Hwang
341-348
Abstract: Early in the fifties, A. J. Lohwater proved that if $f(z)$ is analytic in $\vert z\vert < 1$ and has the radial limit 0 almost everywhere on $\vert z\vert = 1$, then every complex number $ \zeta$ is an asymptotic value of $f(z)$ provided the $\zeta$-points satisfy the following Blaschke condition: $\sum _{k = 1}^\infty (1 - \vert{z_k}\vert) < \infty$, where $ f({z_k}) = \zeta$, $k = 1\,,2, \ldots$. We may, therefore, ask under the hypothesis on $f(z)$ how many complex numbers $\zeta$ are there whose $\zeta $-points can satisfy the Blaschke condition. We show that there is at most one such number and this one number phenomenon can actually occur if the number is zero.
Errata to: ``Linear spaces with an $H\sp{\ast} $-algebra-valued inner product'' [Trans. Amer. Math. Soc. {\bf 262} (1980), no. 2, 543--549; MR 81m:46080]
Parfeny P.
Saworotnow
349