Degenerations of $K3$ surfaces of degree $4$
Jayant
Shah
271-308
Abstract: A generic $ K3$ surface of degree $ 4$ may be embedded as a nonsingular quartic surface in ${{\mathbf{P}}_3}$. Let $ f:X \to \operatorname{Spec} \;{\mathbf{C}}[[t]]$ be a family of quartic surfaces such that the generic fiber is regular. Let ${\Sigma _0}$, $ {\Sigma _2^0}$, ${\Sigma _4}$ be respectively a nonsingular quadric in $ {{\mathbf{P}}_3}$, a cone in $ {{\mathbf{P}}_3}$ over a nonsingular conic and a rational, ruled surface in $ {{\mathbf{P}}_9}$ which has a section with self intersection $- 4$. We show that there exists a flat, projective morphism $f':X' \to {\text{Spec}}\;{\mathbf{C}}[[t]]$ and a map $\rho :{\text{Spec}}\:{\mathbf{C}}[[t]] \to {\text{Spec}}\:{\mathbf{C}}[[t]]$ such that (i) the generic fiber of $ f'$ and the generic fiber of the pull-back of $f$ via $\rho$ are isomorphic, (ii) the fiber $ {X'_0}$ of $f'$ over the closed point of $ {\text{Spec}}\;{\mathbf{C}}[[t]]$ has only insignificant limit singularities and (iii) ${X'_0}$ is either a quadric surface or a double cover of ${\Sigma _0}$, ${\Sigma _2^0}$ or $ {\Sigma _4}$. The theorem is proved using the geometric invariant theory.
Product-convolution operators and mixed-norm spaces
Robert C.
Busby;
Harvey A.
Smith
309-341
Abstract: Conditions for boundedness and compactness of product-convolution operators $g \to {P_h}{C_f}g = h \cdot (f\ast g)$ on spaces $ {L_p}(G)$ are studied. It is necessary for boundedness to define a class of "mixed-norm" spaces $ {L_{(p,q)}}(G)$ interpolating the ${L_p}(G)$ spaces in a natural way $({L_{(p,p)}} = {L_p})$. It is then natural to study the operators acting between ${L_{(p,q)}}(G)$ spaces, where $G$ has a compact invariant neighborhood. The theory of $ {L_{(p,q)}}(G)$ is developed and boundedness and compactness conditions of a nonclassical type are obtained. It is demonstrated that the results extend easily to a somewhat broader class of integral operators. Several known results are strengthened or extended as incidental consequences of the investigation.
Capacity densities and angular limits of quasiregular mappings
Matti
Vuorinen
343-354
Abstract: It is shown that if a bounded quasiregular mapping of the unit ball ${B^n} \subset {R^n}$, $n \geqslant 2$, has a limit at $b \in \partial {B^n}$ through a set $E \subset {B^n}$ with $ b \in \bar E$, then it has an angular limit at $b$ provided that $E$ is contained in an open cone $C \subset {B^n}$ with vertex $b$ and that $E$ is thick enough at $b$. The thickness condition is expressed in terms of the $n$-capacity density.
Spherical bundles adapted to a $G$-fibration
J. P. E.
Hodgson
355-361
Abstract: A spherical fibration $p:E \to B$ is said to be adapted to a $G$-fibration $\pi :E \to E/G$ if there is a fibration $q:E/G \to B$ with fibre the quotient of a sphere by a free $G$-action and such that the composition $q \circ \pi = p$. In this paper it is shown that for spherical bundles in the PL, TOP or Homotopy categories that are adapted to ${Z_2}$- or ${S^1}$-fibrations there is a procedure analogous to the splitting principle for vector bundles that enables one to define characteristic classes for these fibrations and to relate them to the usual characteristic classes. The methods are applied to show that a spherical fibration over a $4$-connected base which is adapted to an $ {S^1}$-fibration admits a PL structure.
The Hewitt realcompactification of products
Haruto
Ohta
363-375
Abstract: For a completely regular Hausdorff space $X$, $\upsilon X$ denotes the Hewitt realcompactification of $X$. Given a topological property $\mathcal{P}$ of spaces, our interest is in characterizing the class $ \mathcal{R}(\mathcal{P})$ of all spaces $X$ such that $\upsilon (X \times Y) = \upsilon X \times \upsilon Y$ holds for each $\mathcal{P}$-space $Y$. In the present paper, we obtain such characterizations in the case that $ \mathcal{P}$ is locally compact and in the case that $ \mathcal{P}$ is metrizable.
Maximum principles for degenerate elliptic-parabolic equations with Venttsel's boundary condition
Kazuo
Amano
377-396
Abstract: In this paper, we first establish interior and boundary maximum principles for degenerate elliptic-parabolic equations; we state both principles in one single theorem in terms of the propagation set (cf. Theorem 1). We next generalize the boundary condition to Venttsel's one and obtain the similar result (cf. Theorem 2). Venttsel''s boundary condition contains Dirichlet, Neumann, oblique derivative and mixed boundary conditions as special cases and, from a probabilistic point of view (cf. Venttsel' [9]), it is the most general admissible boundary condition. We give several examples in the last section.
On computation of eigenvalues for differential equations by Newton's method
Zeev
Nehari
397-409
Abstract: Professor Zeev Nehari, in the last months before his untimely death, had been analyzing and developing a new algorithm for computing eigenvalues of selfadjoint boundary value problems of arbitrary order. Apparently, his main goals were Theorems 3.1 and 3.2 below (which incidentally yield the eigenvalues of the given problem, as well as those of a related problem). Unfortunately, as far as we know, Professor Nehari has not left a proof of the basic Lemma 3.1 which presumably was to be based on $\S 2$ or related results. It seems, however, worthwhile to publish this paper since the results of $ \S \S 1$ and $ 2$ are complete and of independent interest and since the question of the validity of Lemma 3.1 is also of interest. While the presentation in $\S 1$ and $2$ is in most respects complete, there are a few places where some minor clarifications by selected footnotes were felt to be desirable. Also, Professor Nehari's original incomplete $\S 3$ has been replaced by a revised and abbreviated version.
Derivations and automorphisms of nonassociative matrix algebras
G. M.
Benkart;
J. M.
Osborn
411-430
Abstract: This paper studies the derivation algebra and the automorphism group of $ {M_n}(A)$, $n \times n$ matrices over an arbitrary nonassociative algebra $A$ with multiplicative identity $1$. The investigation also includes results on derivations and automorphisms of the algebras obtained from ${M_n}(A)$ using the Lie product $[xy] = xy - yx$, and the Jordan product $x \circ y = \tfrac{1} {2}(xy + yx)$.
Basic sequences and subspaces in Lorentz sequence spaces without local convexity
Nicolae
Popa
431-456
Abstract: After some preliminary results $(\S1)$, we give in $\S2$ another proof of the result of N. J. Kalton [5] concerning the unicity of the unconditional bases of $ {l_p}$, $0 < p < 1$. Using this result we prove in §3 the unicity of certain bounded symmetric block bases of the subspaces of the Lorentz sequence spaces $d(w,p)$, $0 < p < 1$. In $\S4$ we show that every infinite dimensional subspace of $d(w,p)$ contains a subspace linearly homeomorphic to ${l_p}$, $0 < p < 1$. Unlike the case $p \geqslant 1$ there are subspaces of $d(w,p)$, $0 < p < 1$, which contain no complemented subspaces of $d(w,p)$ linearly homeomorphic to $ {l_p}$. In fact there are spaces $d(w,p)$, $0 < p < 1$, which contain no complemented subspaces linearly homeomorphic to ${l_p}$. We conjecture that this is true for every $ d(w,p)$, $0 < p < 1$. The answer to the previous question seems to be important: for example we can prove that a positive complemented sublattice $ E$ of $d(w,p)$, $0 < p < 1$, with a symmetric basis is linearly homeomorphic either to ${l_p}$ or to $d(w,p)$; consequently, a positive answer to this question implies that $E$ is linearly homeomorphic to $ d(w,p)$. In $\S5$ we are able to characterise the sublattices of $d(w,p)$, $ p = {k^{ - 1}}$ (however under a supplementary restriction concerning the sequence $({w_n})_{n = 1}^\infty )$, which are positive and contractive complemented, as being the order ideals of $ d(w,p)$. Finally, in $ \S6$, we characterise the Mackey completion of $d(w,p)$ also in the case $p = {k^{ - 1}}$, $k \in {\mathbf{N}}$.
The lattice of closed congruences on a topological lattice
Dennis J.
Clinkenbeard
457-467
Abstract: Our primary objectives are: (1) if $L$ is a lattice endowed with a topology making both the meet and join continuous then (i) the natural map which associates a congruence with the smallest topologically closed congruence containing it preserves finite meets and arbitrary joins; (ii) the lattice of such closed congruences is a complete Brouwerian lattice; (2) if $L$ is a topological (semi) lattice with the unit interval as a (semi) lattice homomorphic image then the lattice of closed (semi) lattice congruences has no compatible Hausdorff topology.
Boundary crossing probabilities for stationary Gaussian processes and Brownian motion
Jack
Cuzick
469-492
Abstract: Let $X(t)$ be a stationary Gaussian process, $ f(t)$ a continuous function, and $T$ a finite or infinite interval. This paper develops asymptotic estimates for $P(X(t) \geqslant f(t)$, some $t \in T$ when this probability is small. After transformation to an Ornstein Uhlenbeck process the results are also applicable to Brownian motion. In that special case, if $W(t)$ is Brownian motion, $f$ is continuously differentiable, and $T = [0,T]$ our estimate for $P(W(t) \geqslant f(t)$, some $t \in T)$ is $ {\Phi ^ \ast }$ is its upper tail distribution. Our approach is to find an approximate first passage density and then compute crossing probabilities as a one-dimensional integral. In the case of boundaries without cusps, our results unify and extend separate results for crossings of constant levels developed by Pickands, and Qualls-Watanabe, and crossings of rapidly increasing barriers studied by Berman. Applications are also briefly explored.
Representations of functions by derivatives
S. J.
Agronsky;
R.
Biskner;
A. M.
Bruckner;
J.
Mařík
493-500
Abstract: Let $\Delta '$ be the class of all derivatives. The main goal of this paper is the investigation of the vector space generated by $\Delta '$ and O'Malley's class $B_1^ \ast$; this space is identical with our system
The zeros of the second derivative of the reciprocal of an entire function
Simon
Hellerstein;
Jack
Williamson
501-513
Abstract: Let $f$ be a real entire function of finite order with only real zeros. Assuming that $f'$ has only real zeros, we show that the number of nonreal zeros of $f''$ equals the number of real zeros of $F''$, where $F = 1/f$. From this, we show that $F''$ has only real zeros if and only if $ f(z) = \exp(a{z^2} + bz + c)$, $a \geqslant 0$, or $f(z) = {(Az + B)^n}$, $A \ne 0$, $n$ a positive integer.
On the contact between complex manifolds and real hypersurfaces in ${\bf C}\sp{3}$
Thomas
Bloom
515-529
Abstract: Let $m$ be a real ${\mathcal{C}^\infty }$ hypersurface of an open subset of $ {{\mathbf{C}}^3}$ and let $ p \in M$. Let ${a^1}(M,p)$ denote the maximal order of contact of a one-dimensional complex submanifold of a neighborhood of $p$ in $ {{\mathbf{C}}^3}$ with $ M$ at $p$. Let $ {c^1}(M,p)$ denote the $ \sup \{ m \in {\mathbf{Z}}\vert$ for all tangential holomorphic vector fields $ L$ with $L(p) \ne 0$ then $ {L^{{i_0}}}{\bar L^{{j_0}}} \ldots {L^{{i_n}}}{\bar L^{{j_n}}}({\mathfrak{L}_M}(L))(p) = 0\}$ where ${i_0}, \ldots ,{i_n};{j_0}, \ldots ,{j_n}$ are positive integers such that $ \sum\nolimits_{t = 0}^n {{i_t} + {j_t} = m - 3}$ and ${\mathfrak{L}_M}(L)$ denotes the Levi form of $ M$ evaluated on the vector field $L$. Theorem. If $M$ is pseudoconvex near $ p \in M$ then $ {a^1}(M,p) = {c^1}(M,p)$.
A lattice renorming theorem and applications to vector-valued processes
William J.
Davis;
Nassif
Ghoussoub;
Joram
Lindenstrauss
531-540
Abstract: A norm, $ \vert\vert\;\vert\vert$, on a Banach space $E$ is said to be locally uniformly convex if $ \left\Vert {{x_n}} \right\Vert \to \left\Vert x \right\Vert$ and $\left\Vert {{x_n} + x} \right\Vert \to 2\left\Vert x \right\Vert$ implies that ${x_n} \to x$ in norm. It is shown that a Banach lattice has an (order) equivalent locally uniformly convex norm if and only if the lattice is order continuous. This result is used to reduce convergence theorems for (lattice-valued) positive martingales and submartingales to the scalar case.