An algebraic classification of certain simple even-dimensional knots
C.
Kearton
1-53
Abstract: The simple $ 2q$-knots, $q \geqslant 4$, for which ${H_q}(\tilde{K})$ contains no ${\mathbf{Z}}$-torsion, are classified by means of Hermitian duality pairings on their homology and homotopy modules.
On the $\psi $-mixing condition for stationary random sequences
Richard C.
Bradley
55-66
Abstract: For strictly stationary sequences of random variables two mixing conditions are studied which together form the $ \psi$-mixing condition. For the dependence coefficients associated with these two mixing conditions this article gives results on the possible limiting values and possible rates of convergence to these limits.
Syntax and semantics in higher-type recursion theory
David P.
Kierstead
67-105
Abstract: Recursion in higher types was introduced by S. C. Kleene in 1959. Since that time, it has come to be recognized as a natural and important generalization of ordinary recursion theory. Unfortunately, the theory contains certain apparent anomalies, which stem from the fact that higher type computations deal with the intensions of their arguments, rather than the extensions. This causes the failure of the substitution principle (that if $ \varphi ({\alpha ^{j + 1}},\mathfrak{A})$ and $\theta ({\beta ^j},\mathfrak{A})$ are recursive, then there should be a recursive $\psi (\mathfrak{A})$ such that $\psi (\mathfrak{A}) \simeq \varphi (\lambda {\beta ^j}\theta ({\beta ^j},\mathfrak{A}),\mathfrak{A})$ at least whenever $ \lambda {\beta ^j}\theta ({\beta ^j},\mathfrak{A})$ is total), and of the first recursion principle (that if ${\mathbf{F}}(\zeta ;\mathfrak{A})$ is a recursive functional, then the minimal solution $ \zeta$ of the equation ${\mathbf{F}}(\zeta ;\mathfrak{A}) \simeq \zeta (\mathfrak{A})$ should be recursive as well). In an effort to remove--or at least explain--these anomalies, Kleene, in 1978, developed a system for computation in higher types which was based entirely on the syntactic manipulation of formal expressions, called $ j$-expressions. As Kleene pointed out, no adequate semantics for these expressions can be based on the classical (total) type structure $Tp$ over $ {\mathbf{N}}$. In a paper to appear in The Kleene Symposium (North-Holland), we showed that an appropriate semantics could be based on the type structure $\hat Tp$, which is obtained by adding a new object $ \mathfrak{u}$ at level 0 and, at level $(j + 1)$, allowing all monotone, partial functions from type $\hat j$ into $ {\mathbf{N}}$. Over $ \hat Tp$, both of the principles mentioned above do hold. There is a natural embedding to $Tp$ into $\hat Tp$. In this paper, we complement the syntactic structure with a syntax-free definition of recursion over $\hat Tp$, and show that the two notions are equivalent. This system admits an enumeration theorem, in spite of the fact that the presence of partial objects complicates the coding of finite sequences. Indeed, it is not possible to code all finite sequences from type $ \hat j$ as type-$ \hat j$ objects. We use the combination of the syntactic and semantic systems to prove that, for any $ \varphi: Tp^{(\sigma)}\mathop \to \limits_p {\mathbf{N}} $, the following are equivalent: A. $\varphi$ is recursive in the sense of Kleene [1959], B. $\varphi$ is recursive in the sense of Kleene [1978], and C. $\varphi$ is the pull-back in $Tp$ of some recursive $\psi :\hat T{p^{(\sigma )}}\mathop \to \limits_p {\mathbf{N}}$. Using these equivalences, we give a necessary and sufficient condition on $\theta :T{p^{(\sigma )}}\mathop \to \limits_p {\mathbf{N}}$, under which the substitution principle mentioned above will hold for any recursive $\varphi :T{p^{(\tau )}}\mathop \to \limits_p {\mathbf{N}}$. With one trivial exception, the condition is that if $ j \geqslant 1$, then $\mathfrak{A}$ must contain a variable of type greater than $j$. We feel that this result is particularly natural in the current setting.
On homogeneous polynomials on a complex ball
J.
Ryll;
P.
Wojtaszczyk
107-116
Abstract: We prove that there exist $n$-homogeneous polynomials ${p_n}$ on a complex $d$-dimensional ball such that $ {\left\Vert {{p_n}} \right\Vert _\infty} = 1$ and $ {\left\Vert {{p_n}} \right\Vert _2} \geqslant \sqrt \pi {2^{- d}}$. This enables us to answer some questions about ${H_p}$ and Bloch spaces on a complex ball. We also investigate interpolation by $ n$-homogeneous polynomials on a $2$-dimensional complex ball.
CR-hypersurfaces in a space with a pseudoconformal connection
Michael J.
Markowitz
117-132
Abstract: In this paper we study a submanifold in a space with a pseudoconformal connection. We assume that the submanifold $M$ is so situated that it inherits the structure of a $ {\text{CR}}$-hypersurface from the ambient space. $M$ then supports two natural Cartan connections, the normal pseudoconformal connection of Cartan-Chern-Tanaka and an induced pseudoconformal connection. Analogues of the Gauss-Codazzi equations are derived and applied to determine necessary and sufficient conditions for the equivalence of these connections.
Rigidity of pseudoconformal connections
Michael
Markowitz;
Roger
Schlafly
133-135
Abstract: Let ${M^{2n - 1}}(n \geqslant 3)$ be a strictly pseudoconvex abstract $ {\text{CR}}$-hypersurface ${\text{CR}}$-immersed in the unit sphere in ${{\mathbf{C}}^N}$. We show that the pseudoconformal connection induced on $M$ by the standard flat connection agrees with the intrinsic normal connection of Cartan-Chern-Tanaka if and only if $M$ is pseudoconformally flat. In this case $ M$ is a piece of the transverse intersection of $ {S^{2N - 1}}$ with a complex $n$-plane in $ {{\mathbf{C}}^N}$.
On the singular structure of three-dimensional, area-minimizing surfaces
Frank
Morgan
137-143
Abstract: A sufficient condition is given for the union of two three-dimensional planes through the origin in ${{\mathbf{R}}^n}$ to be area-minimizing. The condition is in terms of the three angles $0 \leqslant {\gamma _1} \leqslant {\gamma _2} \leqslant {\gamma _3}$ which characterize the geometric relationship between the planes. If $ {\gamma _3} \leqslant {\gamma _1} + {\gamma _2}$, the union of the planes is area-minimizing.
Unital $l$-prime lattice-ordered rings with polynomial constraints are domains
Stuart A.
Steinberg
145-164
Abstract: It is shown that a unital lattice-ordered ring in which the square of every element is positive must be a domain provided the product of two nonzero $l$-ideals is nonzero. More generally, the same conclusion follows if the condition ${a^2} \geqslant 0$ is replaced by $p(a) \geqslant 0$ for suitable polynomials $ p(x)$; and if it is replaced by $ f(a,b) \geqslant 0$ for suitable polynomials $f(x,y)$ one gets an $l$-domain. It is also shown that if $a \wedge b = 0$ in a unital lattice-ordered algebra which satisfies these constraints, then the $ l$-ideals generated by $ ab$ and $ba$ are identical.
${\rm SL}(2,\,{\bf C})$ actions on compact Kaehler manifolds
James B.
Carrell;
Andrew John
Sommese
165-179
Abstract: Whenever $G = SL(2,C)$ acts holomorphically on a compact Kaehler manifold $X$, the maximal torus $T$ of $G$ has fixed points. Consequently, $X$ has associated Bialynicki-Birula plus and minus decompositions. In this paper we study the interplay between the Bialynicki-Birula decompositions and the $G$-action. A representative result is that the Borel subgroup of upper (resp. lower) triangular matrices in $G$ preserves the plus (resp. minus) decomposition and that each cell in the plus (resp. minus) decomposition fibres $G$-equivariantly over a component of ${X^T}$. We give some applications; e.g. we classify all compact Kaehler manifolds $ X$ admitting a $ G$-action with no three dimensional orbits. In particular we show that if $ X$ is projective and has no three dimensional orbit, and if Pic$(X) \cong {\mathbf{Z}}$, then $X = C{{\mathbf{P}}^n}$. We also show that if $ X$ admits a holomorphic vector field with unirational zero set, and if $\operatorname{Aut}_0(X)$ is reductive, then $ X$ is unirational.
On wave fronts propagation in multicomponent media
M. I.
Freĭdlin
181-191
Abstract: The behavior as $t \to \infty$ of solutions of some parabolic systems of differential equations of the Kolmogorov-Petrovskii-Piskunov type is investigated. The present approach uses the Kac-Feynman formula and estimates on large deviations.
A geometric interpretation of the Chern classes
R.
Sivera Villanueva
193-200
Abstract: Let ${f_\xi}: M \to BU$ be a classifying map of the stable complex bundle $\xi$ over the weakly complex manifold $ M$. If $\tau$ is the stable right homotopical inverse of the infinite loop spaces map $\eta :QBU(1) \to BU$, we define $f_\xi ^{\prime} = \tau \cdot {f_\xi}$ and we prove that the Chern classes ${c_k}(\xi )$ are $ f_\xi^{\prime\ast}(h_k^{\ast}(t_k))$, where ${h_k}$ is given by the stable splitting of $ QBU(1)$ and ${t_k}$ is the Thom class of the bundle ${\gamma ^{(k)}} = E{\Sigma _k}{X_{{\Sigma _k}}}{\gamma ^k}$. Also, we associate to $f^{\prime}$ an immersion $g:N \to M$ and we prove that ${c_k}(\xi )$ is the dual of the image of the fundamental class of the $k$-tuple points manifold of the immersion $ g,g_k^{\ast}([{N_k}])$.
On the dimension of the $l\sp{n}\sb{p}$-subspaces of Banach spaces, for $1\leq p<2$
Gilles
Pisier
201-211
Abstract: We give an estimate relating the stable type $p$ constant of a Banach space $X$ with the dimension of the $ l_p^n$-subspaces of $ X$. Precisely, let $ C$ be this constant and assume $1 < p < 2$. We show that, for each $\varepsilon > 0,X$ must contain a subspace $(1 + \varepsilon )$-isomorphic to $l_p^k$, for every $k$ less than $\delta (\varepsilon ){C^{p^{\prime}}}$ where $ \delta (\varepsilon ) > 0$ is a number depending only on $p$ and $ \varepsilon$.
On the gr\"ossencharacter of an abelian variety in a parametrized family
Robert S.
Rumely
213-233
Abstract: We consider families of abelian varieties parametrized by classical theta-functions, and show that specifying the family and a CM point in Siegel space determines the grössencharacter of the corresponding CM abelian variety. We associate an adelic group to the family, and describe the kernel of the grössencharacter as the pull-back of the group under the map in Shimura's Reciprocity Law.
A general sufficiency theorem for nonsmooth nonlinear programming
R. W.
Chaney
235-245
Abstract: Second-order conditions are given which are sufficient to guarantee that a given point be a local minimizer for a real-valued locally Lipschitzian function over a closed set in $n$-dimensional real Euclidean space. These conditions are expressed in terms of the generalized gradients of Clarke. The conditions provide a very general and unified framework into which many previous first- and second-order theorems fit.
Inequalities for holomorphic functions of several complex variables
Jacob
Burbea
247-266
Abstract: Sharp norm-inequalities, valid for functional Hilbert spaces of holomorphic functions on the polydisk, unit ball and ${{\mathbf{C}}^n}$ are established by using the notion of reproducing kernels. These inequalities extend earlier results of Saitoh and ours.
Applications of variational inequalities to the existence theorem on quadrature domains
Makoto
Sakai
267-279
Abstract: In this paper we shall study quadrature domains for the class of subharmonic functions. By using the theory of variational inequalities, we shall give a new proof of the existence and uniqueness theorem. As an application, we deal with Hele-Shaw flows with a free boundary and show that their two weak solutions, one of which was defined by the author using quadrature domains and the other was defined by Gustafsson [3] using variational inequalities, are identical with each other.
A necessary and sufficient condition for the asymptotic version of Ahlfors' distortion property
Burton
Rodin;
S. E.
Warschawski
281-288
Abstract: Let $f$ be a conformal map of $R = \{w = u + iv \in {\mathbf{C}}\vert{\varphi _0}(u) < v < {\varphi _1}(u)\}$ onto $S = \{z = x + iy \in {\mathbf{C}}\vert < y < 1\}$ where the ${\varphi _j} \in {C^0}( - \infty ,\infty )$ and $ \operatorname{Re} f(w) \to \pm \infty$ as $\operatorname{Re} w \to \pm \infty$. There are well-known results giving conditions on $R$ sufficient for the distortion property $\operatorname{Re} f(u + iv) = \int_0^u ({\varphi _1} - {\varphi _0})^{- 1}du + {\text{const}}. + o(1)$, where $o(1) \to 0$ as $u \to + \infty$. In this paper the authors give a condition on $R$ which is both necessary and sufficient for $ f$ to have this property.
Cartesian-closed coreflective subcategories of uniform spaces
M. D.
Rice;
G. J.
Tashjian
289-300
Abstract: This paper characterizes the coreflective subcategories $\mathcal{C}$ of uniform spaces for which a natural function space structure generates the exponential law $ {X^{Y \otimes Z}} = {({X^Y})^Z}$ on $ \mathcal{C}$. Such categories are cartesian-closed. Specifically, we show that $\mathcal{C}$ is cartesian-closed in this way if and only if $ \mathcal{C}$ is inductively generated by a finitely productive family of locally fine spaces. The results divide naturally into two cases: those subcategories containing the unit interval are generated by precompact spaces, while the subcategories not containing the unit interval are generated by spaces which admit an infinite cardinal. These results may be used to derive the characterizations of cartesian-closed coreflective subcategories of Tychonoff spaces found in [10].
The approximation property for some $5$-dimensional Henselian rings
Joseph
Becker;
J.
Denef;
L.
Lipshitz
301-309
Abstract: Let $k$ be a field of characteristic 0, $k[[{X_1},{X_2}]]$ the ring of formal power series and $R = k[[{X_1},{X_2}]]{[{X_3},{X_4},{X_5}]^ \sim}$ the algebraic closure of $ k[[{X_1},{X_2}]][{X_3},{X_4},{X_5}]$ in $k[[{X_1},\ldots,{X_5}]]$. It is shown that $ R$ has the Approximation Property.
Sufficient conditions for smoothing codimension one foliations
Christopher
Ennis
311-322
Abstract: Let $M$ be a compact ${C^\infty}$ manifold. Let $X$ be a ${C^0}$ nonsingular vector field on $M$, having unique integral curves $ (p,t)$ through $p \in M$. For $f: M \to {\mathbf{R}}$ continuous, call $\left. Xf(p) = df(p,t)/dt\right\vert _{t = 0}$ whenever defined. Similarly, call $ {X^k}f(p)=X(X^{k-1}f)(p)$. For $ 0 \leqslant r < k$, a $ {C^r}$ foliation $\mathcal{F}$ of $M$ is said to be ${C^k}$ smoothable if there exist a ${C^k}$ foliation $\mathcal{G}$, which ${C^r}$ approximates $ \mathcal{F}$, and a homeomorphism $h:M \to M$ such that $h$ takes leaves of $ \mathcal{F}$ onto leaves of $\mathcal{G}$. Definition. A transversely oriented Lyapunov foliation is a pair $(\mathcal{F},X)$ consisting of a $ {C^0}$ codimension one foliation $ \mathcal{F}$ of $ M$ and a ${C^0}$ nonsingular, uniquely integrable vector field $X$ on $M$, such that there is a covering of $M$ by neighborhoods $\{{W_i}\}$, $0 \leqslant i \leqslant N$, on which $\mathcal{F}$ is described as level sets of continuous functions ${f_i}:{W_i} \to {\mathbf{R}}$ for which $ X{f_i}(p)$ is continuous and strictly positive. We prove the following theorems. Theorem 1. Every ${C^0}$ transversely oriented Lyapunov foliation $ (\mathcal{F},X)$ is $ {C^1}$ smoothable to a ${C^1}$ transversely oriented Lyapunov foliation $ (\mathcal{G},X)$. Theorem 2. If $ (\mathcal{F},X)$ is a ${C^0}$ transversely oriented Lyapunov foliation, with $ X \in {C^{k - 1}}$ and $ {X^j}{f_i}(p)$ continuous for $ 1 \leqslant j \leqslant k$ and $ 0 \leqslant i \leqslant N$, then $ (\mathcal{F},X)$ is $ {C^k}$ smoothable to a ${C^k}$ transversely oriented Lyapunov foliation $ (\mathcal{G},X)$. The proofs of the above theorems depend on a fairly deep result in analysis due to F. Wesley Wilson, Jr. With only elementary arguments we obtain the ${C^k}$ version of Theorem 1. Theorem 3. If $ (\mathcal{F},X)$ is a $ {C^{k - 1}}\;(k \geqslant 2)$ transversely oriented Lyapunov foliation, with $X \in {C^{k - 1}}$ and ${X^k}{f_i}(p)$ is continuous, then $(\mathcal{F},X)$ is ${C^k}$ smoothable to a $ {C^k}$ transversely oriented Lyapunov foliation $(\mathcal{G},X)$.
The dispersion of the coefficients of univalent functions
D. H.
Hamilton
323-333
Abstract: The Hayman $ {T_a}$ function for the asymptotic distribution of the coefficients of univalent functions has a continuous derivative which is closely related to the asymptotic behavior of coefficient differences.
On the generalized Seidel class $U$
Jun Shung
Hwang
335-346
Abstract: As usual, we say that a function $f \in U$ if $f$ is meromorphic in $\vert z \vert < 1$ and has radial limits of modulus $ 1$ a.e. (almost everywhere) on an arc $A$ of $ \left\vert z \right\vert = 1$. This paper contains three main results: First, we extend our solution of A. J. Lohwater's problem (1953) by showing that if $f \in U$ and $f$ has a singular point $P$ on $A$, and if $\upsilon$ and $ 1/\bar{\upsilon}$ are a pair of values which are not in the range of $ f$ at $P$, then one of them is an asymptotic value of $f$ at some point of $A$ near $P$. Second, we extend our solution of J. L. Doob's problem (1935) from analytic functions to meromorphic functions, namely, if $f \in U$ and $f(0) = 0$, then the range of $f$ over $\left\vert z \right\vert < 1$ covers the interior of some circle of a precise radius depending only on the length of $A$. Finally, we introduce another class of functions. Each function in this class has radial limits lying on a finite number of rays a.e. on $\left\vert z \right\vert = 1$, and preserves a sector between domain and range. We study the boundary behaviour and the representation of functions in this class.
A generalization of minimal cones
Norio
Ejiri
347-360
Abstract: Let ${R_ +}$ be a positive real line, $ {S^n}$ an $n$-dimensional unit sphere. We denote by ${R_+} \times {S^n}$ the polar coordinate of an $(n + 1)$-dimensional Euclidean space ${R^{n + 1}}$. It is well known that if $M$ is a minimal submanifold in $ {S^n}$, then ${R_ +} \times M$ is minimal in ${R^{n + 1}}$. $ {R_+} \times M$ is called a minimal cone. We generalize this fact and give many minimal submanifolds in real and complex space forms.
Products of powers of nonnegative derivatives
Jan
Mařík;
Clifford E.
Weil
361-373
Abstract: This paper contains some results concerning functions that can be written as $f_1^{{\beta _1}} \cdots f_n^{{\beta _n}}$, where $ n$ is an integer greater than $1$, ${f_j}$ are nonnegative derivatives and ${\beta _j}$ are positive numbers. If we choose $ {\beta _1} = \cdots = {\beta _n} = 1$, we obtain theorems about products of nonnegative derivatives.
Branched coverings. I
R. E.
Stong
375-402
Abstract: This paper analyzes the possible cobordism classes $[M] - (\deg \;\phi )[N]$ for $\phi : M \to N$ a smooth branched covering of closed smooth manifolds. It is assumed that the branch set is a codimension $2$ submanifold. The results are a fairly complete description in the unoriented case, a partial description in the oriented case, and a detailed analysis of the case in which $N$ is a sphere.
Branched coverings. II
R. E.
Stong
403-407
Abstract: This paper improves the analysis of the possible cobordism classes $[M] - (\deg \;\phi )[N]$ for $\phi : M \to N$ a smooth branched covering of closed oriented smooth manifolds. It is assumed that the branch set is a codimension $2$ submanifold.
Convergence of functions: equi-semicontinuity
Szymon
Dolecki;
Gabriella
Salinetti;
Roger J.-B.
Wets
409-430
Abstract: We study the relationship between various types of convergence for extended real-valued functionals engendered by the associated convergence of their epigraphs; pointwise convergence being treated as a special case. A condition of equi-semicontinuity is introduced and shown to be necessary and sufficient to allow the passage from one type of convergence to another. A number of compactness criteria are obtained for families of semicontinuous functions; in the process we give a new derivation of the Arzelá-Ascoli Theorem.