Asymptotic behavior of solutions of linear stochastic differential systems
Avner
Friedman;
Mark A.
Pinsky
1-22
Abstract: Following Kasminski, we investigate asymptotic behavior of solutions of linear time-independent Itô equations. We first give a sufficient condition for asymptotic stability of the zero solution. Then in dimension 2 we determine conditions for spiraling at a linear rate. Finally we give applications to the Cauchy problem for the associated parabolic equation by the use of a tauberian theorem.
Perfect mappings and certain interior images of $M$-spaces
J. M.
Worrell;
H. H.
Wicke
23-35
Abstract: The main theorems of this paper show that certain conditions (called ${\lambda _c},{\lambda _b},{\beta _c}$, and ${\beta _b}$) are invariant, in the presence of ${T_0}$-regularity, under the application of closed continuous peripherally compact mappings. Interest in these conditions lies in the fact that they may be used to characterize certain regular ${T_0}$ open continuous images of some classes of $M$-spaces in the sense of K. Morita, and in the fact that they are preserved by open continuous mappings with certain appropriate additional conditions. For example, the authors have shown that a regular $ {T_0}$-space is an open continuous image of a paracompact Čech complete space if and only if the space satisfies condition ${\lambda _b}$ [Pacific J. Math. 37 (1971), 265-275]. Moreover, in the same paper it is shown that if a completely regular ${T_0}$-space satisfies condition ${\lambda _b}$ then any ${T_0}$ completely regular open continuous image of it also satisfies $ {\lambda _b}$. These results together with the results of the present paper and certain known results lead to the following theorem: The smallest subclass of the class of regular $ {T_0}$-spaces which contains all paracompact Čech complete spaces and which is closed with respect both to the application of perfect mappings and to the application of open continuous mappings preserving ${T_0}$-regularity is the subclass satisfying condition ${\lambda _b}$. Similar results are obtained for the regular ${T_0}$-spaces satisfying ${\lambda _c},{\beta _b}$, and ${\beta _c}$. The other classes of $M$-spaces involved are the regular $ {T_0}$ complete $ M$-spaces (i.e., spaces which are quasi-perfect preimages of complete metric spaces), ${T_2}$ paracompact $M$-spaces, and regular ${T_0}M$-spaces. In the last two cases besides the inferiority of the mappings the notion of uniform $ \lambda$-completeness, which generalizes compactness of a mapping, enters. (For details see General Topology and Appl. 1 (1971), 85-100.) The proofs are accomplished through the use of two basic lemmas on closed continuous mappings satisfying certain additional conditions.
$\alpha_T$ is finite for $\aleph_1$-categorical $T$
John T.
Baldwin
37-51
Abstract: Let $T$ be a complete countable ${\aleph _1}$-categorical theory. Definition. If $\mathcal{A}$ is a model of $T$ and $A$ is a $1$-ary formula in $ L(\mathcal{A})$ then $ A$ has rank 0 if $A(\mathcal{A})$ is finite. $A(\mathcal{A})$ has rank $n$ degree $m$ iff for every set of $m + 1$ formulas $ {B_1}, \cdots ,{B_{m + 1}} \in {S_1}(L(\mathcal{A}))$ which partition $A(\mathcal{A})$ some ${B_i}(\mathcal{A})$ has rank $\leqslant n - 1$. Theorem. If $ T$ is ${\aleph _1}$-categorical then for every $\mathcal{A}$ a model of $T$ and every $A \in {S_1}(L(\mathcal{A})),A(\mathcal{A})$ has finite rank. Corollary. ${\alpha _T}$ is finite. The methods derive from Lemmas 9 and 11 in ``On strongly minimal sets'' by Baldwin and Lachlan. $ {\alpha _T}$ is defined in ``Categoricity in power'' by Michael Morley.
Upper bounds for vertex degrees of planar $5$-chromatic graphs
Lee W.
Johnson
53-59
Abstract: Upper bounds are given for the degrees of vertices in planar $ 5$-chromatic graphs. Some inequalities are derived for irreducible graphs which restrict the type of planar graphs that can be irreducible.
Moment and BV-functions on commutative semigroups
P. H.
Maserick
61-75
Abstract: A general notion of variation of functions on an arbitrary commutative semigroup with identity is introduced. The concept includes Hausdorff's for the additive semigroup of nonnegative integers as well as the more recent notions introduced for semilattices. An abstract moment problem is formulated and solved.
Subgroups of free products with amalgamated subgroups: A topological approach
J. C.
Chipman
77-87
Abstract: The structure of an arbitrary subgroup of the limit of a group system is shown to be itself the limit of a group system, the elements of which can be described in terms of subgroups of the original group system.
On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities
Alan
Landman
89-126
Abstract: We consider a holomorphic family $ {\{ {V_t}\} _{t \in D}}$ of projective algebraic varieties ${V_t}$ parametrized by the unit disc $D = \{ t \in {\mathbf{C}}:\vert t\vert < 1\}$ and where ${V_t}$ is smooth for $t \ne 0$ but ${V_0}$ may have arbitrary singularities. Displacement of cycles around a path $t = {t_0}{e^{i\theta }}(0 \leqslant \theta \leqslant 2\pi )$ leads to the Picard-Lefschetz transformation $T:{H_\ast }({V_{{t_0}}},{\mathbf{Z}}) \to {H_\ast }({V_{{t_0}}},{\mathbf{Z}})$ on the homology of a smooth ${V_{t0}}$. We prove that the eigenvalues of $T$ are roots of unity and obtain an estimate on the elementary divisors of $T$. Moreover, we give a global inductive procedure for calculating $T$ in specific examples, several of which are worked out to illustrate the method.
Regularity properties of the element of closest approximation
Harold S.
Shapiro
127-142
Abstract: Given an element $f \epsilon {L^p}(T),1 < p < \infty$, and a closed translation invariant subspace $S$ of ${L^p}(T)$, we investigate the regularity (smoothness) properties of the element of $S$ which is closest to $f$. The regularity of this element is in general less than that of $f$. The problem reveals a surprising connection with a hitherto unstudied class of extremal Fourier multipliers.
Wreath products and representations of degree one or two
J. M.
Bateman;
Richard E.
Phillips;
L. M.
Sonneborn
143-153
Abstract: ${\mathcal{S}_2}$ denotes all groups $G$ that possess an ascending invariant series whose factors are one- or two-generated Abelian groups. We are interested in the ptoblem (1): For which nontrivial groups $A$ and $B$ is $A$ wr $B$ in $ {\mathcal{S}_2}?$ (1) has been completely solved by D. Parker in the case where $ A$ and $B$ are finite of odd order. Parker's results are partially extended here to cover groups of even order. Our answer to (1) is complete in the case where $ A$ is a finite $ 2$-group: If $ A$ is a finite $ 2$-group, $A$ wr $B$ is in $ {\mathcal{S}_2}$ iff $ B$ is finite and $B/{O_2}(B)$ is isomorphic to a subgroup of a dihedral group of an elementary $3$-group. If $A$ is not a $2$-group, we offer only necessary conditions on $ B$. Problem (1) is closely related to Problem (2): If $F$ is a prime field or the integers, which finite groups $B$ have all their irreducible representations over $F$ of degrees one or two? It is shown that all finite $B$ which satisfy (2) are ${\mathcal{S}_2}$ groups; in particular all such $ B$ are solvable.
On derived functors of limit
Dana May
Latch
155-163
Abstract: If $\mathcal{A}$ is a cocomplete category with enough projectives and $ {\mathbf{C}}$ is a $\downarrow$-finite small category, then there is a spectral sequence which shows that the cardinality of ${\mathbf{C}}$ and colimits over finite initial subcategories ${\mathbf{C}}$ are determining factors for computation of derived functors of colimit. Applying a recent result of Mitchell to this spectral sequence we show that if the cardinality of $ {\mathbf{C}}$ is at most $\aleph _{n}$, and the flat dimension of ${\Delta ^ \ast }Z$ (constant diagram of type $ {{\mathbf{C}}^{{\text{op}}}}$ with value $Z$) is $k$, then the derived functors of $ {\lim _{\mathbf{C}}}:\mathcal{A}{b^{\mathbf{C}}} \to \mathcal{A}b$ vanish above dimension $n + 1 + k$.
Adjoining inverses to commutative Banach algebras
Béla
Bollobás
165-174
Abstract: Let $A$ be a commutative unital Banach algebra. Suppose $G \subset A$ is such that $\vert\vert a\vert\vert \leqslant \vert\vert ga\vert\vert$ for all $g \in G,a \in A$. Two questions are considered in the paper. Does there exist a superalgebra $ B$ of $A$ in which every $g \in G$ is invertible? Can one always have also $ \vert\vert{g^{ - 1}}\vert\vert \leqslant 1$ if $g \in G$? Arens proved that if $G = \{ g\}$ then there is an algebra containing ${g^{ - 1}}$, with $\vert\vert{g^{ - 1}}\vert\vert \leqslant 1$. In the paper it is shown that if $G$ is countable $B$ exists, but if $G$ is uncountable, this is not necessarily so. The answer to the second question is negative even if $G$ consists of only two elements.
Differential geometric structures on principal toroidal bundles
David E.
Blair;
Gerald D.
Ludden;
Kentaro
Yano
175-184
Abstract: Under an assumption of regularity a manifold with an $f$-structure satisfying certain conditions analogous to those of a Kähler structure admits a fibration as a principal toroidal bundle ovet a Kähler manifold. In some natural special cases, additional information about the bundle space is obtained. Finally, curvature relations between the bundle space and the base space are studied.
A law of iterated logarithm for stationary Gaussian processes
Pramod K.
Pathak;
Clifford
Qualls
185-193
Abstract: In this article the following results are established. Theorem A. Let $\{ X(t):0 \leqslant t < \infty \}$ be a stationary Gaussian process with continuous sample functions and $ E[X(t)] \equiv 0$. Suppose that the covariance function $r(t)$ satisfies the following conditions. (a) $r(t) = 1 - \vert t{\vert^\alpha }H(t) + o(\vert t{\vert^\alpha }H(t))$ as $t \to 0$, where $0 < \alpha \leqslant 2$ and $H$ varies slowly at zero, and (b) $ r(t) = O(1/\log t)$ as $t \to \infty$ Then for any nondecreasing positive function $\phi (t)$ defined on $ [a,\infty )$ with $ \phi (\infty ) = \infty ,P[X(t) > \phi (t)$ i.o. for some sequence ${t_n} \to \infty ] = 0or1$ according as the integral $I(\phi ) = \int_a^\infty {g(\phi (t))\phi {{(t)}^{ - 1}}\exp ( - {\phi ^2}(t)/2)dt}$ is finite or infinite, where $g(x) = 1/_\sigma ^{ \sim - 1}(1/x)$ is a regularly varying function with exponent $2/\alpha$ and $_\sigma ^{ \sim 2}(t) = 2\vert t{\vert^\alpha }H(t)$. Theorem C. Let $\{ {X_n}:n \geqslant 1\} $ be a stationary Gaussian sequence with zero mean and unit variance. Suppose that its covariance function satisfies, for some $\gamma > 0,r(n) = O(1/{n^\gamma })\;as\;n \to \infty$. Let $\{ \phi (n):n \geqslant 1\}$ be a nondecreasing sequence of positive numbers with ${\lim _{n \to \infty }}\phi (n) = \infty$; suppose that $\Sigma (1/\phi (n))\exp ( - {\phi ^2}(n)/2) = \infty$. Then $\displaystyle \mathop {\lim }\limits_{n \to \infty } \sum\limits_{1 \leq k \leq n} {{I_k}} /\sum\limits_{1 \leq k \leq n} {E[{I_k}] = 1\quad a.s.,}$ where $ {I_k}$ denotes the indicator function of the event $\{ {X_k} > \phi (k)\} $.
Free $Z\sb{8}$ actions on $S\sp{3}$
Gerhard X.
Ritter
195-212
Abstract: This paper is devoted to the problem of classifying periodic homeomorphisms which act freely on the $3$-sphere. The main result is the classification of free period eight actions and a generalization to free actions whose squares are topologically equivalent to orthogonal transformations. The result characterizes those $3$-manifolds which have the $3$-sphere as universal covering space and the cyclic group of order eight as fundamental group.
On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. I
Chung Wu
Ho
213-233
Abstract: Let $K$ be a proper rectilinear triangulation of a $2$-simplex $S$ in the plane and $L(K)$ be the space of all homeomorphisms of $ S$ which are linear on each simplex of $K$ and are fixed on Bd$ (S)$. The author shows in this paper that $L(K)$ with the compact open topology is simply-connected. This is a generalization of a result of S. S. Cairns in 1944 that the space $L(K)$ is pathwise connected. Both results will be used in Part II of this paper to show that $ {\pi _0}({L_2}) = {\pi _1}({L_2}) = 0$ where ${L_n}$ is a space of p.l. homeomorphisms of an $ n$-simplex, a space introduced by $ {\mathbf{R}}$. Thom in his study of the smoothings of combinatorial manifolds.
On certain homotopy properties of some spaces of linear and piecewise linear homeomorphisms. II
Chung Wu
Ho
235-243
Abstract: In his study of the smoothings of p. l. manifolds, R. Thom considered the homotopy groups of a certain space ${L_n}$ of p.l. homeomorphisms on an $ n$-simplex. N. H. Kuiper showed in 1965 that the higher homotopy groups of $ {L_n}$ were in general nontrivial. The main result in this paper is that $ {\pi _0}({L_2}) = {\pi _1}({L_2}) = 0$. The proof of this result is based on a theorem of S. S. Cairns in 1944 on the deformation of rectilinear complexes in ${R^2}$ and a theorem established in Part I of this paper.
Appell polynomial expansions and biorthogonal expansions in Banach spaces
J. D.
Buckholtz
245-272
Abstract: Let $\{ {p_k}\} _0^\infty$ denote the sequence of Appell polynomials generated by an analytic function $ \phi$ with the property that the power series for $\theta = 1/\phi$ has a larger radius of convergence than the power series for $\phi$. The expansion and uniqueness properties of $\{ {p_k}\}$ are determined completely. In particular, it is shown that the only convergent $\{ {p_k}\}$ expansions are basic series, and that there are no nontrivial representations of 0. An underlying Banach space structure of these expansions is also studied.
Global regularity for $\bar \partial $ on weakly pseudo-convex manifolds
J. J.
Kohn
273-292
Abstract: Let $M'$ be a complex manifold and let $\overline \partial$-closed $ (p,q)$-form $\alpha$, which is ${C^\infty }$ on $ \overline M$ and which is cohomologous to zero on $M$ then for every $m$ there exists a $(p,q - 1)$-form ${u_{(m)}}$ which is ${C^m}$ on $ \overline M$ such that $ \overline \partial {u_{(m)}} = \alpha$.
Fourier analysis on linear metric spaces
J.
Kuelbs
293-311
Abstract: Probability measures on a real complete linear metric space $E$ are studied via their Fourier transform on $E'$ provided $E$ has the approximation property and possesses a real positive definite continuous function $ \Phi (x)$ such that $\vert\vert x\vert\vert > \epsilon$ implies $ \Phi (0) - \Phi (x) > c(\epsilon)$ where $ c(\epsilon) > 0$. In this setting we obtain conditions on the Fourier transforms of a family of tight Borel probabilities which yield tightness of the family of measures. This then is applied to obtain necessary and sufficient conditions for a complex valued function on $E'$ to be the Fourier transform of a tight Borel probability on $E$. An extension of the Levy continuity theorem as given by $ {\text{L}}$. Gross for a separable Hilbert space is obtained for such metric spaces. We also prove that various Orlicz-type spaces are in the class of spaces to which our results apply. Finally we apply our results to certain Orlicz-type sequence spaces and obtain conditions sufficient for tightness of a family of probability measures in terms of uniform convergence of the Fourier transforms on large subsets of the dual. We also obtain a more explicit form of Bochner's theorem for these sequence spaces. The class of sequence spaces studied contains the $ {l_p}$ spaces $(0 < p \leqslant 2)$ and hence these results apply to separable Hilbert space.
Two-norm spaces and decompositions of Banach spaces. II
P. K.
Subramanian;
S.
Rothman
313-327
Abstract: Let $X$ be a Banach space, $Y$ a closed subspace of ${X^\ast }$. One says $X$ is $Y$-reflexive if the canonical imbedding of $ X$ onto ${Y^\ast }$ is an isometry and $ Y$-pseudo reflexive if it is a linear isomorphism onto. If $ X$ has a basis and $ Y$ is the closed linear span of the corresponding biorthogonal functionals, necessary and sufficient conditions for $ X$ to be $Y$-pseudo reflexive are due to I. Singer. To every $B$-space $X$ with a decomposition we associate a canonical two-norm space ${X_s}$ and show that the properties of ${X_s}$, in particular its $ \gamma$-completion, may be exploited to give different proofs of Singer's results and, in particular, to extend them to $ B$-spaces with decompositions. This technique is then applied to a study of direct sum of $B$-spaces with respect to a BK space. Necessary and sufficient conditions for such a space to be reflexive are obtained.
Intersections of quasi-local domains
Bruce
Prekowitz
329-339
Abstract: Let $R = \bigcap {{V_i}}$ be an intersection of quasi-local domains with a common quotient field $ K$. Our goal is to find conditions on the ${V_i}$'s in order to get some or all of $ {V_i}$'s to be localizations of $R$. We show for example that if ${V_1}$ is a $1$-dimensional valuation domain and if ${V_1} \nsupseteq {V_2}$, then both ${V_1}$ and ${V_2}$ are localizations of $R = {V_1} \cap {V_2}$.
Deforming cohomology classes
John J.
Wavrik
341-350
Abstract: Let $\pi :X \to S$ be a flat proper morphism of analytic spaces. $\pi$ may be thought of as providing a family of compact analytic spaces, ${X_s}$, parametrized by the space $S$. Let $ \mathcal{F}$ be a coherent sheaf on $X$ flat over $S$. $ \mathcal{F}$ may be thought of as a family of coherent sheaves, ${\mathcal{F}_s}$, on the family of spaces $ {X_s}$. Let $o \in S$ be a fixed point, ${\xi _o} \in Hq({X_o},{\mathcal{F}_o})$. In this paper, we consider the problem of extending $ {\xi _o}$ to a cohomology class $\xi \in Hq({\pi ^{ - 1}}(U),\mathcal{F})$ where $ U$ is some neighborhood of $ o$ in $S$. Extension problems of this type were first considered by P. A. Griffiths who obtained some results in the case in which the morphism $\pi$ is simple and the sheaf $\mathcal{F}$ is locally free. We obtain generalizations of these results without the restrictions. Among the applications of these results is a necessary and sufficient condition for the existence of a space of moduli for a compact manifold. This application was discussed in an earlier paper by the author. We use the Grauert ``direct image'' theorem, the theory of Stein compacta, and a generalization of a result of M. Artin on solutions of analytic equations to reduce the problem to an algebraic problem. In §2 we discuss obstructions to deforming ${\xi _o}$; in §3 we show that if no obstructions exist, ${\xi _o}$ may be extended; in §4 we give a useful criterion for no obstructions; and in §5 we discuss some examples.
Improbability of collisions in Newtonian gravitational systems. II
Donald G.
Saari
351-368
Abstract: It is shown that the set of initial conditions leading to collision in the inverse square force law has measure zero. For the inverse $q$ force law the behavior of binary collisions for $1 < q < 3$ and the behavior of any collision for $q = 1$ is developed. This information is used to show that collisions are improbable in the inverse $ q$ force law where $q < 17/7$ and that binary collisions are improbable for $q < 3$.
Successive remainders of the Newton series
G. W.
Crofts;
J. K.
Shaw
369-383
Abstract: If $f$ is analytic in the open unit disc $ D$ and $\lambda$ is a sequence of points in $ D$ converging to 0, then $ f$ admits the Newton series expansion $f(z) = f({\lambda _1}) + \sum\nolimits_{n = 1}^\infty {\Delta _\lambda ^nf({\lambda _{n + 1}})(z - {\lambda _1})(z - {\lambda _2}) \cdots (z - {\lambda _n})}$, where $ \Delta _\lambda ^nf(z)$ is the $n$th divided difference of $f$ with respect to the sequence $\lambda$. The Newton series reduces to the Maclaurin series in case ${\lambda _n} \equiv 0$. The present paper investigates relationships between the behavior of zeros of the normalized remainders $\Delta _\lambda ^kf(z) = \Delta _\lambda ^kf({\lambda _{k + 1}}) + \sum\nolimi... ...bda ^nf({\lambda _{n + 1}})(z - {\lambda _{k + 1}}) \cdots (z - {\lambda _n})}$ of the Newton series and zeros of the normalized remainders $\sum\nolimits_{n = k}^\infty {{a_n}{z^{n - k}}}$ of the Maclaurin series for $f$. Let $ {C_\lambda }$ be the supremum of numbers $c > 0$ such that if $f$ is analytic in $D$ and each of $\Delta _\lambda ^kf(z),\;0 \leqslant k < \infty$, has a zero in $ \vert z\vert \leqslant c$, then $f \equiv 0$. The corresponding constant for the Maclaurin series ( ${C_\lambda }$, where ${\lambda _n} \equiv 0$) is called the Whittaker constant for remainders and is denoted by $W$. We prove that ${C_\lambda } \geqslant W$, for all $\lambda$, and, moreover, ${C_\lambda } = W$ if $\lambda \in {l_1}$. In obtaining this result, we prove that functions $f$ analytic in $D$ have expansions of the form $f(z) = \sum\nolimits_{n = 0}^\infty {\Delta _\lambda ^nf({z_n}){C_n}(z)}$, where $\vert{z_n}\vert \leqslant W$, for all $n$, and ${C_n}(z)$ is a polynomial of degree $ n$ determined by the conditions $\Delta _\lambda ^j{C_k}({z_j}) = {\delta _{jk}}$.
Functions automorphic on large domains
David A.
James
385-400
Abstract: For a discontinuous group $\Gamma \subset {\text{SL}}(2,R)$, Poincaré produced a corresponding nonconstant automorphic form, meromorphic on the open upper half plane ${\Pi ^ + }$. When the domain of meromorphicity grows larger than $ {\Pi ^ + }$, the type of group which can support an automorphic form is restricted, and the corresponding forms are generally quite simple. A complete analysis of this phenomenon is presented, with examples which show results are best possible.
The degree of approximation by Chebyshevian splines
R.
DeVore;
F.
Richards
401-418
Abstract: This paper studies the connections between the smoothness of a function and its degree of approximation by Chebyshevian splines. This is accomplished by proving companion direct and inverse theorems which give a characterization of smoothness in terms of degree of approximation. A determination of the saturation properties is included.
Generalized Laplacians and multiple trigonometric series
M. J.
Kohn
419-428
Abstract: V. L. Shapiro gave a $k$-variable analogue for Riemann's theorem on formal integration of trigonometric series. This paper derives Shapiro's results with weaker conditions on the coefficients of the series and extends the results to series which are Bochner-Riesz summable of larger order.
The polynomial identities of the Grassmann algebra
D.
Krakowski;
A.
Regev
429-438
Abstract: By using the theory of codimensions the $T$-ideal of polynomial identities of the Grassmann (exterior) algebra is computed.
Quadratic expressions in a free boson field
Abel
Klein
439-456
Abstract: Quadratic expressions in a massive spinless free Boson field are treated by an appropriate extension of the method of second quantization. A certain class of these second quantized operators is shown to generate semigroups that act on a suitable scale of ${L_p}$-spaces, obtained through the diagonalization of the field at a fixed time, in a particularly regular fashion. The techniques are developed first in an abstract setting, and then applied to the neutral scalar free field. The locally correct generator of Lorentz transformations for $ P{(\varphi )_2}$ is studied in detail, and essential selfadjointness is shown. These techniques are also used to solve explicitly the ${({\varphi ^2})_n}$ model.
The lattice triple packing of spheres in Euclidean space
G. B.
Purdy
457-470
Abstract: We say that a lattice $\Lambda$ in $n$-dimensional Euclidean space ${E_n}$ provides a $k$-fold packing for spheres of radius 1 if, when open spheres of radius 1 are centered at the points of $\Lambda$, no point of space lies in more than $ k$ spheres. The multiple packing constant $ \Delta _k^{(n)}$ is the smallest determinant of any lattice with this property. In the plane, the first three multiple packing constants $\Delta _2^{(2)},\Delta _3^{(2)}$, and $\Delta _4^{(2)}$ are known, due to the work of Blundon, Few, and Heppes. In ${E_3},\Delta _2^{(3)}$ is known, because of work by Few and Kanagasabapathy, but no other multiple packing constants are known. We show that $ \Delta _3^{(3)} \leqslant 8\sqrt {38} /27$ and give evidence that $\Delta _3^{(3)} = 8\sqrt {38} /27$. We show, in fact, that a lattice with determinant $8\sqrt {38} /27$ gives a local minimum of the determinant among lattices providing a $3$-fold packing for the unit sphere in ${E_3}$.
Boundary values in the four color problem
Michael O.
Albertson;
Herbert S.
Wilf
471-482
Abstract: Let $G$ be a planar graph drawn in the plane so that its outer boundary is a $k$-cycle. A four coloring of the outer boundary $ \gamma$ is admissible if there is a four coloring of $G$ which coincides with $\gamma$ on the boundary. If $\psi$ is the number of admissible boundary colorings, we show that the 4CC implies $\psi \geqslant 3 \cdot {2^k}$ for $k = 3, \cdots ,6$. We conjecture this to be true for all $k$ and show $\psi$ is $\geqslant c{((1 + {5^{1/2}})/2)^k}$. A graph is totally reducible (t.r.) if every boundary coloring is admissible. There are triangulations of the interior of a $k$-cycle which are t.r. for anv $k$. We investigate a class of graphs called annuli, characterize t.r. annuli and show that annuli satisfy the above conjecture.
Product integral solutions for hereditary systems
James A.
Reneke
483-493
Abstract: Hereditary systems which satisfy a Lipschitz condition are solved in terms of product integrals. Realizations of this type of hereditary system are provided from functional differential and integral equations.
Topological spaces and lattices of lower semicontinuous functions
M. C.
Thornton
495-506
Abstract: Lower semicontinuous real-valued functions on a space $X$ form a conditionally complete distributive lattice $L(X)$. Those lattices which can be represented as $ L(X)$ for some $ X$ are characterized algebraically. All spaces producing isomorphic lattices ate determined. The class of spaces which are determined by their function lattices is introduced.
Erratum to ``Martingales of strongly measurable Pettis integrable functions''
J. J.
Uhl
507
Erratum to ``Entropy for group endomorphisms and homogeneous spaces''
R.
Bowen
509-510