The weakly coupled Yukawa$\sb{2}$ field theory: cluster expansion and Wightman axioms
Alan
Cooper;
Lon
Rosen
1-88
Abstract: We prove convergence of the Glimm-Jaffe-Spencer cluster expansion for the weakly coupled Yukawa model in two dimensions, thereby verifying the Wightman axioms including a positive mass gap.
P.R.-regulated systems of notation and the subrecursive hierarchy equivalence property
Fred
Zemke
89-118
Abstract: We can attempt to extend the Grzegorczyk Hierarchy transfinitely by defining a sequence of functions indexed by the elements of a system of notation $\mathcal{S}$, using either iteration (majorization) or enumeration techniques to define the functions. (The hierarchy is then the sequence of classes of functions elementary in the functions of the sequence of functions.) In this paper we consider two sequences $ {\{ {F_s}\} _{s \in \mathcal{S}}}$ and ${\{ {G_s}\} _{s \in \mathcal{S}}}$ defined by iteration and a sequence ${\{ {E_s}\} _{s \in \mathcal{S}}}$ defined by enumeration; the corresponding hierarchies are $ \{ {\mathcal{F}_s}\} ,\{ {\mathcal{G}_s}\} ,\{ \mathcal{E}{_s}\}$. We say that $ \mathcal{S}$ has the subrecursive hierarchy equivalence property if these two conditions hold: (I) $ {\mathcal{E}_s} = {\mathcal{F}_s} = {\mathcal{G}_s}$ for all $s \in \mathcal{S}$; (II) ${\mathcal{E}_s} = {\mathcal{E}_t}$ for all $ s,t \in \mathcal{S}$ such that $\vert s\vert = \vert t\vert(\vert s\vert$ is the ordinal denoted by s). We show that a certain type of system of notation, called p.r.-regulated, has the subrecursive hierarchy equivalence property. We present a nontrivial example of such a system of notation, based on Schütte's Klammersymbols. The resulting hierarchy extends those previously in print, which used the so-called standard fundamental sequences for limits $< {\varepsilon _0}$.
A partial surface variation for extremal schlicht functions
T. L.
McCoy
119-138
Abstract: Let a topological sphere be formed from $ \vert z\vert \leqslant 1$ by dissecting the circumference into finitely many pairs $ {I_j},{I'_j}$. In a natural way, Q-polygons become Riemann surfaces, thus can be mapped conformally onto the number sphere. When Q is of the form $Q(z) = \Sigma _{j = - N}^N{B_j}{z^j}$, then the corresponding mapping functions, suitably normalized, become the extremal schlicht functions for the coefficient body $ {V_{N + 1}}$ [3, p. 120]. Suppose that for a given dissection of $\vert z\vert = 1$ there is a family $ Q(z,t)$ of consistent meromorphic functions. For Q sufficiently smooth as a function of $ \varepsilon$, we study the variation of the corresponding normalized mapping functions $ f(p,\varepsilon )$, using results of [2], and show smoothness of f as a function of $ \varepsilon$. Specializing Q to the form above, we deduce the existence of smooth submanifolds of $\partial {V_{N + 1}}$ and obtain a variational formula for the extremal schlicht functions corresponding to motion along these submanifolds.
On $n$-widths in $L\sp{\infty }$
Charles A.
Micchelli;
Allan
Pinkus
139-174
Abstract: The n-width in ${L^\infty }$ of certain sets determined by matrices and integral operators is determined. The notion of total positivity is essential in the analysis.
Weak convergence of the area of nonparametric $L\sb{1}$ surfaces
Kim E.
Michener
175-184
Abstract: The main purpose of this work is to obtain an analogue to a theorem of L. C. Young on the behavior of the nonparametric surface area of continuous functions. The analogue is for $ {L^1}$ functions of generalized bounded variation. By considering arbitrary Borel vector measures and kernels other than the area kernel, results concerning the weak behavior of measures induced by a class of sublinear functionals are obtained.
Inclusion relations between power methods and matrix methods of limitation
Abraham
Ziv
185-211
Abstract: A matrix method of limitation is a generalization of both ordinary Toeplitz methods and semicontinuous methods. A power method is a generalization of both Abel's method and Borel's exponential method. The main concern of this paper is to find necessary and sufficient conditions for the field of a given power method to be included in the field of a given matrix method. The problem is solved for a wide family of power methods which includes all Abel type methods, the logarithmic method, all Borel type methods and others (also nonregular power methods). Preliminary results, which serve as tools in the solution of the main problem, clarify some aspects of the nature of the field of a power method as an FK space.
Group presentations corresponding to spines of $3$-manifolds. II
R. P.
Osborne;
R. S.
Stevens
213-243
Abstract: Let $\phi = \langle {a_1}, \ldots ,{a_n}\vert{R_1}, \ldots ,{R_m}\rangle$ denote a group presentation. Let $ {K_\phi }$ denote the corresponding 2-complex. It is well known that every compact 3-manifold has a spine of the form ${K_\phi }$ for some $\phi$, but that not every $ {K_\phi }$ is a spine of a compact 3-manifold. Neuwirth's algorithm (Proc. Cambridge Philos. Soc. 64 (1968), 603-613) decides whether ${K_\phi }$ can be a spine of a compact 3-manifold. However, it is impractical for presentations of moderate length. In this paper a simple planar graph-like object, called a RR-system (railroad system), is defined. To each RR-system corresponds a whole family of compact orientable 3-manifolds with spines of the form ${K_\phi }$, where $\phi$ has a particular form (e.g., $\langle a,b{a^m}{b^n}{a^p}{b^n},{a^m}{b^n}{a^m}{b^q}\rangle $), subject only to certain requirements of relative primeness of certain pairs of exponents. Conversely, every ${K_\phi }$ which is a spine of some compact orientable 3-manifold can be obtained in this way. An equivalence relation on RR-systems is defined so that equivalent RR-systems determine the same family of manifolds. Results of Zieschang are applied to show that the simplest spine of 3-manifolds arises from a particularly simple kind of RR-system called a reduced RR-system. Following Neuwirth, it is shown how to determine when a RR-system gives rise to a collection of closed 3-manifolds.
Group presentations corresponding to spines of $3$-manifolds. III
R. P.
Osborne;
R. S.
Stevens
245-251
Abstract: Continuing after the previous papers of this series, attention is devoted to RR-systems having two towns (i.e., to compact 3-manifolds with spines corresponding to group presentations having two generators). An interesting kind of symmetry is noted and then used to derive some useful results. Specifically, the following theorems are proved: Theorem 1. Let $ \phi$ be a group presentation corresponding to a spine of a compact orientable 3-manifold, and let w be a relator of $ \phi$ involving just two generators a and b. If w is cyclically reduced, then either (a) w can be ``written backwards" (i.e., if $w = {a^{{m_1}}}{b^{{m_1}}}{a^{{m_2}}}{b^{{n_2}}} \ldots {a^{{m_k}}}{b^{{n_k}}}$, then w is a cyclic conjugate of ${b^{{n_k}}}{a^{{m_k}}} \ldots {b^{{n_2}}}{a^{{m_2}}}{b^{{n_1}}}{a^{{m_1}}}$), or (b) w lies in the commutator subgroup of the free group on a and b. Theorem 2. (Loose translation). If $\phi$ is a group presentation with two generators and if the corresponding 2-complex ${K_\phi }$ is a spine of a closed orientable 3-manifold then, ${K_\phi }$ is a spine of a closed orientable 3-manifold if and (except for two minor cases) only if $\phi$ has two relators and among the six allowable types of syllables (3 in each generator), exactly four occur an odd number of times. Further, each of the two relators can be ``written backwards."
Structure of symmetric tensors of type $(0, 2)$ and tensors of type $(1, 1)$ on the tangent bundle
Kam Ping
Mok;
E. M.
Patterson;
Yung Chow
Wong
253-278
Abstract: The concepts of M-tensor and M-connection on the tangent bundle TM of a smooth manifold M are used in a study of symmetric tensors of type (0, 2) and tensors of type (1, 1) on TM. The constructions make use of certain local frames adapted to an M-connection. They involve extending known results on TM using tensors on M to cases in which these tensors are replaced by M-tensors. Particular attention is devoted to (pseudo-) Riemannian metrics on TM, notably those for which the vertical distribution on TM is null or nonnull, and to the construction of almost product and almost complex structures on TM.
Lattices of convex sets
Mary Katherine
Bennett
279-288
Abstract: If V is a vector space over an ordered division ring, C a convex subset of V and L the lattice of convex subsets of C, then we call L a convexity lattice. We give necessary and sufficient conditions for an abstract lattice to be a convexity lattice in the finite dimensional case.