Transfinite graphs and electrical networks
A. H.
Zemanian
1-36
Abstract: All prior theories of infinite electrical networks assume that such networks are finitely connected, that is, between any two nodes of the network there is a finite path. This work establishes a theory for transfinite electrical networks wherein some nodes are not connected by finite paths but are connected by transfinite paths. Moreover, the voltages at those nodes may influence each other. The main difficulty to surmount for this extension is the construction of an appropriate generalization of the concept of connectedness. This is accomplished by extending the idea of a node to encompass infinite extremities of a graph. The construction appears to be novel and leads to a hierarchy of transfinite graphs indexed by the finite and infinite ordinals. Two equivalent existence and uniqueness theorems are established for transfinite resistive electrical networks based upon Tellegen's equation, one using currents and the other using voltages as the fundamental quantities. Kirchhoff's laws do not suffice for this purpose and indeed need not hold everywhere in infinite networks. Although transfinite countable electrical networks have in general an uncountable infinity of extremities, called "tips," the number of different tip voltages may be radically constrained by both the graph of the network and its resistance values. Conditions are established herein under which various tip voltages are compelled to be the same. Furthermore, a theorem of Shannon-Hagelbarger on the concavity of resistance functions is extended to the driving-point resistance between any two extremities of arbitrary ranks. This is based upon an extension of Thomson's least power principle to transfinite networks.
Isotropy groups of homotopy classes of maps
G.
Triantafillou
37-48
Abstract: Let $\operatorname{aut}(X)$ be the group of homotopy classes of self-homotopy equivalences of a space $ X$ and let $[f] \in [X,Y]$ be a homotopy class of maps from $ X$ to $Y$ . The aim of this paper is to prove that under certain nilpotency and finiteness conditions the isotropy group $ \operatorname{aut}{(X)_{[f]}}$ of $[f]$ under the action of $\operatorname{aut}(X)$ on $[X,Y]$ is commensurable to an arithmetic group. Therefore $ \operatorname{aut}{(X)_{[f]}}$ is a finitely presented group by a result of Borel and Harish-Chandra.
The classification of complete minimal surfaces with total curvature greater than $-12\pi$
Francisco J.
López
49-74
Abstract: We classify complete orientable minimal surfaces with finite total curvature $- 8\pi$.
Subcriticality and gaugeability of the Schr\"odinger operator
Z.
Zhao
75-96
Abstract: We investigate a Schrödinger operator $- \Delta /2 + V$ in ${R^d}\;(d \geq 3)$ with a potential $V$ in the class ${K_d}$ satisfying a similar Kato condition at infinity, and prove an equivalence theorem connecting various conditions on subcriticality, strong positivity and gaugeability of the operator.
QED domains and NED sets in $\overline{\bf R}{}\sp n$
Shan Shuang
Yang
97-120
Abstract: This paper contributes to the theory of quasiextremal distance (or ${\text{QED}}$) domains. We associate with every ${\text{QED}}$ domain $D$ two $ {\text{QED}}$ constants $ M(D)$ and ${M^{\ast} }(D)$ and exhibit how these constants reflect the geometry of $D$. For example, we give a geometric characterization for $ {\text{QED}}$ domains $ D$ with ${M^{\ast}}(D) = 2$ and obtain some sharp estimates of ${\text{QED}}$ constants $M(D)$ and ${M^{\ast} }(D)$ for different kinds of domains.
Mixed groups
Paul
Hill;
Charles
Megibben
121-142
Abstract: As the culmination of a series of several papers, we establish here a combinatorial characterization of Warfield groups (that is, direct summands of simply presented abelian groups) in terms of knice subgroups--a refinement of the concept of nice subgroup appropriate to the study of groups containing elements of infinite order. Central to this theory is the class of $k$-groups, those in which 0 is a knice subgroup, and the proof that this class is closed under the formation of knice isotype subgroups. In particular, a direct summand of a $k$-group is a $k$-group. As an application of our Axiom $ 3$ characterization of Warfield groups, we prove that $k$-groups of cardinality $ {\aleph _1}$ have sequentially pure projective dimension $\leq 1$; or equivalently, if $ H$ is a knice isotype sub-group of the Warfield group $G$ with $ \vert G/H\vert = {\aleph _1}$, then $H$ is itself a Warfield group.
Central extensions of current algebras
Paul
Zusmanovich
143-152
Abstract: The second cohomology group of Lie algebras of kind $L \otimes U$ with trivial coefficients is investigated, where $L$ admits a decomposition with one-dimensional root spaces and $U$ is an arbitrary associative commutative algebra with unit. This paper gives a unification of some recent results of C. Kassel and A. Haddi and provides a determination of central extensions of certain modular semisimple Lie algebras.
Subgroup conditions for groups acting freely on products of spheres
Judith H.
Silverman
153-181
Abstract: Let $d$ and $h$ be integers such that either $d \geq 2$ and $h = {2^d} - 1$, or $d = 4$ and $h = 5$. Suppose that the group $\mathcal{G}$ contains an elementary-abelian $ 2$-subgroup ${E_d}$ of rank $d$ with an element $\sigma$ of order $h$ in its normalizer. We show that if $\mathcal{G}$ admits a free and $ {{\mathbf{F}}_2}$-cohomologically trivial action on $ {({S^n})^d}$, then some nontrivial power of $\sigma$ centralizes ${E_d}$. The cohomology ring $ {H^{\ast} }({E_d};{{\mathbf{F}}_2}) \simeq {{\mathbf{F}}_2}[{y_1}, \ldots ,{y_d}]$ is a module over the Steenrod algebra $\mathcal{A}(2)$. Let $\theta \in {{\mathbf{F}}_2}[{y_1}, \ldots ,{y_d}]$, and let $ c \geq d - 2$ be an integer. We show that $\theta$ divides $S{q^{{2^i}}}(\theta )$ in the polynomial ring for $0 \leq i \leq c \Leftrightarrow \theta = {\tau ^{{2^{c - d + 3}}}}\pi $ , where $\tau$ divides $S{q^{{2^i}}}(\tau )$ for $0 \leq i \leq d - 3$ and $\pi$ is a product of linear forms.
Meromorphic extension of analytic continued fractions across their divergence line with applications to orthogonal polynomials
Hans-J.
Runckel
183-212
Abstract: For the limit periodic $J$-fraction $K( - {a_n}/(\lambda + {b_n}))$, ${a_n}$, ${b_n} \in \mathbb{C}$, $n \in \mathbb{N}$, which is normalized such that it converges and represents a meromorphic function $f(\lambda )$ on $ {\mathbb{C}^{\ast} }: = \mathbb{C}\backslash [ - 1,1]$, the numerators $ {A_n}$ and denominators $ {B_n}$ of its $ n$th approximant are explicitly determined for all $n \in \mathbb{N}$. Under natural conditions on the speed of convergence of ${a_n}$, ${b_n}$, $n \to \infty $, the asymptotic behaviour of the orthogonal polynomials ${B_n}$, $ {A_{n + 1}}$ (of first and second kind) is investigated on ${\mathbb{C}^{\ast} }$ and $[ - 1,1]$. An explicit representation for $f(\lambda )$ yields continuous extension of $ f$ from ${\mathbb{C}^{\ast} }$ onto upper and lower boundary of the cut $( - 1,1)$. Using this and a determinant relation, which asymptotically connects both sequences $ {A_n}$, ${B_n}$, one obtains nontrivial explicit formulas for the absolutely continuous part (weight function) of the distribution functions for the orthogonal polynomial sequences ${B_n}$, ${A_{n + 1}}$, $n \in \mathbb{N}$. This leads to short proofs of results which generalize and supplement results obtained by P. G. Nevai [7]. Under a stronger condition the explicit representation for $ f(\lambda )$ yields meromorphic extension of $f$ from $ {\mathbb{C}^{\ast} }$ across $( - 1,1)$ onto a region of a second copy of $\mathbb{C}$ which there is bounded by an ellipse, whose focal points $\pm 1$ are first order algebraic branch points for $ f$. Then, by substitution, analogous results on continuous and meromorphic extension are obtained for limit periodic continued fractions $K( - {a_n}(z)/(\lambda (z) + {b_n}(z)))$, where $ {a_n}(z)$, ${b_n}(z)$, $ \lambda (z)$ are holomorphic on a region in $ \mathbb{C}$. Finally, for $ T$-fractions $ T(z) = K( - {c_n}z/(1 + {d_n}z))$ with $ {c_n} \to c$, ${d_n} \to d$, $ n \to \infty$, the exact convergence regions are determined for all $ c$, $d \in \mathbb{C}$. Again, explicit representations for $T(z)$ yield continuous and meromorphic extension results. For all $c$, $ d \in \mathbb{C}$ the regions (on Riemann surfaces) onto which $T(z)$ can be extended meromorphically, are described explicitly.
Characteristic cycles of holomorphic discrete series
Jen-Tseh
Chang
213-227
Abstract: The characteristic cycles of standard modules arising from closed $ K$-orbits in a flag variety are studied and those of holomorphic discrete series are determined explicitly. Also the asymptotic expansion of the distribution characters of holomorphic discrete series are computed; the result verifies a conjecture of D. Vogan in this special case.
On conjugacy separability of fundamental groups of graphs of groups
M.
Shirvani
229-243
Abstract: A complete determination of when the elements of a fundamental group of a (countable) graph of profinite groups are conjugacy distinguished is given. By embedding an arbitrary fundamental group $G$ into one with profinite vertex groups and making use of the above result, questions on conjugacy separability of $G$ can be reduced to the solution of equations in the vertex groups of $G$.
Fundamental solutions for hypoelliptic differential operators depending analytically on a parameter
Frank
Mantlik
245-257
Abstract: Let $ P(\lambda ,D) = \sum\nolimits_{\vert\alpha \vert \leq m} {{a_\alpha }(\lambda ){D^\alpha }}$ be a differential operator with constant coefficients ${a_\alpha }$ depending analytically on a parameter $\lambda$. Assume that each $P(\lambda ,D)$ is hypoelliptic and that the strength of $ P(\lambda ,D)$ is independent of $\lambda$. Under this condition we show that there exists a regular fundamental solution of $P(\lambda ,D)$ which also depends analytically on $ \lambda$.
Delta methods in enveloping algebras of Lie superalgebras
Jeffrey
Bergen;
D. S.
Passman
259-280
Abstract: Let $L$ be a Lie superalgebra over a field $ K$ of characteristic $ \ne 2$ . We define $\displaystyle \Delta (L) = \{ l \in L\vert{\dim _K}[L,l] < \infty \}.$ Then $ \Delta (L)$ is a Lie ideal of $L$ and is restricted if $L$ is restricted. $ \Delta (L)$ is the Lie superalgebra analog of the Lie delta ideal, used by the authors in the study of enveloping rings, and also of the finite conjugate center of a group, used in the study of group algebras and crossed products. In this paper we examine $U(L)$, where depending upon $\operatorname{char}K$, $U(L)$ denotes either the enveloping algebra or the restricted enveloping algebra of $L$. We show that $ \Delta (L)$ controls certain properties of $U(L)$. Specifically, we consider semiprimeness, primeness, almost constants, almost centralizers, central closures, and the Artinian condition.
Generalized second-order derivatives of convex functions in reflexive Banach spaces
Chi Ngoc
Do
281-301
Abstract: Generalized second-order derivatives introduced by Rockafellar in finite-dimensional spaces are extended to convex functions in reflexive Banach spaces. Parallel results are shown in the infinite-dimensional case. A result that plays an important role in applications is that the generalized second-order differentiability is preserved under the integral sign.
Nonnegative curvature on piecewise constant curvature spaces
Robert
Peszek
303-315
Abstract: We consider piecewise flat and piecewise spherical spaces. We prove that every piecewise flat cobordism which is a product near the boundary and has nonnegative curvature must be trivial in the metric sense. We also obtain several restrictions for piecewise $p$-spherical manifolds with nonnegative curvature and diameter equal to $\pi p$. We prove that such a manifold must be homeomorphic to a sphere and that it is a disjoint union of minimal paths connecting two points, which have length $\pi p$.
Strong Tutte functions of matroids and graphs
Thomas
Zaslavsky
317-347
Abstract: A strong Tutte function of matroids is a function of finite matroids which satisfies $F({M_1} \oplus {M_2}) = F({M_1})F({M_2})$ and $F(M) = {a_e}F(M\backslash e) + {b_e}F(M/e)$ for $ e$ not a loop or coloop of $ M$, where ${a_e}$, ${b_e}$ are scalar parameters depending only on $ e$. We classify strong Tutte functions of all matroids into seven types, generalizing Brylawski's classification of Tutte-Grothendieck invariants. One type is, like Tutte-Grothendieck invariants, an evaluation of a rank polynomial; all types are given by a Tutte polynomial. The classification remains valid if the domain is any minor-closed class of matroids containing all three-point matroids. Similar classifications hold for strong Tutte functions of colored matroids, where the parameters depend on the color of $e$, and for strong Tutte functions of graphs and edge-colored graphs whose values do not depend on the attachments of loops. The latter classification implies new characterizations of Kauffman's bracket polynomials of signed graphs and link diagrams.
$\Sigma\sb n$ definable sets without $\Sigma\sb n$ induction
C. T.
Chong;
K. J.
Mourad
349-363
Abstract: We prove that the Friedberg-Muchnik Theorem holds in all models of ${\Sigma _1}$ collection under the base theory ${P^- } + I{\Sigma _0}$. Generalizations to higher dimensional analogs are discussed. We also study the splitting of r.e. sets in these weak models of arithmetic.
Cohomological aspects of hypergraphs
F. R. K.
Chung;
R. L.
Graham
365-388
Abstract: By a $k$-graph we will mean a collection of $ k$-element subsets of some fixed set $V$. A $k$-graph can be regarded as a $(k - 1)$-chain on $ {2^V}$, the simplicial complex of all subsets of $V$, over the coefficient group $\mathbb{Z}/2$, the additive group of integers modulo $2$. The induced group structure on the $ (k - 1)$-chains leads to natural definitions of the coboundary $\delta$ of a chain, the cochain complex of $ C = \{ {C^k},\delta \}$ and the usual cohomology groups ${H^k}(C;\mathbb{Z}/2)$. In particular, it is possible to construct what could be called "higher-order" coboundary operators $ {\delta ^{(i)}}$, where ${\delta ^{(i)}}$ increases dimension by $i$ (rather than just $1$). In this paper we will develop various properties of these $ {\delta ^{(i)}}$, and in particular, compute the corresponding cohomology groups for ${2^V}$ over $ \mathbb{Z}/2$. It turns out that these groups depend in a rather subtle way on the arithmetic properties of $i$.
Algebra for Heckoid groups
Robert
Riley
389-409
Abstract: We introduce an infinite collection of (Laurent) polynomials associated with a $2$-bridge knot or link normal form $K = (\alpha ,\beta )$. Experimental evidence suggests that these "Heckoid polynomials" define the affine representation variety of certain groups, the Heckoid groups, for $K$ . We discuss relations which hold in the image of the generic representation for each polynomial. We show that, with a certain change of variable, each Heckoid polynomial divides the nonabelian representation polynomial of $L$ , where $L$ belongs to an infinite collection of $2$-bridge knots/links determined by $ K$ and the Heckoid polynomial. Finally, we introduce a "precusp polynomial" for each $2$-bridge knot normal form, and show it is the product of two (possibly reducible) non-constant polynomials. We are preparing a sequel on the Heckoid groups and the evidence for some of the geometrical assertions stated in the introduction.
Universal spaces for ${\bf R}$-trees
John C.
Mayer;
Jacek
Nikiel;
Lex G.
Oversteegen
411-432
Abstract: ${\mathbf{R}}$-trees arise naturally in the study of groups of isometries of hyperbolic space. An ${\mathbf{R}}$-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. It follows that an ${\mathbf{R}}$-tree is locally arcwise connected, contractible, and one-dimensional. Unique and local arcwise connectivity characterize ${\mathbf{R}}$-trees among metric spaces. A universal ${\mathbf{R}}$-tree would be of interest in attempting to classify the actions of groups of isometries on ${\mathbf{R}}$-trees. It is easy to see that there is no universal $ {\mathbf{R}}$-tree. However, we show that there is a universal separable ${\mathbf{R}}$-tree $ {T_{{\aleph _0}}}$ . Moreover, for each cardinal $\alpha ,3 \leq \alpha \leq {\aleph _0}$ , there is a space ${T_\alpha } \subset {T_{{\aleph _0}}}$ , universal for separable $ {\mathbf{R}}$-trees, whose order of ramification is at most $\alpha$ . We construct a universal smooth dendroid $D$ such that each separable ${\mathbf{R}}$-tree embeds in $D$ ; thus, has a smooth dendroid compactification. For nonseparable $ {\mathbf{R}}$-trees, we show that there is an $ {\mathbf{R}}$-tree ${X_\alpha }$ , such that each ${\mathbf{R}}$-tree of order of ramification at most $\alpha$ embeds isometrically into ${X_\alpha }$ . We also show that each ${\mathbf{R}}$-tree has a compactification into a smooth arboroid (a nonmetric dendroid). We conclude with several examples that show that the characterization of $ {\mathbf{R}}$-trees among metric spaces, rather than, say, among first countable spaces, is the best that can be expected.
Geometric aspects of reduction of order
James
Sherring;
Geoff
Prince
433-453
Abstract: Using the differential geometry of vectorfields and forms we reinterpret and extend the traditional idea of an integrating factor for a first order differential equation with symmetry. In particular, we provide a simple and manifestly geometric approach to reduction of order via symmetry for ordinary differential equations which largely obviates the necessity for canonical coordinates and the associated quotient manifolds. In so doing, some new results which generalise the class of Lie group actions which can be used to solve ordinary differential equations are developed.
Stable-homotopy and homology invariants of boundary links
Michael
Farber
455-477
Abstract: An $n$-dimensional $ (n \geq 5)$ link in the $ (n + 2)$-dimensional sphere is stable if the $i$th homotopy group of its complement $ X$ vanishes for $2 \leq i \leq (n + 1)/3$ and ${\pi _1}(X)$ is freely generated by meridians. In this paper a classification of stable links in terms of stable homotopy theory is given. For simple links this classification gives a complete algebraic description. We also study Poincaré duality in the space of the free covering of the complement of a boundary link. The explicit computation of the corresponding Ext-functors gives a construction of new homology pairings, generalizing the Blanchfield and the torsion pairings for knots.