Existence theorems for matroid designs
H. Peyton
Young
1-35
Abstract: A study is made of matroids in which the hyperplanes have equal cardinality. Fundamental constructions of such matroids are exhibited, and existence theorems are proved for large parametric classes of them.
Branched structures and affine and projective bundles on Riemann surfaces
Richard
Mandelbaum
37-58
Abstract: A classification for analytic branched G-structures on a Riemann surface M is provided by means of a map $ {\phi _G}$, into the moduli spaces of flat G-bundles on $M.\;(G = {\text{GA}}(1,{\text{C}})$ or $ {\text{PL}}(1,{\text{C}}).)$ Conditions are determined under which ${\phi _G}$ is injective and these conditions are related to the total branching order of the G-structures. A decomposition of the space of analytic branched G-structures into a disjoint union of analytic varieties is exhibited and it is shown that ${\phi _G}$ is is fact holomorphic on each such variety.
Torsion in $K$-theory and the Bott maps
Albert T.
Lundell
59-85
Abstract: The nonstable Bott maps
Generalized semigroups of quotients
C. V.
Hinkle
87-117
Abstract: For S a semigroup with 0 and ${M_S}$ a right S-set, certain classes of sub S-sets called right quotient filters are defined. A study of these right quotient filters is made and examples are given including the classes of intersection large and dense sub S-sets respectively. The general semigroup of right quotients Q corresponding to a right quotient filter on a semigroup S is developed and basic properties of this semigroup are noted. A nonzero regular semigroup S is called primitive dependent if each nonzero right ideal of S contains a 0-minimal right ideal of S. The theory developed in the paper enables us to characterize all primitive dependent semigroups having singular congruence the identity in terms of subdirect products of column monomial matrix semigroups over groups.
Lattice points and Lie groups. I
Robert S.
Cahn
119-129
Abstract: Assume that G is a compact semisimple Lie group and $\mathfrak{G}$ its associated Lie algebra. It is shown that the number of irreducible representations of G of dimension less than or equal to n is asymptotic to $ k{n^{a/b}}$, where a = the rank of $ \mathfrak{G}$ and b = the number of positive roots of $\mathfrak{G}$.
Lattice points and Lie groups. II
Robert S.
Cahn
131-137
Abstract: Let C be the Casimir operator on a compact, simple, simply connected Lie group G of dimension n. The number of eigenvalues of C, counted with their multiplicities, of absolute value less than or equal to t is asymptotic to $k{t^{n/2}},\;k$ a constant. This paper shows the error of this estimate to be $O({t^{2b + a(a - 1)/(a + 1)}})$; where a = rank of G and $b = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}(n - a)$.
Involutions on $S\sp{1}\times S\sp{2}$ and other $3$-manifolds
Jeffrey L.
Tollefson
139-152
Abstract: This paper exploits the following observation concerning involutions on nonreducible 3-manifolds: If the dimension of the fixed point set of a PL involution is less than or equal to one then there exists a pair of disjoint 2-spheres that do not bound 3-cells and whose union is invariant under the given involution. The classification of all PL involutions of $ {S^1} \times {S^2}$ is obtained. In particular, ${S^1} \times {S^2}$ admits exactly thirteen distinct PL involutions (up to conjugation). It follows that there is a unique PL involution of the solid torus ${S^1} \times {D^2}$ with 1-dimensional fixed point set. Furthermore, there are just four fixed point free ${Z_{2k}}$-actions and just one fixed point free ${Z_{2k + 1}}$-action on ${S^1} \times {S^2}$ for each positive integer k (again, up to conjugation). The above observation is also used to obtain a general description of compact, irreducible 3-manifolds that admit two-sided embeddings of the projective plane.
Remarks on global hypoellipticity
Stephen J.
Greenfield;
Nolan R.
Wallach
153-164
Abstract: We study differential operators D which commute with a fixed normal elliptic operator E on a compact manifold M. We use eigenfunction expansions relative to E to obtain simple conditions giving global hypoellipticity. These conditions are equivalent to D having parametrices in certain spaces of functions or distributions. An example is given by M = compact Lie group and and E = Casimir operator, with D any invariant differential operator. The connections with global subelliptic estimates are investigated.
Solvable groups having system normalizers of prime order
Gary M.
Seitz
165-173
Abstract: Let G be a solvable group having system normalizer D of prime order. If G has all Sylow groups abelian then we prove that $ l(G) = l({C_G}(D)) + 2$, provided $l(G) \geq 3$ (here $l(H)$ denotes the nilpotent length of the solvable group H). We conjecture that the above result is true without the condition on abelian Sylow subgroups. Other special cases of the conjecture are handled.
Multilinear identities of the matrix ring
Uri
Leron
175-202
Abstract: Let V be a vector space over a field F of zero characteristic, which is acted upon by the symmetric group. Systems of generators for V are constructed, which have special symmetry and skew symmetry properties. This is applied to prove that every multilinear polynomial identity of degree $2n + 1$ which holds in the matrix ring ${F_n}(n > 2)$ is a consequence of the standard identity ${s_{2n}}$. The notions of rigid and semirigid sequences of matrices are defined and treated.
Almost maximal integral domains and finitely generated modules
Willy
Brandal
203-222
Abstract: We present a class of integral domains with all finitely generated modules isomorphic to direct sums of cyclic modules. This class contains all previously known examples (i.e., the principal ideal domains and the almost maximal valuation rings) and, by an example, at least one more domain. The class consists of the integral domains satisfying (1) every finitely generated ideal is principal (obviously a necessary condition) and (2) every proper homomorphic image of the domain is linearly compact. We call an integral domain almost maximal if it satisfies (2). This is one of eleven conditions which, for valuation rings, is equivalent of E. Matlis' ``almost maximal.'' An arbitrary integral domain R is almost maximal if and only if it is h-local and $ {R_M}$ is almost maximal for every maximal ideal M of R. Finally, equivalent conditions for a Prüfer domain to be almost maximal are studied, and in the process some conjectures of E. Matlis are answered.
Sommes de Ces\`aro et multiplicateurs des d\'eveloppements en harmoniques sph\'eriques
Aline
Bonami;
Jean-Louis
Clerc
223-263
Abstract: Nous établissons une inégalité entre les sommes de Cesáro et la fonction maximale associées á une fonction définie sur la sphére, et nous en déduisons divers résultats de convergence en norme ${L^p}$, convergence presque partout, localisation des développements en harmoniques sphériques, ainsi qu'un théorème de multiplicateurs qui généralise le théorème classique de Marcinkiewicz sur les séries trigonométriques. La même étude est faite pour les développements suivant les polynômes ultrasphériques. Nous montrons de plus que les sommes partielles du développement en harmoniques sphériques d'une fonction de ${L^p}({\Sigma _n}),p \ne 2$, ne convergent pas forcément en norme.
Parametrizations of analytic varieties
Joseph
Becker
265-292
Abstract: Let V be an analytic subvariety of an open subset $\Omega$ of ${{\text{C}}^n}$ of pure dimension r; for any $p \in V$, there exists an $n - r$ dim plane T such that ${\pi _T}:V \to {{\text{C}}^r}$, the projection along T to $ {{\text{C}}^r}$, is a branched covering of finite sheeting order $\mu (V,p,T)$ in some neighborhood of V about p. ${\pi _T}$ is called a global parametrization of V if ${\pi _T}$ has all discrete fibers, e.g. ${\dim _p}V \cap (T + p) = 0$ for all $ p \in V$. Theorem. $B = \{ (p,T) \in V \times G(n - r,n)\vert{\dim _p}V \cap (T + p) > 0\}$ is an analytic set. If $ {\pi _2}:V \times G \to G$ is the natural projection, then ${\pi _2}(B)$ is a negligible set in G. Theorem. $\{ (p,T) \in V \times G\vert\mu (V,p,T) \geq k\}$ is an analytic set. For each $p \in V$, there is a least $ \mu (V,p)$ and greatest $m(V,p)$ sheeting multiplicity over all $ T \in G$. If $ \Omega$ is Stein, V is the locus of finitely many holomorphic functions but its ideal in $ \mathcal{O}(\Omega )$ is not necessarily finitely generated. Theorem. If $\mu (V,p)$ is bounded on V, then its ideal is finitely generated.
Discrete $\omega $-sequences of index sets
Louise
Hay
293-311
Abstract: We define a discrete $\omega$-sequence of index sets to be a sequence $ {\{ \theta {A_n}\} _{n \geq 0}}$, of index sets of classes of recursively enumerable sets, such that for each n, $\theta {A_{n + 1}}$ is an immediate successor of $\theta {A_n}$ in the partial order of degrees of index sets under one-one reducibility. The main result of this paper is that if S is any set to which the complete set K is not Turing-reducible, and $ {A^S}$ is the class of recursively enumerable subsets of S, then $\theta {A^S}$ is at the bottom of c discrete $ \omega$-sequences. It follows that every complete Turing degree contains c discrete $\omega$-sequences.
Nilpotent-by-finite groups with isomorphic finite quotients
P. F.
Pickel
313-325
Abstract: Let $\mathcal{F}(G)$ denote the set of isomorphism classes of finite homomorphic images of a group G. We say that groups G and H have isomorphic finite quotients if $\mathcal{F}(G) = \mathcal{F}(H)$. Let $\mathcal{H}$ denote the class of finite extensions of finitely generated nilpotent groups. In this paper we show that if G is in $\mathcal{H}$, then the groups H in $\mathcal{H}$ for which $\mathcal{F}(G) = \mathcal{F}(H)$ lie in only finitely many isomorphism classes.
Slicing and intersection theory for chains modulo $\nu $ associated with real analytic varieties
Robert M.
Hardt
327-340
Abstract: In a real analytic manifold a k dimensional (real) analytic chain is a locally finite sum of integral multiples of chains given by integration over certain k dimensional analytic submanifolds (or strata) of some k dimensional real analytic variety. In this paper, for any integer $\nu \geq 2$, the concepts and results of [6] on the continuity of slicing and the intersection theory for analytic chains are fully generalized to the modulo $\nu$ congruence classes of such chains.
Curvature tensors in Kaehler manifolds
Malladi
Sitaramayya
341-353
Abstract: Curvature tensors of Kaehler type (or type K) are defined on a hermitian vector space and it has been proved that the real vector space $ {\mathcal{L}_K}(V)$ of curvature tensors of type K on V is isomorphic with the vector space of sym metric endomorphisms of the symmetric product of ${V^ + }$, where ${V^{\text{C}}} = {V^ + } \oplus {V^ - }$ (Theorem 3.6). Then it is shown that ${\mathcal{L}_K}(V)$ admits a natural orthogonal decomposition (Theorem 5.1) and hence every $L \in {\mathcal{L}_K}(V)$ is expressed as $L = {L_1} + {L_W} + {L_2}$. These components are explicitly determined and then it is observed that ${L_W}$ is a certain formal tensor introduced by Bochner. We call ${L_W}$ the Bochner-Weyl part of L and the space of all these ${L_W}$ is called the Weyl subspace of $ {\mathcal{L}_K}(V)$.
Schur multipliers of finite simple groups of Lie type
Robert L.
Griess
355-421
Abstract: This paper presents results on Schur multipliers of finite groups of Lie type. Specifically, let p denote the characteristic of the finite field over which such a group is defined. We determine the p-part of the multiplier of the Chevalley groups ${G_2}(4),{G_2}(3)$ and ${F_4}(2)$ the Steinberg variations; the Ree groups of type ${F_4}$ and the Tits simple group
On factorized groups
David C.
Buchthal
423-430
Abstract: The effect on a finite group G by the imposition of the condition that G is factorized by each of its maximal subgroups has been studied by Huppert, Deskins, Kegel, and others. In this paper, the effect on G brought about by the condition that G is factorized by a normalizer of a Sylow p-subgroup for each $p \in \pi (G)$ is studied. Through an extension of a classical theorem of Burnside, it is shown that certain results in the case where the factors are maximal subgroups continue to hold under the new conditions. Definite results are obtained in the case where the supplements of the Sylow normalizers are cyclic groups of prime power order or are abelian Hall subgroups of G.
Finite groups with nicely supplemented Sylow normalizers
David
Perin
431-435
Abstract: This paper considers finite groups G whose Sylow normalizers are supplemented by groups D having a cyclic Hall $2'$-subgroup. G is solvable and all odd order composition factors of G are cyclic. If $S \in {\text{Syl}_2}(D)$ is cyclic, dihedral, semidihedral, or generalized quaternion, then G is almost super-solvable.
On sequences containing at most $3$ pairwise coprime integers
S. L. G.
Choi
437-440
Abstract: It has been conjectured by Erdös that the largest number of natural numbers not exceeding n from which one cannot select $k + 1$ pairwise coprime integers, where $k \geq 1$ and $n \geq {p_k}$, with $ {p_k}$ denoting the kth prime, is equal to the number of natural numbers not exceeding n which are multiples of at least one of the first k primes. It is known that the conjecture holds for k = 1 and 2. In this paper we establish the truth of the conjecture for k = 3.
Mielnik's probability spaces and characterization of inner product spaces
C. V.
Stanojevic
441-448
Abstract: A characterization of inner product spaces is given in terms of Mielnik's probability function. The generalized parallelogram law is related to the functional equation $f + f \circ g = 1$.
Bicohomology theory
Donovan H.
Van Osdol
449-476
Abstract: Given a triple T and a cotriple G on a category $\mathcal{D}$ such that T preserves group objects in $ \mathcal{D}$, let P and M be in $ \mathcal{D}$ with M an abelian group object. Applying the ``hom functor'' $\mathcal{D}( - , - )$ to the (co)simplicial resolutions ${G^ \ast }P$ and ${T^ \ast }M$ yields a double complex $ \mathcal{D}({G^ \ast }P,{T^ \ast }M)$. The nth homology group of this double complex is denoted $ {H^n}(P,M)$, and this paper studies ${H^0}$ and ${H^1}$. When $ \mathcal{D}$ is the category of bialgebras arising from a triple, cotriple, and mixed distributive law, a complete description of $ {H^0}$ and ${H^1}$ is given. The applications include a solution of the singular extension problem for sheaves of algebras.
Bessel series expansions of the Epstein zeta function and the functional equation
Audrey A.
Terras
477-486
Abstract: For the Epstein zeta function of an n-ary positive definite quadratic form, $n - 1$ generalizations of the Selberg-Chowla formula (for the binary case) are obtained. Further, it is shown that these $n - 1$ formulas suffice to prove the functional equation of the Epstein zeta function by mathematical induction. Finally some generalizations of Kronecker's first limit formula are obtained.
Irreducible representations of the $C\sp{\ast} $-algebra generated by a quasinormal operator
John W.
Bunce
487-494
Abstract: For A a quasinormal operator on Hilbert space, we determine the irreducible representations of ${C^ \ast }(A)$, the $ {C^ \ast }$-algebra generated by A and the identity. We also explicitly describe the topology on the space of unitary equivalence classes of irreducible representations of ${C^ \ast }(A)$ and calculate the regularized transform of $ {C^ \ast }(A)$, thus exhibiting an isomorphic copy of ${C^ \ast }(A)$.
On the isotropic group of a homogeneous polynomial
Siu Ming
Ho
495-498
Abstract: Let G be the linear group leaving a homogeneous polynomial of degree k fixed. The author shows that either the polynomial is a polynomial in fewer than the assigned number of variables or that the $(k - 1)$st prolongation of G is 0. The author also shows that this result is optimal.
Hull subordination and extremal problems for starlike and spirallike mappings
Thomas H.
MacGregor
499-510
Abstract: Let $\mathfrak{F}$ be a compact subset of the family $\mathcal{A}$ of functions analytic in $\Delta = \{ z:\;\vert z\vert < 1\}$, and let $\mathcal{L}$ be a continuous linear operator of order zero on $ \mathcal{A}$. We show that if the extreme points of the closed convex hull of $\mathcal{F}$ is the set $\{ {f_0}(xz)\} (\vert x\vert = 1)$, then $\mathcal{L}(f)$ is hull subordinate to $\mathcal{L}({f_0})$ in $\Delta$. This generalizes results of R. M. Robinson corresponding to families $\mathcal{F}$ of functions that are subordinate to $(1 + z)/(1 - z)$ or to $1/{(1 - z)^2}$. Families $ \mathcal{F}$ to which this theorem applies are discussed and we identify each such operator $ \mathcal{L}$ with a suitable sequence of complex numbers. Suppose that $ \Phi$ is a nonconstant entire function and that $ 0 < \vert{z_0}\vert < 1$. We show that the maximum of $ \operatorname{Re} \{ \Phi [\log (f({z_0})/{z_0})]\}$ over the class of starlike functions of order a is attained only by the functions $f(z) = z/{(1 - xz)^{2 - 2\alpha }},\;\vert x\vert = 1$. A similar result is obtained for spirallike mappings. Both results generalize a theorem of G. M. Golusin corresponding to the family of starlike mappings.