On subgroups of $M\sb{24}$. I. Stabilizers of subsets
Chang
Choi
1-27
Abstract: In this paper we study the orbits of the Mathieu group ${M_{24}}$ on sets of n points, $1 \leqq n \leqq 12$. For $n \geqq 6,{M_{24}}$ is not transitive on these sets, so we may classify the sets into types corresponding to the orbits of ${M_{24}}$ and then show how to construct a set of each type from smaller sets. We determine the stabilizer of a set of each type and describe its representation on the 24 points. From the conclusions, the class of subgroups which are maximal among the intransitives of $ {M_{24}}$ can be read off. This work forms the first part of a study which yields, in particular, a complete list of the primitive representations of ${M_{24}}$.
On subgroups of $M\sb{24}$. II. The maximal subgroups of $M\sb{24}$
Chang
Choi
29-47
Abstract: In this paper we effect a systematic study of transitive subgroups of $ {M_{24}}$, obtaining 5 transitive maximal subgroups of ${M_{24}}$ of which one is primitive and four imprimitive. These results, along with the results of the paper, On subgroups of ${M_{24}}$. I, enable us to enumerate all the maximal subgroups of ${M_{24}}$. There are, up to conjugacy, nine of them. The complete list includes one more in addition to those listed by J. A. Todd in his recent work on $ {M_{24}}$. The two works were done independently employing completely different methods.
A geometry for $E\sb{7}$
John R.
Faulkner
49-58
Abstract: A geometry is defined by the 56-dimensional representation $\mathfrak{M}$ of a Lie algebra of type $ {E_7}$. Every collineation is shown to be induced by a semisimilarity of $\mathfrak{M}$, and the image of the automorphism group of $ \mathfrak{M}$ in the collineation group is shown to be simple.
A reciprocity theorem for ergodic actions
Kenneth
Lange
59-78
Abstract: An analogue of the Frobenius Reciprocity Theorem is proved for virtual groups over a locally compact separable group G. Specifically, an ergodic analytic Borel G-space $M(V\pi )$ is constructed from a virtual group V and a homomorphism $ \pi :V \to G$ of V into G. This construction proves to be functorial for the category of virtual groups over G; in fact, it is a left adjoint of the functor which takes an ergodic analytic Borel G-space T into the virtual group $T \times G$ together with projection $ \rho :T \times G \to G$ onto G. Examples such as Kakutani's induced transformation and flows under functions show the scope of this construction. A method for constructing the product of two virtual groups is also presented. Some of the structural properties of the product virtual group are deduced from those of the components. Finally, for virtual groups ${\pi _1}:{V_1} \to {G_1}$ and ${\pi _2}:{V_2} \to {G_2}$ over groups ${G_1}$ and ${G_2}$ respectively, the adjoint functor construction applied to ${\pi _1} \times {\pi _2}:{V_1} \times {V_2} \to {G_1} \times {G_2}$ is shown to give the product of the ${G_1}$-space derived from ${\pi _1}:{V_1} \to {G_1}$ and the $ {G_2}$-space derived from $ {\pi _2}:{V_2} \to {G_2}$, up to suitably defined isomorphism.
Some classes of flexible Lie-admissible algebras
Hyo Chul
Myung
79-88
Abstract: Let $\mathfrak{A}$ be a finite-dimensional, flexible, Lie-admissible algebra over a field of characteristic $\ne 2$. Suppose that ${\mathfrak{A}^ - }$ has a split abelian Cartan subalgebra $\mathfrak{H}$ which is nil in $\mathfrak{A}$. It is shown that if every nonzero root space of $ {\mathfrak{A}^ - }$ for $\mathfrak{H}$ is one-dimensional and the center of $ {\mathfrak{A}^ - }$ is 0, then $ \mathfrak{A}$ is a Lie algebra isomorphic to $ {\mathfrak{A}^ - }$. This generalizes the known result obtained by Laufer and Tomber for the case that ${\mathfrak{A}^ - }$ is simple over an algebraically closed field of characteristic 0 and $\mathfrak{A}$ is power-associative. We also give a condition that a Levi-factor of ${\mathfrak{A}^ - }$ be an ideal of $\mathfrak{A}$ when the solvable radical of ${\mathfrak{A}^ - }$ is nilpotent. These results yield some interesting applications to the case that ${\mathfrak{A}^ - }$ is classical or reductive.
Stochastic integral representation of multiplicative operator functionals of a Wiener process
Mark A.
Pinsky
89-104
Abstract: Let M be a multiplicative operator functional of (X, L) where X is a d-dimensional Wiener process and L is a separable Hilbert space. Sufficient conditions are given in order that M be equivalent to a solution of the linear Itô equation $\displaystyle M(t) = I + \sum\limits_{j = 1}^d {\int_0^t {M(s){B_j}} } (x(s))d{x_j}(s) + \int_0^t {M(s){B_0}(x(s))ds,}$ where ${B_0}, \ldots ,{B_d}$ are bounded operator functions on ${R^d}$. The conditions require that the equation $ T(t)f = E[M(t)f(x(t))]$ define a semigroup on $ {L^2}({R^d})$ whose infinitesimal generator has a domain which contains all linear functions of the coordinates $({x_1}, \ldots ,{x_d})$. The proof of this result depends on an a priori representation of the semigroup $T(t)$ in terms of the Wiener semigroup and a first order matrix operator. A second result characterizes solutions of the above Itô equation with $ {B_0} = 0$. A sufficient condition that M belong to this class is that $ E[M(t)]$ be the identity operator on L and that $M(t)$ be invertible for each $t > 0$. The proof of this result uses the martingale stochastic integral of H. Kunita and S. Watanabe.
Some remarks on quasi-analytic vectors
Paul R.
Chernoff
105-113
Abstract: Recently a number of authors have developed conditions of a generalized quasi-analytic nature which imply essential selfadjointness for semibounded symmetric operators in Hilbert space. We give a unified derivation of these results by reducing them to the basic theorems of Nelson and Nussbaum. In addition we present an extension of Nussbaum's quasi-analytic vector theorem to the setting of semigroups in Banach spaces.
Multipliers for spherical harmonic expansions
Robert S.
Strichartz
115-124
Abstract: Sufficient conditions are given for an operator on the sphere that commutes with rotations to be bounded in ${L^p}$. The conditions are analogous to those of Hörmander's well-known theorem on Fourier multipliers.
Hyperbolic limit sets
Sheldon E.
Newhouse
125-150
Abstract: Many known results for diffeomorphisms satisfying Axiom A are shown to be true with weaker assumptions. It is proved that if the negative limit set $ {L^ - }(f)$ of a diffeomorphism f is hyperbolic, then the periodic points of f are dense in ${L^ - }(f)$. A spectral decomposition theorem and a filtration theorem for such diffeomorphisms are obtained and used to prove that if $ {L^ - }(f)$ is hyperbolic and has no cycles, then f satisfies Axiom A, and hence is $\Omega$-stable. Examples are given where ${L^ - }(f)$ is hyperbolic, there are cycles, and f fails to satisfy Axiom A.
Geodesic flow in certain manifolds without conjugate points
Patrick
Eberlein
151-170
Abstract: A complete simply connected Riemannian manifold H without conjugate points satisfies the uniform Visibility axiom if the angle subtended at a point p by any geodesic $ \gamma$ of H tends uniformly to zero as the distance from p to $ \gamma$ tends uniformly to infinity. A complete manifold M is a uniform Visibility manifold if it has no conjugate points and if the simply connected covering H satisfies the uniform Visibility axiom. We derive criteria for the existence of uniform Visibility manifolds. Let M be a uniform Visibility manifold, SM the unit tangent bundle of M and $ {T_t}$ the geodesic flow on SM. We prove that if every point of SM is nonwandering with respect to ${T_t}$ then ${T_t}$ is topologically transitive on SM. We also prove that if $M'$ is a normal covering of M then $ {T_t}$ is topologically transitive on $SM'$ if ${T_t}$ is topologically transitive on SM.
Some theorems on the cos $\pi \lambda $ inequality
John L.
Lewis
171-189
Abstract: In this paper we consider subharmonic functions $u \leqq 1$ in the unit disk whose minimum modulus and maximum modulus satisfy a certain inequality. We show the existence of an extremal member of this class with largest maximum modulus. We then obtain an upper bound for the maximum modulus of this function in terms of the logarithmic measure of a certain set. We use this upper bound to prove theorems about subharmonic functions in the plane.
Subharmonic functions in certain regions
John L.
Lewis
191-201
Abstract: In a recent paper Hellsten, Kjellberg, and Norstad considered bounded subharmonic functions u in $\vert z\vert < 1$ which satisfy a certain inequality. They obtained an exact upper bound for the maximum modulus of u. We first show that this bound still holds when u satisfies less restrictive hypotheses. We then give an application of this result.
On character sums and power residues
Karl K.
Norton
203-226
Abstract: Sharp estimates are given for a double sum involving Dirichlet characters. These are applied to the problem of estimating certain sums whose values give a measure of the average distance between successive power residues to an arbitrary modulus. A particularly good result of the latter type is obtained when the modulus is prime.
Correspondence between Lie algebra invariant subspaces and Lie group invariant subspaces of representations of Lie groups
Joel
Zeitlin
227-242
Abstract: Let G be a Lie group with Lie algebra $ \mathfrak{g}$ and $\mathfrak{B} = \mathfrak{u}(\mathfrak{g})$, the universal enveloping algebra of $\mathfrak{g}$; also let U be a representation of G on H, a Hilbert space, with dU the corresponding infinitesimal representation of $\mathfrak{g}$ and $\mathfrak{B}$. For G semisimple Harish-Chandra has proved a theorem which gives a one-one correspondence between $ dU(\mathfrak{g})$ invariant subspaces and $U(G)$ invariant subspaces for certain representations U. This paper considers this theorem for more general Lie groups. A lemma is proved giving such a correspondence without reference to some of the concepts peculiar to semisimple groups used by Harish-Chandra. In particular, the notion of compactly finitely transforming vectors is supplanted by the notion of ${\Delta _f}$, the $\Delta$ finitely transforming vectors, for $ \Delta \in \mathfrak{B}$. The lemma coupled with results of R. Goodman and others immediately yields a generalization to Lie groups with large compact subgroup. The applicability of the lemma, which rests on the condition $\mathfrak{g}{\Delta _f} \subseteq {\Delta _f}$, is then studied for nilpotent groups. The condition is seen to hold for all quasisimple representations, that is representations possessing a central character, of nilpotent groups of class $\leqq 2$. However, this condition fails, under fairly general conditions, for $\mathfrak{g} = {N_4}$, the 4-dimensional class 3 Lie algebra. ${N_4}$ is shown to be a subalgebra of all class 3 $\mathfrak{g}$ and the condition is seen to fail for all $ \mathfrak{g}$ which project onto an algebra where the condition fails. The result is then extended to cover all $\mathfrak{g}$ of class 3 with general dimension 1. Finally, it is conjectured that $ \mathfrak{g}{\Delta _f} \subseteq {\Delta _f}$ for all quasisimple representations if and only if class $\mathfrak{g} \leqq 2$.
Temperatures in several variables: Kernel functions, representations, and parabolic boundary values
John T.
Kemper
243-262
Abstract: This work develops the notion of a kernel function for the heat equation in certain regions of $n + 1$-dimensional Euclidean space and applies that notion to the study of the boundary behavior of nonnegative temperatures. The regions in question are bounded between spacelike hyperplanes and satisfy a parabolic Lipschitz condition at points on the lateral boundary. Kernel functions (normalized, nonnegative temperatures which vanish on the parabolic boundary except at a single point) are shown to exist uniquely. A representation theorem for nonnegative temperatures is obtained and used to establish the existence of finite parabolic limits at the boundary (except for a set of heat-related measure zero).
Bounded linear operators on Banach function spaces of vector-valued functions
N. E.
Gretsky;
J. J.
Uhl
263-277
Abstract: Representations of bounded linear operators on Banach function spaces of vector-valued functions to Banach spaces are given in terms of operator-valued measures. Then spaces whose duals are Banach function spaces are characterized. With this last information, reflexivity of this type of space is discussed. Finally, the structure of compact operators on these spaces is studied, and an observation is made on the approximation problem in this context.
Orientation-preserving mappings, a semigroup of geometric transformations, and a class of integral operators
Antonio O.
Farias
279-289
Abstract: A Titus transformation $T = \langle \alpha ,v\rangle$ is a linear operator on the vector space of ${C^\infty }$ mappings from the circle into the plane given by ${C^\infty }$ function on the circle ${S^1}$. Let $\tau$ denote the semigroup generated by finite compositions of Titus transformations. A Titus mapping is the image by an element of $\tau$ of a degenerate curve, ${\alpha _0}{v_0}$, where $ {\alpha _0}$ is a ${C^\infty }$ function on ${S^1}$ and ${v_0}$ is fixed in the plane ${R^2}$. A $ {C^\infty }$ mapping $f:{S^1} \to {R^2}$ is called properly extendable if there is a $ {C^\infty }$ mapping $F:{D^ - } \to {R^2}$, D the open unit disk and ${D^ - }$ its closure, such that ${J_F} \geqq 0$ on $ D,{J_F} > 0$ near the boundary ${S^1}$ of ${D^ - }$ and $ F{\vert _{{s^1}}} = f$. A ${C^\infty }$ mapping $f:{S^1} \to {R^2}$ is called normal if it is an immersion with no triple points and all its double points are transversal. The main result of this paper can be stated: a normal mapping is extendable if and only if it is a Titus mapping. An application is made to a class of integral operators of the convolution type, $y(t) = - \smallint_0^{2\pi } {k(s)x(t - s)ds}$. It is proved that, under certain technical conditions, such an operator is topologically equivalent to Hilbert's transform of potential theory, $y(t) = \smallint_0^{2\pi } {\cot (s/2)x(t - s)ds}$, which gives the relation between the real and imaginary parts of the restriction to the boundary of a function holomorphic inside the unit disk.
Noetherian intersections of integral domains
William
Heinzer;
Jack
Ohm
291-308
Abstract: Let $D < R$ be integral domains having the same quotient field K and suppose that there exists a family $ {\{ {V_i}\} _{i \in I}}$ of 1-dim quasi-local domains having quotient field K such that $D = R \cap \{ {V_i}\vert i \in I\}$. The goal of this paper is to find conditions on R and the ${V_i}$ in order for D to be noetherian and, conversely, conditions on D in order for R and the ${V_i}$ to be noetherian. An important motivating case is when the set $ \{ {V_i}\}$ consists of a single element V and V is a valuation ring. It is shown, for example, in this case that (i) if V is centered on a finitely generated ideal of D, then V is noetherian and (ii) if V is centered on a maximal ideal of D, then D is noetherian if and only if R and V are noetherian.
Interpolation to analytic data on unbounded curves
Maynard
Thompson
309-318
Abstract: This paper provides a method for constructing a family of sets of points on the boundary (assumed suitably smooth) of an unbounded Jordan region in the complex plane which is useful for certain interpolation problems. It is proved that if these sets are used as nodes for Lagrange interpolation to analytic data, then the resulting polynomials converge in the region, and the limit function is related in a natural way to the boundary data. Subsidiary results include an approximate quadrature formula for slowly decreasing functions on an infinite interval.
The dominion of Isbell
Barry
Mitchell
319-331
Abstract: A well-known characterization of epimorphisms in the category of rings with identity is imitated to give a similar characterization of epimorphisms in the category of small pre-additive categories. From this one deduces Isbell's ``Zigzag Theorem'' concerning epimorphisms in Cat.
On the equivalence of multiplicity and the generalized topological degree
T.
O’Neil;
J. W.
Thomas
333-345
Abstract: In this paper we first extend the definition of the multiplicity (as defined by J. Cronin-Scanlon) of operators of the form $I + C + T$ to operators of the form $H + C + T$. We then show that the generalized topological degree (as defined by F. E. Browder and W. V. Petryshyn) of operators of the form $H + C + T$ is also defined. Finally, we show that when both the multiplicity and generalized topological degree of $H + C + T$ are defined, they are equal.
An equality for $2$-sided surfaces with a finite number of wild points
Michael D.
Taylor;
Harvey
Rosen
347-358
Abstract: Let S be a 2-sided surface in a 3-manifold that is wild from one side U at just m points. It is shown that the minimal genus possible for all members of a sequence of surfaces in U converging to S (where these surfaces each separate the same point from S in $U \cup S$) is equal to the sum of the genus of S and a certain multiple of the sum of m special topological invariants associated with the wild points. In this equality, the sum of these invariants is multiplied by just one of the numbers 0, 1, or 2, dependent upon the genus and orientability class of S and the value of m. As an application, an upper bound is given for the number of nonpiercing points that a 2-sided surface has with respect to one side.
Transversals to the flow induced by a differential equation on compact orientable $2$-dimensional manifolds
Carl S.
Hartzman
359-368
Abstract: Every treatment of the theory of differential equations on a torus uses the fact that given a differential equation on a torus of class ${C^k}$, there is a non-null-homotopic closed Jordan curve $\Gamma$ of class ${C^k}$ which is transverse to the trajectories of the differential equation that pass through points of $\Gamma$. Such a curve necessarily cannot separate the torus. Here, we prove that given a differential equation on an n-fold torus ${T_n}$ of class ${C^k}$, possessing only ``simple'' singularities of negative index there is a non-null-homotopic closed Jordan curve $\Gamma$ of class ${C^k}$ which is a transversal. The nonseparating property, however, does not follow immediately. For the particular case ${T_2}$, we prove the existence of such a transversal that does not separate ${T_2}$.
Martingales of strongly measurable Pettis integrable functions
J. J.
Uhl
369-378
Abstract: This paper deals with convergence theorems for martingales of strongly measurable Pettis integrable functions. First, a characterization of those martingales which converge in the Pettis norm is obtained. Then it is shown that a martingale which is convergent in the Pettis norm converges to its limit strongly in measure and, if the index set is the positive integers, it converges strongly almost everywhere to its limit. The second part of the paper deals with the strong measure and strong almost everywhere convergence of martingales which are not necessarily convergent in the Pettis norm. The resulting theorems here show that $ {L^1}$-boundedness can be considerably relaxed to a weaker control condition on the martingale by the use of some facts on finitely additive vector measures.
The envelope of holomorphy of Riemann domains over a countable product of complex planes
Mário C.
Matos
379-387
Abstract: This paper deals with the problem of constructing envelopes of holomorphy for Riemann domains over a locally convex space. When this locally convex space is a countable product of complex planes the existence of the envelope of holomorphy is proved and the domains of holomorphy are characterized.
Two theorems in the commutator calculus
Hermann V.
Waldinger
389-397
Abstract: Let $F = \langle a,b\rangle$. Let ${F_n}$ be the nth subgroup of the lower central series. Let p be a prime. Let $ {c_3} < {c_4} < \cdots < {c_z}$ be the basic commutators of dimension $ > 1$ but $< p + 2$. Let ${P_1} = (a,b),{P_m} = ({P_{m - 1}},b)$ for $m > 1$. Then $ (a,{b^p}) \equiv \prod\nolimits_{i = 3}^z {c_i^{{\eta _i}}\bmod {F_{p + 2}}}$. It is shown in Theorem 1 that the exponents $ {\eta _i}$ are divisible by p, except for the exponent of ${P_p}$ which $= 1$. Let the group $ \mathcal{G}$ be a free product of finitely many groups each of which is a direct product of finitely many groups of order p, a prime. Let $\mathcal{G}$-simple basic commutators'' of dimension $> 1$ defined below are free generators of
Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations
C. V.
Coffman;
J. S. W.
Wong
399-434
Abstract: This paper treats the ordinary differential equation $y'' + yF({y^2},x) = 0,x > 0$ , where $yF({y^2},x)$ is continuous in (y, x) for $x > 0,\vert y\vert < \infty$, and $ F(t,x)$ is non-negative; the equation is assumed to be either of sublinear or superlinear type. Criteria are given for the equation to be oscillatory, to be nonoscillatory, to possess oscillatory solutions or to possess nonoscillatory solutions. An attempt has been made to unify the methods of treatment of the sublinear and superlinear cases. These methods consist primarily of comparison with linear equations and the use of ``energy'' functions. An Appendix treats the questions of continuability and uniqueness of solutions of the equation considered in the main text.
Locally $B\sp{\ast} $-equivalent algebras
Bruce A.
Barnes
435-442
Abstract: Let A be a Banach $^ \ast$-algebra. A is locally ${B^ \ast }$-equivalent if, for every selfadjoint element $t \in A$, the closed $^ \ast$-subalgebra of A generated by t is $^\ast$-isomorphic to a $ {B^ \ast }$-algebra. In this paper it is shown that when A is locally $ {B^\ast}$-equivalent, and in addition every selfadjoint element in A has at most countable spectrum, then A is $^ \ast$-isomorphic to a ${B^ \ast }$-algebra.
On modification theorems
Murali
Rao
443-450
Abstract: Given a right continuous family ${F_t}$ of complete $\sigma$-fields and a bounded right continuous family ${X_t}$ of random variables, we show in this paper that it is possible to modify the conditional expectations $ E({X_t}\vert{F_t})$ to be right continuous. When ${X_t} = X$, this reduces to a result of J. L. Doob.
Restricted mean values and harmonic functions
John R.
Baxter
451-463
Abstract: A function h defined on a region R in ${{\mathbf{R}}^n}$ will be said to possess a restricted mean value property if the value of the function at each point is equal to the mean value of the function over one open ball in R, with centre at that point. It is proved here that this restricted mean value property implies h is harmonic under certain conditions.
A sufficient condition for the lower semicontinuity of parametric integrals
Edward
Silverman
465-469
Abstract: We use simple convex functions and standard techniques in area theory to treat Morrey's extension of McShane's lower semicontinuity theorem for parametric integrals. This enables us to eliminate some technical hypotheses, simplify the proof and obtain a more general result.
On time-free functions
Gideon
Schwarz
471-478
Abstract: By regarding as equivalent any two real-valued functions of a real variable that can be obtained from each other by a monotone continuous transformation of the independent variable, time-free functions are defined. A convenient maximal invariant is presented, and applied to some time-free functional equations.
Mappings from $3$-manifolds onto $3$-manifolds
Alden
Wright
479-495
Abstract: Let f be a compact, boundary preserving mapping from the 3-manifold $ {M^3}$ onto the 3-manifold $ {N^3}$. Let ${Z_p}$ denote the integers mod a prime p, or, if $p = 0$, the integers. (1) If each point inverse of f is connected and strongly 1-acyclic over $ {Z_p}$, and if $ {M^3}$ is orientable for $p > 2$, then all but a locally finite collection of point inverses of f are cellular. (2) If the image of the singular set of f is contained in a compact set each component of which is strongly acyclic over ${Z_p}$, and if ${M^3}$ is orientable for $p \ne 2$, then ${N^3}$ can be obtained from ${M^3}$ by cutting out of $ \operatorname{Int} \;{M^3}$ a compact 3-manifold with 2-sphere boundary, and replacing it by a ${Z_p}$-homology 3-cell. (3) If the singular set of f is contained in a 0-dimensional set, then all but a locally finite collection of point inverses of f are cellular.