On the detonation of a combustible gas
Robert A.
Gardner
431-468
Abstract: This paper is concerned with the existence of detonation waves for a combustible gas. The equations are those of a viscous, heat conducting, polytropic gas coupled with an additional equation which governs the evolution of the mass fraction of the unburned gas (see (1)). The reaction is assumed to be of the simplest form: $A \to B$, i.e., there is a single product and a single reactant. The main result (see Theorem 2.1) is a rigorous existence theorem for strong, and under certain conditions, weak detonation waves for explicit ranges of the viscosity, heat conduction, and species diffusion coefficients. In other words, a class of admissible "viscosity matrices" is determined. The problem reduces to finding an orbit of an associated system of four ordinary differential equations which connects two distinct critical points. The proof employs topological methods, including Conley's index of isolated invariant sets.
The Atiyah-Singer invariant, torsion invariants, and group actions on spheres
Donald E.
Smith
469-488
Abstract: This paper deals with the classification of cyclic group actions on spheres using the Atiyah-Singer invariant and Reidemeister-type torsion. Our main tool is the computation of the group of relative homotopy triangulations of the product of a disk and a lens space. These results are applied to obtain lower bounds on the image of an equivariant $J$-homomorphism.
Extension of Wiener's Tauberian identity and multipliers on the Marcinkiewicz space
Ka-Sing
Lau
489-506
Abstract: This is a continuation of the work of Bertrandias, Lee and Lau on Wiener's generalized harmonic analysis. Among the other results, we extend Wiener's Tauberian identity to cover a larger class of functions; we characterize the multipliers on the Marcinkiewicz space ${\mathcal{M}^2}$, and we obtain a Tauberian theorem on $ {\mathcal{M}^2}$ with full generality.
Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium
Juan Luis
Vázquez
507-527
Abstract: The one-dimensional porous media equation ${u_t} = {({u^m})_{xx}}$, $m > 1$, is considered for $x \in R$, $t > 0$ with initial conditions $u(x,0) = {u_0}(x)$ integrable, nonnegative and with compact support. We study the behaviour of the solutions as $t \to \infty $ proving that the expressions for the density, pressure, local velocity and interfaces converge to those of a model solution. In particular the first term in the asymptotic development of the free-boundary is obtained.
Linear superpositions with mappings which lower dimension
Y.
Sternfeld
529-543
Abstract: It is shown that for every $n$-dimensional compact metric space $X$, there exist $2n + 1$ functions $\{ {\varphi _j}\}_{j = 1}^{2n + 1}$ in $ C(X)$ and $n$ mappings $\{ {\psi _i}\}_{i = 1}^n$ on $X$ with $1$-dimensional range each, with the following property: for every $0 \leqslant k \leqslant n$, every $k$ tuple $\{ {\psi_{i_l}}\}_{l = 1}^k$ of the $ {\psi _i}$'s and every $2(n - k) + 1$ tuple $\{ {\varphi _{{j_m}}}\}_{m = 1}^{2(n - k) + 1}$ of the $ {\varphi_j}$'s, each $f \in C(X)$ can be represented as $f(x) = \Sigma _{l = 1}^k{g_l}({\psi _{{i_l}}}(x)) + \Sigma_{m = 1}^{2(n - k) + 1}{h_m}({\varphi_{{j_m}}}(x))$, with ${g_l} \in C({\psi _{{i_l}}}(X))$ and ${h_m} \in C(R)$. It is also shown that in many cases the number $ 2(n - k) + 1$ is the smallest possible.
Meromorphic functions that share four values
Gary G.
Gundersen
545-567
Abstract: An old theorem of ${\text{R}}$. Nevanlinna states that if two distinct nonconstant meromorphic functions share four values counting multiplicities, then the functions are Möbius transformations of each other, two of the shared values are Picard values for both functions, and the cross ratio of a particular permutation of the shared values equals -1. In this paper we show that if two nonconstant meromorphic functions share two values counting multiplicities and share two other values ignoring multiplicities, then the functions share all four values counting multiplicities.
Quadratic spaces over Laurent extensions of Dedekind domains
Raman
Parimala
569-578
Abstract: Let $R$ be a Dedekind domain in which $ 2$ is invertible. We show in this paper that any isotropic quadratic space over $R[T,{T^{ - 1}}]$ is isometric to ${q_1} \perp T{q_2}$ where ${q_1},{q_2}$ are quadratic spaces over $ R$. We give an example to show that this result does not hold for anisotropic spaces.
Independence results on the global structure of the Turing degrees
Marcia J.
Groszek;
Theodore A.
Slaman
579-588
Abstract: From CON(ZFC) we obtain: 1. CON$ ($ZFC$+ 2^\omega$ is arbitrarily large $ +$ there is a locally finite upper semilattice of size $ {\omega_2}$ which cannot be embedded into the Turing degrees as an upper semilattice). 2. CON$($ZFC$+ 2^\omega$ is arbitrarily large $ +$ there is a maximal independent set of Turing degrees of size ${\omega _1}$).
The symmetric derivative
Lee
Larson
589-599
Abstract: It is shown that all symmetric derivatives belong to Baire class one, and a condition characterizing all measurable symmetrically differentiable functions is presented. A method to find a well-behaved primitive for any finite symmetric derivative is introduced, and several of the standard theorems of differential calculus are extended to include the symmetric derivative.
Tangent cones and quasi-interiorly tangent cones to multifunctions
Lionel
Thibault
601-621
Abstract: R. T. Rockafellar has proved a number of rules of subdifferential calculus for nonlocally lipschitzian real-valued functions by investigating the Clarke tangent cones to the epigraphs of such functions. Following these lines we study in this paper the tangent cones to the sum and the composition of two multifunctions. This will be made possible thanks to the notion of quasi-interiorly tangent cone which has been introduced by the author for vector-valued functions in [29] and whose properties in the context of multifunctions are studied. The results are strong enough to cover the cases of real-valued or vector-valued functions.
Some examples of square integrable representations of semisimple $p$-adic groups
George
Lusztig
623-653
Abstract: We construct irreducible representations of the Hecke algebra of an affine Weyl group analogous to Kilmoyer's reflection representation corresponding to finite Weyl groups, and we show that in many cases they correspond to a square integrable representation of a simple $p$-adic group.
Ambiently universal sets in $E\sp{n}$
David G.
Wright
655-664
Abstract: For each closed set $ X$ in ${E^n}$ of dimension at most $ n - 3$, we show that $ X$ fails to be ambiently universal with respect to Cantor sets in $ {E^n}$; i.e., we find a Cantor set $Y$ in ${E^n}$ so that for any self-homeomorphism $ h$ of ${E^n}$, $h(Y)$ is not contained in $X$. This result answers a question posed by H. G. Bothe and completes the understanding of ambiently universal sets in ${E^n}$.
Stable orbits of differentiable group actions
Dennis
Stowe
665-684
Abstract: We prove that a compact orbit of a smooth Lie group action is stable provided the first cohomology space vanishes for the normal representation at some (equivalently, every) point of the orbit. When the orbit is a single point, the acting group need only be compactly generated and locally compact for this conclusion to hold. Applied to foliations, this provides a sufficient condition for the stability of a compact leaf and includes the stability theorems of Reeb and Thurston and of Hirsch as cases.
Rotation hypersurfaces in spaces of constant curvature
M.
do Carmo;
M.
Dajczer
685-709
Abstract: Rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed. A local characterization of such hypersurfaces, with dimensions greater than two, is given in terms of principal curvatures. Some special cases of rotation hypersurfaces, with constant mean curvature, in hyperbolic space are studied. In particular, it is shown that the well-known conjugation between the belicoid and the catenoid in euclidean three-space extends naturally to hyperbolic three-space $H^3$; in the latter case, catenoids are of three different types and the explicit correspondence is given. It is also shown that there exists a family of simply-connected, complete, embedded, nontotally geodesic stable minimal surfaces in $H^3$.
Gauss sums and Fourier analysis on multiplicative subgroups of $Z\sb{q}$
Harold G.
Diamond;
Frank
Gerth;
Jeffrey D.
Vaaler
711-726
Abstract: Let $G(q)$ denote the multiplicative group of invertible elements in $ {{\mathbf{Z}}_q}$, the ring of integers modulo $q$. Let $H \subseteq G(q)$ be a multiplicative subgroup with cosets $aH$ and $bH$. If $f: {\mathbf{Z}}_q \to {\mathbf{C}}$ is supported in $aH$ we show that $f$ can be recovered from the values of $\hat f$ restricted to $bH$ if and only if Gauss sums for $ H$ are nonvanishing. Here $ \hat f$ is the (finite) Fourier transform of $f$ with respect to the additive group ${{\mathbf{Z}}_q}$. The main result is a simple criterion for deciding when these Gauss sums are nonvanishing. If $H = G(q)$ then $f$ can be recovered from $\hat f$ restricted to $G(q)$ by a particularly elementary formula. This formula provides some inequalities and extremal functions.
Rees matrix covers for locally inverse semigroups
D. B.
McAlister
727-738
Abstract: A regular semigroup $ S$ is locally inverse if each local submonoid $eSe$, $e$ an idempotent, is an inverse semigroup. It is shown that every locally inverse semigroup is an image of a regular Rees matrix semigroup, over an inverse semigroup, by a homomorphism $\theta$ which is one-to-one on each local submonoid; such a homomorphism is called a local isomorphism. Regular semigroups which are locally isomorphic images of regular Rees matrix semigroups over semilattices are also characterized.
Strongly Cohen-Macaulay schemes and residual intersections
Craig
Huneke
739-763
Abstract: This paper studies the local properties of closed subschemes $ Y$ in Cohen-Macaulay schemes $X$ such that locally the defining ideal of $ Y$ in $X$ has the property that its Koszul homology is Cohen-Macaulay. Whenever this occurs $ Y$ is said to be strongly Cohen-Macaulay in $X$. This paper proves several facts about such embeddings, chiefly with reference to the residual intersections of $Y$ in $X$. The main result states that any residual intersection of $Y$ in $X$ is again Cohen-Macaulay.
Isomorphism types in wreath products and effective embeddings of periodic groups
Kenneth K.
Hickin;
Richard E.
Phillips
765-778
Abstract: For any finitely generated group $ Y,\omega (Y)$ denotes the Turing degree of the word problem of $Y$. Let $G$ be any non-Abelian $2$-generator group and $B$ an infinite group generated by $k \geqslant 1$ elements. We prove that if $ \tau$ is any Turing degree with $\tau \geqslant 1.{\text{u.b.}}\{ {\omega (G),\omega (B)} \}$ then the unrestricted wreath product $ W = G{\text{Wr}}\,B$ has a $( {k + 1} )$-generator subgroup $H$ with $ \omega (H) = \tau$. If $ B$ is also periodic, then $ W$ has a $k$-generator subgroup $H$ such that $\tau = 1.{\text{u.b.}}\{ {\omega (B),\omega (H)} \}$. Easy consequences include: $ G{\text{Wr}}\,{\mathbf{Z}}$ has ${2^{{\aleph _0}}}$ pairwise nonembeddable $2$-generator subgroups and if $B$ is periodic then $G{\text{Wr}}\,B$ has ${2^{{\aleph _0}}}$ pairwise nonembeddable $ k$-generator subgroups. Using similar methods, we prove an effective embedding theorem for embedding countable periodic groups in $ 2$-generator periodic groups.
The optimal accuracy of difference schemes
Arieh
Iserles;
Gilbert
Strang
779-803
Abstract: We consider difference approximations to the model hyperbolic equation ${u_{t}} = {u_x}$ which compute each new value $ U(x,t + \Delta t)$ as a combination of the known values $ U(x - r\Delta x,t),\ldots,U(x + s\Delta x,\Delta t)$. For such schemes we find the optimal order of accuracy: stability is possible for small $ \Delta t/\Delta x$ if and only if $p \leqslant \min \{ {r + s,2r + 2,2s} \}$. A similar bound is established for implicit methods. In this case the most accurate schemes are based on Padé approximations $P(z)/Q(z)$ to $ {z^\lambda }$ near $ z = 1$, and we find an expression for the difference $\vert Q{\vert^2} - \vert P{\vert^2}$; this allows us to test the von Neumann condition $\vert P/Q\vert \leqslant 1$. We also determine the number of zeros of $Q$ in the unit circle, which decides whether the implicit part is uniformly invertible.
Nonanalytic solutions of certain linear PDEs
E. C.
Zachmanoglou
805-814
Abstract: It is shown that if $ P$ is a linear partial differential operator with analytic coefficients, and if $M$ is an analytic submanifold of codimensions $ 3$ in ${{\mathbf{R}}^n}$, which is partially characteristic with respect to $P$ and satisfies certain additional conditions, then one can find, in a neighborhood of any point of $ M$, solutions of the equation $Pu = 0$ which are flat or singular precisely on $ M$. The additional condition requires that a nonhomogeneous Laplace equation in two variables possesses a solution with a strong extremum at the origin. The right side of this nonhomogeneous equation is a homogeneous polynomial in two variables with coefficients being repeated Poisson brackets of the real and imaginary parts of the principal symbol of $P$.
Projections onto translation-invariant subspaces of $L\sb{1}({\bf R})$
Dale E.
Alspach;
Alec
Matheson
815-823
Abstract: The complemented translation-invariant subspaces of ${L_1}({\mathbf{R}})$ are characterized. This completes an investigation begun by H. P. Rosenthal.
Small into-isomorphisms between spaces of continuous functions. II
Yoav
Benyamini
825-833
Abstract: We construct two compact Hausdorff spaces, $X$ and $Y$, so that $C(X)$ does not embed isometrically into $ C(Y)$, but for each $\varepsilon > 0$, there is an isomorphism ${T_\varepsilon }$ from $C(X)$ into $C(Y)$ satisfying $\parallel f\parallel \leqslant \parallel {T_\varepsilon }f\;\parallel \leqslant (1 + \varepsilon)\parallel f\parallel$ for all $f \in C(X)$.
Borel functions of bounded class
D. H.
Fremlin;
R. W.
Hansell;
H. J. K.
Junnila
835-849
Abstract: Let $X$ and $Y$ be metric spaces and $f:X \to Y$ a Borel measurable function. Does $ f$ have to be of bounded class, i.e. are the sets $ {f^{ - 1}}[ H ]$, for open $H \subseteq Y$, of bounded Baire class in $X?$ This is an old problem of A. H. Stone. Positive answers have been given under a variety of extra hypotheses and special axioms. Here we show that (i) unless something similar to a measurable cardinal exists, then $f$ is of bounded class and (ii) if $f$ is actually a Borel isomorphism, then $f\,({\text{and}} {f^{ - 1}})$ are of bounded class.
Spectral decomposition with monotonic spectral resolvents
I.
Erdélyi;
Sheng Wang
Wang
851-859
Abstract: The spectral decomposition problem of a Banach space over the complex field entails two kinds of constructive elements: (1) the open sets of the field and (2) the invariant subspaces (under a given linear operator) of the Banach space. The correlation between these two structures, in the framework of a spectral decomposition, is the spectral resolvent concept. Special properties of the spectral resolvent determine special types of spectral decompositions. In this paper, we obtain conditions for a spectral resolvent to have various monotonic properties.