Interaction de deux chocs pour un syst\`eme de deux lois de conservation, en dimension deux d'espace
Guy
Métivier
431-479
Abstract: The existence of shock front solutions to a system of conservation laws in several space variables has been proved by A. Majda, solving a Cauchy problem, with a suitable discontinuous Cauchy data. But, in general, the solution to such a Cauchy problem will present $N$ singularities, $ N$ being the number of laws. In this paper we solve (locally) this Cauchy problem, with a Cauchy data which is piecewise smooth, in the case where all the singularities are expected to be shock waves. Actually the construction is written for a system of two laws, with two space variables and similarly, for such a system, the same method enables us to study the interaction of two shock waves. The key point, in the construction below, is the study of a nonlinear, free boundary Goursat problem.
Peano arithmetic and hyper-Ramsey logic
James H.
Schmerl
481-505
Abstract: It is known that $ {\text{PA}}({Q^2})$, Peano arithmetic in a language with the Ramsey quantifier, is complete and compact and that its first-order consequences are the same as those of $\Pi _1^1{\text{-CA}_0}$. A logic $ \mathcal{H}{\mathcal{R}_\omega }$, called hyper-Ramsey logic, is defined; it is the union of an increasing sequence $\mathcal{H}{\mathcal{R}_1} \subseteq {\mathcal{H}_{\mathcal{R}2}} \subseteq \mathcal{H}{\mathcal{R}_3} \subseteq \cdots$ of sublogics, and $ \mathcal{H}{\mathcal{R}_1}$ contains $L({Q^2})$. It is proved that $ {\text{PA}}(\mathcal{H}{\mathcal{R}_n})$, which is Peano arithmetic in the context of $ \mathcal{H}{\mathcal{R}_n}$, has the same first-order consequences as $\Pi _n^1{\text{-CA}_0}$. A by-product and ingredient of the proof is, for example, the existence of a model of $ {\text{CA}}$ having the form $(\mathcal{N}, {\text{Class}}(\mathcal{N}))$.
Hypothesis testing in integral geometry
Peter
Waksman
507-520
Abstract: Probability distributions are defined relative to a fixed plane domain and are calculated explicitly when the domain is a union of coordinate rectangles. The theory of approximating step functions by the resulting special functions gives an interpretation of the problem of guessing a domain given a random sample of observations.
The Riemann hypothesis and the Tur\'an inequalities
George
Csordas;
Timothy S.
Norfolk;
Richard S.
Varga
521-541
Abstract: A solution is given to a fifty-eight year-old open problem of G Pólya, involving the Taylor coefficients of the Riemann $ \xi$-function.
Equivariant intersection forms, knots in $S\sp 4$, and rotations in $2$-spheres
Steven P.
Plotnick
543-575
Abstract: We consider the problem of distinguishing the homotopy types of certain pairs of nonsimply-connected four-manifolds, which have identical three-skeleta and intersection pairings, by the equivariant isometry classes of the intersection pairings on their universal covers. As applications of our calculations, we: (i) construct distinct homology four-spheres with the same three-skeleta, (ii) generalize a theorem of Gordon to show that any nontrivial fibered knot in ${S^4}$ with odd order monodromy is not determined by its complement, and (iii) give a more constructive proof of a theorem of Hendriks concerning rotations in two-spheres embedded in threemanifolds.
On the depth of the symmetric algebra
J.
Herzog;
M. E.
Rossi;
G.
Valla
577-606
Abstract: Let $(R,\mathfrak{m})$ be a local ring. Assume that $R = A/I$, where $(A,\mathfrak{n})$ is a regular local ring and $I \subseteq {\mathfrak{n}^2}$ is an ideal. The depth of the symmetric algebra $S(\mathfrak{m})$ of $ \mathfrak{m}$ over $ R$ is computed in terms of the depth of the associated graded module $ {\text{gr}_\mathfrak{n}}(I)$ and the so-called "strong socle condition." Explicit results are obtained, for instance, if $I$ is generated by a super-regular sequence, if $I$ has a linear resolution or if $ I$ has projective dimension one.
The dual of the Bergman space $A\sp 1$ in symmetric Siegel domains of type ${\rm II}$
David
Békollé
607-619
Abstract: An affirmative answer is given to the following conjecture of R. Coifman and R. Rochberg: in any symmetric Siegel domain of type II, the dual of the Bergman space ${A^1}$ coincides with the Bloch space of holomorphic functions and can be realized as the Bergman projection of $ {L^\infty }$.
The Bergman projection of $L\sp \infty$ in tubes over cones of real, symmetric, positive-definite matrices
David
Békollé
621-639
Abstract: We determine a defining kernel for the Bergman projection of ${L^\infty }$ in tubes over cones of real, symmetric, positive-definite matrices.
On the homology of associative algebras
David J.
Anick
641-659
Abstract: We present a new free resolution for $k$ as an $G$-module, where $G$ is an associative augmented algebra over a field $k$. The resolution reflects the combinatorial properties of $G$.
On secondary bifurcations for some nonlinear convolution equations
F.
Comets;
Th.
Eisele;
M.
Schatzman
661-702
Abstract: On the $ d$-dimensional torus ${{\mathbf{T}}^d} = {({\mathbf{R}}/{\mathbf{Z}})^d}$, we study the nonlinear convolution equation $\displaystyle u(t) = g\{ \lambda \cdot w \ast u(t)\} , \quad t \in {{\mathbf{T}}^d}, \lambda > 0.$ where $ \ast$ is the convolution on $ {{\mathbf{T}}^d}$, $ w$ is an integrable function which is not assumed to be even, and $g$ is bounded, odd, increasing, and concave on $ {{\mathbf{R}}^ + }$. A typical example is $ g = {\text{th}}$. For a general function $w$, we show by the standard theory of local bifurcation that, if the eigenspace of the linearized problem is of dimension $2$, a branch of solutions bifurcates at $\lambda = {(g\prime(0)\hat w(p))^{ - 1}}$ from the zero solution, and we show that it can be extended to infinity. For special simple forms of $ w$, we show that the first bifurcating branch has no secondary bifurcation, but the other branches can. These results are related to the theory of spin models on ${{\mathbf{T}}^d}$ in statistical mechanics, where they allow one to show the existence of a secondary phase transition of first order, and to some models of periodic structures in the brain in neurophysiology.
Optimal-partitioning inequalities for nonatomic probability measures
John
Elton;
Theodore P.
Hill;
Robert P.
Kertz
703-725
Abstract: Suppose ${\mu _1}, \ldots ,{\mu _n}$ are nonatomic probability measures on the same measurable space $(S,\mathcal{B})$. Then there exists a measurable partition $ \{ {S_i}\} _{i = 1}^n$ of $ S$ such that $ {\mu _i}({S_i}) \geq {(n + 1 - M)^{ - 1}}$ for all $i = 1, \ldots ,n$, where $M$ is the total mass of $\vee _{i = 1}^n\,{\mu _i}$ (the smallest measure majorizing each ${\mu _i}$). This inequality is the best possible for the functional $M$, and sharpens and quantifies a well-known cake-cutting theorem of Urbanik and of Dubins and Spanier. Applications are made to ${L_1}$-functions, discrete allocation problems, statistical decision theory, and a dual problem.
Playful Boolean algebras
Boban
Veličković
727-740
Abstract: We show that for an atomless complete Boolean algebra $\mathcal{B}$ of density $\leq {2^{{\aleph _0}}}$, the Banach-Mazur, the split and choose, and the Ulam game on $\mathcal{B}$ are equivalent. Moreover, one of the players has a winning strategy just in trivial cases: Empty wins iff $ \mathcal{B}$ adds a real; Nonempty wins iff $ \mathcal{B}$ has a $ \sigma$-closed dense set. This extends some previous results of Foreman, Jech, and Vojtáš
Brownian motion at a slow point
Martin T.
Barlow;
Edwin A.
Perkins
741-775
Abstract: If $c > 1$ there are points $ T(\omega)$ such that the piece of a Brownian path $B,X(t) = B(T + t) - B(T)$, lies within the square root boundaries $ \pm c\sqrt t$. We study probabilistic and sample path properties of $X$. In particular, we show that $ X$ is an inhomogeneous Markov process satisfying a certain stochastic differential equation, and we analyze the local behaviour of its local time at zero.
On the covering dimension of the set of solutions of some nonlinear equations
P. M.
Fitzpatrick;
I.
Massabò;
J.
Pejsachowicz
777-798
Abstract: We prove an abstract theorem whose sole hypothesis is that the degree of a certain map is nonzero and whose conclusions imply sharp, multidimensional continuation results. Applications are given to nonlinear partial differential equations.
Distant future and analytic measures
Jun-ichi
Tanaka
799-814
Abstract: Using a representation of analytic measures in terms of a flow built under a function, it is shown that a positive measure is the total variation measure of an analytic measure if and only if the distant future is the zero subspace. This settles a problem posed by Forelli in connection with his generalization of F. and M. Riesz theorems. We also provide another version of Helson's existence theorem.
Stable finitely homogeneous structures
G.
Cherlin;
A. H.
Lachlan
815-850
Abstract: Let $L$ be a finite relational language and $ \operatorname{Hom}(L,\omega)$ denote the class of countable $L$-structures which are stable and homogeneous. The main result of the paper is that there exists a natural number $c(L)$ such that for any transitive $\mathcal{M} \in \operatorname{Hom}(L;\omega)$, if $E$ is a maximal 0-definable equivalence relation on $ \mathcal{M}$, then either $ \vert\mathcal{M}/E\vert < c(L)$, or $ \mathcal{M}/E$ is coordinatizable. In an earlier paper the second author analyzed certain subclasses $\operatorname{Hom}(L, r) (r < \omega)$ of $ \operatorname{Hom}(L,\omega)$ for all sufficiently small $r$. Thus the earlier analysis now applies to $ \operatorname{Hom}(L,\omega)$.
Word maps, isotopy and entropy
David
Fried
851-859
Abstract: We find diffeomorphisms of low entropy in each isotopy class on ${S^3} \times {S^3}$. These arise as word maps, a nonabelian analogue of toral automorphisms. Hyperbolic examples of equal entropy are also found. The group ${\pi _0}\;\operatorname{Diff}\,({S^3} \times {S^3})$ is computed.