Stopping times and $\Gamma$-convergence
J.
Baxter;
G.
Dal Maso;
U.
Mosco
1-38
Abstract: The equation $ \partial u/\partial t = \Delta u - \mu u$ represents diffusion with killing. The strength of the killing is described by the measure $ \mu$, which is not assumed to be finite or even $\sigma$-finite (to illustrate the effect of infinite values for $\mu$, it may be noted that the diffusion is completely absorbed on any set $A$ such that $ \mu (B) = \infty$ for every nonpolar subset $B$ of $A$). In order to give rigorous mathematical meaning to this general diffusion equation with killing, one may interpret the solution $u$ as arising from a variational problem, via the resolvent, or one may construct a semigroup probabilistically, using a multiplicative functional. Both constructions are carried out here, shown to be consistent, and applied to the study of the diffusion equation, as well as to the study of the related Dirichlet problem for the equation $\Delta u - \mu u = 0$. The class of diffusions studied here is closed with respect to limits when the domain is allowed to vary. Two appropriate forms of convergence are considered, the first being $\gamma $-convergence of the measures $\mu$, which is defined in terms of the variational problem, and the second being stable convergence in distribution of the multiplicative functionals associated with the measures $\mu$. These two forms of convergence are shown to be equivalent.
Character table and blocks of finite simple triality groups $\sp 3D\sb 4(q)$
D. I.
Deriziotis;
G. O.
Michler
39-70
Abstract: Based on recent work of Spaltenstein [14] and the Deligne-Lusztig theory of irreducible characters of finite groups of Lie type, in this paper the character table of the finite simple groups ${}^3{D_4}(q)$ is given. As an application we obtain a classification of the irreducible characters of ${}^3{D_4}(q)$ into $r$-blocks for all primes $r > 0$. This enables us to verify Brauer's height zero conjecture, his conjecture on the bound of irreducible characters belonging to a give block, and the Alperin-McKay conjecture for the simple triality groups $ {}^3{D_4}(q)$. It also follows that for every prime $r$ there are blocks of defect zero in ${}^3{D_4}(q)$.
An \'etale cohomology duality theorem for number fields with a real embedding
Mel
Bienenfeld
71-96
Abstract: The restriction on $ 2$-primary components in the Artin-Verdier duality theorem [2] has been eliminated by Zink [9], who has shown that the sheaf of units for the étale topology over the ring of integers of any number field acts as a dualizing sheaf for a modified cohomology of sheaves. The present paper provides an alternate means of removing the $2$-primary restriction. Like Zink's, it involves a topology which includes infinite primes, but it avoids modified cohomology and will be more directly applicable in the proof of a theorem of Lichtenbaum regarding zeta- and $L$-functions [4, 5]. Related results--including the cohomology of units sheaves, the norm theorem, and punctual duality theorem of Mazur [6]--are also affected by the use of a topology including the infinite primes. The corresponding results in the new setting are included here.
Braids and the Jones polynomial
John
Franks;
R. F.
Williams
97-108
Abstract: An important new invariant of knots and links is the Jones polynomial, and the subsequent generalized Jones polynomial or two-variable polynomial. We prove inequalities relating the number of strands and the crossing number of a braid with the exponents of the variables in the generalized Jones polynomial which is associated to the link formed from the braid by connecting the bottom ends to the top ends. We also relate an exponent in the polynomial to the number of components of this link.
Where does the $L\sp p$-norm of a weighted polynomial live?
H. N.
Mhaskar;
E. B.
Saff
109-124
Abstract: For a general class of nonnegative weight functions $w(x)$ having bounded or unbounded support $ \Sigma \subset {\mathbf{R}}$, the authors have previously characterized the smallest compact set $ {\mathfrak{S}_w}$, having the property that for every $n = 1,\,2, \ldots$ and every polynomial $ P$ of degree $\leqslant n$, $\displaystyle \vert\vert{[w(x)]^n}P(x)\vert{\vert _{{L^\infty }(\Sigma )}} = \vert\vert{[w(x)]^n}P(x)\vert{\vert _{{L^\infty }({\mathfrak{S}_w})}}$ . In the present paper we prove that, under mild conditions on $w$, the ${L^p}$-norms $ (0 < p < \infty )$ of such weighted polynomials also "live" on ${\mathfrak{S}_w}$ in the sense that for each $ \eta > 0$ there exist a compact set $\Delta$ with Lebesgue measure $m(\Delta ) < \eta$ and positive constants $ {c_1}$, ${c_2}$ such that $\displaystyle \vert\vert{w^n}P\vert{\vert _{{L^p}(\Sigma )}} \leqslant (1 + {c_... ... - {c_2}n))\vert\vert{w^n}P\vert{\vert _{{L^p}({\mathfrak{S}_w} \cup \Delta )}}$ . As applications we deduce asymptotic properties of certain extremal polynomials that include polynomials orthogonal with respect to a fixed weight over an unbounded interval. Our proofs utilize potential theoretic arguments along with Nikolskii-type inequalities.
The Milne problem for the radiative transfer equations (with frequency dependence)
François
Golse
125-143
Abstract: We study the following stationary frequency dependent transport equation: \begin{displaymath}\begin{array}{*{20}{c}} {\mu {\partial _x}f + \sigma (\nu ,\,... ... (\mu ,\,\nu ),\qquad \nu > 0,\;u \in ]0;\,1[,} \end{array} \end{displaymath} where ${B_\nu }$ is the well-known Planck function appearing in astrophysics. We are able to describe the asymptotic behavior of $f$ and $T$ for $x$ large, when $ \sigma (\nu ,\,T)$ is of the special form $\sigma (\nu ,\,T) = \sigma (\nu )k(T)$. Our method relies mainly on the monotonicity of the nonlinearity. The proof does not use any linearization of the equation; in particular, no smallness assumption on the data $\varphi$ (in any sense) is required. Résumé. Nous étudions l'équation de transport stationnaire avec dépendance en fréquence: \begin{displaymath}\begin{array}{*{20}{c}} {\mu {\partial _x}f + \sigma (\nu ,\,... ...\mu ,\,\nu );\qquad \nu > 0,\;\mu \in ]0;\,1[.} \end{array} \end{displaymath} Lorsque $ \sigma (\nu ,\,T)$ est de la forme particulière $\sigma (\nu ,\,T) = \sigma (\nu )k(T)$, nous savons décrire le comportement asymptotique de $ f$ et $T$ pour $x$ grand. Notre méthode repose principalement sur la monotonie de la non-linéarité. La preuve n'utilise aucune linéarisation de l'équation; en particulier, nous n'avons besoin d'aucune hypothèse de petitesse (d'aucune sorte) sur la donnée $\varphi$.
The structure of groups which are almost the direct sum of countable abelian groups
Alan H.
Mekler
145-160
Abstract: The notion of being in standard form is defined for the groups described in the title of the paper which are of cardinality ${\omega _1}$. Being in "standard form" is a structural description of the group. The consequences of being in standard form are explored, sometimes with the use of additional set-theoretic axioms. It is shown that it is consistent that a large class of these groups, including every weakly ${\omega _1}$-separable $ {\omega _1}$-$ \Sigma$-cyclic group of cardinality $ {\omega _1}$, can be put in standard form.
Scalar curvature and warped products of Riemann manifolds
F.
Dobarro;
E.
Lami Dozo
161-168
Abstract: We establish the relationship between the scalar curvature of a warped product $M \times {}_fN$ of Riemann manifolds and those ones of $M$ and $N$. Then we search for weights $f$ to obtain constant scalar curvature on $M \times {}_fN$ when $M$ is compact.
Global existence for $1$D, compressible, isentropic Navier-Stokes equations with large initial data
David
Hoff
169-181
Abstract: We prove the global existence of weak solutions of the Cauchy problem for the Navier-Stokes equations of compressible, isentropic flow of a polytropic gas in one space dimension. The initial velocity and density are assumed to be in $ {L^2}$ and ${L^2} \cap BV$ respectively, modulo additive constants. In particular, no smallness assumptions are made about the intial data. In addition, we prove a result concerning the asymptotic decay of discontinuities in the solution when the adiabatic constant exceeds $ 3/2$.
Strong Ramsey theorems for Steiner systems
Jaroslav
Nešetřil;
Vojtěch
Rödl
183-192
Abstract: It is shown that the class of partial Steiner $(k,\,l)$-systems has the edge Ramsey property, i.e., we prove that for every partial Steiner $ (k,\,l)$-system $\mathcal{G}$ there exists a partial Steiner $ (k,\,l)$-system $\mathcal{H}$ such that for every partition of the edges of $ \mathcal{H}$ into two classes one can find an induced monochromatic copy of $\mathcal{G}$. As an application we get that the class of all graphs without cycles of lengths $ 3$ and $4$ has the edge Ramsey property. This solves a longstanding problem in the area.
Polynomial approximation in the mean with respect to harmonic measure on crescents
John
Akeroyd
193-199
Abstract: For $1 \leqslant s < \infty $ and "nice" crescents $ G$, this paper gives a necessary condition (Theorem 2.6) and a sufficient condition (Theorem 2.5) for density of the polynomials in the generalized Hardy space ${H^s}(G)$. These conditions are easily tested and almost equivalent.
On a property of Castelnuovo varieties
Ciro
Ciliberto
201-210
Abstract: Castelnuovo varieties are those irreducible complete varieties in a projective space whose geometric genus is maximal according to their dimension, degree and embedding dimension. In this paper, extending results by Severi and Accola, we prove that, under suitable conditions, such varieties are birational if and only if they are projectively equivalent.
Bounded weak solutions of an elliptic-parabolic Neumann problem
J.
Hulshof
211-227
Abstract: In this paper we establish existence and uniqueness for bounded weak solutions of an elliptic-parabolic Neumann problem. We also describe the asymptotic behavior as $t \to \infty$.
On the differentiability of Lipschitz-Besov functions
José R.
Dorronsoro
229-240
Abstract: ${L^r}$ and ordinary differentiability is proved for functions in the Lipschitz-Besov spaces $ B_a^{p,q},\;1 \leqslant p < \infty ,\;1 \leqslant q \leqslant \infty ,\;a > 0$, using certain maximal operators measuring smoothness. These techniques allow also the study of lacunary directional differentiability and of tangential convergence of Poisson integrals.
Thue equations with few coefficients
Wolfgang M.
Schmidt
241-255
Abstract: Let $F(x,\,y)$ be a binary form of degree $r \geqslant 3$ with integer coefficients, and irreducible over the rationals. Suppose that only $s + 1$ of the $r + 1$ coefficients of $ F$ are nonzero. Then the Thue equations $ F(x,\,y) = 1$ has $\ll {(rs)^{1/2}}$ solutions. More generally, the inequality $\vert F(x,\,y)\vert \leqslant h$ has $\ll {(rs)^{1/2}}{h^{2/r}}(1 + \log {h^{1/r}})$ solutions.
A new proof that Teichm\"uller space is a cell
A. E.
Fischer;
A. J.
Tromba
257-262
Abstract: A new proof is given, using the energy of a harmonic map, that Teichmüller space is a cell.
Riccati techniques and variational principles in oscillation theory for linear systems
G. J.
Butler;
L. H.
Erbe;
A. B.
Mingarelli
263-282
Abstract: We consider the seond order differential system $(1)\,Y'' + Q(t)Y = 0$, where $Q$, $Y$ are $n \times n$ matrices with $Q = Q(t)$ a continuous symmetric matrix-valued function, $t \in [a,\, + \infty ]$. We obtain a number of sufficient conditions in order that all prepared solutions $ Y(t)$ of $(1)$ are oscillatory. Two approaches are considered, one based on Riccati techniques and the other on variational techniques, and involve assumptions on the behavior of the eigenvalues of $ Q(t)$ (or of its integral). These results extend some well-known averaging techniques for scalar equations to system $(1)$.
Smale flows on the three-sphere
Ketty
de Rezende
283-310
Abstract: In this paper, a complete classification of Smale flows on $ {S^3}$ is obtained. This classification is presented by means of establishing a concise set of properties that must be satisfied by an (abstract) Lyapunov graph associated to a Smale flow and a Lyapunov function. We show that these properties are necessary, that is, given a Smale flow and a Lyapunov function, its Lyapunov graph satisfies this set of properties. We also show that these properties are sufficient, that is, given an abstract Lyapunov graph $ L'$ satisfying this set of properties, it is possible to realize a Smale flow on $ {S^3}$ that has a graph $ L$ as its Lyapunov graph where $L$ is equal to $L'$ up to topological equivalence. The techniques employed in proving that the conditions imposed on the graph are necessary involve some use of homology theory. Geometrical methods are used to construct the flow on ${S^3}$ associated to the given graph and therefore establish the sufficiency of the above conditions. The main theorem in this paper generalizes a result of Franks [8] who classified nonsingular Smale flows on $ {S^3}$.
Sporadic and irrelevant prime divisors
Stephen
McAdam;
L. J.
Ratliff
311-324
Abstract: Let $I$ represent a regular ideal in a Noetherian ring $R$. If $W$ is a finite set of prime ideals in $ R$, some conditions on $ W$ are given assuring that an $I$ can be found such that $W$ is exactly the set of primes which are in $ \operatorname{Ass} R/I$ but not in $\operatorname{Ass} R/{I^n}$ for all large $ n$. Furthermore, if $ I$ is fixed, and if $ P$ is a prime ideal containing $I$, some conditions are given assuring that in the Rees ring ${\mathbf{R}} = R[u,\,It],\,(u,\,P,\,It){\mathbf{R}}$ is a prime divisor of $u{\mathbf{R}}$.
A formula for the resolvent of $(-\Delta)\sp m+M\sp {2m}\sb q$ with applications to trace class
Peter
Takáč
325-344
Abstract: We derive a formula for the resolvent of the elliptic operator $H = {( - \Delta )^m} + M_q^{2m}$ on $ {L_2}({\mathbb{R}^N})$ in terms of bounded integral operators ${S_\lambda }$ and $ {T_\lambda }$ whose kernels we know explicitly. We use this formula to specify the domain of the operator ${A_\lambda } = (H + \lambda I){M_p}$ on ${L_2}({\mathbb{R}^N})$, and to estimate the Hilbert-Schmidt norm of its inverse $A_\lambda ^{ - 1}$, for $\lambda \geqslant 0$. Finally we exploit the last two results to prove a trace class criterion for an integral operator $K$ on $ {L_2}({\mathbb{R}^N})$.
On the generalized spectrum for second-order elliptic systems
Robert Stephen
Cantrell;
Chris
Cosner
345-363
Abstract: We consider the system of homogeneous Dirichlet boundary value problems $ ({\ast})$ $\displaystyle {L_1}u = \lambda [{a_{11}}(x)u + {a_{12}}(x)v],\quad {L_2}v = \mu [{a_{12}}(x)u + {a_{22}}(x)v]$ in a smooth bounded domain $ \Omega \subseteq {{\mathbf{R}}^N}$, where ${L_1}$ and ${L_2}$ are formally self-adjoint second-order strongly uniformly elliptic operators. Using linear perturbation theory, continuation methods, and the Courant-Hilbert variational eigenvalue characterization, we give a detailed qualitative and quantitative description of the real generalized spectrum of $ ({\ast})$, i.e., the set $(\lambda ,\,\mu ) \in {{\mathbf{R}}^2}:\,({\ast})$ has a nontrivial solution. The generalized spectrum, a term introduced by Protter in 1979, is of considerable interest in the theory of linear partial differential equations and also in bifurcation theory, as it is the set of potential bifurcation points for associated semilinear systems.
Scalar curvatures on $S\sp 2$
Wen Xiong
Chen;
Wei Yue
Ding
365-382
Abstract: A theorem for the existence of solutions of the nonlinear elliptic equation $- \Delta u + 2 = R(x){e^u},\;x \in {S^2}$, is proved by using a "mass center" analysis technique and by applying a continuous "flow" in ${H^1}({S^2})$ controlled by $\nabla R$.
A GCH example of an ordinal graph with no infinite path
Jean A.
Larson
383-393
Abstract: It is hard to find nontrivial positive partition relations which hold for many ordinals in ordinary set theory, or even ordinary set theory with the additional assumption of the Generalized Continuum Hypothesis. Erdös, Hajnal and Milner have proved that limit ordinals $ \alpha < \omega _1^{\omega + 2}$ satisfy a positive partition relation that can be expressed in graph theoretic terms. In symbols one writes $\alpha \to {(\alpha ,\,\operatorname{infinite} \operatorname{path} )^2}$ to mean that every graph on an ordinal $\alpha$ either has a subset order isomorphic to $ \alpha$ in which no two points are joined by an edge or has an infinite path. This positive result generalizes to ordinals of cardinality $ {\aleph _m}$ for $ m$ a natural number. However, the argument, based on a set mapping theorem, works only on the initial segment of the limit ordinals of cardinality ${\aleph _m}$ for which the set mapping theorem is true. In this paper, the Generalized Continuum Hypothesis is used to construct counterexamples for a cofinal set of ordinals of cardinality ${\aleph _m}$, where $m$ is a natural number at least two.
Convergence in distribution of products of random matrices: a semigroup approach
Arunava
Mukherjea
395-411
Abstract: The problem of weak convergence of the sequence of convolution powers of a probability measure has been considered in this paper in the general context of a noncompact semigroup and in particular, in the semigroup of nonnegative and real matrices. Semigroup methods have been used to give simple proofs of some recent results of Kesten and Spitzer in nonnegative matrices. It has been also shown that these methods often lead to similar results in the more general context of real matrices.
Homological stability for ${\rm O}\sb {n,n}$ over a local ring
Stanisław
Betley
413-429
Abstract: Let $R$ be a local ring, ${V^{2n}}$ a free module over $ R$ of rank $2n$ and $q$ a bilinear form on ${V^{2n}}$ which has in some basis the matrix $ \left\vert {\begin{array}{*{20}{c}} 0 & 1 1 & 0 \end{array} } \right\vert\,$. Let ${O_{n,n}}$ be the group of automorphisms of $ {V^{2n}}$ which preserve $ q$. We prove the following theorem: if $n$ is big enough with respect to $k$ then the inclusion homomorphism $ i:{O_{n,n}} \to {O_{n + 1,n + 1}}$ induces an isomorphism $ {i_{\ast}}:{H_k}({O_{n,n}};\,Z) \to {H_k}({O_{n + 1,n + 1}};Z)$.