Variational problems with two phases and their free boundaries
Hans Wilhelm
Alt;
Luis A.
Caffarelli;
Avner
Friedman
431-461
Abstract: The problem of minimizing $\int {[\nabla \upsilon {\vert^2}} + {q^2}(x){\lambda ^2}(\upsilon )]dx$ in an appropriate class of functions $\upsilon$ is considered. Here $q(x) \ne 0$ and ${\lambda ^2}(\upsilon ) = \lambda _1^2$if $\upsilon < 0, = \lambda _2^2$ if $\upsilon > 0$. Any minimizer $u$ is harmonic in $\{ u \ne 0\}$ and $ \vert\nabla u{\vert^2}$ has a jump $\displaystyle {q^2}(x)\left( {\lambda _1^2 - \lambda _2^2} \right)$ across the free boundary $\{ u \ne 0\} $. Regularity and various properties are established for the minimizer $ u$ and for the free boundary.
Codimension $1$ orbits and semi-invariants for the representations of an oriented graph of type $\mathcal{A}_n$
S.
Abeasis
463-485
Abstract: We consider the Dynkin diagram $ \mathcal{A}_n$ with an arbitrary orientation $\Omega$. For a given dimension $d = ({d_1}, \ldots ,{d_n})$ we consider the corresponding variety ${L_d}$ of all the representations of $ (\mathcal{A}_n,\Omega )$ on which a group ${G_d}$ acts naturally. In this paper we determine the maximal orbit and the codim. $1$ orbits of this action, giving explicitly their decomposition in terms of the irreducible representations of $ \mathcal{A}_n$. We also deduce a set of algebraically independent semi-invariant polynomials which generate the ring of semi-invariants.
Some properties of viscosity solutions of Hamilton-Jacobi equations
M. G.
Crandall;
L. C.
Evans;
P.-L.
Lions
487-502
Abstract: Recently M. G. Crandall and P. L. Lions introduced the notion of "viscosity solutions" of scalar nonlinear first order partial differential equations. Viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics. The value of this concept is established by the fact that very general existence, uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application. The notion of a " viscosity solution" admits several equivalent formulations. Here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner. Moreover, by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper.
Matrix localizations of $n$-firs. I
Peter
Malcolmson
503-518
Abstract: An $n$-fir is an associative ring in which every $n$-generator right ideal is free of unique rank. Matrix localization of a ring involves the adjunction of universal inverses to certain matrices over the ring, so that a new ring results over which the matrices have inverses, but so that the minimum of additional relations is imposed. A full matrix is a square matrix which, when considered as an endomorphism of a free module, cannot be factored through a free module of smaller rank. The main result of this paper is that if the original ring is an $n$-fir with $n > 2k$ and if we form a matrix localization by adjoining universal inverses to all full matrices of size $ k$, then the resulting ring is an $(n - 2k)$-fir. This generalizes an announced result of V. N. Gerasimov. There are related results on the structure of the universal skew field of fractions of a semifir.
Matrix localizations of $n$-firs. II
Peter
Malcolmson
519--527
Abstract: In a previous paper by this author and with a similar title, it was shown that adjoining universal inverses for all $k \times k$ full matrices over an $n$-fir results in the localized ring being an $(n - 2k)$-fir. In this note a counterexample is used to show that the result is best possible in general. Techniques of the previous paper are strengthened and a result on a kind of finite inertia of certain rings within their localizations is obtained.
Haefliger structures and linear homotopy
Javier
Bracho
529-538
Abstract: The notion of linear-homotopy into a classifying space is introduced and used to give a precise classification of Haefliger structures. Appendix on the product theorem for simplicial spaces and realizations of bisimplicial spaces.
Degrees of recursively saturated models
Angus
Macintyre;
David
Marker
539-554
Abstract: Using relativizations of results of Goncharov and Peretyat'kin on decidable homogeneous models, we prove that if $M$ is $S$-saturated for some Scott set $S$, and $F$ is an enumeration of $S$, then $M$ has a presentation recursive in $F$. Applying this result we are able to classify degrees coding (i) the reducts of models of PA to addition or multiplication, (ii) internally finite initial segments and (iii) nonstandard residue fields. We also use our results to simplify Solovay's characterization of degrees coding nonstandard models of Th(N).
Orthogonal polynomials on the sphere with octahedral symmetry
Charles F.
Dunkl
555-575
Abstract: For any finite reflection group $G$ acting on $ {{\mathbf{R}}^N}$ there is a family of $G$-invariant measures ( $({h^2}d\omega$, where $h$ is a certain product of linear functions whose zero-sets are the reflecting hyperplanes for $ G$) on the unit sphere and an associated partial differential operator ( $ {L_h}f: = \Delta (fh) - f\Delta h$; $\Delta$ is the Laplacian). Analogously to spherical harmonics, there is an orthogonal (with respect to ${h^2}d\omega $) decomposition of homogeneous polynomials, that is, if $p$ is of degree $n$ then $\displaystyle p(x) = \sum\limits_{j = 0}^{[n/2]} {\vert x{\vert^{2j}}{p_{n - 2j}}(x),}$ where ${L_h}{p_i} = 0$ and ${\operatorname{deg}}{p_i} = i$ for each $ i$, but with the restriction that $p$ and ${p_i}$ must all be $G$-invariant. The main topic is the hyperoctahedral group with $\displaystyle h(x) = {({x_1}{x_2} \cdots {x_N})^\alpha }{\left( {\prod\limits_{i < j} {(x_i^2 - x_j^2)} } \right)^\beta }.$ The special case $N = 2$ leads to Jacobi polynomials. A detailed study of the case $N = 3$ is made; an important result is the construction of a third-order differential operator that maps polynomials associated to $h$ with indices $ (\alpha ,\beta )$ to those associated with $(\alpha + 2,\beta + 1)$.
Asymptotic behavior of solutions of second order differential equations with integrable coefficients
Manabu
Naito
577-588
Abstract: The differential equation $ x'' + a(t)f(x) = 0$, $t > 0$, is considered under the condition that ${\lim_{t \to \infty }}{\int ^t}a(s)ds$ exists and is finite, and necessary and/or sufficient conditions are given for this equation to have solutions which behave asymptotically like nontrivial linear functions ${c_1} + {c_2}t$.
Entropy via random perturbations
Yuri
Kifer
589-601
Abstract: The entropy of a dynamical system ${S^t}$ on a hyperbolic attractor with respect to the Bowen-Ruelle-Sinai measure is obtained as a limit of entropy characteristics of small random perturbations $x_t^\varepsilon $ of ${S^t}$. Both the case of perturbations only in some neighborhood of an attractor and global perturbations of a flow with hyperbolic attracting sets are considered.
Jordan domains and the universal Teichm\"uller space
Barbara Brown
Flinn
603-610
Abstract: Let $L$ denote the lower half plane and let $ B(L)$ denote the Banach space of analytic functions $f$ in $L$ with ${\left\Vert f \right\Vert _L} < \infty$, where $ {\left\Vert f \right\Vert _L}$ is the suprenum over $z \in L$ of the values $\left\vert {f(z)} \right\vert{(text{Im} z)^2}$. The universal Teichmüller space, $ T$, is the subset of $ B(L)$ consisting of the Schwarzian derivatives of conformal mappings of $ L$ which have quasiconformal extensions to the extended plane. We denote by $ J$ the set $\displaystyle \left\{ {{S_f}:f{\text{is conformal in }}L{\text{and }}f(L){\text{is a Jordan domain}}} \right\},$ which is a subset of $B(L)$ contained in the Schwarzian space $ S$. In showing $S - \bar T \ne \emptyset$, Gehring actually proves $ S - \bar J \ne \emptyset$. We give an example which demonstrates that $J - \bar T \ne \emptyset$.
Moduli of continuity in ${\bf R}\sp{n}$ and $D\subset {\bf R}\sp{n}$
Z.
Ditzian
611-623
Abstract: The $r$ modulus of continuity for $f \in C({R^n})$ is expressed in terms of $ r$ moduli of continuity in $ n$ independent directions. Generalizations to other spaces of functions on $ {R^n}$ or $D \subset {R^n}$ are also given.
On maximal rearrangement inequalities for the Fourier transform
W. B.
Jurkat;
G.
Sampson
625-643
Abstract: Suppose that $ w$ is a measurable function on $ {{\mathbf{R}}^n}$ and denote by $W = {w^ \ast }$ the decreasing rearrangement of $\left\vert w \right\vert$ (provided that it exists). We show that the $n$-dimensional Fourier transform $ \hat f$ satisfies (1) $\displaystyle {\left\Vert {w\hat f} \right\Vert _q} \leqslant {\left\Vert {W{{(... ...t)\int_0^{1/t} {{f^ \ast }} } \right\Vert\quad (C {\text{absolute constant}}),$ if $ 1 < q < \infty$ and $ {t^{2/q - 1}}W(t) \searrow$ for $t > 0$. We also show that (2) $\displaystyle {\left\Vert {w\hat f} \right\Vert _q} \geqslant {c_{n,q}}{\left\V... ...\vert x \right\vert} {f(y)} dy} \right\Vert _q}\quad (f {\text{nonnegative),}}$ if $ 1 < q < \infty$ and $ w$ is nonnegative and symmetrically decreasing. Inequality (2) implies that (1) is maximal in the sense that the left side reaches the right side if $f$ is nonnegative and symmetrically decreasing. Hence, (1) implies all other possible estimates in terms of $W$ and ${f^ \ast }$. The cases $q \ne 2$ of (1) can be derived from the case $q = 2$ (and same $f$) by a convexity principle which does not involve interpolation. The analogue of (1) for Fourier series is due to H. L. Montgomery if $q \geqslant 2$ (then the extra condition on $ W$ is automatically satisfied).
Intermediate normalizing extensions
A. G.
Heinicke;
J. C.
Robson
645-667
Abstract: Relationships between the prime ideals of a ring $R$ and of a normalizing extension $ S$ have been studied by several authors recently. In this work, most of these known results are extended to give relationships between the prime ideals of $R$ and of $T$ where $T$ is a ring with $R \subset T \subset S$, and $S$ is a normalizing extension of $ R$: such rings $ T$ are called intermediate normalizing extensions of $ R$. One result ("Cutting Down") asserts that for any prime ideal $ J$ of $T$, $J \cap R$ is the intersection of a finite set of prime ideals ${P_i}$ of $R$, uniquely defined by $J$, whose corresponding factor rings $R/{P_i}$ are mutually isomorphic. The minimal members of the family of ${P_i}$'s are the primes of $R$ minimal over $J \cap R$, and an "incomparability" theorem is proved which shows that no two comparable primes of $ T$ can have their intersections with $R$ share a common minimal prime. Other results include versions of the "lying over" and "going up" theorems, proofs that chain conditions such as right Goldie or right Noetherian pass between $T/J$ and each of the rings $R/{P_i}$, and a demonstration that the "additivity principle" holds.
Mean convergence of Lagrange interpolation. III
Paul
Nevai
669-698
Abstract: Necessary and sufficient conditions are found for weighted mean convergence of Lagrange and quasi-Lagrange interpolation based at the zeros of generalized Jacobi polynomials.
Monotone decompositions of IUC continua
W. Dwayne
Collins
699-709
Abstract: For the class of hereditarily unicoherent metric continua a spectrum of monotone decompositions has been developed by several authors which "improves" the quotient spaces. This spectrum is developed for a broader class of continua, namely continua with property IUC. A metric continuum $ M$ has property IUC provided each proper subcontinuum of $M$ with interior is unicoherent. One important result which develops is that semiaposyndetic IUC continua are hereditarily arcwise connected. Also the notion of smoothness is studied for IUC continua.
Projectively equivalent metrics subject to constraints
William
Taber
711-737
Abstract: This work examines the relationship between pairs of projectively equivalent Riemannian or Lorentz metrics $g$ and $ {g^ \ast }$ on a manifold $ M$ that induce the same Riemannian metric on a hypersurface $H$. In general such metrics must be equal. In the case of distinct metrics, the structure of the metrics and the manifold are strongly determined by the set, $C$, of points at which $g$ and ${g^ \ast }$ are conformally related. The space $(M - C,g)$ is locally a warped product manifold over the hypersurface $H$. In the Lorentz setting, $C$ is empty. In the Riemannian case, $ C$ contains at most two points. If $C$ is nonempty, then $H$ is isometric to a standard sphere. Furthermore, in the case that $C$ contains one point, natural hypotheses imply $ M$ is diffeomorphic to $ {R^n}$. If $C$ contains two points $M$ is diffeomorphic to $ {S^n}$.
On specializations of curves. I
A.
Nobile
739-748
Abstract: The following is proved: Given a family of projective reduced curves $X \to T$ ($T$ irreducible), if $ {X_t}$ (the general curve) is integral and ${X_0}$ is a special curve (having irreducible components ${X_1}, \ldots ,{X_r}$), then $\sum\nolimits_{i = 1}^r {{g_i}({X_i}) \leqslant g({X_t})}$, where $g(Z) =$ geometric genus of $Z$. Conversely, if $A$ is a reduced plane projective curve, of degree $ n$ with irreducible components $ {X_1}, \ldots ,{X_r}$, and $ g$ satisfies $ \sum\nolimits_{i = 1}^r {{g_i}({X_i}) \leqslant g \leqslant \frac{1} {2}(n - 1)(n - 2)}$, then a family of plane curves $ X \to T$ (with $ T$ integral) exists, where for some $ {t_0} \in T,{X_{{t_0}}} = Z$ and for $t$ generic, ${X_t}$ is integral and has only nodes as singularities. Results of this type appear in an old paper by G. Albanese, but the exposition is rather obscure.
Stable viscosities and shock profiles for systems of conservation laws
Robert L.
Pego
749-763
Abstract: Wide classes of high order "viscosity" terms are determined, for which small amplitude shock wave solutions of a nonlinear hyperbolic system of conservation laws ${u_t} + f{(u)_x} = 0$ are realized as limits of traveling wave solutions of a dissipative system ${u_t} + f{(u)_x} = \nu {({D_1}{u_x})_x} + \cdots + {\nu ^n}{({D_n}{u^{(n)}})_x}$. The set of such "admissible" viscosities includes those for which the dissipative system satisfies a linearized stability condition previously investigated in the case $n = 1$ by A. Majda and the author. When $n = 1$ we also establish admissibility criteria for singular viscosity matrices $ {D_1}(u)$, and apply our results to the compressible Navier-Stokes equations with viscosity and heat conduction, determining minimal conditions on the equation of state which ensure the existence of the "shock layer" for weak shocks.
Brauer factor sets and simple algebras
Louis H.
Rowen
765-772
Abstract: It is shown that the Brauer factor set $ ({c_{ijk}})$ of a finite-dimensional division algebra of odd degree $n$ can be chosen such that $ {c_{iji}} = {c_{iij}} = {c_{jii}} = 1$ for all $i,j$ and ${c_{ijk}} = c_{kji}^{ - 1}$. This implies at once the existence of an element $a \ne 0$ with ${\text{tr}}(a) = {\text{tr}}({a^2}) = 0$; the coefficients of $ {x^{n - 1}}$ and ${x^{n - 2}}$ in the characteristic polynomial of $ a$ are thus 0. Also one gets a generic division algebra of degree $ n$ whose center has transcendence degree $ n + (n - 1)(n - 2)/2$, as well as a new (simpler) algebra of generic matrices. Equations are given to determine the cyclicity of these algebras, but they may not be tractable.
Realizability of branched coverings of surfaces
Allan L.
Edmonds;
Ravi S.
Kulkarni;
Robert E.
Stong
773-790
Abstract: A branched covering $M \to N$ of degree $ d$ between closed surfaces determines a collection $ \mathfrak{D}$ of partitions of $d$--its "branch data"--corresponding to the set of branch points. The collection of partitions must satisfy certain obvious conditions implied by the Riemann-Hurwitz formula. This paper investigates the extent to which any such finite collection $\mathfrak{D}$ of partitions of $d$ can be realized as the branch data of a suitable branched covering. If $N$ is not the $2$-sphere, such data can always be realized. If $\mathfrak{D}$ contains sufficiently many elements compared to $d$, then it can be realized. And whenever $ d$ is nonprime, examples are constructed of nonrealizable data.
Strong martingale convergence of generalized conditional expectations on von Neumann algebras
Fumio
Hiai;
Makoto
Tsukada
791-798
Abstract: Accardi and Cecchini generalized the concept of conditional expectations on von Neumann algebras. In this paper we give some conditions for strong convergence of increasing or decreasing martingales of Accardi and Cecchini's conditional expectations.
A sharp form of the Ahlfors' distortion theorem, with applications
D. H.
Hamilton
799-806
Abstract: The constant appearing in the asymptotic version of the Ahlfors' distortion theorem is $1$. Also it is shown that for mean $1$-valent functions $f = z + {a_2}{z^2} \cdots \left\Vert {{a_{n + 1}}\vert - \vert{a_n}} \right\Vert \leqslant 1$ for $n \geqslant N(f)$.
On the proper holomorphic equivalence for a class of pseudoconvex domains
M.
Landucci
807-811
Abstract: A complete and explicit description of the holomorphic proper mappings between weakly pseudoconvex domains of the class ${\Delta _p}$ (see ( *) below) is given.