Linearized stability of extreme shock profiles in systems of conservation laws with viscosity
Robert L.
Pego
431-461
Abstract: For a genuinely nonlinear hyperbolic system of conservation laws with added artificial viscosity, ${u_t} + f{(u)_x} = \varepsilon {u_{xx}}$, we prove that traveling wave profiles for small amplitude extreme shocks (the slowest and fastest) are linearly stable to perturbations in initial data chosen from certain spaces with weighted norm; i.e., we show that the spectrum of the linearized equation lies strictly in the left-half plane, except for a simple eigenvalue at the origin (due to phase translations of the profile). The weight ${e^{cx}}$ is used in components transverse to the profile, where, for an extreme shock, the linearized equation is dominated by unidirectional convection.
Complex and integral laminated lattices
J. H.
Conway;
N. J. A.
Sloane
463-490
Abstract: In an earlier paper we studied real laminated lattices (or ${\mathbf{Z}}$-modules) $ {\Lambda_n}$, where ${\Lambda_1}$ is the lattice of even integers, and ${\Lambda_n}$ is obtained by stacking layers of a suitable $(n - 1)$-dimensional lattice ${\Lambda_{n - 1}}$ as densely as possible, keeping the same minimal norm. The present paper considers two generalizations: (a) complex and quaternionic lattices, obtained by replacing $ {\mathbf{Z}}$-module by $ J$-module, where $ J$ may be the Eisenstein, Gaussian or Hurwitzian integers, etc., and (b) integral laminated lattices, in which ${\Lambda_n}$ is required to be an integral lattice with the prescribed minimal norm. This enables us to give a partial answer to a question of J. G. Thompson on integral lattices, and to explain some of the computer-generated results of Plesken and Pohst on this problem. Also a number of familiar lattices now arise in a canonical way. For example the Coxeter-Todd lattice is the $6$-dimensional integral laminated lattice over ${\mathbf{Z}}[ \omega ]$ of minimal norm $ 2$. The paper includes tables of the best real integral lattices in up to $ 24$ dimensions.
A topological group having no homeomorphisms other than translations
Jan
van Mill
491-498
Abstract: We give an example of a (separable metric) connected and locally connected topological group, the only autohomeomorphisms of which are group translations.
Secant functions, the Reiss relation and its converse
Mark L.
Green
499-507
Abstract: Generalizing a classical Euclidean theorem for the circle, certain meromorphic functions on $ {{\mathbf{P}}_1}$ relating to the geometry of algebraic plane curves are shown to be constant. Differentiated twice, this gives a new proof of the Reiss relation and its converse. The relation of these functions to Abel's Theorem is discussed, and a generalization of secant functions to space curves is given, for which the Chow form arises in a natural way.
Multi-invariant sets on tori
Daniel
Berend
509-532
Abstract: Given a compact metric group $G$, we are interested in those semigroups $ \Sigma$ of continuous endomorphisms of $G$, possessing the following property: The only infinite, closed, $\Sigma$-invariant subset of $G$ is $G$ itself. Generalizing a one-dimensional result of Furstenberg, we give here a full characterization--for the case of finitedimensional tori--of those commutative semigroups with the aforementioned property.
Degenerate elliptic operators as regularizers
R. N.
Pederson
533-553
Abstract: The spaces ${\mathcal{K}_{mk}}$, introduced in the Nehari Volume of Journal d'Analyse Mathématique, for nonnegative integer values of $m$ and arbitrary real values of $k$ are extended to negative values of $m$. The extension is consistent with the equivalence $ \parallel {\zeta ^j}u{\parallel_{m,k}}\sim\parallel u{\parallel_{m,k - j}}$, the inequality $ \parallel {D^\alpha }u{\parallel_{m,k}} \leqslant {\text{const}}\parallel u{\parallel_{m + \vert\alpha \vert,k + \vert\alpha \vert}}$, and the generalized Cauchy-Schwarz inequality $\vert\langle {u,v} \rangle \vert \leqslant \parallel u\,{\parallel_{m,k}}\parallel v\parallel_{ - m, - k}$. (Here $\langle u, \upsilon \rangle$ is the $ {L_2}$ scalar product.) There exists a second order degenerate elliptic operator which maps ${\mathcal{K}_{m,k}}\,1 - 1$ onto ${\mathcal{K}_{m - 2,k}}$. These facts are used to simplify proof of regularity theorems for elliptic and hyperbolic problems and to give new results concerning growth rates at the boundary for the coefficients of the operator and the forcing function. (See Notices Amer. Math. Soc. 28 (1981), 226.)
Real vs. complex rational Chebyshev approximation on an interval
Lloyd N.
Trefethen;
Martin H.
Gutknecht
555-561
Abstract: If $f \in C[ - 1,1]$ is real-valued, let ${E^{r}(f)}$ and $ {E^{c}(f)}$ be the errors in best approximation to $f$ in the supremum norm by rational functions of type $(m,n)$ with real and complex coefficients, respectively. It has recently been observed that ${E^c}(f) < {E^r}(f)$ can occur for any $n \geqslant 1$, but for no $n \geqslant 1$ is it known whether ${\gamma_{mn}} = \inf_f\,{E^c}(f)/{E^{r}(f)}$ is zero or strictly positive. Here we show that both are possible: $ {\gamma_{01}} > 0$, but ${\gamma_{mn}} = 0$ for $n \geqslant m + 3$. Related results are obtained for approximation on regions in the plane.
Szeg\H o limit theorems for the harmonic oscillator
A. J. E. M.
Janssen;
Steven
Zelditch
563-587
Abstract: Let $H = - \frac{1}{2}{d^2}/d{x^2} + \frac{1}{2}{x^2}$ be the harmonic oscillator Hamiltonian on $ {L^2}( {\mathbf{R}})$, and let $A$ be a selfadjoint $DO$ of order $O$ in the Beals-Fefferman class with weights $\varphi = 1,\Phi (x,\xi ) = {(1 + \vert\xi \,{\vert^2} + \vert x\,{\vert^2})^{1/2}}$. Form the measure $\mu(f) = {\lim_{\lambda \to \infty }}(1/{\text{rank}}\;{\pi_\lambda })\,{\text{tr}}\,f({\pi_\lambda }\,A{\pi_\lambda })$ where ${\pi_\lambda }\,A{\pi_\lambda }$ is the compression of $A$ onto the span of the Hermite functions with eigenvalue less than or equal to $\lambda$. Then one has the following Szegö limit theorem: $\displaystyle \mu (f) = \mathop {\lim }\limits_{T \to \infty } \;\frac{1} {{2\,... ...qslant T} {f(a(x,\xi ))\;dx} \;d\xi \qquad {\text{for}} f \in C({\mathbf{R}}).$ For the special case where $f(x) = x$, this will be proved for a considerably wider class of operators by employing the Weyl correspondence. Moreover, by using estimates on Wigner functions of Hermite functions we are able to prove the full Szegö theorem for a fairly general class of multiplication operators.
Lower semicontinuity, existence and regularity theorems for elliptic variational integrals of multiple valued functions
Pertti
Mattila
589-610
Abstract: Let $A$ be an open set in ${{\mathbf{R}}^m}$ with compact smooth boundary, and let $ {\mathbf{Q}}$ be the space of unordered $Q$ tuples of points of ${{\mathbf{R}}^n}$. F. J. Almgren, Jr. has developed a theory for functions $f:A \to {\mathbf{Q}}$ and used them to prove regularity theorems for area minimizing integral currents. In particular, he has defined in a natural way the space $ {\mathcal{Y}_2}(A,{\mathbf{Q}})$ of functions $f:A \to {\mathbf{Q}}$ with square summable distributional partial derivatives and the Dirichlet integral $\operatorname{Dir}(f;A)$ of such functions. In this paper we study more general constant coefficient quadratic integrals $ {\mathbf{G}}(f;A)$ which are $Q$ elliptic in the sense that there is $c > 0$ such that ${\mathbf{G}}(f;A) \geqslant c\,\operatorname{Dir}(f;A)$ for $f \in {\mathcal{Y}_2}(A;{\mathbf{Q}})$ with zero boundary values. We prove a lower semicontinuity theorem which leads to the existence of a ${\mathbf{G}}$ minimizing function with given reasonable boundary values. In the case $m = 2$ we also show that such a function is Hölder continuous and regular on an open dense set. In the case $ m \geqslant 3$ the regularity problem remains open.
On certain sums of Fourier-Stieltjes coefficients
J. B.
Twomey
611-621
Abstract: We obtain estimates for certain sums of Fourier-Stieltjes (and hence also Fourier) coefficients of continuous functions $ f$ of bounded variation in terms of the modulus of continuity of $f$. As a consequence of one of our results we obtain an improvement on a theorem of Zygmund on the absolute convergence of Fourier series of functions of bounded variation. We also consider absolutely continuous functions and show by examples that a number of the results we obtain are "best possible".
All varieties of central completely simple semigroups
Mario
Petrich;
Norman R.
Reilly
623-636
Abstract: Completely simple semigroups may be considered as a variety of algebras with the binary operation of multiplication and the unary operation of inversion. A completely simple semigroup is central if the product of any two idempotents lies in the centre of the containing maximal subgroup. Central completely simple semigroups form a subvariety $\mathcal{C}$ of the variety of all completely simple semigroups. We find an isomorphic copy of $ \mathcal{L}(\mathcal{C})$ as a subdirect product of the lattices $ \mathcal{L}(\mathcal{R}\,\mathcal{B})$, $ \mathcal{L}(\mathcal{A}\,\mathcal{G})$, and $\mathcal{L}(\mathcal{G})$ of all varieties of rectangular bands, abelian groups, and groups, respectively. We consider also several homomorphisms and study congruences they induce.
Determination of bounds for the solutions to those binary Diophantine equations that satisfy the hypotheses of Runge's theorem
David Lee
Hilliker;
E. G.
Straus
637-657
Abstract: In 1887 Runge [13] proved that a binary Diophantine equation $F(x,y) = 0$, with $F$ irreducible, in a class including those in which the leading form of $F$ is not a constant multiple of a power of an irreducible polynomial, has only a finite number of solutions. It follows from Runge's method of proof that there exists a computable upper bound for the absolute value of each of the integer solutions $ x$ and $y$. Runge did not give such a computation. Here we first deduce Runge's Theorem from a more general theorem on Puiseux series that may be of interest in its own right. Second, we extend the Puiseux series theorem and deduce from the generalized version a generalized form of Runge's Theorem in which the solutions $x$ and $y$ of the polynomial equation $F(x,y) = 0$ are integers, satisfying certain conditions, of an arbitrary algebraic number field. Third, we compute bounds for the solutions $(x,y) \in {{\mathbf{Z}}^2}$ in terms of the height of $F$ and the degrees in $x$ and $y$ of $F$.
The rank of a Hardy field
Maxwell
Rosenlicht
659-671
Abstract: A Hardy field is a field of germs of real-valued functions on positive half-lines that is closed under differentiation. Its rank is the rank of the associated ordered abelian group, the value group of the canonical valuation of the field. The properties of this rank are worked out, its relevance to asymptotic expansions indicated, examples provided, and applications given to the order of growth of solutions of ordinary differential equations.
Invariant theory and the lambda algebra
William M.
Singer
673-693
Abstract: Let $A$ be the Steenrod algebra over the field ${F_2}$. In this paper we construct for any left $ A$-module $M$ a chain complex whose homology groups are isomorphic to the groups $ \operatorname{Tor}_s^A({F_2},M)$. This chain complex in homological degree $ s$ is built from a ring of invariants associated with the action of the linear group $ G{L_s}({F_2})$ on a certain algebra of Laurent series. Thus, the homology of the Steenrod algebra (and so the Adams spectral sequence for spheres) is seen to have a close relationship to invariant theory. A key observation in our work is that the Adem relations can be described in terms of the invariant theory of $ G{L_2}({F_2})$. Our chain complex is not new: it turns out to be isomorphic to the one constructed by Kan and his coworkers from the dual of the lambda algebra. Thus, one effect of our work is to give an invariant-theoretic interpretation of the lambda algebra. As a consequence we find that the dual of lambda supports an action of the Steenrod algebra that commutes with the differential. The differential itself appears as a kind of "residue map". We are also able to describe the coalgebra structure of the dual of lambda using our invariant-theoretic language.
Strong Fatou-$1$-points of Blaschke products
C. L.
Belna;
F. W.
Carroll;
G.
Piranian
695-702
Abstract: This paper shows that to every countable set $M$ on the unit circle there corresponds a Blaschke product whose set of strong Fatou-$1$-points contains $M$. It also shows that some Blaschke products have an uncountable set of strong Fatou-$ 1$-points.
Forcing positive partition relations
Stevo
Todorčević
703-720
Abstract: We show how to force two strong positive partition relations on ${\omega_1}$ and use them in considering several well-known open problems.
Inverses and parametrices for right-invariant pseudodifferential operators on two-step nilpotent Lie groups
Kenneth G.
Miller
721-736
Abstract: Let $P$ be a right-invariant pseudodifferential operator with principal part $ {P_0}$ on a simply connected two-step nilpotent Lie group $G$ of type $H$. It will be shown that if $\pi (P_0)$ is injective in ${\mathcal{S}_\pi }$ for every nontrivial irreducible unitary representation $\pi$ of $G$, then $P$ has a pseudodifferential left parametrix. For such groups this generalizes the Rockland-Helffer-Nourrigat criterion for the hypoellipticity of a homogeneous right-invariant partial differential operator on $G$. If, in addition, $\pi (P)$ is injective in ${\mathcal{S}_\pi }$ for every irreducible unitary representation of $G$, it will be shown that $P$ has a pseudodifferential left inverse. The constructions of the inverse and parametrix make use of the Kirillov theory, their symbols being obtained on the orbits individually and then pieced together.
Reye congruences
François R.
Cossec
737-751
Abstract: This paper studies the congruences of lines which are included in two distinct quadrics of a given generic three-dimensional projective space of quadrics in ${{\mathbf{P}}^3}$.
Representations of generic algebras and finite groups of Lie type
R. B.
Howlett;
G. I.
Lehrer
753-779
Abstract: The complex representation theory of a finite Lie group $G$ is related to that of certain "generic algebras". As a consequence, formulae are derived ("the Comparison Theorem"), relating multiplicities in $ G$ to multiplicities in the Weyl group $W$ of $G$. Applications include an explicit description of the dual (see below) of an arbitrary irreducible complex representation of $G$.
Systems of conservation laws with invariant submanifolds
Blake
Temple
781-795
Abstract: Systems of conservation laws with coinciding shock and rarefaction curves arise in the study of oil reservoir simulation, multicomponent chromatography, as well as in the study of nonlinear motion in elastic strings. Here we characterize this phenomenon by deriving necessary and sufficient conditions on the geometry of a wave curve in order that the shock wave curve coincide with its associated rarefaction wave curve for a system of conservation laws. This coincidence is the one dimensional case of a submanifold of the state variables being invariant for the system of equations, and the necessary and sufficient conditions are derived for invariant submanifolds of arbitrary dimension. In the case of $2 \times 2$ systems we derive explicit formulas for the class of flux functions that give rise to the coupled nonlinear conservation laws for which the shock and rarefaction wave curves coincide.
Uniqueness of torsion free connection on some invariant structures on Lie groups
Michel Nguiffo
Boyom;
Georges
Giraud
797-808
Abstract: Let $\mathcal{G}$ be a connected Lie group with Lie algebra $ \mathfrak{g}$. Let $ \operatorname{Int}(\mathfrak{g})$ be the group of inner automorphisms of $\mathfrak{g}$. The group $ \mathcal{G}$ is naturally equipped with $ \operatorname{Int}(\mathfrak{g})$-reductions of the bundle of linear frames on $\mathcal{G}$. We investigate for what kind of Lie group the 0-connection of E. Cartan is the unique torsion free connection adapted to any of those $ \operatorname{Int}(\mathfrak{g})$-reductions.
Length dependence of solutions of FitzHugh-Nagumo equations
Clyde
Collins
809-832
Abstract: We investigate the behavior of the solutions of the problem \begin{displaymath}\begin{array}{*{20}{c}} {{u_t} = {u_{xx}} - \alpha u - \upsil... ... h(t),} & {{u_x}(L,t) = {\upsilon _x}(L,t) = 0} \end{array} \end{displaymath} where $t \geqslant 0$ and $0 < x < L \leqslant \infty$. Solutions of the above equations are considered a qualitative model of conduction of nerve axon impulses. Using explicit constructions and semigroup methods, we obtain decay results on the norms of differences between the solution for $L$ infinite and the solutions when $ L$ is large but finite. We conclude that nerve impulses for long finite nerves become uniformly close to those of the semi-infinite nerves away from the right endpoint of the finite nerve.
Diffusion dependence of the FitzHugh-Nagumo equations
Clyde
Collins
833-839
Abstract: We investigate the behavior of the solutions of \begin{displaymath}\begin{array}{*{20}{c}} {{u_t} = {u_{x\,x}} - \alpha \,u - v ... ...} = \eta \,{v_{x\,x}} + \sigma \,u - \gamma v,} \end{array} \end{displaymath} as $\eta$ tends to zero from above.
Correction to: ``The stable geometric dimension of vector bundles over real projective spaces'' [Trans. Amer. Math. Soc. {\bf 268} (1981), no. 1, 39--61; MR0628445 (83c:55006)]
Donald M.
Davis;
Sam
Gitler;
Mark
Mahowald
841-843
Abstract: The theory of $ bo$-resolutions as utilized in The stable geometric dimension of vector bundles over real projective spaces did not give adequate care to the $ K{{\mathbf{Z}}_2}$'s occurring at each stage of the resolution. This restricts somewhat the set of integers $e$ for which we can prove that the geometric dimension of vector bundles of order ${2^e}$ on large real projective spaces is precisely $2\,e + \delta $.