Errata to: ``Hypersurfaces with constant mean curvature in the complex hyperbolic space'' [Trans. Amer. Math. Soc. {\bf 339} (1993), no. 2, 685--702; MR1123452 (93m:53065)]
S.
Fornari;
K.
Frensel;
J.
Ripoll
The large deviation principle for a general class of queueing systems. I
Paul
Dupuis;
Richard S.
Ellis
2689-2751
Abstract: We prove the existence of a rate function and the validity of the large deviation principle for a general class of jump Markov processes that model queueing systems. A key step in the proof is a local large deviation principle for tubes centered at a class of piecewise linear, continuous paths mapping [0,1] into $[0,1]$. In order to prove certain large deviation limits, we represent the large deviation probabilities as the minimal cost functions of associated stochastic optimal control problems and use a subadditivity--type argument. We give a characterization of the rate function that can be used either to evaluate it explicitly in the cases where this is possible or to compute it numerically in the cases where an explicit evaluation is not possible.
On Lam\'e operators which are pull-backs of hypergeometric ones
Bruno
Chiarellotto
2753-2780
Abstract: We give a method that would allow one to calculate the number of Lamé operators, $ {\mathcal{L}_n}$, $n \in {\mathbf{N}}$, with prescribed finite monodromy and do the calculation for the case $n = 1$. We find a bound for the degree over $ {\mathbf{Q}}$ of the field of definition of the coefficients of a Lamé operator with prescribed finite monodromy and give examples of Lamé operators with finite monodromy. Finally we study Lamé operators with infinite monodromy and generic second order differential operators which are pull-backs of hypergeometric ones under algebraic maps.
On the $K$-theory of crystallographic groups
Georgios
Tsapogas
2781-2794
Abstract: For any crystallographic group $\Gamma$ we show that the groups ${K_i}(\Gamma )$ are isomorphic, via the forget control map, to the controlled $K$-groups ${K_i}{(\Gamma )_c}$, for all $i \leqslant 1$ and for an appropriate choice of the control map. By using this result and under a mild hypothesis on the crystallographic group $ \Gamma$, it is proved that ${K_i}(\Gamma ) = 0$ for all $i \leqslant - 2$ and ${N^j}{K_i}(\Gamma ) = 0$ for all $i \leqslant - 1$ and $j > 0$.
The index of determinacy for measures and the $l\sp 2$-norm of orthonormal polynomials
Christian
Berg;
Antonio J.
Duran
2795-2811
Abstract: For determinate measures $\mu$ having moments of every order we define and study an index of determinacy which checks the stability of determinacy under multiplication by even powers of $\vert t - z\vert$ for $z$ a complex number. Using this index of determinacy, we solve the problem of determining for which $z \in \mathbb{C}$ the sequence $ {(p_n^{(m)}(z))_n}(m \in \mathbb{N})$ belongs to ${\ell ^2}$, where $ {({p_n})_n}$ is the sequence of orthonormal polynomials associated with the measure $\mu$.
Immersions and embeddings in domains of holomorphy
Avner
Dor
2813-2849
Abstract: Let ${D_1}$ be a bounded smooth strongly pseudoconvex domain in $ {\mathbb{C}^N}$ and let $ {D_2}$ be a domain of holomorphy in $ {\mathbb{C}^M}(2 \leqslant N,5 \leqslant M,2N \leqslant M)$. There exists then a proper holomorphic immersion from ${D_1}$ to ${D_2}$. Furthermore if $ {\mathbf{PI}}({D_1},{D_2})$ is the set of proper holomorphic immersions from $ {D_1}$ to ${D_2}$ and $A({D_1},{D_2})$ is the set of holomorphic maps from $ {D_1}$ to ${D_2}$ that are continuous on the boundary, then the closure of $ {\mathbf{PI}}({D_1},{D_2})$ in the topology of uniform convergence on compacta contains $ A({D_1},{D_2})$. The approximating proper maps can be made tangent to any finite order of contact at a given point. The same result was obtained for proper holomorphic maps, in one codimension, when the target domain has a plurisubharmonic exhaustion function with no saddle critical points. This includes the case where the target domain is convex. Density in a weaker sense was derived in one codimension when the critical points are contained in a compact subset of the target domain. This occurs (for example) when the target domain is bounded weakly pseudoconvex with ${C^2}$-smooth boundary. If the target domain is strongly pseudoconvex then the approximating proper holomorphic maps can also be made continuous on the boundary. A lesser degree of pseudoconvexity is required from the target domain when the codimension is larger than the minimal. A domain in ${\mathbb{C}^L}$ is called "$ M$dimensional-pseudoconvex" (where $ L \geqslant M$) if it has a smooth exhaustion function $r$ such that every point $w$ in this domain has some $ M$-dimensional complex affine subspace going through this point for which $ r$, restricted to this subspace, is strictly plurisubharmonic in $ w$. In the result mentioned above the assumption that the target domain is pseudoconvex in ${\mathbb{C}^M}(M \geqslant 2N,5)$ can be substituted for the assumption that the domain is "$ M$-dimensional-pseudoconvex". Similarly, the assumption that the target domain ${D_2}$ is "$(N + 1)$-dimensional-pseudoconvex" and all the critical points of some appropriate exhaustion function are "$(N + 1)$-dimensional-convex" (defined in a similar manner) yields that the closure of the set of proper holomorphic maps from ${D_1}$ to ${D_2}$ contains $ A({D_1},{D_2})$. All the results are obtained with embeddings when the Euclidean dimensions are such that ${\dim _\mathbb{C}}({D_2}) \geqslant 2{\dim _\mathbb{C}}({D_1}) + 1$. Thus, in this case, when one of the assumptions mentioned above is fulfilled, then the closure of the set of embeddings from ${D_1}$ to ${D_2}$ contains $ A({D_1},{D_2})$.
A principle of linearized stability for nonlinear evolution equations
Nobuyuki
Kato
2851-2868
Abstract: We present a principle of linearized stability of stationary solutions to nonlinear evolution equation in Banach spaces. The well-known semilinear case is shown to fit into our framework. Applications to nonlinear population dynamics and to functional differential equations are also considered.
When do the upper Kuratowski topology (homeomorphically, Scott topology) and the co-compact topology coincide?
Szymon
Dolecki;
Gabriele H.
Greco;
Alojzy
Lechicki
2869-2884
Abstract: A topology is called consonant if the corresponding upper Kuratowski topology on closed sets coincides with the co-compact topology, equivalently if each Scott open set is compactly generated. It is proved that Čechcomplete topologies are consonant and that consonance is not preserved by passage to $ {G_\delta }$-sets, quotient maps and finite products. However, in the class of the regular spaces, the product of a consonant topology and of a locally compact topology is consonant. The latter fact enables us to characterize the topologies generated by some $\Gamma$-convergences.
Induced $C\sp *$-algebras and Landstad duality for twisted coactions
John C.
Quigg;
Iain
Raeburn
2885-2915
Abstract: Suppose $ N$ is a closed normal subgroup of a locally compact group $G$. A coaction $:A \to M(A \otimes {C^ * }(N))$ of $N$ on a ${C^ * }$-algebra $A$ can be inflated to a coaction $\delta$ of $G$ on $A$, and the crossed product $A{ \times_\delta }G$ is then isomorphic to the induced ${C^ * }$-algebra Ind$_N^G A{\times_\epsilon}N$. We prove this and a natural generalization in which $A{ \times_\epsilon}N$ is replaced by a twisted crossed product $A{ \times _{G/N}}G$; in case $G$ is abelian, we recover a theorem of Olesen and Pedersen. We then use this to extend the Landstad duality of the first author to twisted crossed products, and give several applications. In particular, we prove that if $\displaystyle 1 \to N \to G \to G/N \to 1$ is topologically trivial, but not necessarily split as a group extension, then every twisted crossed product $A{ \times _{G/N}}G$ is isomorphic to a crossed product of the form $A \times N$.
Comparison of certain $H\sp \infty$-domains of holomorphy
Ulf
Backlund
2917-2926
Abstract: We study open sets defined by certain global plurisubharmonic functions in $ {\mathbb{C}^N}$. We examine how the fact that the connected components of the sets are $ {H^\infty }$-domains of holomorphy is related to the structure of the set of discontinuity points of the global defining functions and to polynomial convexity.
Orbits of rank one and parallel mean curvature
Carlos
Olmos
2927-2939
Abstract: Let ${M^n}(n \geqslant 2)$ be a (extrinsic) homogeneous irreducible full submanifold of Euclidean space with $rank(M) = k \geqslant 1$ (i.e., it admits $k \geqslant 1$ locally defined, linearly independent parallel normal vector fields). We prove that $M$ must be contained in a sphere. This result toghether with previous work of the author about homogeneous submanifolds of higher rank imply, in particular, the following theorem: A homogeneous irreducible submanifold of Euclidean space with parallel mean curvature vector is either minimal, or minimal in a sphere, or an orbit of the isotropy representation of a simple symmetric space.
The structure of the reverse H\"older classes
David
Cruz-Uribe;
C. J.
Neugebauer
2941-2960
Abstract: In this paper we study the structure of the class of functions $ (R{H_s})$ which satisfy the reverse Hölder inequality with exponent $ s > 1$. To do so we introduce a new operator, the minimal operator, which is analogous to the Hardy-Littlewood maximal operator, and a new class of functions, $(R{H_\infty })$, which plays the same role for $ (R{H_s})$ that the class $ ({A_1})$ does for the $ ({A_p})$ classes.
An analogue of the Jacobson-Morozov theorem for Lie algebras of reductive groups of good characteristics
Alexander
Premet
2961-2988
Abstract: Let $\mathfrak{g}$ be the Lie algebra of a connected reductive group $G$ over an algebraically closed field of characteristic $p > 0$. Suppose that ${G^{(1)}}$ is simply connected and $p$ is good for the root system of $ G$. Given a one-dimensional torus $ \lambda \subset G$ let $ \mathfrak{g}(\lambda ,1)$ denote the weight component of ${\text{Ad(}}\lambda {\text{)}}$ corresponding to weight $i \in X(\lambda ) \cong \mathbb{Z}$. It is proved in the paper that, for any nonzero nilpotent element $e \in \mathfrak{g}$, there is a one-dimentional torus ${\lambda _e} \subset G$ such that $e \in \mathfrak{g}({\lambda _e},2)$ and ${\text{Ker}}{\text{ad}}e \subseteq { \oplus _{i \geqslant 0}}\mathfrak{g}({\lambda _e},i)$.
Test problems for operator algebras
Edward A.
Azoff
2989-3001
Abstract: Kaplansky's test problems, originally formulated for abelian groups, concern the relationship between isomorphism and direct sums. They provide a "reality check" for purported structure theories. The present paper answers Kaplansky's problems in operator algebraic contexts including unitary equivalence of von Neumann algebras and equivalence of representations of (non self-adjoint) matrix algebras. In particular, it is shown that matrix algebras admitting similar ampliations are themselves similar.
An explicit theory of heights
E. V.
Flynn
3003-3015
Abstract: We consider the problem of explicitly determining the naive height constants for Jacobians of hyperelliptic curves. For genus $> 1$, it is impractical to apply Hilbert's Nullstellensatz directly to the defining equations of the duplication law; we indicate how this technical difficulty can be overcome by use of isogenies. The height constants are computed in detail for the Jacobian of an arbitrary curve of genus $2$, and we apply the technique to compute generators of $\mathcal{J}(\mathbb{Q})$, the Mordell-Weil group for a selection of rank $1$ examples.
Periodic orbits for the planar Newtonian three-body problem coming from the elliptic restricted three-body problems
Jaume
Llibre;
Donald G.
Saari
3017-3030
Abstract: Through the introduction of a new coordinate system followed by a Poincaré compactification, a new relationship is developed to connect the planar three-body problem with the various planar restricted three-body systems. This framework is further used to develop new conditions for the continuation of symmetric periodic orbits from elliptic restricted systems to the full planar three-body problem.
Local uniqueness in the inverse conductivity problem with one measurement
G.
Alessandrini;
V.
Isakov;
J.
Powell
3031-3041
Abstract: We prove local uniqueness of a domain $D$ entering the conductivity equation $ {\text{div}}((1 + \chi (D))\nabla u) = 0$ in a bounded planar domain $\Omega$ given the Cauchy data for $ u$ on a part of $\partial \Omega$. The main assumption is that $ \nabla u$ has zero index on $\partial \Omega $ which is easy to guarantee by choosing special boundary data for $ u$. To achieve our goals we study index of critical points of $u$ on $ \partial \Omega$.
Hadamard convexity and multiplicity and location of zeros
Faruk F.
Abi-Khuzam
3043-3051
Abstract: We consider certain questions arising from a paper of Hayman concerning quantitative versions of the Hadamard three-circle theorem for entire functions. If $b(r)$ denotes the second derivative of $\log M(r)$ with respect to $ \log r$, the principal contributions of this work are (i) a characterization of those entire $f$ with nonnegative Maclaurin coefficients for which $ \lim \sup b(r) = \frac{1} {4}$ and (ii) some exploration of the relationship between multiple zeros of $f$ and the growth of $b(r)$.
On the relational basis of Cayley's theorem and of similar representations for algebras
Hassan
Sedaghat
3053-3060
Abstract: Considering a binary operation as a ternary relation permits certain sections of the latter (which are functions) to be used in representing an abstract semigroup as a family of the self-maps of its underlying set under function composition. The idea is thus seen to be entirely similar to the way that the sections of a partial ordering under set inclusion represent the (abstract) partially ordered set. An extension of this procedure yields a uniform set of representation theorems for a large class of associative algebras.
Multiple viscous solutions for systems of conservation laws
A. V.
Azevedo;
D.
Marchesin
3061-3077
Abstract: We exhibit an example of mechanism responsible for multiple solutions in the Riemann problem for a mixed elliptic-hyperbolic type system of two quadratic polynomial conservation laws. In this example, multiple solutions result from folds in the set of Riemann solutions. The multiple solutions occur despite the fact that they all satisfy the viscous profile entropy criterion. The failure of this criterion to provide uniqueness is evidence in support of a need for conceptual change in the theory of shock waves for a system of conservation laws.
On the number of solutions of a third-order boundary value problem
Eva
Rovderová
3079-3092
Abstract: This paper deals with the number of solutions of the third-order boundary value problem $ y''' = f(t,y,y',y'')$, $y(0) = {A_0}$, $y''(T) = B$. This number of solutions is investigated in connection with the number of zeros of a solution for the corresponding variational problem.
Irreducible semigroups of functionally positive nilpotent operators
Yong
Zhong
3093-3100
Abstract: For each irrational number $ \theta \in (0,1)$, we construct a semigroup $ {\mathcal{S}_\theta }$ of nilpotent operators on ${\mathcal{S}^2}([0,1])$ that are also partial isometries and positive in the sense that the operator maps nonnegative functions to nonnegative functions. We prove that each semigroup ${\mathcal{S}_\theta }$ is discrete in the norm topology and hence norm-closed and that the weak closure of $ {\mathcal{S}_\theta }$ is independent of $ {\mathcal{S}_\theta }$. We show that each semigroup ${\mathcal{S}_\theta }$ has no nontrivial invariant subspaces.
Power regular operators
Aharon
Atzmon
3101-3109
Abstract: We show that for a wide class of operators $T$ on a Banach space, including the class of decomposable operators, the sequence $\left\{ {{{\left\Vert {{T^n}x} \right\Vert}^{1/n}}} \right\}_{n = 1}^\infty $ converges for every $ x$ in the space to the spectral radius of the restriction of $ T$ to the subspace $ \vee _{n = 0}^\infty \{ {T^n}x\}$.
A new functional equation of Pexider type related to the complex exponential function
Hiroshi
Haruki;
Themistocles M.
Rassias
3111-3119
Abstract: The purpose of this paper is to solve a new functional equation, characteristic for the complex exponential function, which contains four unknown entire functions and to solve, as an application, three further functional equations.
Sub-self-similar sets
K. J.
Falconer
3121-3129
Abstract: A compact set $E \subseteq {{\mathbf{R}}^n}$ is called sub-self-similar if $E \subseteq \bigcup\nolimits_{i = 1}^m {{S_i}(E)}$, where the ${S_i}$ are similarity transfunctions. We consider various examples and constructions of such sets and obtain formulae for their Hausdorff and box dimensions, generalising those for self-similar sets.
A right countably sigma-CS ring with ACC or DCC on projective principal right ideals is left Artinian and QF-$3$
Dinh Van Huynh
3131-3139
Abstract: A module $ M$ is called a CS module if every submodule of $M$ is essential in a direct summand of $ M$. A ring $R$ is said to be right (countably) $ \Sigma$-CS if any direct sum of (countably many) copies of the right $ R$-module $R$ is CS. It is shown that for a right countably $\Sigma$-CS ring $R$ the following are equivalent: (i) $ R$ is right $\Sigma$-CS, (ii) $R$ has ACC or DCC on projective principal right ideals, (iii) $R$ has finite right uniform dimension and ACC or DCC holds on projective uniform principal right ideals of $R$, (iv) $R$ is semiperfect. From results of Oshiro [12], [13], under these conditions, $R$ is left artinian and QF-$3$. As a consequence, a ring $R$ is quasi-Frobenius if it is right countably $\Sigma$-CS, semiperfect and no nonzero projective right ideals are contained in its Jacobson radical.
Stable range one for rings with many idempotents
Victor P.
Camillo;
Hua-Ping
Yu
3141-3147
Abstract: An associative ring $ R$ is said to have stable range $1$ if for any $a$, $b \in R$ satisfying $ aR + bR = R$, there exists $y \in R$ such that $ a + by$ by is a unit. The purpose of this note is to prove the following facts. Theorem $3$: An exchange ring $R$ has stable range $1$ if and only if every regular element of $ R$ is unit-regular. Theorem $5$: If $R$ is a strongly $\pi$-regular ring with the property that all powers of every regular element are regular, then $ R$ has stable range $ 1$. The latter generalizes a recent result of Goodearl and Menal [$5$].
Convergence of diagonal Pad\'e approximants for functions analytic near $0$
D. S.
Lubinsky
3149-3157
Abstract: For functions analytic in a neighbourhood of 0, we show that at least for a subsequence of the diagonal Padé approximants, the point 0 attracts a zero proportion of the poles. The same is true for every "sufficiently dense" diagonal subsequence. Consequently these subsequences have a convergence in capacity type property, which is possibly the correct analogue of the Nuttall-Pommerenke theorem in this setting.
Left annihilators characterized by GPIs
Tsiu Kwen
Lee
3159-3165
Abstract: Let $R$ be a semiprime ring with extended centroid $C$, $U$ the right Utumi quotient ring of $ R$, $S$ a subring of $U$ containing $R$ and ${\rho _1}$, ${\rho _2}$ two right ideals of $R$. In the paper we show that $ {l_S}({\rho _1}) = {l_S}({\rho _2})$ if and only if ${\rho _1}$ and ${\rho _2}$ satisfy the same generalized polynomial identities (GPIs) with coefficients in $ SC$, where ${l_S}({\rho _i})$ denotes the left annihilator of ${\rho _i}$ in $S$. As a consequence of the result, if $ \rho$ is a right ideal of $ R$ such that ${l_R}(\rho ) = 0$, then $\rho$ and $U$ satisfy the same GPIs with coefficients in the two-sided Utumi quotient ring of $R$.
Construction of homomorphisms of $M$-continuous lattices
Xiao Quan
Xu
3167-3175
Abstract: We present a direct approach to constructing homomorphisms of $ M$-continuous lattices, a generalization of continuous lattices, into the unit interval, and show that an $M$-continuous lattice has sufficiently many homomorphisms into the unit interval to separate the points.