Groups of piecewise linear homeomorphisms
Melanie
Stein
477-514
Abstract: In this paper we study a class of groups which may be described as groups of piecewise linear bijections of a circle or of compact intervals of the real line. We use the action of these groups on simplicial complexes to obtain homological and combinatorial information about them. We also identify large simple subgroups in all of them, providing examples of finitely presented infinite simple groups.
The Gauss map for K\"ahlerian submanifolds of ${\bf R}\sp n$
Marco
Rigoli;
Renato
Tribuzy
515-528
Abstract: We introduce a Gauss map for Kähler submanifolds of Euclidean space and study its geometry in relation to that of the given immersion. In particular we generalize a number of results of the classical theory of minimal surfaces in Euclidean space.
Hessenberg varieties
F.
De Mari;
C.
Procesi;
M. A.
Shayman
529-534
Abstract: Numerical algorithms involving Hessenberg matrices correspond to dynamical systems which evolve on the subvariety of complete flags ${S_1} \subset {S_2} \subset \cdots \subset {S_{n - 1}}$ in $ {\mathbb{C}^n}$ satisfying the condition $s({S_i}) \subset {S_{i + 1}}$, $\forall i$, where $s$ is an endomorphism of ${\mathbb{C}^n}$. This paper describes the basic topological features of the generalization to subvarieties of $G/B$, $G$ a complex semisimple algebraic group, which are indexed by certain subsets of negative roots. In the special case where the subset consists of the negative simple roots, the variety coincides with the torus embedding associated to the decomposition into Weyl chambers.
The structure of solutions of a semilinear elliptic equation
Kuo-Shung
Cheng;
Tai Chia
Lin
535-554
Abstract: We give a complete classification of solutions of the elliptic equation $ \Delta u + K(x){e^{2u}} = 0$ in $ \mathbb{R}^n, n \geq 3$, for some interesting cases of $K$.
Automorphisms of the lattice of recursively enumerable sets: promptly simple sets
P.
Cholak;
R.
Downey;
M.
Stob
555-570
Abstract: We show that for every coinfinite r.e. set $A$ there is a complete r.e. set $B$ such that ${\mathcal{L}^{\ast} }(A){ \approx _{{\text{eff}}}}{\mathcal{L}^{\ast} }(B)$ and that every promptly simple set is automorphic (in ${\mathcal{E}^{\ast} }$) to a complete set.
$N$-body observables in the Calkin algebra
Jan
Dereziński
571-582
Abstract: The commutators of many operators which are used in the phase space analysis of the $N$-body scattering are compact. This fact makes it possible to give a description of certain classes of such operators in terms of commutative ${C^{\ast} }$-algebras inside the Calkin algebra.
Admissible boundary values of bounded holomorphic functions in wedges
Franc
Forstnerič
583-593
Abstract: If $M \subset {\mathbb{C}^N}$ is a generic Cauchy-Riemann manifold and $\mathcal{W} \subset {\mathbb{C}^N}$ is a wedge domain with edge $M$, then every bounded holomorphic function on $\mathcal{W}$ has an admissible limit at almost every point of $M$. Moreover, if a holomorphic function $ f$ on $\mathcal{W}$ has a distribution boundary value $ (\operatorname{bv}\;f)$ on $ M$ that is a bounded measurable function, then $f$ is bounded on every finer wedge near $ M$ , and its admissible limit equals $ (\operatorname{bv}\;f)(p)$ at almost every point $p \in M$.
Isometries of CSL algebras
Baruch
Solel
595-606
Abstract: We show that every Jordan isomorphism of CSL algebras, whose restriction to the diagonal of the algebra is a selfadjoint map, is the sum of an isomorphism and an anti-isomorphism. It follows that every surjective linear isometry of CSL algebras is the sum of an isomorphism and an anti-isomorphism, followed by a unitary multiplication.
Harmonic maps into hyperbolic $3$-manifolds
Yair N.
Minsky
607-632
Abstract: High-energy degeneration of harmonic maps of Riemann surfaces into a hyperbolic $3$-manifold target is studied, generalizing results of [M1] in which the target is two-dimensional. The Hopf foliation of a high-energy map is mapped to an approximation of its geodesic representative in the target, and the ratio of the squared length of that representative to the extremal length of the foliation in the domain gives an estimate for the energy. The images of harmonic maps obtained when the domain degenerates along a Teichmüller ray are shown to converge generically to pleated surfaces in the geometric topology or to leave every compact set of the target when the limiting foliation is unrealizable.
Automorphisms of torsion-free nilpotent groups of class two
Manfred
Dugas;
Rüdiger
Göbel
633-646
Abstract: We construct $ 2$-divisible, torsion-free abelian groups $G$ admitting an alternating bilinear map. We use these groups $G$ to find nilpotent groups $N$ of class $2$ such that $ \operatorname{Aut}(N)$ modulo a natural normal subgroup is a prescribed group.
Finite semilattices whose monoids of endomorphisms are regular
M. E.
Adams;
Matthew
Gould
647-665
Abstract: A classification is obtained for the finite semilattices $S$ such that the monoid of endomorphisms of $S$ is regular in the semigroup-theoretic sense.
Homological theory of idempotent ideals
M.
Auslander;
M. I.
Platzeck;
G.
Todorov
667-692
Abstract: Let $\Lambda$ be an artin algebra $\mathfrak{A}$ and a two-sided ideal of $ \Lambda$. Then $\mathfrak{A}$ is the trace of a projective $ \Lambda$-module $ P$ in $\Lambda$. We study how the homological properties of the categories of finitely generated modules over the three rings $\Lambda /\mathfrak{A}$, $\Lambda$ and the endomorphism ring of $ P$ are related. We give some applications of the ideas developed in the paper to the study of quasi-hereditary algebras.
Classification of finite-dimensional universal pseudo-boundaries and pseudo-interiors
J. J.
Dijkstra;
J.
van Mill;
J.
Mogilski
693-709
Abstract: Let $n$ and $k$ be fixed integers such that $n \geq 1$ and $0 \leq k \leq n$. Let $B_k^n$ and $s_k^n$ denote the $k$-dimensional universal pseudo-boundary and the $ k$-dimensional universal pseudo-interior in $ {{\mathbf{R}}^n}$, respectively. The aim of this paper is to prove that $ B_k^n$ is homeomorphic to $ B_k^m$ if and only if $ s_k^n$ is homeomorphic to $ s_k^m$ if and only if $ n = m$ or $n$, $ m \geq 2k + 1$.
Accessible points of hereditarily decomposable chainable continua
Piotr
Minc;
W. R. R.
Transue
711-727
Abstract: In this paper it is proven that a chainable continuum $X$ can be embedded in the plane in such a way that every point is accessible from its complement if and only if it is Suslinean. An example is shown of an hereditarily decomposable chainable continuum which cannot be embedded in the plane in such a way that each endpoint is accessible.
Expansions of chromatic polynomials and log-concavity
Francesco
Brenti
729-756
Abstract: In this paper we present several results and open problems about logconcavity properties of sequences associated with graph colorings. Five polynomials intimately related to the chromatic polynomial of a graph are introduced and their zeros, combinatorial and log-concavity properties are studied. Four of these polynomials have never been considered before in the literature and some yield new expansions for the chromatic polynomial.
Quasidiagonality of direct sums of weighted shifts
Sivaram K.
Narayan
757-774
Abstract: Let $\mathcal{H}$ be a separable Hilbert space. A bounded linear operator $A$ defined on $ \mathcal{H}$ is said to be quasidiagonal if there exists a sequence $\{ {P_n}\}$ of projections of finite rank such that ${P_n} \to I$ strongly and $\left\Vert A{P_n} - {P_n}A\right\Vert \to 0$ as $n \to \infty $. We give a necessary and sufficient condition for a finite direct sum of weighted shifts to be quasidiagonal. The condition is stated using a marked graph (a graph with a $ (0)$, $( + )$ or $( - )$ attached to its vertices) that can be associated with the direct sum.
Existence of positive nonradial solutions for nonlinear elliptic equations in annular domains
Song-Sun
Lin
775-791
Abstract: We study the existence of positive nonradial solutions of equation $\Delta u + f(u) = 0$ in ${\Omega _a}$, $u = 0$ on $ \partial {\Omega _a}$, where ${\Omega _a} = \{ x \in {\mathbb{R}^n}:a < \vert x\vert < 1\}$ is an annulus in ${\mathbb{R}^n}$, $n \geq 2$, and $f$ is positive and superlinear at both 0 and $ \infty$. We use a bifurcation method to show that there is a nonradial bifurcation with mode $k$ at $ {a_k} \in (0,1)$ for any positive integer $k$ if $f$ is subcritical and for large $k$ if $f$ is supercritical. When $f$ is subcritical, then a Nehari-type variational method can be used to prove that there exists $ {a^{\ast} } \in (0,1)$ such that for any $ a \in ({a^{\ast} },1)$, the equation has a nonradial solution on ${\Omega _a}$.
$L\sp p$ estimates for the X-ray transform restricted to line complexes of Kirillov type
Hann Tzong
Wang
793-821
Abstract: Let there be given a piecewise continuous rectifiable curve $ \phi :{\mathbf{R}} \to {{\mathbf{R}}^n}$. Let ${G_{1,n}}({M_{1,n}})$ be the usual Grassmannian (bundle) in $ {{\mathbf{R}}^n}$. Define an $n$-dimensional submanifold ${M_\phi }({{\mathbf{R}}^n})$ of $ {M_{1,n}}$ as the set of all copies of ${G_{1,n}}$ along the curve $\phi$. Following Kirillov, we know that a nice function $f(x)$ can be recovered from its X-ray transform ${R_{1,n}}f$ on ${M_\phi }({{\mathbf{R}}^n})$ if and only if the curve $\phi$ intersects almost every affine hyperplane. Define a measure on ${M_\phi }({{\mathbf{R}}^n})$ by $ d\mu = d{\mu _x}(\pi )d\lambda (x)$, where $d{\mu _x}$ is the probability measure on $ {M_{1,n}}$ carried by the set of lines passing through the point $x$ and invariant under the stabilizer of $x$ in $O(n)$ and $d\lambda$ is the usual measure on $\phi$. We show that, if $n > 2$ and $\phi$ is unbounded, then $\left\Vert {R_{1,n}}f\right\Vert _{{L^q}({M_\phi }({{\mathbf{R}}^n}),d\mu )} \leq C\left\Vert f\right\Vert _{{L^p}({{\mathbf{R}}^n})}$ if and only if $p = q = n - 1$ and $\phi$ is line-like, that is, $\lambda (\phi \cap B(0;R)) = O(R)$. This result gives a classification of Kirillov line complexes in terms of ${L^p}$ estimates.
Harmonic localization of algebraic $K$-theory spectra
Stephen A.
Mitchell
823-837
Abstract: The Lichtenbaum-Quillen conjectures hold for the harmonic localization of the $K$-theory spectrum of a nice scheme. Various consequences of this fact are explored; for example, the harmonic localization of the $K$-theory of the integers at a regular prime is explicitly identified.
Existence of smooth solutions to the classical moment problems
Palle E. T.
Jorgensen
839-848
Abstract: Let $s(0),s(1), \ldots$ be a given sequence, and define $ s(n) = \overline {s( - n)}$ for $n < 0$. If $\Sigma \Sigma {\overline \xi _n}{\xi _m}s(m - n) \geq 0$ holds for all finite sequences $ {({\xi _n})_{n \in \mathbb{Z}}}$, then it is known that there is a positive Borel measure $\mu$ on the circle $ \mathbb{T}$ such that $s(n) = \smallint_{ - \pi }^\pi {{e^{int}}d\mu (t)}$, and conversely. Our main theorem provides a necessary and sufficient condition on the sequence $(s(n))$ that the measure $\mu$ may be chosen to be smooth. A measure $\mu$ is said to be smooth if it has the same spectral type as the operator $id/dt$ acting on $ {L^2}(\mathbb{T})$ with respect to Haar measure $dt$ on $ \mathbb{T}$: Equivalently, $ \mu$ is a superposition (possibly infinite) of measures of the form $\vert w(t){\vert^2}dt$ with $w \in {L^2}(\mathbb{T})$ such that $dw/dt \in {L^2}(\mathbb{T})$. The condition is stated purely in terms of the initially given sequence $(s(n))$: We show that a smooth representation exists if and only if, for some $\varepsilon \in {\mathbb{R}_ + }$, the a priori estimate $\displaystyle \sum {\sum {s(m - n){{\overline \xi }_n}{\xi _m} \geq \varepsilon {{\left\vert {\sum {ns(n){\xi _n}} } \right\vert}^2}} } $ is valid for all finite double sequences $({\xi _n})$. An analogous result is proved for the determinate (Hamburger) moment problem on the line. But the corresponding result does not hold for the indeterminate moment problem.
Orbits in unimodular Hermitian lattices
Donald G.
James
849-860
Abstract: Let $L$ be a unimodular indefinite hermitian lattice over the integers $\mathfrak{o}$ of an algebraic number field, and $ N(L,c)$ the number of primitive representations of $c \in \mathfrak{o}$ by $L$ that are inequalivant modulo the action of the integral special unitary group $SU(L)$ on $L$. The value of $N(L,c)$ is determined from the local representations via a product formula.
An extension of Attouch's theorem and its application to second-order epi-differentiation of convexly composite functions
René A.
Poliquin
861-874
Abstract: In 1977, Hedy Attouch established that a sequence of (closed proper) convex functions epi-converges to a convex function if and only if the graphs of the subdifferentials converge (in the Mosco sense) to the subdifferential of the limiting function and (roughly speaking) there is a condition that fixes the constant of integration. We show that the theorem is valid if instead one considers functions that are the composition of a closed proper convex function with a twice continuously differentiable mapping (in addition a constraint qualification is imposed). Using Attouch's Theorem, Rockafellar showed that second-order epi-differentiation of a convex function and proto-differentiability of the subdifferential set-valued mapping are equivalent, moreover the subdifferential of one-half the second-order epi-derivative is the proto-derivative of the subdifferential mapping; we will extend this result to the convexly composite setting.
Spectral multiplicity for ${\rm Gl}\sb n({\bf R})$
Jonathan
Huntley
875-888
Abstract: We study the behavior of the cuspidal spectrum of $\Gamma \backslash \mathcal{H}$, where $\mathcal{H}$ is associated to $\operatorname{Gl}_n(R)$ and $\Gamma$ is cofinite but not compact. By a technique that modifies the Lax-Phillips technique and uses ideas from wave equation techniques, if $ r$ is the dimension of $\mathcal{H}$, $ {N_\alpha }(\lambda )$ is the counting function for the Laplacian attached to a Hilbert space $ {H_\alpha }$, ${M_\alpha }(\lambda )$ is the multiplicity function, and ${H_0}$ is the space of cusp forms, we obtain the following results: Theorem 1. There exists a space of functions ${H^1}$, containing all cusp forms, such that $\displaystyle N\prime(\lambda ) = {C_r}({\text{Vol}}\;X){\lambda ^{\frac{r} {2}... ...{\frac{{r - 1}} {2}}}{\lambda ^{\frac{1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).$ Theorem 2. $\displaystyle {M_0}(\lambda ) = O({\lambda ^{\frac{{r - 1}} {2}}}{\lambda ^{\frac{1} {{n + 1}}}}{(\log \lambda )^{n - 1}}).$
Nonsingular affine $k\sp *$-surfaces
Jean
Rynes
889-921
Abstract: Nonsingular affine ${k^{\ast} }$-surfaces are classified as certain invariant open subsets of projective ${k^{\ast}}$-surfaces. A graph is defined which is an equivariant isomorphism invariant of an affine ${k^{\ast}}$-surface. Over the complex numbers, it is proved that the only acyclic affine surface which admits an effective action of the group ${{\mathbf{C}}^{\ast} }$ is ${{\mathbf{C}}^2}$ which admits only linear actions of $ {{\mathbf{C}}^{\ast}}$.
Elementary proofs of the abstract prime number theorem for algebraic function fields
Wen-Bin
Zhang
923-937
Abstract: Elementary proofs of the abstract prime number theorem of the form $ \Lambda (m) = {q^m} + O({q^m}{m^{ - 1}})$ for algebraic function fields are given. The proofs use a refinement of a tauberian theorem of Bombieri.