On the core of a cone-preserving map
Bit Shun
Tam;
Hans
Schneider
479-524
Abstract: This is the third of a sequence of papers in an attempt to study the Perron-Frobenius theory of a nonnegative matrix and its generalizations from the cone-theoretic viewpoint. Our main object of interest here is the core of a cone-preserving map. If A is an $n \times n$ real matrix which leaves invariant a proper cone K in $ {\mathbb{R}^n}$, then by the core of A relative to K, denoted by $ {\text{core}}_K(A)$, we mean the convex cone $\bigcap\nolimits_{i = 1}^\infty {{A^i}K}$. It is shown that when $ {\text{core}}_K(A)$ is polyhedral, which is the case whenever K is, then $ {\text{core}}_K(A)$ is generated by the distinguished eigenvectors of positive powers of A. The important concept of a distinguished A-invariant face is introduced, which corresponds to the concept of a distinguished class in the nonnegative matrix case. We prove a significant theorem which describes a one-to-one correspondence between the distinguished A-invariant faces of K and the cycles of the permutation induced by A on the extreme rays of ${\text{core}}_K(A)$, provided that the latter cone is nonzero, simplicial. By an interplay between cone-theoretic and graph-theoretic ideas, the extreme rays of the core of a nonnegative matrix are fully described. Characterizations of K-irreducibility or A-primitivity of A are also found in terms of $ {\text{core}}_K(A)$. Several equivalent conditions are also given on a matrix with an invariant proper cone so that its spectral radius is an eigenvalue of index one. An equivalent condition in terms of the peripheral spectrum is also found on a real matrix A with the Perron-Schaefer condition for which there exists a proper invariant cone K suchthat $ {\text{core}}_K(A)$ is polyhedral, simplicial, or a single ray. A method of producing a large class of invariant proper cones for a matrix with the Perron-Schaefer condition is also offered.
Minimal models and degenerations of surfaces with Kodaira number zero
Bruce
Crauder;
David R.
Morrison
525-558
Abstract: For Ueno's degenerations of abelian surfaces and for analytic flowerpot degenerations of surfaces with Kodaira number zero, we find analytically $ \mathbb{Q}$-factorial terminal minimal models. All contractions, singularities and flips are given by explicit geometric constructions.
A maximally pathological Brouwer homeomorphism
Edward Warwick
Daw
559-573
Abstract: This paper constructs a Brouwer homeomorphism which does not act properly discontinuously on any nonempty invariant closed connected set in $ {\mathbb{R}^2}$ .
Minimal displacement of points under holomorphic mappings and fixed point properties for unions of convex sets
Tadeusz
Kuczumow;
Simeon
Reich;
Adam
Stachura
575-586
Abstract: Let D be an open convex bounded subset of a complex Banach space $ (X,\left\Vert \cdot \right\Vert)$, and let C be the union of a finite number of closed convex sets lying strictly inside D. Using the Kuratowski measure of noncompactness with respect to the Kobayashi distance in D, we first show that if $f:D \to D$ is a holomorphic mapping which leaves C invariant, and if the Lefschetz number $\lambda ({f_{\vert C}}) \ne 0$, then $\inf \{ \left\Vert {x - f(x)} \right\Vert:x \in C\} = 0$. We then deduce several new fixed point theorems for holomorphic and nonexpansive mappings.
The orders of solutions of the Kummer system of congruences
Ladislav
Skula
587-607
Abstract: A new method concerning solutions of the Kummer system of congruences (K) (modulo an odd prime l) is developed. This method is based on the notion of the Stickelberger ideal. By means of this method a new proof of Pollaczek's and Morishima's assertion on solutions of (K) of orders 3, 6 and 4 $\bmod\; l$ is given. It is also shown that in case there is a solution of $(K)\, \not\equiv \, 0, \pm 1\;\pmod l$, then for the index of irregularity $i(l)$ of the prime l we have $i(l) \geq [\sqrt[3]{{l/2}}]$.
Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis
Boris
Mordukhovich
609-657
Abstract: In this paper we develop a stability theory for broad classes of parametric generalized equations and variational inequalities in finite dimensions. These objects have a wide range of applications in optimization, nonlinear analysis, mathematical economics, etc. Our main concern is Lipschitzian stability of multivalued solution maps depending on parameters. We employ a new approach of nonsmooth analysis based on the generalized differentiation of multivalued and nonsmooth operators. This approach allows us to obtain effective sufficient conditions as well as necessary and sufficient conditions for a natural Lipschitzian behavior of solution maps. In particular, we prove new criteria for the existence of Lipschitzian multivalued and single-valued implicit functions.
Fractal properties of invariant subsets for piecewise monotonic maps on the interval
Franz
Hofbauer;
Mariusz
Urbański
659-673
Abstract: Let T be a piecewise monotonic transformation on [0, 1] and let A be a T-invariant subset, which has positive topological entropy and satisfies the Darboux property. A general existence theorem for conformal measures on A is proved. This is then used to show equality of the dynamical dimension of A and the minimal zero of a certain pressure function.
Weak solutions of hyperbolic-parabolic Volterra equations
Gustaf
Gripenberg
675-694
Abstract: The existence of a global weak solution, satisfying certain a priori ${L^\infty }$-bounds, of the equation ${u_t}(t,x) = \int _0^tk(t - s){(\sigma ({u_x}))_x}(s,x)ds + f(t,x)$ is established. The kernel k is locally integrable and log-convex, and $\sigma \prime$ has only one local minimum which is positive.
On realization of Bj\"orner's ``continuous partition lattice'' by measurable partitions
Mark D.
Haiman
695-711
Abstract: Björner [1] showed how a construction by von Neumann of examples of continuous geometries can be adapted to construct a continuous analogue of finite partition lattices. Björner's construction realizes the continuous partition lattice abstractly, as a completion of a direct limit of finite lattices. Here we give an alternative construction realizing a continuous partition lattice concretely as a lattice of measurable partitions. This new lattice contains the Björner lattice and shares its key properties. Furthermore its automorphism group is the full automorphism group $ \pmod 0$ of the unit interval with Lebesgue measure, whereas, as we show, the Björner lattice possesses only a proper subgroup of these automorphisms.
Le th\'eor\`eme de Fermat-Goss
Laurent
Denis
713-726
Abstract: The analogue of the Fermat equation and of the Fermat conjecture is studied by Goss [G], on the rational function fields in characteristic $p > 0$. We prove here that this equation has no nontrivial solutions. When $q = 2$, the method uses the canonical height on the t-module constructed in [D]. This method also gives finiteness theorems for some generalization of the Fermat equation in higher dimension.
Powers of ideals generated by quadratic sequences
K.
Raghavan
727-747
Abstract: Huneke's conjecture that weak d-sequences generate ideals of quadratic type is proved. The proof suggests the definition of quadratic sequences, which are more general than weak d-sequences yet simpler to define and handle, in addition to being just as useful. We extend the theory of d-sequences and weak d-sequences to quadratic sequences. Results of Costa on sequences of linear type are generalized. An example of a two-dimensional local domain in which every system of parameters is a d-sequence in some order but which nevertheless fails to be Buchsbaum is given. A criterion is established for when equality holds in Burch's inequality for an ideal generated by a quadratic sequence.
On a two-dimensional elliptic problem with large exponent in nonlinearity
Xiaofeng
Ren;
Juncheng
Wei
749-763
Abstract: A semilinear elliptic equation on a bounded domain in ${R^2}$ with large exponent in the nonlinear term is studied in this paper. We investigate positive solutions obtained by the variational method. It turns put that the constrained minimizing problem possesses nice asymptotic behavior as the nonlinear exponent, serving as a parameter, gets large. We shall prove that $ {c_p}$, the minimum of energy functional with the nonlinear exponent equal to p, is like ${(8\pi e)^{1/2}}{p^{ - 1/2}}$ as p tends to infinity. Using this result, we shall prove that the variational solutions remain bounded uniformly in p. As p tends to infinity, the solutions develop one or two peaks. Precisely the solutions approach zero except at one or two points where they stay away from zero and bounded from above. Then we consider the problem on a special class of domains. It turns out that the solutions then develop only one peak. For these domains, the solutions enlarged by a suitable quantity behave like a Green's function of $- \Delta$. In this case we shall also prove that the peaks must appear at a critical point of the Robin function of the domain.
Hodge decompositions and Dolbeault complexes on normal surfaces
Jeffrey
Fox;
Peter
Haskell
765-778
Abstract: Give the smooth subset of a normal singular complex projective surface the metric induced from the ambient projective space. The ${L^2}$ cohomology of this incomplete manifold is isomorphic to the surface's intersection cohomology, which has a natural Hodge decomposition. This paper identifies Dolbeault complexes whose $\bar \partial$-closed and $\bar \partial$-coclosed forms represent the classes of pure type in the corresponding Hodge decomposition of ${L^2}$ cohomology.
Flows in fibers
Jun-ichi
Tanaka
779-804
Abstract: Let ${H^\infty }(\Delta )$ be the algebra of all bounded analytic functions on the open unit disc $ \Delta$, and let $ \mathfrak{M}({H^\infty }(\Delta ))$ be the maximal ideal space of ${H^\infty }(\Delta )$. Using a flow, we represent a reasonable portion of a fiber in $ \mathfrak{M}({H^\infty }(\Delta ))$. This indicates a relation between the corona theorem and the individual ergodic theorem. As an application, we answer a question of Forelli [3] by showing that there exists a minimal flow on which the induced uniform algebra is not a Dirichlet algebra. The proof rests on the fact that the closure of a nonhomeomorphic part in $\mathfrak{M}({H^\infty }(\Delta ))$ may contain a homeomorphic copy of $\mathfrak{M}({H^\infty }(\Delta ))$. Taking suitable factors, we may construct a lot of minimal flows which are not strictly ergodic.
Geometry of $2$-step nilpotent groups with a left invariant metric. II
Patrick
Eberlein
805-828
Abstract: We obtain a partial description of the totally geodesic submanifolds of a 2-step, simply connected nilpotent Lie group with a left invariant metric. We consider only the case that N is nonsingular; that is, $ {\text{ad}}\xi :\mathcal{N} \to \mathcal{Z}$ is surjective for all elements $\xi \in \mathcal{N} - \mathcal{Z}$, where $\mathcal{N}$ denotes the Lie algebra of N and $\mathcal{Z}$ denotes the center of $\mathcal{N}$. Among other results we show that if H is a totally geodesic submanifold of N with $ \dim H \geq 1 + \dim \mathcal{Z}$, then H is an open subset of $ g{N^\ast}$, where g is an element of H and ${N^\ast}$ is a totally geodesic subgroup of N. We find simple and useful criteria that are necessary and sufficient for a subalgebra ${\mathcal{N}^\ast}$ of $ \mathcal{N}$ to be the Lie algebra of a totally geodesic subgroup $ {N^\ast}$. We define and study the properties of a Gauss map of a totally geodesic submanifold H of N. We conclude with a characterization of 2-step nilpotent Lie groups N of Heisenberg type in terms of the abundance of totally geodesic submanifolds of N.
Entropy solutions for diffusion-convection equations with partial diffusivity
M.
Escobedo;
J. L.
Vázquez;
Enrike
Zuazua
829-842
Abstract: We consider the Cauchy problem for the following scalar conservation law with partial viscosity $\displaystyle {u_t} = {\Delta _x}u + {\partial _y}(f(u)),\quad (x,y) \in {{\mathbf{R}}^N},t > 0.$ The existence of solutions is proved by the vanishing viscosity method. By introducing a suitable entropy condition we prove uniqueness of solutions. This entropy condition is inspired by the entropy criterion introduced by Kruzhkov for hyperbolic conservation laws but it takes into account the effect of diffusion.
Escape rate for $2$-dimensional Brownian motion conditioned to be transient with application to Zygmund functions
Elizabeth Ann
Housworth
843-852
Abstract: The escape rate of a 2-dimensional Brownian motion conditioned to be transient is determined to be $P\{ X(t) < f(t)$ i.o. as $t \uparrow \infty \} = 0$ or 1 according as $\sum\nolimits_{n = 1}^\infty {{e^{ - n}}\log f({e^{{e^n}}}) < }$ or $= \infty$. The result is used to construct a complex-valued Zygmund function--as a lacunary series--whose graph does not have $\sigma$-finite linear Hausdorff measure. This contrasts the result of Mauldin and Williams that the graphs of all real-valued Zygmund functions have $ \sigma$-finite linear Hausdorff measure.
Quasiconformal mappings and Ahlfors-David curves
Paul
MacManus
853-881
Abstract: We show that if $ \rho$ is a quasiconformal mapping of the plane whose dilatation satisfies a certain quadratic Carleson measure condition relative to an Ahlfors-David curve $\Gamma$ then $\rho$ is differentiable almost everywhere on $ \Gamma$ and $ \log \vert\rho \prime \vert \in {\text{BMO}}$. When $\Gamma$ is chord-arc we show that its image is a Bishop-Jones curve. If the Carleson norm is small then we show that $\rho$ is absolutely continuous on $\Gamma$, the image of $\Gamma$ is an Ahlfors-David curve, and $\rho \prime = {e^a}$, where $a \in {\text{BMO}}$ with a small norm.
Une dichotomie de Hopf pour les flots g\'eod\'esiques associ\'es aux groupes discrets d'isom\'etries des arbres
M.
Coornaert;
A.
Papadopoulos
883-898
Abstract: Let X be a complete locally compact metric tree and $ \Gamma$ a group of isometries of X acting properly on this space. The space of bi-infinite geodesics in X constitutes a space GX on which $\Gamma$ acts properly. Let $\Omega$ be the quotient of GX by this action. The geodesic flow associated to $ \Gamma$ is the flow on $ \Omega$ which is the quotient of the geodesic flow on GX, defined by the time-shift on geodesics. To any $\Gamma $-conformal measure on the boundary $ \partial X$ there is an associated measure m on $\Omega$ which is invariant by the geodesic flow. We prove the following results: The geodesic flow on $(\Omega ,m)$ is either conservative or dissipative. If it is conservative, then it is ergodic, If it is dissipative, then it is not ergodic unless it is measurably conjugate to the action of $\mathbb{R}$ on itself by conjugation. We prove also a dichotomy in terms of the conical limit set $ {\Lambda _c} \subset \partial X$ of $\Gamma$: the flow on $ (\Omega ,m)$ is conservative if and only if $\mu ({\Lambda _c}) = \mu (\partial X)$, and it is dissipative if and only if $\mu ({\Lambda _c}) = 0$. The results are analogous to results of E. Hopf and D. Sullivan in the case of Riemannian manifolds of constant negative curvature.
On the force between rotating co-axial black holes
Gilbert
Weinstein
899-906
Abstract: We study the force between rotating coaxial black holes, as it was defined in [9 and 10]. We show that under a certain limit, the force is attractive, and in fact tends to infinity. This lends support to the conjecture that the force is always positive.
Radially symmetric solutions to a Dirichlet problem involving critical exponents
Alfonso
Castro;
Alexandra
Kurepa
907-926
Abstract: In this paper we answer, for $N = 3,4$, the question raised in [1] on the number of radially symmetric solutions to the boundary value problem $- \Delta u(x) = \lambda u(x) + u(x)\vert u(x){\vert^{4/(N - 2)}}$, $x \in B: = \{ x \in {R^N}:\left\Vert x \right\Vert < 1\}$, $u(x) = 0$, $x \in \partial B$, where $ \Delta$ is the Laplacean operator and $ \lambda > 0$. Indeed, we prove that if $N = 3,4$, then for any $ \lambda > 0$ this problem has only finitely many radial solutions. For $ N = 3,4,5$ we show that, for each $\lambda > 0$, the set of radially symmetric solutions is bounded. Moreover, we establish geometric properties of the branches of solutions bifurcating from zero and from infinity.
Holomorphic motions and Teichm\"uller spaces
C. J.
Earle;
I.
Kra;
S. L.
KrushkalЬ
927-948
Abstract: We prove an equivariant form of Slodkowski's theorem that every holomorphic motion of a subset of the extended complex plane $\hat{\mathbb{C}}$ extends to a holomorphic motion of $ \widehat{\mathbb{C}}$. As a consequence we prove that every holomorphic map of the unit disc into Teichmüller space lifts to a holomorphic map into the space of Beltrami forms. We use this lifting theorem to study the Teichmüller metric.