On quasiconvexity and relatively hyperbolic structures on groups

Let G be a group which is hyperbolic relative to a collection of subgroups H₁{\mathcal{H}_1}, and it is also hyperbolic relative to a collection of subgroups H₂{\mathcal{H}_2}. Suppose that H₁ ì H₂{\mathcal{H}_1 \subset \mathcal{H}_2}. We characterize when a relative quasiconvex subgroup of (G, H₂){(G, \mathcal H_2)} is still relatively quasiconvex in (G, H₁){(G, \mathcal H_1)}. We also show that relative quasiconvexity is preserved when passing from (G, H₁){(G, \mathcal H_1)} to (G, H₂){(G, \mathcal H_2)}. Applications are discussed.     

Counting Ω-ideals

Let ${\mathbb{A}}Let \mathbbA{\mathbb{A}} be a universal algebra of signature Ω, and let I{\mathcal{I}} be an ideal in the Boolean algebra P_\mathbbA{\mathcal{P}_{\mathbb{A}}} of all subsets of \mathbbA{\mathbb{A}} . We say that I{\mathcal{I}} is an Ω-ideal if I{\mathcal{I}} contains all finite subsets of \mathbbA{\mathbb{A}} and f(Aⁿ) ? I{f(A^{n}) \in \mathcal{I}} for every n-ary operation f ? W{f \in \Omega} and every A ? I{A \in \mathcal{I}} . We prove that there are 2^2à₀{2^{2^{\aleph_0}}} Ω-ideals in P_\mathbbA{\mathcal{P}_{\mathbb{A}}} provided that \mathbbA{\mathbb{A}} is countably infinite and Ω is countable.     

On the Right (Left) Invertible Completions for Operator Matrices

Let ${\mathcal {H}_{1}}Let H₁{\mathcal {H}_{1}} and H₂{\mathcal {H}_{2}} be separable Hilbert spaces, and let A ? B(H₁), B ? B(H₂){A \in \mathcal {B}(\mathcal {H}_{1}),\, B \in \mathcal {B}(\mathcal {H}_{2})} and C ? B(H₂, H₁){C \in \mathcal {B}(\mathcal {H}_{2},\, \mathcal {H}_{1})} be given operators. A necessary and sufficient condition is given for ${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)}${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)} to be a right (left) invertible operator for some X ? B(H₁, H₂){X \in \mathcal {B}(\mathcal {H}_{1},\, \mathcal {H}_{2})}. Furthermore, some related results are obtained.     

Stability and Instability of Extreme Reissner�CNordstr?m Black Hole Spacetimes for Linear Scalar Perturbations II

This paper contains the second part of a two-part series on the stability and instability of extreme Reissner–Nordstr?m spacetimes for linear scalar perturbations. We continue our study of solutions to the linear wave equation \square_gy = 0{\square_{g}\psi=0} on a suitable globally hyperbolic subset of such a spacetime, arising from regular initial data prescribed on a Cauchy hypersurface Σ₀ crossing the future event horizon H⁺{\mathcal{H}^{+}}. We here obtain definitive energy and pointwise decay, non-decay and blow-up results. Our estimates hold up to and including the horizon H⁺{\mathcal{H}^{+}}. A hierarchy of conservations laws on degenerate horizons is also derived.     

Periodicity of Non-Central Integral Arrangements Modulo Positive Integers

An integral coefficient matrix determines an integral arrangement of hyperplanes in \mathbbR^m{\mathbb{R}^m} . After modulo q reduction ${(q \in {\mathbb{Z}_{ >0 }})}${(q \in {\mathbb{Z}_{ >0 }})} , the same matrix determines an arrangement A_q{\mathcal{A}_q} of “hyperplanes” in \mathbbZ^m_q{\mathbb{Z}^m_q} . In the special case of central arrangements, Kamiya, Takemura, and Terao [J. Algebraic Combin. 27(3), 317–330 (2008)] showed that the cardinality of the complement of A_q{\mathcal{A}_q} in \mathbbZ^m_q{\mathbb{Z}^m_q} is a quasi-polynomial in ${q \in {\mathbb{Z}_{ >0 }}}${q \in {\mathbb{Z}_{ >0 }}} . Moreover, they proved in the central case that the intersection lattice of A_q{\mathcal{A}_q} is periodic from some q on. The present paper generalizes these results to the case of non-central arrangements. The paper also studies the arrangement [^(B)]_m^[0,a]{\hat{\mathcal{B}}_m^{[0,a]}} of Athanasiadis [J. Algebraic Combin. 10(3), 207–225 (1999)] to illustrate our results.     

Ergodic Subequivalence Relations Induced by a Bernoulli Action

Let Γ be a countable group and denote by S{\mathcal{S}} the equivalence relation induced by the Bernoulli action G\curvearrowright [0, 1]^G{\Gamma\curvearrowright [0, 1]^{\Gamma}}, where [0, 1]^Γ is endowed with the product Lebesgue measure. We prove that, for any subequivalence relation R{\mathcal{R}} of S{\mathcal{S}}, there exists a partition {X _i } _i≥0 of [0, 1]^Γ into R{\mathcal{R}}-invariant measurable sets such that R_|X0{\mathcal{R}_{\vert X_{0}}} is hyperfinite and R_|Xi{\mathcal{R}_{\vert X_{i}}} is strongly ergodic (hence ergodic and non-hyperfinite), for every i ≥ 1.     

Topology and Smooth Structure for Pseudoframes

Given a closed subspace ${\mathcal{S}}Given a closed subspace S{\mathcal{S}} of a Hilbert space H{\mathcal{H}}, we study the sets F_S{\mathcal{F}_\mathcal{S}} of pseudo-frames, CF_S{\mathcal{C}\mathcal{F}_\mathcal{S}} of commutative pseudo-frames and \mathfrakX_S{\tiny{\mathfrak{X}}_{\mathcal{S}}} of dual frames for S{\mathcal{S}}, via the (well known) one to one correspondence which assigns a pair of operators (F, H) to a frame pair ({f_n}_{n ? \mathbbN},{h_n}_{n ? \mathbbN}){(\{f_n\}_{n\in\mathbb{N}},\{h_n\}_{n\in\mathbb{N}})},

F:l2? H,     F({cn}n ? \mathbbN )=?n cn fn,F:\ell^2\to\,\mathcal{H}, \quad F\left(\{c_n\}_{n\in\mathbb{N}} \right)=\sum_n c_n f_n,      8. The Ground State and the Long-Time Evolution in the CMC Einstein Flow      Martin Reiris 《Annales Henri Poincare》2010,10(8):1559-1604 Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)3Volg(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by Vinf=(\frac-16Y(S))\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g i , K i )} satisfying: (i) k i  = −3, (ii) Viˉ Vinf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q 0((g i , K i )) ≤ Λ, where Q 0 is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)2g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E1)\tilde]=E1(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E1=Q0+Q1{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ Vinf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.      9. On Hankel Nonnegative Definite Sequences, the Canonical Hankel Parametrization, and Orthogonal Matrix Polynomials      Bernd Fritzsche  Bernd Kirstein  Conrad M?dler 《Complex Analysis and Operator Theory》2011,5(2):447-511 This paper continues recent investigations started in Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) into the structure of the set Hq,2n 3 {\mathcal{H}_{q,2n}^{\ge}} of all Hankel nonnegative definite sequences, (sj)j=02n{(s_{j})_{j=0}^{2n}}, of complex q × q matrices and its important subclasses Hq,2n 3 ,e{\mathcal{H}_{q,2n}^{\ge,{\rm e}}} and ${\mathcal{H}_{q,2n}^>}${\mathcal{H}_{q,2n}^>} of all Hankel nonnegative definite extendable sequences and of all Hankel positive definite sequences, respectively. These classes of sequences arise quite naturally in the framework of matrix versions of the truncated Hamburger moment problem. In Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) a canonical Hankel parametrization [(Ck)k=1n, (Dk)k=0n]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} consisting of two sequences of complex q × q matrices was associated with an arbitrary sequence (sj)j=02n{(s_{j})_{j=0}^{2n}} of complex q × q matrices. The sequences belonging to each of the classes Hq,2n 3 , Hq,2n 3 ,e{\mathcal{H}_{q,2n}^{\ge}, \mathcal{H}_{q,2n}^{\ge,{\rm e}}}, and ${\mathcal{H}_{q,2n}^>}${\mathcal{H}_{q,2n}^>} were characterized in terms of their canonical Hankel parametrization (see, Dyukarev et al. in Complex anal oper theory 3(4):759–834, 2009; Proposition 2.30). In this paper, we will study further aspects of the canonical Hankel parametrization. Using the canonical Hankel parametrization [(Ck)k=1n, (Dk)k=0n]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} of a sequence (sj)j=02n ? Hq,2n 3 {(s_{j})_{j=0}^{2n} \in \mathcal{H}_{q,2n}^{\ge}}, we give a recursive construction of a monic right (resp. left) orthogonal system of matrix polynomials with respect to (sj)j=02n{(s_{j})_{j=0}^{2n}} (see Theorem 5.5). The matrices [(Ck)k=1n, (Dk)k=0n]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} will be expressed in terms of an arbitrary monic right (resp. left) orthogonal system with respect to (sj)j=02n{(s_{j})_{j=0}^{2n}} (see Theorem 5.11). This result will be reformulated in terms of nonnegative Hermitian Borel measures on \mathbbR{\mathbb{R}}. In this way, integral representations for the matrices [(Ck)k=1n, (Dk)k=0n]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} will be obtained (see Theorem 6.9). Starting from the monic orthogonal polynomials with respect to some classical probability distributions on \mathbbR{\mathbb{R}}, Theorem 6.9 is used to compute the canonical Hankel parametrization of their moment sequences. Moreover, we discuss important number sequences from enumerative combinatorics using the canonical Hankel parametrization.      10. Around splitting and reaping for partitions of ω      Hiroaki Minami 《Archive for Mathematical Logic》2010,49(4):501-518 We investigate splitting number and reaping number for the structure (ω) ω of infinite partitions of ω. We prove that \mathfrakrd ￡ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and \mathfraksd 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency \mathfrakrd < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and \mathfraksd < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants \mathfrakrpair{\mathfrak{r}_{pair}} and \mathfrakspair{\mathfrak{s}_{pair}} . We also study the relation between \mathfrakrpair, \mathfrakspair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that cov(M),cov(N) ￡ \mathfrakrpair ￡ \mathfraksd,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and \mathfraks ￡ \mathfrakspair ￡ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} .      11. The Asymptotic Schottky Problem      Lizhen Ji  Enrico Leuzinger 《Geometric And Functional Analysis》2010,19(6):1693-1712 Let ${\mathcal{M}_g}Let Mg{\mathcal{M}_g} denote the moduli space of compact Riemann surfaces of genus g and let Ag{\mathcal{A}_g} be the moduli space of principally polarized abelian varieties of dimension g. Let J : Mg ? Ag{J : \mathcal{M}_g \rightarrow \mathcal{A}_g} be the map which associates to a Riemann surface its Jacobian. The map J is injective, and the image Jg : = J(Mg){\mathcal{J}_g := J(\mathcal{M}_g)} is contained in a proper subvariety of Ag{\mathcal{A}_g} when g ≥  4. The classical and long-studied Schottky problem is to characterize the Jacobian locus Jg{\mathcal{J}_g} in Ag{\mathcal{A}_g}. In this paper we address a large scale version of this problem posed by Farb and called the coarse Schottky problem: What does Jg{\mathcal{J}_g} look like “from far away”, or how “dense” is Jg{\mathcal{J}_g} in the sense of coarse geometry? The large scale geometry of Ag{\mathcal{A}_g} is encoded in its asymptotic cone, Cone￥(Ag){{\rm Cone}_\infty(\mathcal{A}_g)}, which is a Euclidean simplicial cone of real dimension g. Our main result asserts that the Jacobian locus Jg{\mathcal{J}_g} is “coarsely dense” in Ag{\mathcal{A}_g}, which implies that the subset of Cone￥(Ag){{\rm Cone}_\infty(\mathcal{A}_g)} determined by Jg{\mathcal{J}_g} actually coincides with this cone. The proof shows that the Jacobian locus of hyperelliptic curves is coarsely dense in Ag{\mathcal{A}_g} as well. We also study the boundary points of the Jacobian locus Jg{\mathcal{J}_g} in Ag{\mathcal{A}_g} and in the Baily–Borel and the Borel–Serre compactification. We show that for large genus g the set of boundary points of Jg{\mathcal{J}_g} in these compactifications is “small”.      12. Desingularization of singular Riemannian foliation      Marcos M. Alexandrino 《Geometriae Dedicata》2010,149(1):397-416 Let F{\mathcal{F}} be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F{\mathcal{F}} we construct a regular Riemannian foliation [^(F)]{\hat{\mathcal{F}}} on a compact Riemannian manifold [^(M)]{\hat{M}} and a desingularization map [^(r)]:[^(M)]? M{\hat{\rho}:\hat{M}\rightarrow M} that projects leaves of [^(F)]{\hat{\mathcal{F}}} into leaves of F{\mathcal{F}}. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F{\mathcal{F}} are compact, then, for each small ${\epsilon >0 }${\epsilon >0 }, we can find [^(M)]{\hat{M}} and [^(F)]{\hat{\mathcal{F}}} so that the desingularization map induces an e{\epsilon}-isometry between M/F{M/\mathcal{F}} and [^(M)]/[^(F)]{\hat{M}/\hat{\mathcal{F}}}. This implies in particular that the space of leaves M/F{M/\mathcal{F}} is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {([^(M)]n/[^(F)]n)}{\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}.      13. Uniformization of {\mathcal {G}}-bundles      Jochen Heinloth 《Mathematische Annalen》2010,347(3):499-528 We show some of the conjectures of Pappas and Rapoport concerning the moduli stack BunG{{\rm Bun}_\mathcal {G}} of G{\mathcal {G}}-torsors on a curve C, where G{\mathcal {G}} is a semisimple Bruhat-Tits group scheme on C. In particular we prove the analog of the uniformization theorem of Drinfeld-Simpson in this setting. Furthermore we apply this to compute the connected components of these moduli stacks and to calculate the Picard group of BunG{{\rm Bun}_\mathcal {G}} in case G{\mathcal {G}} is simply connected.      14. Iterates and Hypoellipticity of Partial Differential Operators on Non-Quasianalytic Classes      Jordi Juan-Huguet 《Integral Equations and Operator Theory》2010,68(2):263-286 Let P be a linear partial differential operator with constant coefficients. For a weight function ω and an open subset Ω of \mathbbRN{\mathbb{R}^N} , the class EP,{w}(W){\mathcal{E}_{P,\{\omega\}}(\Omega)} of Roumieu type involving the successive iterates of the operator P is considered. The completeness of this space is characterized in terms of the hypoellipticity of P. Results of Komatsu and Newberger-Zielezny are extended. Moreover, for weights ω satisfying a certain growth condition, this class coincides with a class of ultradifferentiable functions if and only if P is elliptic. These results remain true in the Beurling case EP,(w)(W){\mathcal{E}_{P,(\omega)}(\Omega)}.      15. Augmented Teichmüller spaces and orbifolds      Vladimir Hinich  Arkady Vaintrob 《Selecta Mathematica, New Series》2010,16(3):533-629 We study complex analytic properties of the augmented Teichmüller spaces [`(T)]g,n{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces Tg,n{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike Tg,n{\mathcal{T}_{g,n}}, the space [`(T)]g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.      16. Distortio n theorems o n the Lie ball $$ \mathcal{R}_{IV} $$ ( n) i n ℂ n      JianFei Wang  TaiShun Liu  HuiMing Xu 《中国科学A辑(英文版)》2009,52(12):2743-2750 In this paper, we introduce the subfamilies H m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) of holomorphic mappings defined on the Lie ball $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n) which reduce to the family of holomorphic mappings and the family of locally biholomorphic mappings when m = 1 and m → +∞, respectively. Various distortion theorems for holomophic mappings H m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are established. The distortion theorems coincide with Liu and Minda’s as the special case of the unit disk. When m = 1 and m → +∞, the distortion theorems reduce to the results obtained by Gong for $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n), respectively. Moreover, our method is different. As an application, the bounds for Bloch constants of H m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are given.      17. Arithmetic of semigroups of series in multiplicative systems      I. P. Il’inskaya 《Ukrainian Mathematical Journal》2009,61(7):1113-1122 We study the arithmetic of a semigroup MP\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?n = 0￥ ancn(x)    ( an 3 0,?n = 0￥ an = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { cn }n = 0￥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup MP\mathcal{M}_{\mathcal{P}}, analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R n are true. We describe the class I0(MP)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in MP\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence.      18. Matching Points with Squares   总被引：1，自引：0，他引：1   Bernardo M. Ábrego  Esther M. Arkin  Silvia Fernández-Merchant  Ferran Hurtado  Mikio Kano  Joseph S. B. Mitchell  Jorge Urrutia 《Discrete and Computational Geometry》2009,41(1):77-95 Given a class of geometric objects and a point set P, a -matching of P is a set of elements of such that each C i contains exactly two elements of P and each element of P lies in at most one C i . If all of the elements of P belong to some C i , M is called a perfect matching. If, in addition, all of the elements of M are pairwise disjoint, we say that this matching M is strong. In this paper we study the existence and characteristics of -matchings for point sets in the plane when is the set of isothetic squares in the plane. A consequence of our results is a proof that the Delaunay triangulations for the L ∞ metric and the L 1 metric always admit a Hamiltonian path.      19. Unique Ergodicity of Harmonic Currents On Singular Foliations of {\mathbb{P}^2}      John Erik Fornæss  Nessim Sibony 《Geometric And Functional Analysis》2010,19(5):1334-1377 Let F{\mathcal{F}} be a holomorphic foliation of \mathbbP2{\mathbb{P}^2} by Riemann surfaces. Assume all the singular points of F{\mathcal{F}} are hyperbolic. If F{\mathcal{F}} has no algebraic leaf, then there is a unique positive harmonic (1, 1) current T of mass one, directed by F{\mathcal{F}}. This implies strong ergodic properties for the foliation F{\mathcal{F}}. We also study the harmonic flow associated to the current T.      20. Corrigendum to “Resource-monotonicity for house allocation problems”      Lars Ehlers  Bettina Klaus 《International Journal of Game Theory》2011,40(2):281-287 Ehlers and Klaus (Int J Game Theory 32:545–560, 2003) study so-called allocation problems and claim to characterize all rules satisfying efficiency, independence of irrelevant objects, and resource-monotonicity on two preference domains (Ehlers and Klaus 2003, Theorem 1). They explicitly prove Theorem 1 for preference domain R0{\mathcal{R}_0} which requires that the null object is always the worst object and mention that the corresponding proofs for the larger domain R{\mathcal{R}} of unrestricted preferences “are completely analogous.” In Example 1 and Lemma 1, this corrigendum provides a counterexample to Ehlers and Klaus (2003, Theorem 1) on the general domain R{\mathcal{R}} . We also propose a way of correcting the result on the general domain R{\mathcal{R}} by strengthening independence of irrelevant objects: in addition to requiring that the chosen allocation should depend only on preferences over the set of available objects (which always includes the null object), we add a situation in which the allocation should also be invariant when preferences over the null object change. Finally, we offer a short proof of the corrected result that uses the established result of Theorem 1 for the restricted domain R0{\mathcal{R}_0}.

The Ground State and the Long-Time Evolution in the CMC Einstein Flow

Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)³Vol_g(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by V_inf=(\frac-16Y(S))^\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g _i , K _i )} satisfying: (i) k _i  = −3, (ii) V_iˉ V_inf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q ₀((g _i , K _i )) ≤ Λ, where Q ₀ is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)²g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E₁)\tilde]=E₁(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E₁=Q₀+Q₁{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ V_inf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.     

On Hankel Nonnegative Definite Sequences, the Canonical Hankel Parametrization, and Orthogonal Matrix Polynomials

This paper continues recent investigations started in Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) into the structure of the set H_q,2n ³ {\mathcal{H}_{q,2n}^{\ge}} of all Hankel nonnegative definite sequences, (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}}, of complex q × q matrices and its important subclasses H_q,2n ^{3 ,e}{\mathcal{H}_{q,2n}^{\ge,{\rm e}}} and ${\mathcal{H}_{q,2n}^>}${\mathcal{H}_{q,2n}^>} of all Hankel nonnegative definite extendable sequences and of all Hankel positive definite sequences, respectively. These classes of sequences arise quite naturally in the framework of matrix versions of the truncated Hamburger moment problem. In Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) a canonical Hankel parametrization [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} consisting of two sequences of complex q × q matrices was associated with an arbitrary sequence (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} of complex q × q matrices. The sequences belonging to each of the classes H_q,2n ³ , H_q,2n ^{3 ,e}{\mathcal{H}_{q,2n}^{\ge}, \mathcal{H}_{q,2n}^{\ge,{\rm e}}}, and ${\mathcal{H}_{q,2n}^>}${\mathcal{H}_{q,2n}^>} were characterized in terms of their canonical Hankel parametrization (see, Dyukarev et al. in Complex anal oper theory 3(4):759–834, 2009; Proposition 2.30). In this paper, we will study further aspects of the canonical Hankel parametrization. Using the canonical Hankel parametrization [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} of a sequence (s_j)_j=0²ⁿ ? H_q,2n ³ {(s_{j})_{j=0}^{2n} \in \mathcal{H}_{q,2n}^{\ge}}, we give a recursive construction of a monic right (resp. left) orthogonal system of matrix polynomials with respect to (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} (see Theorem 5.5). The matrices [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} will be expressed in terms of an arbitrary monic right (resp. left) orthogonal system with respect to (s_j)_j=0²ⁿ{(s_{j})_{j=0}^{2n}} (see Theorem 5.11). This result will be reformulated in terms of nonnegative Hermitian Borel measures on \mathbbR{\mathbb{R}}. In this way, integral representations for the matrices [(C_k)_k=1ⁿ, (D_k)_k=0ⁿ]{[(C_k)_{k=1}^n, (D_k)_{k=0}^n]} will be obtained (see Theorem 6.9). Starting from the monic orthogonal polynomials with respect to some classical probability distributions on \mathbbR{\mathbb{R}}, Theorem 6.9 is used to compute the canonical Hankel parametrization of their moment sequences. Moreover, we discuss important number sequences from enumerative combinatorics using the canonical Hankel parametrization.     

Around splitting and reaping for partitions of ω

We investigate splitting number and reaping number for the structure (ω) ^ω of infinite partitions of ω. We prove that \mathfrakr_d ￡ non(M),non(N),\mathfrakd{\mathfrak{r}_{d}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N}),\mathfrak{d}} and \mathfraks_d 3 \mathfrakb{\mathfrak{s}_{d}\geq\mathfrak{b}} . We also show the consistency results ${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})}${\mathfrak{r}_{d} > \mathfrak{b}, \mathfrak{s}_{d} < \mathfrak{d}, \mathfrak{s}_{d} < \mathfrak{r}, \mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and ${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})}${\mathfrak{s}_{d} > \mathsf{cof}(\mathcal{M})} . To prove the consistency \mathfrakr_d < add(M){\mathfrak{r}_{d} < \mathsf{add}(\mathcal{M})} and \mathfraks_d < cof(M){\mathfrak{s}_{d} < \mathsf{cof}(\mathcal{M})} we introduce new cardinal invariants \mathfrakr_pair{\mathfrak{r}_{pair}} and \mathfraks_pair{\mathfrak{s}_{pair}} . We also study the relation between \mathfrakr_pair, \mathfraks_pair{\mathfrak{r}_{pair}, \mathfrak{s}_{pair}} and other cardinal invariants. We show that cov(M),cov(N) ￡ \mathfrakr_pair ￡ \mathfraks_d,\mathfrakr{\mathsf{cov}(\mathcal{M}),\mathsf{cov}(\mathcal{N})\leq\mathfrak{r}_{pair}\leq\mathfrak{s}_{d},\mathfrak{r}} and \mathfraks ￡ \mathfraks_pair ￡ non(M),non(N){\mathfrak{s}\leq\mathfrak{s}_{pair}\leq\mathsf{non}(\mathcal{M}),\mathsf{non}(\mathcal{N})} .     

The Asymptotic Schottky Problem

Let ${\mathcal{M}_g}Let M_g{\mathcal{M}_g} denote the moduli space of compact Riemann surfaces of genus g and let A_g{\mathcal{A}_g} be the moduli space of principally polarized abelian varieties of dimension g. Let J : M_g ? A_g{J : \mathcal{M}_g \rightarrow \mathcal{A}_g} be the map which associates to a Riemann surface its Jacobian. The map J is injective, and the image J_g : = J(M_g){\mathcal{J}_g := J(\mathcal{M}_g)} is contained in a proper subvariety of A_g{\mathcal{A}_g} when g ≥  4. The classical and long-studied Schottky problem is to characterize the Jacobian locus J_g{\mathcal{J}_g} in A_g{\mathcal{A}_g}. In this paper we address a large scale version of this problem posed by Farb and called the coarse Schottky problem: What does J_g{\mathcal{J}_g} look like “from far away”, or how “dense” is J_g{\mathcal{J}_g} in the sense of coarse geometry? The large scale geometry of A_g{\mathcal{A}_g} is encoded in its asymptotic cone, Cone_￥(A_g){{\rm Cone}_\infty(\mathcal{A}_g)}, which is a Euclidean simplicial cone of real dimension g. Our main result asserts that the Jacobian locus J_g{\mathcal{J}_g} is “coarsely dense” in A_g{\mathcal{A}_g}, which implies that the subset of Cone_￥(A_g){{\rm Cone}_\infty(\mathcal{A}_g)} determined by J_g{\mathcal{J}_g} actually coincides with this cone. The proof shows that the Jacobian locus of hyperelliptic curves is coarsely dense in A_g{\mathcal{A}_g} as well. We also study the boundary points of the Jacobian locus J_g{\mathcal{J}_g} in A_g{\mathcal{A}_g} and in the Baily–Borel and the Borel–Serre compactification. We show that for large genus g the set of boundary points of J_g{\mathcal{J}_g} in these compactifications is “small”.     

Desingularization of singular Riemannian foliation

Let F{\mathcal{F}} be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F{\mathcal{F}} we construct a regular Riemannian foliation [^(F)]{\hat{\mathcal{F}}} on a compact Riemannian manifold [^(M)]{\hat{M}} and a desingularization map [^(r)]:[^(M)]? M{\hat{\rho}:\hat{M}\rightarrow M} that projects leaves of [^(F)]{\hat{\mathcal{F}}} into leaves of F{\mathcal{F}}. This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of F{\mathcal{F}} are compact, then, for each small ${\epsilon >0 }${\epsilon >0 }, we can find [^(M)]{\hat{M}} and [^(F)]{\hat{\mathcal{F}}} so that the desingularization map induces an e{\epsilon}-isometry between M/F{M/\mathcal{F}} and [^(M)]/[^(F)]{\hat{M}/\hat{\mathcal{F}}}. This implies in particular that the space of leaves M/F{M/\mathcal{F}} is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {([^(M)]_n/[^(F)]_n)}{\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}.     

Uniformization of {\mathcal {G}}-bundles

We show some of the conjectures of Pappas and Rapoport concerning the moduli stack Bun_G{{\rm Bun}_\mathcal {G}} of G{\mathcal {G}}-torsors on a curve C, where G{\mathcal {G}} is a semisimple Bruhat-Tits group scheme on C. In particular we prove the analog of the uniformization theorem of Drinfeld-Simpson in this setting. Furthermore we apply this to compute the connected components of these moduli stacks and to calculate the Picard group of Bun_G{{\rm Bun}_\mathcal {G}} in case G{\mathcal {G}} is simply connected.     

Iterates and Hypoellipticity of Partial Differential Operators on Non-Quasianalytic Classes

Let P be a linear partial differential operator with constant coefficients. For a weight function ω and an open subset Ω of \mathbbR^N{\mathbb{R}^N} , the class E_P,{w}(W){\mathcal{E}_{P,\{\omega\}}(\Omega)} of Roumieu type involving the successive iterates of the operator P is considered. The completeness of this space is characterized in terms of the hypoellipticity of P. Results of Komatsu and Newberger-Zielezny are extended. Moreover, for weights ω satisfying a certain growth condition, this class coincides with a class of ultradifferentiable functions if and only if P is elliptic. These results remain true in the Beurling case E_P,(w)(W){\mathcal{E}_{P,(\omega)}(\Omega)}.     

Augmented Teichmüller spaces and orbifolds

We study complex analytic properties of the augmented Teichmüller spaces [`(T)]<sub>g,n</sub>{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces T<sub>g,n</sub>{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike T<sub>g,n</sub>{\mathcal{T}_{g,n}}, the space [`(T)]_g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.     

Distortio n theorems o n the Lie ball $$ \mathcal{R}_{IV} $$ ( n) i n ℂ n

In this paper, we introduce the subfamilies H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) of holomorphic mappings defined on the Lie ball $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n) which reduce to the family of holomorphic mappings and the family of locally biholomorphic mappings when m = 1 and m → +∞, respectively. Various distortion theorems for holomophic mappings H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are established. The distortion theorems coincide with Liu and Minda’s as the special case of the unit disk. When m = 1 and m → +∞, the distortion theorems reduce to the results obtained by Gong for $ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n), respectively. Moreover, our method is different. As an application, the bounds for Bloch constants of H _m ($ \mathcal{R}_{IV} $ \mathcal{R}_{IV} (n)) are given.     

Arithmetic of semigroups of series in multiplicative systems

We study the arithmetic of a semigroup M_P\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?_{n = 0}^￥ a_nc_n(x)    ( a_n 3 0,?_{n = 0}^￥ a_n = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { c_n }_{n = 0}^￥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup M_P\mathcal{M}_{\mathcal{P}}, analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R ⁿ are true. We describe the class I₀(M_P)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in M_P\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence.     

Matching Points with Squares

Given a class of geometric objects and a point set P, a -matching of P is a set of elements of such that each C _i contains exactly two elements of P and each element of P lies in at most one C _i . If all of the elements of P belong to some C _i , M is called a perfect matching. If, in addition, all of the elements of M are pairwise disjoint, we say that this matching M is strong. In this paper we study the existence and characteristics of -matchings for point sets in the plane when is the set of isothetic squares in the plane. A consequence of our results is a proof that the Delaunay triangulations for the L _∞ metric and the L ₁ metric always admit a Hamiltonian path.     

Unique Ergodicity of Harmonic Currents On Singular Foliations of {\mathbb{P}^2}

Let F{\mathcal{F}} be a holomorphic foliation of \mathbbP²{\mathbb{P}^2} by Riemann surfaces. Assume all the singular points of F{\mathcal{F}} are hyperbolic. If F{\mathcal{F}} has no algebraic leaf, then there is a unique positive harmonic (1, 1) current T of mass one, directed by F{\mathcal{F}}. This implies strong ergodic properties for the foliation F{\mathcal{F}}. We also study the harmonic flow associated to the current T.     

Corrigendum to “Resource-monotonicity for house allocation problems”

Ehlers and Klaus (Int J Game Theory 32:545–560, 2003) study so-called allocation problems and claim to characterize all rules satisfying efficiency, independence of irrelevant objects, and resource-monotonicity on two preference domains (Ehlers and Klaus 2003, Theorem 1). They explicitly prove Theorem 1 for preference domain R₀{\mathcal{R}_0} which requires that the null object is always the worst object and mention that the corresponding proofs for the larger domain R{\mathcal{R}} of unrestricted preferences “are completely analogous.” In Example 1 and Lemma 1, this corrigendum provides a counterexample to Ehlers and Klaus (2003, Theorem 1) on the general domain R{\mathcal{R}} . We also propose a way of correcting the result on the general domain R{\mathcal{R}} by strengthening independence of irrelevant objects: in addition to requiring that the chosen allocation should depend only on preferences over the set of available objects (which always includes the null object), we add a situation in which the allocation should also be invariant when preferences over the null object change. Finally, we offer a short proof of the corrected result that uses the established result of Theorem 1 for the restricted domain R₀{\mathcal{R}_0}.