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})\}}.     

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,      3. Swan-like results for binomials and trinomials over finite fields of odd characteristic      B. Hanson  D. Panario  D. Thomson 《Designs, Codes and Cryptography》2011,61(3):273-283 Swan (Pac. J. Math. 12:1099–1106, 1962) gives conditions under which the trinomial x n + x k + 1 over \mathbbF2{\mathbb{F}_{2}} is reducible. Vishne (Finite Fields Appl. 3:370–377, 1997) extends this result to trinomials over extensions of \mathbbF2{\mathbb{F}_{2}}. In this work we determine the parity of the number of irreducible factors of all binomials and some trinomials over the finite field \mathbbFq{\mathbb{F}_{q}}, where q is a power of an odd prime.      4. Lifting of convex functions on Carnot groups and lack of convexity for a gauge function      Andrea Bonfiglioli 《Archiv der Mathematik》2009,93(3):277-286 Let ${\mathbb{G}}Let \mathbbG{\mathbb{G}} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \mathbbFm,r{\mathbb{F}_{m,r}} denote the free Lie group of step r and m generators. Let also p:\mathbbFm,r?\mathbbG{\pi:\mathbb{F}_{m,r}\to\mathbb{G}} be a lifting map. We show that any horizontally convex function u on \mathbbG{\mathbb{G}} lifts to a horizontally convex function u°p{u\circ \pi} on \mathbbFm,r{\mathbb{F}_{m,r}} (with respect to a suitable horizontal frame on \mathbbFm,r{\mathbb{F}_{m,r}}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian L=?j=1m Xj2{\mathcal{L}=\sum_{j=1}^m X_j^2} on a Carnot group of step two such that the relevant L{\mathcal{L}}-gauge function d (i.e., d 2-Q is the fundamental solution for L{\mathcal{L}}) is not h-convex with respect to the horizontal frame {X 1, . . . , X m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).      5. A trade-off between collision probability and key size in universal hashing using polynomials      Palash Sarkar 《Designs, Codes and Cryptography》2011,58(3):271-278 Let \mathbbF{\mathbb{F}} be a finite field and suppose that a single element of \mathbbF{\mathbb{F}} is used as an authenticator (or tag). Further, suppose that any message consists of at most L elements of \mathbbF{\mathbb{F}}. For this setting, usual polynomial based universal hashing achieves a collision bound of (L-1)/|\mathbbF|{(L-1)/|\mathbb{F}|} using a single element of \mathbbF{\mathbb{F}} as the key. The well-known multi-linear hashing achieves a collision bound of 1/|\mathbbF|{1/|\mathbb{F}|} using L elements of \mathbbF{\mathbb{F}} as the key. In this work, we present a new universal hash function which achieves a collision bound of mélogm Lù/|\mathbbF|, m 3 2{m\lceil\log_m L\rceil/|\mathbb{F}|, m\geq 2}, using 1+élogm Lù{1+\lceil\log_m L\rceil} elements of \mathbbF{\mathbb{F}} as the key. This provides a new trade-off between key size and collision probability for universal hash functions.      6. An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}      Vitaly Bergelson  Terence Tao  Tamar Ziegler 《Geometric And Functional Analysis》2010,19(6):1539-1596 Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U k (X) for an ergodic action (Tg)g ? \mathbb Fw{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb Fw{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S1{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.      7. 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.      8. Normal families of meromorphic functions and shared values      Xiangzhong Wu  Yan Xu 《Monatshefte für Mathematik》2012,9(6):569-578 Let k be a positive integer, b ≠ 0 be a finite complex number, let P be a polynomial with either deg P ≥ 3 or deg P = 2 and P having only one distinct zero, and let F{\mathcal{F}} be a family of functions meromorphic in a domain D, all of whose zeros have multiplicities at least k. If, each pair of functions f and g in F, P(f)f(k){\mathcal{F}, P(f)f^{(k)}} and P(g)g (k) share b in D, then F{\mathcal{F}} is normal in D.      9. Contraction and Expansion of Convex Sets      Michael Langberg  Leonard J. Schulman 《Discrete and Computational Geometry》2009,42(4):594-614 Let S{\mathcal{S}} be a set system of convex sets in ℝ d . Helly’s theorem states that if all sets in S{\mathcal{S}} have empty intersection, then there is a subset S￠ ì S{\mathcal{S}}'\subset{\mathcal{S}} of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in S{\mathcal{S}} are not convex or if S{\mathcal{S}} does not have empty intersection. Nevertheless, in this work we present Helly-type theorems relevant to these cases with the aid of a new pair of operations, affine-invariant contraction, and expansion of convex sets. These operations generalize the simple scaling of centrally symmetric sets. The operations are continuous, i.e., for small ε>0, the contraction C −ε and the expansion C ε are close (in the Hausdorff distance) to C. We obtain two results. The first extends Helly’s theorem to the case of set systems with nonempty intersection:      10. Two Inequalities for Convex Functions   总被引：1，自引：0，他引：1   Ping Zhi Yuan  Hai Bo Chen 《数学学报(英文版)》2005,21(1):193-196 Let a 0 < a 1 < ··· < a n be positive integers with sums $ {\sum\nolimits_{i = 0}^n {\varepsilon _{i} a_{i} {\left( {\varepsilon _{i} = 0,1} \right)}} } Let a 0 < a 1 < ··· < a n be positive integers with sums distinct. P. Erd?s conjectured that The best known result along this line is that of Chen: Let f be any given convex decreasing function on [A, B] with α 0, α 1, ... , α n , β 0, β 1, ... , β n being real numbers in [A, B] with α 0 ≤ α 1 ≤ ··· ≤ α n , Then In this paper, we obtain two generalizations of the above result; each is of special interest in itself. We prove: Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α 0, α 1, ... , α n , β 0, β 1, ... , β n , α' 0, α' 1, ... , α' n , β' 0 , β' 1 , ... , β' n be real numbers in [A, B] with α 0 ≤ α 1 ≤ ··· ≤ α n , α' 0 ≤ α' 1 ≤ ··· ≤ α' n , Then Theorem 2 Let f be any given convex decreasing function on [A, B] with k 0, k 1, ... , k n being nonnegative real numbers and α 0, α 1, ... , α n , β 0, β 1, ... , β n being real numbers in [A, B] with α 0 ≤ α 1 ≤ ··· ≤ α n , Then       11. A Splitting of the Frobenius Morphism on the Whole Algebra of Distributions of SL 2      Michel Gros 《Algebras and Representation Theory》2012,15(1):109-118 We define, over k = \BbbFpk = {\Bbb{F}}_{p}, a splitting of the Frobenius morphism Fr : \textDist (G) ? \textDist (G)Fr : {\text{Dist}}\,(G) \rightarrow {\text{Dist}}\,(G) on the whole \textDist (G){\text{Dist}}\,(G), the algebra of distributions of the k-algebraic group G: = SL 2. This splitting is compatible (and lifts) the theory of Frobenius descent for arithmetic D{\cal{D}}-modules over X:=\BbbPk1X:={\Bbb{P}}_{k}^{1}.      12. A Generalization of the Erdos - Szekeres Theorem to Disjoint Convex Sets   总被引：2，自引：0，他引：2   J. Pach  G. Tóth 《Discrete and Computational Geometry》1998,19(3):437-445 Let F denote a family of pairwise disjoint convex sets in the plane. F is said to be in convex position if none of its members is contained in the convex hull of the union of the others. For any fixed k≥ 3 , we estimate P k (n) , the maximum size of a family F with the property that any k members of F are in convex position, but no n are. In particular, for k=3 , we improve the triply exponential upper bound of T. Bisztriczky and G. Fejes Tóth by showing that P 3 (n) < 16 n . <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p437.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received March 27, 1997, and in revised form July 10, 1997.      13. Approximate Subgroups of Linear Groups      Emmanuel Breuillard  Ben Green  Terence Tao 《Geometric And Functional Analysis》2011,21(4):774-819 We establish various results on the structure of approximate subgroups in linear groups such as SL n (k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of SLn(\mathbb Fq){{\rm SL}_{n}({\mathbb {F}}_{q})} which generates the group must be either very small or else nearly all of SLn(\mathbb Fq){{\rm SL}_{n}({\mathbb {F}}_{q})}. The argument generalises to other absolutely almost simple connected (and non-commutative) algebraic groups G over an arbitrary field k and yields a classification of approximate subgroups of G(k). In a subsequent paper, we will give applications of this result to the expansion properties of Cayley graphs.      14. A normality criterion involving rotations and dilations in the argument      Jürgen Grahl 《Arkiv f?r Matematik》2012,50(1):89-110 We show that a family F\mathcal{F} of analytic functions in the unit disk \mathbbD{\mathbb{D}} all of whose zeros have multiplicity at least k and which satisfy a condition of the form fn(z)f(k)(xz) 1 1f^n(z)f^{(k)}(xz)\ne1      15. A correspondence for the supercuspidal representations of $\overline {SL_2(F)}$, F p-adic      D. Joyner 《Archiv der Mathematik》1999,73(5):332-340 Following D. Manderscheid, we describe the supercuspidal representations of the n-fold metaplectic cover [`(SL2(F))]\overline {SL_2(F)}, where F is a p-adic field with (p, 2n) = 1. We prove a "Frobenius formula" for the character of a supercuspidal representation of [`(SL2(F))]\overline {SL_2(F)}. Using this formula, we obtain a character relation between corresponding supercuspidal representations of [`(SL2(F))]\overline {SL_2(F)} and of SL2(F)> in the case n = 2.      16. Line Transversals to Translates of a Convex Body      J. Jerónimo-Castro  E. Roldán-Pensado 《Discrete and Computational Geometry》2011,45(2):329-339 Let K be a convex body in the plane. Define λ(K,t) as the smallest number satisfying the following: if F\mathcal{F} is any family of translates of K such that every t members of F\mathcal{F} have a common transversal, then all the members of l(K,t)F\lambda(K,t)\mathcal{F} have a common transversal. We give bounds for λ(K,3) and λ(K,4) for a general convex figure K. In particular, we obtain that λ(K,3)≤1.79 when K is the Euclidean disc.      17. Modular Lie algebras and the Gelfand–Kirillov conjecture      Alexander Premet 《Inventiones Mathematicae》2010,181(2):395-420 Let ${\mathfrak{g}}Let \mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field \mathbbK\mathbb{K} of characteristic 0. Let \mathfrakg\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of \mathfrakg{\mathfrak{g}} and \mathfrakg\Bbbk=\mathfrakg\mathbbZ?\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where \Bbbk\Bbbk is the algebraic closure of  \mathbbFp{\mathbb{F}}_{p}. Let G\BbbkG_{\Bbbk} be a simple, simply connected algebraic \Bbbk\Bbbk-group with \operatornameLie(G\Bbbk)=\mathfrakg\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(\mathfrakg\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for \mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions \Bbbk (\mathfrakg\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield \Bbbk(\mathfrakg\Bbbk)G\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions \mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield \mathbbK(\mathfrakg)\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if \mathfrakg{\mathfrak{g}} is of type B n , n≥3, D n , n≥4, E6, E7, E8 or F4. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if \mathfrakg{\mathfrak{g}} is of type B n , n≥3, D n , n≥4, E6, E7, E8 or F4, then the Lie field of \mathfrakg{\mathfrak{g}} is more complicated than expected.      18. On standard Auslander-Reiten sequences for finite groups      S. Kawata 《Archiv der Mathematik》2000,75(2):92-97 Let G be a finite group and O{\cal O} a complete discrete valuation ring of characteristic zero with maximal ideal (p)(\pi ) and residue field k = O/(p)k = {\cal O}/(\pi ) of characteristic p > 0. Let S be a simple kG-module and QS a projective O G{\cal O} G-lattice such that QS / pQSQ_S / \pi Q_S is a projective cover of S. We show that if S is liftable and QS belongs to a block of O G{\cal O} G of infinite representation type, then the standard Auslander-Reiten sequence terminating in W-1S\Omega ^{-1}S is a direct summand of the short exact sequence obtained from some Auslander-Reiten sequence of OG{\cal O}G-lattices by reducing each term mod (p)(\pi ).      19. Hilbert–Chow Morphism for Non Commutative Hilbert Schemes and Moduli Spaces of Linear Representations      Federica Galluzzi  Francesco Vaccarino 《Algebras and Representation Theory》2010,13(4):491-509       20. 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.

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Swan-like results for binomials and trinomials over finite fields of odd characteristic

Swan (Pac. J. Math. 12:1099–1106, 1962) gives conditions under which the trinomial x ⁿ + x ^k + 1 over \mathbbF₂{\mathbb{F}_{2}} is reducible. Vishne (Finite Fields Appl. 3:370–377, 1997) extends this result to trinomials over extensions of \mathbbF₂{\mathbb{F}_{2}}. In this work we determine the parity of the number of irreducible factors of all binomials and some trinomials over the finite field \mathbbF_q{\mathbb{F}_{q}}, where q is a power of an odd prime.     

Lifting of convex functions on Carnot groups and lack of convexity for a gauge function

Let ${\mathbb{G}}Let \mathbbG{\mathbb{G}} be a Carnot group of step r and m generators and homogeneous dimension Q. Let \mathbbF_m,r{\mathbb{F}_{m,r}} denote the free Lie group of step r and m generators. Let also p:\mathbbF_m,r?\mathbbG{\pi:\mathbb{F}_{m,r}\to\mathbb{G}} be a lifting map. We show that any horizontally convex function u on \mathbbG{\mathbb{G}} lifts to a horizontally convex function u°p{u\circ \pi} on \mathbbF_m,r{\mathbb{F}_{m,r}} (with respect to a suitable horizontal frame on \mathbbF_m,r{\mathbb{F}_{m,r}}). One of the main aims of the paper is to exhibit an example of a sub-Laplacian L=?_j=1^m X_j²{\mathcal{L}=\sum_{j=1}^m X_j^2} on a Carnot group of step two such that the relevant L{\mathcal{L}}-gauge function d (i.e., d ^2-Q is the fundamental solution for L{\mathcal{L}}) is not h-convex with respect to the horizontal frame {X ₁, . . . , X _m }. This gives a negative answer to a question posed in Danielli et al. (Commun. Anal. Geom. 11 (2003), 263–341).     

A trade-off between collision probability and key size in universal hashing using polynomials

Let \mathbbF{\mathbb{F}} be a finite field and suppose that a single element of \mathbbF{\mathbb{F}} is used as an authenticator (or tag). Further, suppose that any message consists of at most L elements of \mathbbF{\mathbb{F}}. For this setting, usual polynomial based universal hashing achieves a collision bound of (L-1)/|\mathbbF|{(L-1)/|\mathbb{F}|} using a single element of \mathbbF{\mathbb{F}} as the key. The well-known multi-linear hashing achieves a collision bound of 1/|\mathbbF|{1/|\mathbb{F}|} using L elements of \mathbbF{\mathbb{F}} as the key. In this work, we present a new universal hash function which achieves a collision bound of mélog_m Lù/|\mathbbF|, m 3 2{m\lceil\log_m L\rceil/|\mathbb{F}|, m\geq 2}, using 1+élog_m Lù{1+\lceil\log_m L\rceil} elements of \mathbbF{\mathbb{F}} as the key. This provides a new trade-off between key size and collision probability for universal hash functions.     

An Inverse Theorem for the Uniformity Seminorms Associated with the Action of {{\mathbb {F}^{\infty}_{p}}}

Let ${\mathbb {F}}Let \mathbb F{\mathbb {F}} a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U ^k (X) for an ergodic action (T_g)_{g ? \mathbb F^w}{(T_{g})_{{g} \in \mathbb {F}^{\omega}}} of the infinite abelian group \mathbb F^w{\mathbb {F}^{\omega}} on a probability space X = (X, B, m){X = (X, \mathcal {B}, \mu)} is generated by phase polynomials f: X ? S¹{\phi : X \to S^{1}} of degree less than C(k) on X, where C(k) depends only on k. In the case where k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} we obtain the sharp result C(k) = k. This is a finite field counterpart of an analogous result for \mathbb Z{\mathbb {Z}} by Host and Kra [HK]. In a companion paper [TZ] to this paper, we shall combine this result with a correspondence principle to establish the inverse theorem for the Gowers norm in finite fields in the high characteristic case k ￡ char(\mathbb F){k \leq {\rm char}(\mathbb {F})} , with a partial result in low characteristic.     

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.     

Normal families of meromorphic functions and shared values

Let k be a positive integer, b ≠ 0 be a finite complex number, let P be a polynomial with either deg P ≥ 3 or deg P = 2 and P having only one distinct zero, and let F{\mathcal{F}} be a family of functions meromorphic in a domain D, all of whose zeros have multiplicities at least k. If, each pair of functions f and g in F, P(f)f^(k){\mathcal{F}, P(f)f^{(k)}} and P(g)g ^(k) share b in D, then F{\mathcal{F}} is normal in D.     

Contraction and Expansion of Convex Sets

Let S{\mathcal{S}} be a set system of convex sets in ℝ ^d . Helly’s theorem states that if all sets in S{\mathcal{S}} have empty intersection, then there is a subset S￠ ì S{\mathcal{S}}'\subset{\mathcal{S}} of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in S{\mathcal{S}} are not convex or if S{\mathcal{S}} does not have empty intersection. Nevertheless, in this work we present Helly-type theorems relevant to these cases with the aid of a new pair of operations, affine-invariant contraction, and expansion of convex sets. These operations generalize the simple scaling of centrally symmetric sets. The operations are continuous, i.e., for small ε>0, the contraction C ^−ε and the expansion C ^ε are close (in the Hausdorff distance) to C. We obtain two results. The first extends Helly’s theorem to the case of set systems with nonempty intersection:     

Two Inequalities for Convex Functions

Let a ₀ < a ₁ < ··· < a _n be positive integers with sums $ {\sum\nolimits_{i = 0}^n {\varepsilon _{i} a_{i} {\left( {\varepsilon _{i} = 0,1} \right)}} } Let a ₀ < a ₁ < ··· < a _n be positive integers with sums distinct. P. Erd?s conjectured that The best known result along this line is that of Chen: Let f be any given convex decreasing function on [A, B] with α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then In this paper, we obtain two generalizations of the above result; each is of special interest in itself. We prove: Theorem 1 Let f and g be two given non-negative convex decreasing functions on [A, B], and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n , α' ₀, α' ₁, ... , α' _n , β' ₀ , β' ₁ , ... , β' _n be real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , α' ₀ ≤ α' ₁ ≤ ··· ≤ α' _n , Then Theorem 2 Let f be any given convex decreasing function on [A, B] with k ₀, k ₁, ... , k _n being nonnegative real numbers and α ₀, α ₁, ... , α _n , β ₀, β ₁, ... , β _n being real numbers in [A, B] with α ₀ ≤ α ₁ ≤ ··· ≤ α _n , Then      

A Splitting of the Frobenius Morphism on the Whole Algebra of Distributions of SL 2

We define, over k = \BbbF_pk = {\Bbb{F}}_{p}, a splitting of the Frobenius morphism Fr : \textDist (G) ? \textDist (G)Fr : {\text{Dist}}\,(G) \rightarrow {\text{Dist}}\,(G) on the whole \textDist (G){\text{Dist}}\,(G), the algebra of distributions of the k-algebraic group G: = SL ₂. This splitting is compatible (and lifts) the theory of Frobenius descent for arithmetic D{\cal{D}}-modules over X:=\BbbP_k¹X:={\Bbb{P}}_{k}^{1}.     

A Generalization of the Erdos - Szekeres Theorem to Disjoint Convex Sets

Let F denote a family of pairwise disjoint convex sets in the plane. F is said to be in convex position if none of its members is contained in the convex hull of the union of the others. For any fixed k≥ 3 , we estimate P _k (n) , the maximum size of a family F with the property that any k members of F are in convex position, but no n are. In particular, for k=3 , we improve the triply exponential upper bound of T. Bisztriczky and G. Fejes Tóth by showing that P ₃ (n) < 16 ⁿ . <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p437.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received March 27, 1997, and in revised form July 10, 1997.     

Approximate Subgroups of Linear Groups

We establish various results on the structure of approximate subgroups in linear groups such as SL _n (k) that were previously announced by the authors. For example, generalising a result of Helfgott (who handled the cases n = 2 and 3), we show that any approximate subgroup of SL_n(\mathbb F_q){{\rm SL}_{n}({\mathbb {F}}_{q})} which generates the group must be either very small or else nearly all of SL_n(\mathbb F_q){{\rm SL}_{n}({\mathbb {F}}_{q})}. The argument generalises to other absolutely almost simple connected (and non-commutative) algebraic groups G over an arbitrary field k and yields a classification of approximate subgroups of G(k). In a subsequent paper, we will give applications of this result to the expansion properties of Cayley graphs.     

A normality criterion involving rotations and dilations in the argument

We show that a family F\mathcal{F} of analytic functions in the unit disk \mathbbD{\mathbb{D}} all of whose zeros have multiplicity at least k and which satisfy a condition of the form

A correspondence for the supercuspidal representations of $\overline {SL_2(F)}$, F p-adic

Following D. Manderscheid, we describe the supercuspidal representations of the n-fold metaplectic cover [`(SL<sub>2</sub>(F))]\overline {SL_2(F)}, where F is a p-adic field with (p, 2n) = 1. We prove a "Frobenius formula" for the character of a supercuspidal representation of [`(SL₂(F))]\overline {SL_2(F)}. Using this formula, we obtain a character relation between corresponding supercuspidal representations of [`(SL₂(F))]\overline {SL_2(F)} and of SL₂(F)> in the case n = 2.     

Line Transversals to Translates of a Convex Body

Let K be a convex body in the plane. Define λ(K,t) as the smallest number satisfying the following: if F\mathcal{F} is any family of translates of K such that every t members of F\mathcal{F} have a common transversal, then all the members of l(K,t)F\lambda(K,t)\mathcal{F} have a common transversal. We give bounds for λ(K,3) and λ(K,4) for a general convex figure K. In particular, we obtain that λ(K,3)≤1.79 when K is the Euclidean disc.     

Modular Lie algebras and the Gelfand–Kirillov conjecture

Let ${\mathfrak{g}}Let \mathfrakg{\mathfrak{g}} be a finite dimensional simple Lie algebra over an algebraically closed field \mathbbK\mathbb{K} of characteristic 0. Let \mathfrakg_\mathbbZ{\mathfrak{g}}_{{\mathbb{Z}}} be a Chevalley ℤ-form of \mathfrakg{\mathfrak{g}} and \mathfrakg_\Bbbk=\mathfrakg_\mathbbZ?_\mathbbZ\Bbbk{\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk, where \Bbbk\Bbbk is the algebraic closure of  \mathbbF_p{\mathbb{F}}_{p}. Let G_\BbbkG_{\Bbbk} be a simple, simply connected algebraic \Bbbk\Bbbk-group with \operatornameLie(G_\Bbbk)=\mathfrakg_\Bbbk\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra U(\mathfrakg_\Bbbk)U({\mathfrak{g}}_{\Bbbk}) to show that if the Gelfand–Kirillov conjecture (from 1966) holds for \mathfrakg{\mathfrak{g}}, then for all p≫0 the field of rational functions \Bbbk (\mathfrakg_\Bbbk)\Bbbk ({\mathfrak{g}}_{\Bbbk}) is purely transcendental over its subfield \Bbbk(\mathfrakg_\Bbbk)^G_\Bbbk\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions \mathbbK(\mathfrakg){\mathbb{K}}({\mathfrak{g}}) is not purely transcendental over its subfield \mathbbK(\mathfrakg)^\mathfrakg{\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}} if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if \mathfrakg{\mathfrak{g}} is of type B _n , n≥3, D _n , n≥4, E₆, E₇, E₈ or F₄, then the Lie field of \mathfrakg{\mathfrak{g}} is more complicated than expected.     

On standard Auslander-Reiten sequences for finite groups

Let G be a finite group and O{\cal O} a complete discrete valuation ring of characteristic zero with maximal ideal (p)(\pi ) and residue field k = O/(p)k = {\cal O}/(\pi ) of characteristic p > 0. Let S be a simple kG-module and Q_S a projective O G{\cal O} G-lattice such that Q_S / pQ_SQ_S / \pi Q_S is a projective cover of S. We show that if S is liftable and Q_S belongs to a block of O G{\cal O} G of infinite representation type, then the standard Auslander-Reiten sequence terminating in W^-1S\Omega ^{-1}S is a direct summand of the short exact sequence obtained from some Auslander-Reiten sequence of OG{\cal O}G-lattices by reducing each term mod (p)(\pi ).     

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.