ON THE COHOMOLOGY OF A SPACE CURVE CONTAINING A PLANE CURVE

In this paper we study the Hartshorne–Rao module of curves in P ³ of degree d and genus g, containing plane curves of degree d ? p, p ≥ 1. We prove an optimal upper bound for the Rao function of these curves and we show that the curves attaining the bound are obtained from an extremal curve by an elementary biliaison of height min(p, d ? p) ? 1 on a quadric surface.     

Estimates for Hyperbolic Equations of Space Dimension 3

We discuss the problem of boundedness fromL^p(Rⁿ) toL^p′(Rⁿ) (1/p+1/p′=1, 1?p?2) of operators of the typeM=F⁻¹e^i?(ξ)a(ξ) F, which is related to the study of hyperbolic equations with constant coefficients. The boundedness is dependent on a geometrical property ofΣ=?⁻¹(1), and its dependence has been exactly determined in the casesn=2, 1?p?2 andn?3,p=1, 2 (M. Sugimoto,Math. Z.215(1994), 519–531;222(1996), 521–531). This paper is devoted to the unsolved case 1<p<2, and a strange phenomenon is exhibited in the simplest casen=3.     

LIMITS OF SOLUTIONS TO THE GENERALIZED GINZBURG-LANDAU FUNCTIONAL

ABSTRACT

We discuss the convergence issue for the generalized Ginzburg-Landau functional which penalizes the L^p -energy. We prove the strong convergence for non-integer p and obtain the obstruction for strong convergence for integer p. In the presence of obstruction, we show the limiting couple varifold is a (generalized) stationary (n ? p)-varifold.     

Generators of Simple Lie Algebras and the Lower Rank of Some Pro-P Groups

Abstract

Let  be a simple classical Lie algebra over a field F of characteristic p > 7. We show that > d () = 2, where d() is the number of generators of . Let G be a profinite group. We say that G has lower rank ≤ l, if there are {G _α} open subgroups which from a base for the topology at the identity and each G _α is generated (topologically) by no more than l elements. There is a standard way to associate a Lie algebra L(G) to a finitely generated (filtered) pro-p group G. Suppose L(G) ?  ? tF _p [t], where  is a simple Lie algebra over F _p , the field of p elements. We show that the lower rank of G is ≤ d () + 1. We also show that if  is simple classical of rank r and p > 7 or p 2r ² ? r, then the lower rank is actually 2.     

Lacunary Series in Weighted Hyperholomorphic B p,q (G) Spaces

The goal of this article is two-fold. First, we consider a class of hyperholomorphic functions, the so called B ^{p, q} (G) space in ?³. Then, we use the B ^{p, q} (G) space to characterize the hyperholomorphic α-Bloch space. Second, we obtain characterizations of the weighted hyperholomorphic B ^{p, q} (G)-functions by the coefficients of certain lacunary series expansions in Clifford Analysis.     

On the Number of Galois Points for a Plane Curve in Positive Characteristic

We study Galois points for a plane smooth curve C ? P ² of degree d ≥ 4 in characteristic p > 2. We generalize Yoshihara's result on the number of inner (resp., outer) Galois points to positive characteristic under the assumption that d ? 1 (resp., d ? 0) modulo p. As an application, we also find the number of Galois points in the case that d = p.     

Lipschitz Continuity Properties for p−Adic Semi-Algebraic and Subanalytic Functions

We prove that a (globally) subanalytic function ${f : X \subset {\bf Q}^{n}_{p} \rightarrow {\bf Q}_{p}}We prove that a (globally) subanalytic function f : X ì Qⁿ_p ? Q_p{f : X \subset {\bf Q}^{n}_{p} \rightarrow {\bf Q}_{p}} which is locally Lipschitz continuous with some constant C is piecewise (globally on each piece) Lipschitz continuous with possibly some other constant, where the pieces can be taken to be subanalytic. We also prove the analogous result for a subanalytic family of functions f_y : X_y ì Qⁿ_p ? Q_p{f_{y} : X_{y} \subset {\bf Q}^{n}_{p} \rightarrow {\bf Q}_{p}} depending on p−adic parameters. The statements also hold in a semi-algebraic set-up and also in a finite field extension of Q _p . These results are p−adic analogues of results of K. Kurdyka over the real numbers. To encompass the total disconnectedness of p−adic fields, we need to introduce new methods adapted to the p−adic situation.     

The (p,q)-mixed geominimal surface areas1

Abstract

The (p, q)-mixed geominimal surface areas are introduced. A special case of the new concept is the L_p geominimal surface area introduced by Lutwak. Related inequalities, such as a?ne isoperimetric inequality, monotonous inequality, cyclic inequality, and Brunn-Minkowski inequality, are established. These new inequalities strengthen some well-known inequalities related to the L_p geominimal surface area.     

THREE NOTES ON NICE EXTENSION DOMAINS

The first note shows that the integral closure L′ of certain localities L over a local domain R are unmixed and analytically unramified, even when it is not assumed that R has these properties. The second note considers a separably generated extension domain B of a regular domain A, and a sufficient condition is given for a prime ideal p in A to be unramified with respect to B (that is, p B is an intersection of prime ideals and B/P is separably generated over A/p for all P ∈ Ass (B/p B)). Then, assuming that p satisfies this condition, a sufficient condition is given in order that all but finitely many q ∈ S = {q ∈ Spec(A), p ? q and height(q/p) = 1} are unramified with respect to B, and a form of the converse is also considered. The third note shows that if R′ is the integral closure of a semi-local domain R, then I(R) = ∩{R′ _p′ ;p′ ∈ Spec(R′) and altitude(R′/p′) = altitude(R′) ? 1} is a quasi-semi-local Krull domain such that: (a) height(N ^*) = altitude(R) for each maximal ideal N ^* in I(R); and, (b) I(R) is an H-domain (that is, altitude(I(R)/p ^*) = altitude(I(R)) ? 1 for all height one p ^* ∈ Spec(I(R))). Also, K = ∩{R _p ; p ∈ Spec(R) and altitude(R/p) = altitude(R) ? 1} is a quasi-semi-local H-domain such that height (N) = altitude(R) for all maximal ideals N in K.     

Strange bundles onP 2 and elementary transformations

Summary Here we begin the study of moduli of vector bundles on a surfaceS with a fixed restriction to a divisorD. Here we stress the caseD≊P ¹. In this way we construct many families of stable rank-2 bundles onP ² with unbalanced general splitting type (in characteristicp>0).

---

Riassunto Si comincia qui lo studio dei moduli di fibrati vettoriali su una superficieS con una assegnata restrizione ad un divisoreD (quasi sempre qui conD≊P ¹). In caratteristicap si ottengono così molte famiglie di fibrati stabili suP ² con ?strana? restrizione ad una retta generica.

     

On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Abstract

Let Ω be a bounded Lipschitz domain in ? ⁿ , n ≥ 3 with connected boundary. We study the Robin boundary condition ?u/?N + bu = f ∈ L ^p (?Ω) on ?Ω for Laplace's equation Δu = 0 in Ω, where b is a non-negative function on ?Ω. For 1 < p < 2 + ?, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ‖(?u)*‖ _p  ≤ C‖f‖ _p , as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.     

Vanishing Theorems for Conformally Compact Manifolds

Abstract

We study L ² harmonic p-forms on conformally compact manifolds with a rather weak boundary regularity assumption. We proved that if the lower bound of the curvature operator is great than or equal to ?1 and the infimum of the L ² spectrum of the Laplacian great than p(n ? p) for some p ≤ n/2, then there is no nontrivial L ² harmonic p-form.     

Large orbits of elements centralized by a Sylow subgroup

If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist ${v_1, v_2 \in V}If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist v₁, v₂ ? V{v_1, v_2 \in V} such that C_G(v₁)?C_G(v₂) = P{{\bf C}_{G}{(v_1)}\cap {\bf C}_{G}{(v_2)} = P} , where P ? Syl_p(G){P \in {\rm Syl}_p(G)} . We hence deduce that, if the normal p-complement K is nontrivial, there exists v ? C_V(P){v \in {\bf C}_{V}(P)} such that |K : C _K (v)|² > |K|.     

The Structure of Hereditary Properties and Colourings of Random Graphs

in the probability space ? Second, does there exist a constant such that the -chromatic number of the random graph is almost surely ? The second question was posed by Scheinerman (SIAM J. Discrete Math. 5 (1992) 74–80). The two questions are closely related and, in the case p=1/2, have already been answered. Pr?mel and Steger (Contemporary Mathematics 147, Amer. Math. Soc., Providence, 1993, pp. 167-178), Alekseev (Discrete Math. Appl. 3 (1993) 191-199) and the authors ( Algorithms and Combinatorics 14 Springer-Verlag (1997) 70–78) provided an approximation which was used by the authors (Random Structures and Algorithms 6 (1995) 353–356) to answer the -chromatic question for p=1/2. However, the approximating properties that work well for p=1/2 fail completely for . In this paper we describe a class of properties that do approximate in , in the following sense: for any desired accuracy of approximation, there is a property in our class that approximates to this level of accuracy. As may be expected, our class includes the simple properties used in the case p=1/2. The main difficulty in answering the second of our two questions, that concerning the -chromatic number of , is that the number of small -graphs in has, in general, large variance. The variance is smaller if we replace by a simple approximation, but it is still not small enough. We overcome this by considering instead a very rigid non-hereditary subproperty of the approximating property; the variance of the number of small -graphs is small enough for our purpose, and the structure of is sufficiently restricted to enable us to show this by a fine analysis. Received April 20, 1999     

Quasicentralizer of LCM-Homomorphisms

Let A _S and A _T be Artin–Tits groups and let ? _p : A _S  →  A _T be an LCM -homomorphism constructed using the folding method. We characterize the quasicentralizer of ? _p (S) in A _T in the case when T is of spherical type.     

On the U-Invariant of P-Adic Function Fields

ABSTRACT

We show that a quadratic form defined over the rational function field ?(x ₁ , …, _{x n} ) of dimension at least 4.2 ⁿ  + 1 is isotropic over all fields ? _p (x ₁ , …, _{x n} ), except for finitely many primes. Partial results concerning the u-invariant of p-adic function fields are also shown.     

Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques

In this paper we study formal moduli for wildly ramified Galois covering. We prove a local-global principle. We then focus on the infinitesimal deformations of the Z/p Z-covers. We explicitly compute a deformation of an automorphism of order p which implies a universal obstruction for p>2. By deforming Artin-Schreier equations we obtain a lower bound on the dimension of the local versal deformation ring. At last, by comparing the global versal deformation ring to the complete local ring in a point of a moduli space, we determine the dimensions of the global and local versal deformation rings.

---

Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques

---

Oblatum 22-II-1999 & 15-XII-1999?Published online: 29 March 2000     

Estimate of the Hölder constant for solutions to <Emphasis Type="Italic">m</Emphasis>-Hessian equations

An a priori estimate for the H?lder constant with exponent 0 < α < 1 is obtained for smooth solutions to m-Hessian equations in a closed domain with (m − 1)-convex boundary of class C ². The H?lder constant depends on the L ^p -norm, p > [n(n + 1)]/2, of the right-hand side of the equation, and the estimate remains valid for approximate solutions. Bibliography: 4 titles. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 69–76.