Infinitesimal extensions of ℙ1 and their Hilbert schemes

 In order to calculate the multiplicity of an isolated rational curve C on a local complete intersection variety X, i.e. the length of the local ring of the Hilbert Scheme of X at [C], it is important to study infinitesimal neighborhoods of the curve in X. This is equivalent to infinitesimal extensions of ℙ¹ by locally free sheaves. In this paper we study infinitesimal extensions of ℙ¹, determine their structure and obtain upper and lower bounds for the length of the local rings of their Hilbert schemes at [ℙ¹]. Received: 11 June 2001 / Revised version: 28 January 2002     

On the Automorphism Group of a Free Pro-l Meta-abelian Group and an Application to Galois Representations

In § l of this article, we study group-theoretical properties of some automorphism group Ψ^* of the meta-abelian quotient § of a free pro-l group § of rank two, and show that the conjugacy class of some element of order two of Ψ^* is not determined by the action induced on the abelian quotient § of § in the case of § = 2. In § 2 we apply the results to the outer Galois representation § attached to the curve C deleted one point from an elliptic curve E, and give an example that §_c does not factor through the l-adic representation attached to E.     

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.     

Pyramids in the complex projective plane

We study the critical points of the diameter functional on the n-fold Cartesian product of the complex projective plane C P ² with the Fubini-Study metric. Such critical points arise in the calculation of a metric invariant called the filling radius, and are akin to the critical points of the distance function. We study a special family of such critical points, P _kC P ¹C P ², k=1,2... We show that P _k is a local minimum of by verifying the positivity of the Hessian of (a smooth approximation to) at P _k. For this purpose, we use Shirokov's law of cosines and the holonomy of the normal bundle of C P ¹C P ². We also exhibit a critical point of , given by a subset which is not contained in any totally geodesic submanifold of C P ².     

Inertial relaxed CQ algorithms for solving a split feasibility problem in Hilbert spaces

The split feasibility problem is to find a point x^? with the property that x^?∈ C and Ax^?∈ Q, where C and Q are nonempty closed convex subsets of real Hilbert spaces X and Y, respectively, and A is a bounded linear operator from X to Y. The split feasibility problem models inverse problems arising from phase retrieval problems and the intensity-modulated radiation therapy. In this paper, we introduce a new inertial relaxed CQ algorithm for solving the split feasibility problem in real Hilbert spaces and establish weak convergence of the proposed CQ algorithm under certain mild conditions. Our result is a significant improvement of the recent results related to the split feasibility problem.

     

Normal forms for singularities of pedal curves produced by non-singular dual curve germs in <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

For an n-dimensional spherical unit speed curve r and a given point P, we can define naturally the pedal curve of r relative to the pedal point P. When the dual curve germs are non-singular, singularity types of such pedal curves depend only on locations of pedal points. In this paper, we give a complete list of normal forms for singularities and locations of pedal points when the dual curve germs are non-singular. As an application of our list, we characterize C ^∞ left equivalence classes of pedal curve germs (I, s ₀) → S ⁿ produced by non-singular dual curve germ from the viewpoint of the relation between tangent space and tangent space.      

Iterative approximation to common fixed points of a countable family of nonexpansive mappings

We proved several strong convergence results by using the conception of a uniformly asymptotically regular sequence {T _n } of nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping J _?(l ^p (1?p?t)?=?t ^p?1. The results presented develop and complement the corresponding ones by Song, Y. and Chen, R., 2007 [Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces. Nonlinear Analysis, 66, 591–603], Song, Y., Chen, R. and Zhou, H., 2007 [Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear Analysis, 66, 1016–1024] and O'Hara, J.G., Pillay, P. and Xu, H.K., 2006 [Iterative approaches to convex feasibility problem in Banach Space. Nonlinear Analysis, 64, 2022–2042], O'Hara, J.G., Pillay, P. and Xu, H.K., 2003 [Iterative approaches to fineding nearest common fixed point of nonexpansive mappings in Hilbert spaces. Nonlinear Analysis, 54, 1417–1426] and Jung, J.S., 2005 [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications, 302, 509–520] and many other existing literatures.     

Reducibility of the families of 0-dimensional schemes on a variety

IfA is a regular local ring of dimensionr>2, over an algebraically closed fieldk, we show that the Hilbert scheme Hilb ⁿ A parametrizing ideals of colengthn inA(dim _k A/I=n) has dimension>cn ^2?2/r and is reducible, for alln>c′, wherec andc′ depend only onr. We conclude that ifV is a nonsingular projective variety of dimensionr>2, the Hilbert scheme Hilb ⁿ V parametrizing the 0-dimensional subschemes ofV having lengthn, is reducible for alln>c″(r). We may takec″(r) to be (1) $$102 ifr = 3,25 ifr = 4,35 ifr = 5,and\left( {1 + r} \right)\left( {{{1 + r} \mathord{\left/ {\vphantom {{1 + r} 4}} \right. \kern-\nulldelimiterspace} 4}} \right)ifr > 5.$$ The result answers in the negative a conjecture of Fogarty [1] but leaves open the question of the conjectured irreducibility of Hilb ⁿ A, whereA has dimension 2. Hilb ⁿ V is known to be irreducible ifV is a nonsingular surface (Hartshorne forP ₂, and Fogarty [1]). In all cases Hilb ⁿ V and Hilb ⁿ A are known to be connected (Hartshorne forP _r, and Fogarty [1]). The author is indebted to Hartshorne for suggesting that Hilb ⁿ A might be reducible ifr>2. The proof has 3 steps. We first show that ifV is a variety of dimensionr, then Hilb ⁿ V is irreducible only if it has dimensionr n. We then show that ifA is a regular local ring of dimensionr, Hilb ⁿ A can be irreducible only if it has dimension (r?1)(n?1). Finally in § 3 we construct a family of graded ideals of colengthn in the local ringA, and having dimensionc′ n^2?2/r. Since for largen this dimension is greater thanr n, and since Hilb ⁿ A?Hilb ⁿ V whenA is the local ring of a closed point onV, the proof is complete, except for (1), which follows from § 3, and the monotonicity of (dim Hilb ⁿ V?r n) (see (2)). In § 4, we comment on some related questions.     

Strong Convergence Theorems for Maximal Monotone Operators with Nonlinear Mappings in Hilbert Spaces

Let C be a closed and convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself, A be an α-inverse strongly-monotone mapping of C into H and let B be a maximal monotone operator on H, such that the domain of B is included in C. We introduce an iteration scheme of finding a point of F (T)∩(A+B)⁻¹0, where F (T) is the set of fixed points of T and (A+B)⁻¹0 is the set of zero points of A+B. Then, we prove a strong convergence theorem, which is different from the results of Halpern’s type. Using this result, we get a strong convergence theorem for finding a common fixed point of two nonexpansive mappings in a Hilbert space. Further, we consider the problem for finding a common element of the set of solutions of a mathematical model related to equilibrium problems and the set of fixed points of a nonexpansive mapping.     

Homotopy Type of Complexes of Graph Homomorphisms between Cycles

In this paper we study the homotopy type of Hom(C_m,C_n), where C_k is the cyclic graph with k vertices. We enumerate connected components of Hom(C_m,C_n) and show that each such component is either homeomorphic to a point or homotopy equivalent to S¹. Moreover, we prove that Hom(C_m,L_n) is either empty or is homotopy equivalent to the union of two points, where L_n is an n-string, i.e., a tree with n vertices and no branching points.     

Some remarks on hilbert functions of veronese algebras

We study the Hilbert polynomials of finitely generated graded algebras R, with generators not all of degree one (i.e. non-standard). Given an expression P(R,t)=a(t)/(1-t^l ) ⁿ for the Poincare series of R as a rational function, we study for 0 ≤ i ≤ l the graded subspaces ? _kR_kl+i (which we denote R[l;i]) of R, in particular their Poincaré series and Hilbert functions. We prove, for example, that if R is Cohen-Macaulay then the Hilbert polynomials of all non-zeroR[l;i] share a common degree. Furthermore, if R is also a domain then these Hilbert polynomials have the same leading coefficient.     

Closing Aubry Sets II

Given a Tonelli Hamiltonian H:T*M → ? of class C^k, with k ≥ 4, we prove the following results: (1) Assume there is a critical viscosity subsolution that is of class C^{k + 1} in an open neighborhood of a positive orbit of a recurrent point of the projected Aubry set. Then there exists a potential V : M → ? of class C^k?1, small in the C² topology, for which the Aubry set of the new Hamiltonian H + V is either an equilibrium point or a periodic orbit. (2) For every ? > 0 there exists a potential V : M → ? of class C^k?2, with for which the Aubry set of the new Hamiltonian H + V is either an equilibrium point or a periodic orbit. The latter result solves in the affirmative the Mañé density conjecture in the C¹ topology. © 2015 Wiley Periodicals, Inc.     

A Dual Approach to Constrained Interpolationfrom a Convex Subset of Hilbert Space

Many interesting and important problems of best approximationare included in (or can be reduced to) one of the followingtype: in a Hilbert spaceX, find the best approximationP_K(x) to anyxXfrom the setKC∩A⁻¹(b),whereCis a closed convex subset ofX,Ais a bounded linearoperator fromXinto a finite-dimensional Hilbert spaceY, andbY. The main point of this paper is to show thatP_K(x)isidenticaltoP_C(x+A*y)—the best approximationto a certain perturbationx+A*yofx—from the convexsetCor from a certain convex extremal subsetC_bofC. Thelatter best approximation is generally much easier to computethan the former. Prior to this, the result had been known onlyin the case of a convex cone or forspecialdata sets associatedwith a closed convex set. In fact, we give anintrinsic characterizationof those pairs of setsCandA⁻¹(b) for which this canalways be done. Finally, in many cases, the best approximationP_C(x+A*y) can be obtained numerically from existingalgorithms or from modifications to existing algorithms. Wegive such an algorithm and prove its convergence     

Cantor exchange systems and renormalization

We prove a one-to-one correspondence between (i) C¹⁺ conjugacy classes of C^1+H Cantor exchange systems that are C^1+H fixed points of renormalization and (ii) C¹⁺ conjugacy classes of C^1+H diffeomorphisms f with a codimension 1 hyperbolic attractor Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. However, we prove that there is no C^1+α Cantor exchange system, with bounded geometry, that is a C^1+α fixed point of renormalization with regularity α greater than the Hausdorff dimension of its invariant Cantor set.     

On the distribution of typical shortest-path lengths in connected random geometric graphs

Stationary point processes in ?² with two different types of points, say H and L, are considered where the points are located on the edge set G of a random geometric graph, which is assumed to be stationary and connected. Examples include the classical Poisson–Voronoi tessellation with bounded and convex cells, aggregate Voronoi tessellations induced by two (or more) independent Poisson processes whose cells can be nonconvex, and so-called β-skeletons being subgraphs of Poisson–Delaunay triangulations. The length of the shortest path along G from a point of type H to its closest neighbor of type L is investigated. Two different meanings of “closeness” are considered: either with respect to the Euclidean distance (e-closeness) or in a graph-theoretic sense, i.e., along the edges of G (g-closeness). For both scenarios, comparability and monotonicity properties of the corresponding typical shortest-path lengths C ^e? and C ^g? are analyzed. Furthermore, extending the results which have recently been derived for C ^e?, we show that the distribution of C ^g? converges to simple parametric limit distributions if the edge set G becomes unboundedly sparse or dense, i.e., a scaling factor κ converges to zero and infinity, respectively.     

The number of nearest and farthest points to a compactum in Euclidean space

A typical (in the sense of Baire category) compactA inE, whereE is either the Euclidean spaceE ⁸,s≧2, or the separable Hilbert space ℍ, generates a dense subsetC ^n,m(A) of the underlying space, such that everyx∈C ^n,m(A) has exactlyn nearest andm farthest points fromA, whenevern andm are positive integers satisfyingn+m≦ dimE+2. Research of this author is in part supported by Consiglio Nazionale delle Ricerche, G.N.A.F.A., Italy.     

Projective models of Enriques surfaces in scrolls

We study Cossec's ? ‐function, which is defined for divisors with positive self‐intersection on an Enriques surface. In this paper we study the existence of pairs (C ², ? (C )) with C an irreducible curve. The ? ‐function gives in a natural way scrolls containing Enriques surfaces. We compute scroll types to some of these scrolls. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Norm Retrievable Frames and Their Perturbation in Finite Dimensional Complex Hilbert Spaces

In this article, we study the property of norm retrievability of spanning vectors in a finite dimensional complex Hilbert space ?. Using the set of zero trace operators on ? and two sets of self-adjoint operators on ? denoted by 𝒮^1,0 and 𝒮^1,1, we present some equivalent conditions to the norm retrievable frames in ?. We will also show that the property of norm retrievability for n-dimensional complex Hilbert space ? with n≠2 is stable under enough small perturbation of the frame set only for phase retrievable frames.     

An existence theorem for invariant manifolds

For systems of the form =Ax +F ₁ (x, y, z), =By +F ₂ (x, y,z), =Cz +F ₃ (x, y, z) possessingP = {(0,0,z)} as invariant manifold we present sufficient conditions for the extension ofP to an invariant manifold of the form (x, s (x, z), z). Hereby we assume that the spectrum _A ofA is located to the left and the spectrum _b ofB to the right of a vertical straight linel in . In the case where the spectrum _C ofC lies to the left ofl too such an extension ofP is rather simple. We consider the situation where _{A C} cannot be separated from _B by a vertical line in .

---

Zusammenfassung Für Systeme der Form =Ax +F ₁ (x, y, z), =By +F ₂ (x, y,z), =Cz +F ₃ (x, y, z) mitP = {(0,0,z)} als invarianter Mannigfaltigkeit geben wir Bedingungen an, unter welchen sichP zu einer invarianten Mannigfaltigkeit der Form (x, s (x, z), z) fortsetzen läßt. Wir gehen stets davon aus, daß das Spektrum _A vonA links und das Spektrum _B vonB rechts einer vertikalen Geradenl in liegen. Eine derartige Fortsetzung vonP ist einfach, falls das Spektrum _C vonC ebenfalls links vonl liegt. Wir untersuchen den Fall, in welchem _{A C} sich nicht durch eine vertikale Gerade von _B trennen läßt.

---

---

Dedicated to H. W. Knobloch on the occasion of his sixtieth birthday

---

Supported by the Volkswagenwerk foundation