On the Lüroth semigroup of curves lying on a smooth quadric

For a smooth irreducible complete algebraic curveC the “gaps” are the integersn such that every linear series of degreen has at least a base point. The Lüroth semigroup S_C of a curveC is the subsemigroup ofN whose elements are not gaps. In this paper we deal with irreducible smooth curves of type (a, b) on a smooth quadricQ. The main result is an algorithm by which we can say if some integer λ∈N is a gap or is in S_C. In the general case there are integers λ which are undecidable. For curves such as complete intersection, arithmetically Cohen-Macaulay or Buchsbaum, we are able to describe explicitly “intervals” of gaps and “intervals” of integers which belong to S_C. For particular cases we can completely determine S_C, by giving just the type of the curve (in particular the degree and the genus). Work done with financial support of M.U.R.S.T. while the authors were members of G.N.S.A.G.A. of C.N.R.     

Convergence of the efficient sets

LetA _n,n=1, 2, ... be nonempty subsets of a linear metric spaceE andC _n, n=1, 2, ... convex cones ofE. We consider the efficient sets Min(A _n, C_n) and the aim of this paper is to show that under suitable conditions, the convergence ofA _n andC _n toA andC respectively, implies the convergence of Min(A _n,C _n) to Min(A, C). Several illustrative examples are given which clarify the results.     

A banach space containing non-trivial limited sets but no non-trivial bounding sets

An example of a Banach spaceE is given with the following properties: Every bounding setA⊂E (i.e.f(A) is bounded for each holomorphic functionf:E →C) is relatively compact but there are relatively non-compact limited setsA (i.e.T(A) is relatively compact for each bounded linear mapT:E →c ₀).     

Isoperimetric inequalities on minimal submanifolds of space forms

For a domainU on a certaink-dimensional minimal submanifold ofS ⁿ orH ⁿ, we introduce a “modified volume”M(U) ofU and obtain an optimal isoperimetric inequality forU k ^k ω _k M (D) ^k-1 ≤Vol(∂D) ^k , where ω _k is the volume of the unit ball ofR ^k . Also, we prove that ifD is any domain on a minimal surface inS ₊ ⁿ (orH ⁿ, respectively), thenD satisfies an isoperimetric inequality2π A≤L ²+A² (2π A≤L²−A² respectively). Moreover, we show that ifU is ak-dimensional minimal submanifold ofH ⁿ, then(k−1) Vol(U)≤Vol(∂U). Supported in part by KME and GARC     

A non-reflexive Grothendieck space that does not containl ∞

A compact spaceS is constructed such that, in the dual Banach spaceC(S)^*, every weak^* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol _∞. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact spaceT is constructed such thatC(T) does not containl _∞ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.     

Interaction of Legendre curves and Lagrangian submanifolds

It is proved in [8] that there exist no totally umbilical Lagrangian submanifolds in a complex-space-form ,n≥2, except the totally geodesic ones. In this paper we introduce the notion of LagrangianH-umbilical submanifolds which are the “simplest” Lagrangian submanifolds next to the totally geodesic ones in complex-space-forms. We show that for each Legendre curve in a 3-sphereS ³ (respectively, in a 3-dimensional antide Sitter space-timeH ₁ ³ ), there associates a LagrangianH-umbilical submanifold in ℂP ⁿ (respectively, in ℂH ⁿ ) via warped products. The main part of this paper is devoted to the classification of LagrangianH-umbilical submanifolds in ℂP ⁿ and in ℂH ⁿ . Our classification theorems imply in particular that “except some exceptional classes”, LagrangianH-umbilical submanifolds of ℂP ⁿ and of ℂH ⁿ are obtained from Legendre curves inS ³ or inH ₁ ³ via warped products. This provides us an interesting interaction of Legendre curves and LagrangianH-umbilical submanifolds in non-flat complex-space-forms. As an immediate by-product, our results provide us many concrete examples of LagrangianH-umbilical isometric immersions of real-space-forms into non-flat complex-space-forms.     

On the normal bundle of curves on nodal quartic surfaces

Let k be an algebraically closed field, char k = 0. Let C be an irreducible nonsingular curve such that 2C = S ? F, where S and F are two surfaces and all the singularities of F are rational double points (if any). We prove that C can never pass through rational singularities of types A _2n n∈N, E₆ and E₈. We give conditions for C to pass through rational singularities of types. A _2k+1 k∈Z⁺ D_n n≥4 and E₇, (0.8).     

Relations between some constants associated with finite dimensional Banach spaces

We define the asymmetry constants(E) of a Banach spaceE, and show examples of finite-dimensional spaces with “large” asymmetry constants. IfE isn-dimensional,λ(E)17its projection constant and π ₁(I _E ) the absolutely summing norm of the identity operatorI _E , thenn≦λ(E)π₁(I _E )≤n(s(E))². Similar equations linking thep-absolutely summing and the nuclear norms ofI _E are established. We also obtain estimates on these norms, for example π₂(I _E )=√n. The contribution of this author is a part of a Ph.D. Thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss whose guidance and valuable suggestions are gratefully acknowledged.     

On the description of overgroups of E(m,R)?E(n,R)

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL _m ⊗GL _n is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A ² ≤ B ≤ A and nA,A ² ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N _G (E(m,n,R, A,B,C)). Bibliography: 46 titles.     

Compactifying moduli of hyperelliptic curves

We construct a new compactification of the moduli spaceH _g of smooth hyperelliptic curves of genusg. We compare our compactification with other well-known remarkable compactifications ofH _g. The author was partially supported byCNP _q, Proc. 151610/2005-3, and by Faperj, Proc. E-26/152-629/2005.     

On the range of random walk

Let {S _n , n=0, 1, 2, …} be a random walk (S _n being thenth partial sum of a sequence of independent, identically distributed, random variables) with values inE _d , thed-dimensional integer lattice. Letf _n =Prob {S ₁ ≠ 0, …,S _n −1 ≠ 0,S _n =0 |S ₀=0}. The random walk is said to be transient if and strongly transient if . LetR _n =cardinality of the set {S ₀,S ₁, …,S _n }. It is shown that for a strongly transient random walk with p<1, the distribution of [R _n −np]/σ √n converges to the normal distribution with mean 0 and variance 1 asn tends to infinity, where σ is an appropriate positive constant. The other main result concerns the “capacity” of {S ₀, …,S _n }. For a finite setA inE _d , let C(A=Σ _x∈A ) Prob {S _n ∉A, n≧1 |S ₀=x} be the capacity ofA. A strong law forC{S ₀, …,S _n } is proved for a transient random walk, and some related questions are also considered. This research was partially supported by the National Science Foundation.     

Rates of convex approximation in non-hilbert spaces

This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis” subsetS of a function space. Specifically, the focus is on therate at which the lowest achievable error can be reduced as larger subsets ofS are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including, in particular, a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces; in particular, forL _p spaces with 1<p<∞, theO (n ^−1/2) bounds of Barron and Jones are recovered whenp=2. One motivation for the questions studied here arises from the area of “artificial neural networks,” where the problem can be stated in terms of the growth in the number of “neurons” (the elements ofS) needed in order to achieve a desired error rate. The focus on non-Hilbert spaces is due to the desire to understand approximation in the more “robust” (resistant to exemplar noise)L _p, 1 ≤p<2, norms. The techniques used borrow from results regarding moduli of smoothness in functional analysis as well as from the theory of stochastic processes on function spaces.     

Sausages are good packings

LetB ^d be thed-dimensional unit ball and, for an integern, letC _n ={x ¹,...,x ⁿ } be a packing set forB ^d , i.e.,|x ⁱ −x ^j |≥2, 1≤i<j≤n. We show that for every a dimensiond(ρ) exists such that, ford≥d(ρ),V(conv(C _n )+ρB ^d )≥V(conv(S _n )+ρB ^d ), whereS _n is a “sausage” arrangement ofn balls, holds. This gives considerable improvement to Fejes Tóth's “sausage” conjecture in high dimensions. Further, we prove that, for every convex bodyK and ρ<1/32d ⁻²,V(conv(C _n )+ρK)≥V(conv(S _n )+ρK), whereC _n is a packing set with respect toK andS _n is a minimal “sausage” arrangement ofK, holds.     

Intersections by hyperplanes

LetA be a finite nonempty family of nonempty disjoint closed and bounded sets in a Banach spaceE which is either separable and the conjugate of some Banach spaceX (i.e.E=X*) or, reflexive and locally uniformly convex. IfC denotes the weak*-closed convex hull of ∪ {A:A ∈A} then the set of points inE ∼C through which there is no hyperplane intersecting exactly one member ofA is discrete (or empty). This research was supported by the National Research Council of Canada, Grant A-3999.     

A note on the abstract cauchy problem

LetA be the infinitesimal generator of aC ₀ semigroup in a Banach spaceE. We obtain necessary conditions for a solution of the Cauchy problem {fx112-1} to be classical for arbitrary ϕ εC([0,T]) andf εE.     

Quantum anchor:U q (sl(2)) case

We introduce the tangent spaceT(H _q ) on the quantum hyperboloid (A _0,q ^c ) and equip it with an action onA _0,q ^c being a deformation of the action of vectors fields on functions. An embeddingsl(2) _q →T(H _q ) ofq-deformed Lie algebrasl(2) being an analogue of the anchorsl(2)→Vect (H) is called “quantum anchor”.     

Perturbations of regularizing maximal monotone operators

We consideru′(t)+Au(t)∋f(t), whereA is maximal monotone in a Hilbert spaceH. AssumeA is continuous or A=ϱφ or intD(A)≠∅ or dimH<∞. For suitably boundedf′s, it is shown that the solution mapf→u is continuous, even if thef′s are topologized much more weakly than usual. As a corollary we obtain the existence of solutions ofu′(t)+Au(t)∋B(u(t)), whereB is a compact mapping inH. An erratum to this article is available at .     

The structure of injective covers of special modules

In this paper, we prove that the injective cover of theR-moduleE(R/B)/R/B for a prime ideal B ofR is the direct sum of copies ofE(R/B) for prime ideals D ⊃ B, and if B is maximal, the injective cover is a finite sum of copies ofE(R/B). For a finitely generatedR-moduleM withn generators andG an injectiveR-module, we argue that the natural mapG ⁿ →G ⁿ/Hom _R (M, G) is an injective precover if Ext _R ¹ (M, R) = 0, and that the converse holds ifG is an injective cogenerator ofR. Consequently, for a maximal ideal R ofR, depth_R R ≧ 2 if and only if the natural mapE(R/R) →E(R/R)/R/R is an injective cover and depth_R R > 0.     

A new simplicial variable dimension algorithm to find equilibria on the product space of unit simplices

A new simplicial variable dimension restart algorithm is introduced to solve the nonlinear complementarity problem on the product spaceS of unit simplices. The triangulation which underlies the algorithm differs from the triangulations ofS used thus far. Moreover, the number of rays along which the algorithm can leave the arbitrarily chosen starting point is much larger. More precisely, there is a ray leading from the starting point to each vertex ofS. In caseS is the product ofn one-dimensional unit simplices the alogrithm is similar to the octahedral algorithm onR ⁿ having 2 ⁿ rays. Also, the accuracy of an approximate solution in the terminal simplex of the algorithm is in general better than for the other algorithms onS. Computational results will show that the number of iterations for the new algorithm is much less. The examples concern the computation of equilibria in noncooperative games, exchange economies and trade models. This author is financially supported by the Netherlands Organisation for the Advancement of Pure Research (ZWO), Grant 46-98. This research is part of the VF-program “Equilibrium and Disequilibrium in Demand and Supply” which has been approved by the Netherlands ministry of education and sciences.