On the biggest maximally generated ideal as the conductor in the blowing up ring

Let (R, M) be a Noetherian one-dimensional local ring.C Gottlieb calls anM-primary idealI maximally generated ifμ(I)=?(R/(r)), or which is the same, ifIM=rI for somer∈M, and he also proves that if there is a maximally generated ideal inR then there is a unique biggest one (see [4]). In this paper each ring (R, M) is a local one-dimensional Cohen-Macaulay ring. LetQ be the total ring of fractions ofR, and letB(M) be the ring obtained by blowing upM, i.e.B (M)=U _i≥1 (M ⁱ :M ⁱ ) _Q . We prove in Theorem 1 that if there are maximally generated ideals inR then they are theM-primary ideals ofR which are ideals ofB(M) too. And the biggest maximally generated idealÎ ofR is the conductor ofR inB(M), i.e.(R∶B(M)) _R . We give in Theorem 3 an algorithm for findingÎ when the integral closure ofR is a local domain with the same residue field asR. In section 3 there are applications to semigroup rings. We prove thatÎ is generated by monomials in Proposition 7, and therefore semigroups are considered in the continuation. Let σ be the reduction exponent ofM, i.e. δ=min{i∶?(M ⁱ /M ⁱ⁺¹) =e(M)} wheree(M) denotes the multiplicity ofM. In Proposition 10, δ is determined, and there is also given a sufficient condition forÎ not to be a power ofM. In Propositions 11 and 12Î is determined for two special cases of semigroup rings whereÎ is a power ofM.     

Asset Pricing with Stochastic Volatility

In this paper we study the asset pricing problem when the volatility is random. First, we derive a PDE for the risk-minimizing price of any contingent claim. Secondly, we assume that the volatility process \si _t is observed through an observation process Y _t subject to random error. A price formula and a PDE are then derived regarding the stock price S _t and the observation process Y _t as parameters. Finally, we assume that S _t is observed. In this case we have a complete market and any contingent claim is then priced by an arbitrage argument instead of by risk-minimizing. Accepted 15 August 2000. Online publication 8 December 2000.     

Contact points of convex bodies

LetB be a convex body in ? ⁿ and let ? be an ellipsoid of minimal volume containingB. By contact points ofB we mean the points of the intersection between the boundaries ofB and ?. By a result of P. Gruber, a generic convex body in ? ⁿ has (n+3)·n/2 contact points. We prove that for every ?>0 and for every convex bodyB ? ? ⁿ there exists a convex bodyK having $$m \leqslant C(\varepsilon ) \cdot n\log ^3 n$$ contact points whose Banach-Mazur distance toB is less than 1+?. We prove also that for everyt>1 there exists a convex symmetric body Γ ? ? ⁿ so that every convex bodyD ? ? ⁿ whose Banach-Mazur distance to Γ is less thant has at least (1+c ₀/t ²)·n contact points for some absolute constantc ₀. We apply these results to obtain new factorizations of Dvoretzky-Rogers type and to estimate the size of almost orthogonal submatrices of an orthogonal matrix.     

Essential and inessential points of local nonconvexity

LetS be a closed connected subset of a Hausdorff linear topological space,Q the points of local nonconvexity ofS, E the essential members ofQ, N the inessential. IfS～Q is connected, then the following are true: Theorem 1.If Q ?Øis countable, then S is planar. Theorem 2.If Q is finite and nonempty, then cardE≧cardN+1. Theorem 3.If SυR ² and N is infinite, then E is infinite.     

Finite projective spaces and intersecting hypergraphs

Let ? be a family ofk-subsets on ann-setX andc be a real number 0 <c<1. Suppose that anyt members of ? have a common element (t ≧ 2) and every element ofX is contained in at mostc|?| members of ?. One of the results in this paper is (Theorem 2.9): If $$c = {{(q^{t - 1} + ... + q + 1)} \mathord{\left/ {\vphantom {{(q^{t - 1} + ... + q + 1)} {(q^t + ... + q + 1)}}} \right. \kern-\nulldelimiterspace} {(q^t + ... + q + 1)}}$$ . whereq is a prime power andn is sufficiently large, (n >n (k, c)) then The corresponding lower bound is given by the following construction. LetY be a (q ^t + ... +q + 1)-subset ofX andH ₁,H ₂, ...,H _|Y| the hyperplanes of thet-dimensional projective space of orderq onY. Let ? consist of thosek-subsets which intersectY in a hyperplane, i.e., ?={F∈( _k ^X ): there exists ani, 1≦i≦|Y|, such thatY∩F=H _i }.     

On the power of the zeros of Bessel functions

For ν≥0 let c_νk be the k-th positive zero of the cylinder functionC _v(t)=J _v(t)cosα-Y _v(t)sinα, 0≤α>π whereJ _ν(t) andY _ν(t) denote the Bessel functions of the first and the second kind, respectively. We prove thatC _v,k ^1+H(x) is convex as a function of ν, ifc _νk≥x>0 and ν≥0, whereH(x) is specified in Theorem 1.1.     

Matroids with given restrictions and contractions

If M is a matroid on a set S and if X is a subset of S, then there are two matroids on X induced by M: namely, the restriction and the contraction of M onto X. Necessary and sufficient conditions are obtained for two matroids on the same set to be of this form and an analogous result is obtained when (X₁,…, X_t) is a partition of S. The corresponding results when all the matroids are binary are also obtained.     

Farthest points in weakly compact sets

LetS be a weakly compact subset of a Banach spaceB. We show that of all points inB which have farthest points inS contains a denseG ₅ ofB. Also, we give a necessary and sufficient condition for bounded closed convex sets to be the closed convex hull of their farthest points in reflexive Banach spaces.     

Metrics with harmonic spinors

We show that every closed spin manifold of dimensionn 3 mod 4 with a fixed spin structure can be given a Riemannian metric with harmonic spinors, i.e. the corresponding Dirac operator has a non-trivial kernel (Theorem A). To prove this we first compute the Dirac spectrum of the Berger spheresS ⁿ ,n odd (Theorem 3.1). The second main ingredient is Theorem B which states that the Dirac spectrum of a connected sumM ₁#M ₂ with certain metrics is close to the union of the spectra ofM ₁ and ofM ₂.Partially supported by SFB 256 and by the GADGET program of the EU     

Answer to a conjecture of J.M. Cohen

The origin of Gelfand rings comes from [9] where the Jacobson topology and the weak topology are compared. The equivalence of these topologies defines a regular Banach algebra. One of the interests of these rings resides in the fact that we have an equivalence of categories between vector bundles over a compact manifold and finitely generated projective modules over C(M), the ring of continuous real functions on M [17].These rings have been studied by R. Bkouche (soft rings [3]) C.J. Mulvey (Gelfand rings [15]) and S. Teleman (harmonic rings [19]).Firstly we study these rings geometrically (by sheaves of modules (Theorem 2.5)) and then introduce the ?ech covering dimension of their maximal spectrums. This allows us to study the stable rank of such a ring A (Theorem 6.1), the nilpotence of the nilideal of K₀(A) - The Grothendieck group of the category of finitely generated projective A-modules - (Theorem 9.3), and an upper limit on the maximal number of generators of a finitely generated A-module as a function of the afore-mentioned dimension (Theorem 4.4).Moreover theorems of stability are established for the group K₀(A), depending on the stable rank (Theorems 8.1 and 8.2). They can be compared to those for vector bundles over a finite dimensional paracompact space [18].Thus there is an analogy between finitely generated projective modules over Gelfand rings and ?ech dimension, and finitely generated projective modules over noetherian rings and Krull dimension.     

Properties of Hida processes on . 2. Prediction and interpolation problems for processes on

If E is an ordered set, we study the processes Y_t, t E, for which the vectorial spaces _t generated by all the conditional expectations E(Y_sβ _t) for s ≥ t have finite dimensions d(t) ≤ N. ( _t is some convenient filtration.) We first develop a geometrical approach in the general situation and give a “Goursat's representation” Y_t = Σf_i(t)M_i(t), where the M_i(t) are martingales. We then restrict us to the cases E = or E = ² and give representations of the processes by the mean of stochastic integrals of “Goursat's kernels.” The special case when Y_t is the solution of a differential equation is considered.     

Spatiality of Derivations on the Algebra of τ-Compact Operators

This paper is devoted to derivations on the algebra S ₀(M, τ) of all τ-compact operators affiliated with a von Neumann algebra M and a faithful normal semi-finite trace τ. The main result asserts that every t _τ -continuous derivation ${D : S_0(M, \tau) \rightarrow S_{0}(M, \tau)}$ is spatial and implemented by a τ-measurable operator affiliated with M, where t _τ denotes the measure topology on S ₀(M, τ). We also show the automatic t _τ -continuity of all derivations on S ₀(M, τ) for properly infinite von Neumann algebras M. Thus in the properly infinite case the condition of t _τ -continuity of the derivation is redundant for its spatiality.     

Bounding the number of bases of a matroid

Letb(M) andc(M), respectively, be the number of bases and circuits of a matroidM. For any given minor closed class? of matroids, the following two questions, are investigated in this paper. (1) When is there a polynomial functionp(x) such thatb(M)≤p(c(m)|E(M)|) for every matroidM in?? (2) When is there a polynomial functionp(x) such thatb(M)≤p(|E(M)|) for every matroidM in?? Let us denote byM _Mn the direct sum ofn copies ofU _1,2. We prove that the answer to the first question is affirmative if and only if someM _Mn is not in?. Furthermore, if all the members of? are representable over a fixed finite field, then we prove that the answer to the second question is affirmative if and only if, also, someM _Mn is not in?.     

An exchange theorem for bases of matroids

Theorem. Given two basesB₁andB₂of a matroid ($\text{M}$, r), and a partitionB₁ = X₁ ∪ Y₁, there is a partitionB₂ = X₂ ∪ Y₂such thatX₁ ∪ Y₂andX₂ ∪ Y₁are both bases of$\text{M}$.     

Asymptotic convergence of nonlinear contraction semigroups in Hilbert space

Let S be a contraction semigroup on a closed convex subset C of a Hilbert space. If the generator of S satisfies a strengthened monotonicity condition then the weak lim_{t → ∞}S(t)x exists for all x in C. As one consequence, the method of steepest descent converges weakly for convex functions in Hilbert space; and it converges strongly for even convex functions.     

Noncommutative classical invariant theory

In this thesis, we consider some aspects ofnoncommutative classical invariant theory, i.e., noncommutative invariants ofthe classical group SL(2, k). We develop asymbolic method for invariants and covariants, and we use the method to compute some invariant algebras. The subspace? _d ^m of the noncommutative invariant algebra? _d consisting of homogeneous elements of degreem has the structure of a module over thesymmetric group S _m . We find the explicit decomposition into irreducible modules. As a consequence, we obtain theHilbert series of the commutative classical invariant algebras. TheCayley—Sylvester theorem and theHermite reciprocity law are studied in some detail. We consider a new power series H(? _d,t) whose coefficients are the number of irreducibleS _m -modules in the decomposition of? _d ^m , and show that it is rational. Finally, we develop some analogues of all this for covariants.     

Generalized contraction mapping principles in probabilistic metric spaces

We give an improvement of Theorem 1 from [2] with a quite different approach, which enable us to prove that the fixed point is also globally attractive. In Theorem 2.11 a further generalization is obtained for a complete Menger space (S,F,T), where T belongs to a more general class of continuous t-norms than in the previous case where T=T _M (=min). Theorem 3.2 is a generalization of Theorem 2 from [2]. Thereafter the notion of a generalized C-contraction of Krasnoselski's type is introduced and a fixed point theorem for such mappings is proved. An application in the space S(Ω, K, P) is given.     

Voisinages dérivés ε-barycentriques

Let Y be a finite full subcomplex of a simplicial complex X. For any subdivision X′ of X keeping Y invariant, and for ε small enough relatively to X′, we define the ε-barycentric derived neighbourhood V_ε(X′,Y) of Y in X′. Theorem: for small enoughε, and for any simplexKofY, the transverse stars ofKinV_ε(X,Y) andV_ε(X′,Y) have the same support. As a consequence, we derive at the end of the paper a decomposition theorem for p.l. homeomorphisms of a polyhedron keeping a finite subpolyhedron invariant. Keywords: Polyhedron; Simplicial complex; Derived neighbourhood; p.l. homeomorphism     

Hypersurfaces through higher-codimensional submanifolds of ℂ n with preserved Levi-kernelwith preserved Levi-kernel

For a “generic” submanifoldS of a complex manifoldX, we show that there exists a hypersurfaceM⊃S which has the same number of negative (or positive) Levi-eigenvalues asS at one prescribed conormal (cf. also [9]). When ranksL _S is constant, thenM may be found such thatL _M andL _S have the same number of negative eigenvalues at any common conormal. Assuming the existence of a hypersurfaceM with the above property, we then discuss the link between complex submanifolds ofS whose tangent plane belongs to the null-space of the Levi-formL _S ofS (of all complex submanifolds whenL _S is semi-definite), and complex submanifolds ofT _S ^* X. As an application we give a simple result on propagation of microanalyticity for CR-hyperfunctions along complex,L _S -null, curves (cf. [3]).