Storage capacity of a dam with gamma type inputs

Summary Consider mutually independent inputsX ₁,...,X _n onn different occasions into a dam or storage facility. The total input isY=X ₁+...+X _n. This sum is a basic quantity in many types of stochastic process problems. The distribution ofY and other aspects connected withY are studied by different authors when the inputs are independently and identically distributed exponential or gamma random variables. In this article explicit exact expressions for the density ofY are given whenX ₁,...,X _n are independent gamma distributed variables with different parameters. The exact density is written as a finite sum, in terms of zonal polynomials and in terms of confluent hypergeometric functions. Approximations whenn is large and asymptotic results are also given.     

On the multiples of the adjoint divisor to an ample divisor on ad-fold

Summary LetX be a complex manifold of dimensiond≥3 andH an ample divisor onX. In this paper the global sections of the sheafA _X(n(K _X +(d-2)H)) are studied whenn∈Z, this generalizing a result by A. J. Sommese.

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Riassunto SiaX una varietà complessa di dimensioned≥3 eH un divisore ampio suX. In questo articolo vengono studiate le sezioni globali del fascioA _X(n(K _X +(d-2)H)), doven∈Z. Cià permette di generalizzare un risultato di A. J. Sommese.

     

On the local minimaxity of a test of independence in incomplete samples

Summary LetX=(X ₁,X ₂) be ap-dimensional normal variable. Suppose that we haven independent observations onX andm, independent observations onX _i ,i=1,2, respectively. For testing the independence betweenX ₁ andX ₂, Eaton and Kariya (1983),Ann. Statist.,11, 654–665) obtains a locally most powerful test among the class of tests invariant under the group of affine transformations. In this paper, this LMPI test is shown to be locally minimax within the class of all tests of the same level.     

Estimation of the extreme value and the extreme points

Summary Letf be a continuous function defined on some domainA andX ₁,X ₂, ... be iid random variables. We estimate the extreme value off onA by studying the limiting distribution of min {f(X ₁), ...,f(X _n )} or max {f(X ₁), ...,f(X _n )} properly normalized. Sufficient conditions for the existence of the limiting distribution as well as a characterization of the limiting distribution relative to the extreme points off will be provided. A discussion of the multidimensional case is also carried out. Partially supported by CNPq-No. 301508/84.     

Cotype 2 estimates for spaces of polynomials on sequence spaces

We give asymptotically correct estimations for the cotype 2 constant C₂(P(^m X _n )) ofthe spaceP(^m X _n ) of allm-homogenous polynomials onX _n , the span of the firstn sequencese _k =(\gd _kj ) _j in a Banach sequence spaceX. Applications to Minkowski, Orlicz and Lorentz sequence spaces are given. The second author was supported by DGESIC pr. no. 96-0758. The third author was supported by a grant of the Ministerio de Educación y Cultura (Spain), FP-97 29183763.     

Representability of Hom implies flatness

LetX be a projective scheme over a noetherian base schemeS, and letF be a coherent sheaf onX. For any coherent sheaf ε onX, consider the set-valued contravariant functor Hom(ε,F)S-schemes, defined by Hom(ε,F) (T)= Hom(ε _T ,F _T) where ε _T andF _T are the pull-backs of ε andF toX _T =X x _S T. A basic result of Grothendieck ([EGA], III 7.7.8, 7.7.9) says that ifF is flat over S then Kom_ε,F) is representable for all ε. We prove the converse of the above, in fact, we show that ifL is a relatively ample line bundle onX over S such that the functor Hom(L ^-n ,F) is representable for infinitely many positive integersn, thenF is flat overS. As a corollary, takingX =S, it follows that ifF is a coherent sheaf on S then the functorT ↦H°(T, F _t) on the category ofS-schemes is representable if and only ifF is locally free onS. This answers a question posed by Angelo Vistoli. The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author’s earlier result (see [N1]) that the automorphism group functor of a coherent sheaf onS is representable if and only if the sheaf is locally free.     

Two-gate formula

The two gate formula for a diffusionX _t inR ⁿ is considered. The formula gives an expression of the density function ofX _t in terms of the path integral of the Brownian bridge starting fromx and ending ony at timet.     

Functions of ann-dimensional Brownian motion that are Markovian

Letf be a continuous function fromR ⁿ toR and letX(t)=(X ₁ (t), …, X _n (t)) be a Brownian motion onR ⁿ . The explicit form off necessary in order to makef(X(t)) a Markov process is determined.     

Sobolev embeddings into BMO, VMO, andL ∞

LetX be a rearrangement-invariant Banach function space onR ⁿ and letV ¹ X be the Sobolev space of functions whose gradient belongs toX. We give necessary and sufficient conditions onX under whichV ¹ X is continuously embedded into BMO or intoL _∞. In particular, we show thatL _{n, ∞} is the largest rearrangement-invariant spaceX such thatV ¹ X is continuously embedded into BMO and, similarly,L _{n, 1} is the largest rearrangement-invariant spaceX such thatV ¹ X is continuously embedded intoL _∞. We further show thatV ¹ X is a subset of VMO if and only if every function fromX has an absolutely continuous norm inL _{n, ∞} . A compact inclusion ofV ¹ X intoC ⁰ is characterized as well.     

Kernel-type density and failure rate estimation for associated sequences

Let {X _n ,n1} be a strictly stationary sequence of associated random variables defined on a probability space (,B, P) with probability density functionf(x) and failure rate functionr(x) forX ₁. Letf _n (x) be a kerneltype estimator off(x) based onX ₁,...,X _n . Properties off _n (x) are studied. Pointwise strong consistency and strong uniform consistency are established under a certain set of conditions. An estimatorr _n (x) ofr(x) based onf _n (x) andF _n (x), the empirical survival function, is proposed. The estimatorr _n (x) is shown to be pointwise strongly consistent as well as uniformly strongly consistent over some sets.     

On the average distance property and certain energy integrals

One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceR ⁿ andr(X, d ₂) the rendezvous number ofX, whered ₂ denotes the Euclidean distance inR ⁿ . (The rendezvous numberr(X, d ₂) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x ₁,x ₂,...,x _n inX, there exists somex inX such that .) Then there exists some regular Borel probability measure μ₀ onX such that the value of ∫ _X d ₂(x, y)dμ₀ (y) is independent of the choicex inX, if and only ifr(X, d ₂) = sup_μ ∫ _X ∫ _X d ₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX.     

Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces

LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let X_hol be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on X_hol (X normal algebraic surface) by algebraic vector bundles.     

Stability of testing hypotheses

The stability of testing hypotheses is discussed. Differing from the usual tests measured by Neyman-Pearson lemma, the regret and correction of the tests are considered. After the decision is made based on the observationsX ₁,X ₂, ⋅⋅⋅,X _n, one more piece of datumX _n+1 is picked and the test is done again in the same way but based onX ₁,X ₂, ⋅⋅⋅,X _n,X _n+l There are three situations: (i) The previous decision is right but the new decision is wrong; (ii) the previous decision is wrong but the new decision is right; (iii) both of them are right or both of them are wrong. Of course, it is desired that the probability of the occurrence of (i) is as small as possible and the probability of the occurrence of (ii) is as large as possible. Since the sample size is sometimes not chosen very precisely after the type I error and the type II error are determined in practice, it seems more urgent to consider the above problem. Some optimal plans are also given. Project supported by the National Natural Science Foundation of China and the Doctoral Programme Foundation.     

On multivariate Gaussian tails

Let {X _{n, n} ≥1} be a sequence of standard Gaussian random vectors in ℝ _d ,d ≥ 2. In this paper we derive lower and upper bounds for the tail probabilityP{X _n >t _n } witht _n ∈ ℝ _d some threshold. We improve for instance bounds on Mills ratio obtained by Savage (1962,J. Res. Nat. Bur. Standards Sect. B,66, 93–96). Furthermore, we prove exact asymptotics under fairly general conditions on bothX _n andt _n , as ‖t _n ‖→∞ where the correlation matrix Σ _n ofX _n may also depend onn.     

Symbolic representations of nonexpansive group automorphisms

Ifα is an irreducible nonexpansive ergodic automorphism of a compact abelian groupX (such as an irreducible nonhyperbolic ergodic toral automorphism), thenα has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts inX. In spite of this we are able to construct a symbolic spaceV and a class of shift-invariant probability measures onV each of which corresponds to anα-invariant probability measure onX. Moreover, everyα-invariant probability measure onX arises essentially in this way. The last part of the paper deals with the connection between the two-sided beta-shiftV _β arising from a Salem numberβ and the nonhyperbolic ergodic toral automorphismα arising from the companion matrix of the minimal polynomial ofβ, and establishes an entropy-preserving correspondence between a class of shift-invariant probability measures onV _β and certainα-invariant probability measures onX. This correspondence is much weaker than, but still quite closely modelled on, the connection between the two-sided beta-shifts defined by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms.     

A note on sub-multiplicative norms

Summary IfX is a finite-dimensional linear space andL(X) the linear space of linear operators onX thenL(X) may be represented asXX ^*. IfE={e ₁, ...,e _n } is a basis forX and e _j y _j ^* is a typical element ofXX ^*, then norms can be introduced onL(X) in the form y _j ^* e _j . Given that the norm onX isE-absolute we derive a necessary and sufficient condition for the norm onL(X) to be submultiplicative.     

On the singularities of foliations and of vector bundle maps

LetF be a (smooth) Γ _q ^∞ -stucture (often called a codimension-q Haefliger structure) on a compact manifoldX ⁿ . Cohomological invariants associated to the singularities ofF are defined whose vanishing is shown to be a necessary condition for deformingF to a codimension-q foliation onX ⁿ . An analagous approach to vector bundle maps is then utilized to prove a general theorem concerning the possibility of embedding a vector bundle in the tangent bundle ofX ⁿ , and applications to the planefield problem are given. In the final section geometric realizations of the singularity classes associated toF are constructed.     

The nearest neighbor random walk on subspaces of a vector space and rate of convergence

LetX be the collection ofk-dimensional subspaces of ann-dimensional vector spaceV _n overGF(q). A metric may be defined onX by letting

Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(X _ω ), where X _ω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(X _ω ), where X _ω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. Research partially supported by an Australian Research Council Small Grant. Second author also partially supported by KBN 610/P3/94.     

Degenerations of the moduli spaces of vector bundles on curves II (generalized Gieseker moduli spaces)

LetX ₀ be a projective curve whose singularity is one ordinary double point. We construct a birational modelG(n, d) of the moduli spaceU(n, d) of stable torsion free sheaves in the case (n, d)= 1, such that G(n, d) has normal crossing singularities and behaves well under specialization i.e. if a smooth projective curve specializes toX ₀, then the moduli space of stable vector bundles of rankn and degreed onX specializes toG(n, d). This generalizes an earlier work of Gieseker in the rank two case.