Additive Properties of the Dufresne Laws and Their Multivariate Extension

The Dufresne laws are defined on the positive line by their Mellin transform , where the a _i and b _j are positive numbers, with pq, and where (x) _s denotes (x+s)/(x). Typical examples are the laws of products of independent random variables with gamma and beta distributions. They occur as the stationary distribution of certain Markov chains (X _n) on defined by where X ₀, (A ₁, B ₁),..., (A _n, B _n),... are independent. This paper gives some explicit examples of such Markov chains. One of them is surprisingly related to the golden number. While the properties of the product of two independent Dufresne random variables are trivial, we give several properties of their sum: the hypergeometric functions are the main tool here. The paper ends with an extension of these Dufresne laws to the space of positive definite matrices and to symmetric cones.     

A Feynman integral in Lifshitz‐point and Lorentz‐violating theories in

We evaluate a 1‐loop, 2‐point, massless Feynman integral I_D,m(p,q) relevant for perturbative field theoretic calculations in strongly anisotropic d=D+m dimensional spaces given by the direct sum . Our results are valid in the whole convergence region of the integral for generic (noninteger) codimensions D and m. We obtain series expansions of I_D,m(p,q) in terms of powers of the variable X:=4p²/q⁴, where p=| p |, q=| q |, , , and in terms of generalised hypergeometric functions ₃F₂(−X), when X<1. These are subsequently analytically continued to the complementary region X≥1. The asymptotic expansion in inverse powers of X^1/2 is derived. The correctness of the results is supported by agreement with previously known special cases and extensive numerical calculations.     

Embedding of a Maximal Curve in a Hermitian Variety

Let X be a projective, geometrically irreducible, non-singular, algebraic curve defined over a finite field F q ² of order q ². If the number of F q ²-rational points of X satisfies the Hasse–Weil upper bound, then X is said to be F q ²-maximal. For a point P ₀ X(F q ²), let be the morphism arising from the linear series D: = |(q + 1)P ₀|, and let N: = dim(D). It is known that N 2 and that is independent of P ₀ whenever X is F q ²-maximal.     

Barth type vanishing theorems

Let X be a smooth, projective, d-dimensional subvariety of ⁿ (). Barth's theorem says that H ^q (X, ^p _X )=0 when pq and q+p2d–n (if p=0 we must have q>0). It is very interesting to look for analogous vanishing theorems for H ^q (X, ^p _X (m)), m (see [S-S], [F], [S]). In this paper we prove some vanishing theorems for H ^q (X, ^p _X (1)), for H ^q (X, ^p _X (m)) when m–1, and, if dim(X)=n–2, for H ^q (X, ² _X (m)) and H ^q (X, S ^{k 1} _X (m)). We use standard techniques and some of our previous results.     

A space <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(<Emphasis Type="Italic">X</Emphasis>) is dominated by irrationals if and only if it is<Emphasis Type="Italic"> K</Emphasis>-analytic

Summary We prove that, for any Tychonoff X, the space C_p(X) is K-analytic if and only if it has a compact cover {K_p: p } such that K_p subset K_q whenever p,q and p q. Applying this result we show that if C_p(X) is K-analytic then C_p(X) is K-analytic as well. We also establish that a space C_p(X) is K-analytic and Baire if and only if X is countable and discrete.     

Fields with Exceptional Tangent Fields

The asymptotic self-similarity property describes the local structure of a random field. In this paper, we introduce a locally asymptotically self-similar second order field X_H, whose local structures at x=0 and at x0 are very far from each other. More precisely, whereas its tangent field at x0 is a Fractional Brownian Motion, its tangent field at x=0 is a Fractional Stable Motion. In addition, X_H, is asymptotically self-similar at infinity with a Gaussian field, which is not a Fractional Brownian Motion, as tangent field. Then, its trajectories regularity is studied. Finally, the Hausdorff dimension of its graphs is given.     

Multivariate model with a Kronecker product covariance structure: S. N. Roy method of estimation

In this paper we consider a (p × q)-matrix X = (X ₁, ..., X _q ), where a pq-vector vec (X) = (X ₁ ^T , ...,X _q ^T ) ^T is assumed to be distributed normally with mean vector vec (M) = (M ₁ ^T , ...,M _q ^T ) ^T and a positive definite covariance matrix Λ. Suppose that Λ follows a Kronecker product covariance structure, that is Λ = Φ?Σ, where Φ = (? _ij ) is a (q × q)-matrix and Σ = (σ _ij ) is a (p × p)-matrix and the matrices Φ, Σ are positive definite. Such a model is considered in [4], where the maximum likelihood estimates of the parameters M, Φ, Σ are obtained. Using S. N. Roy’s technique (see, e.g., [3]) of the multivariate statistical analysis, we obtain consistent and unbiased estimates of M, Φ, Σ as in [4], but with less calculations.     

Subspaces of Lp Isometric to Subspaces of ℓ p

We present three results on isometric embeddings of a (closed, linear) subspace X of L_p=L_p[0,1] into _p . First we show that if p 2N, then X is isometrically isomorphic to a subspace of _p if and only if some, equivalently every, subspace of L_p which contains the constant functions and which is isometrically isomorphic to X, consists of functions having discrete distribution. In contrast, if p 2N; and X is finite-dimensional, then X is isometrically isomorphic to a subspace of _p , where the positive integer N depends on the dimension of X, on p , and on the chosen scalar field. The third result, stated in local terms, shows in particular that if p is not an even integer, then no finite-dimensional Banach space can be isometrically universal for the 2-dimensional subspaces of L_p .     

Relative Position,Sequences and Operators in Banach Spaces

A (p + q) × (p + q) matrix-valued inner function S in the unit disc ?? is called (p, q)-type Arov-inner if in the block partition . the p × p diagonal block S₁₁ and the q × q diagonal block S₂₂ are outer matrix-valued functions. A holomorphic p × q matrix-valued function f in ?? is called Arov-completable if there is a (p, q)-type Arov-inner function S such that S₁₂ = f Arov-completability of a given p × q Schur function f is characterized in terms of a (p + q)-variate stationary sequence (X_{n) ? Z}) in Hilbert space which is naturally associated with f. The necessary and sufficient condition for Arov-completability is an orthogonality condition for certain backward and forward innovation vectors generated by (X_{n) ? Z}.     

Upper semicontinuity of parametric projections

In this paper we study upper semicontinuity of the metric projection P _(p)(x) with respect to (x, p), where x is a point in a normed linear space X and (p) is an approximatively compact subset of X depending on a parameter p. An application to parametric spline approximation is given.     

A Study of the Equivalence of the BLUEs between a Partitioned Singular Linear Model and Its Reduced Singular Linear Models

Abstract Consider the partitioned linear regression model and its four reduced linear models, where y is an n × 1 observable random vector with E(y) = Xβ and dispersion matrix Var(y) = σ² V, where σ² is an unknown positive scalar, V is an n × n known symmetric nonnegative definite matrix, X = (X ₁ : X ₂) is an n×(p+q) known design matrix with rank(X) = r ≤ (p+q), and β = (β′ ₁: β′₂ )′ with β₁ and β₂ being p×1 and q×1 vectors of unknown parameters, respectively. In this article the formulae for the differences between the best linear unbiased estimators of M ₂ X ₁β₁under the model and its best linear unbiased estimators under the reduced linear models of are given, where M ₂ = I -X ₂ X ₂ ⁺ . Furthermore, the necessary and sufficient conditions for the equalities between the best linear unbiased estimators of M ₂ X ₁β₁ under the model and those under its reduced linear models are established. Lastly, we also study the connections between the model and its linear transformation model. *This work is supported by the National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 10226024), Postdoctoral Foundation of China and Lab. of Math. for Nonlinear Sciences at Fudan University. This research is supported in part by The International Organizing Committee and The Local Organizing Committee at the University of Tampere for this Workshop **The work is supported in part by an NSF grant of China. Results in this paper were presented by the first author at The Eighth International Workshop on Matrices and Statistics: Tampere, Finland, August 1999     

Clifford Fourier Transformation and Uncertainty Principle for the Clifford Geometric Algebra Cl 3,0

First, the basic concept of the vector derivative in geometric algebra is introduced. Second, beginning with the Fourier transform on a scalar function we generalize to a real Fourier transform on Clifford multivector-valued functions Third, we show a set of important properties of the Clifford Fourier transform on Cl_3,0 such as differentiation properties, and the Plancherel theorem. Finally, we apply the Clifford Fourier transform properties for proving an uncertainty principle for Cl_3,0 multivector functions.     

On periodic points under the iteration of additive polynomials

Let F be a field and f(X) F[X]. An element P of F is called a (pre)periodic point of f if the sequence P, f(P), f(f(P)), is (eventually) periodic. In the case where F is a function field of characteristic p>0 and f(X)=aX ^q +bX or f(X)=aX ^{q 2}+bX ^q +cX with q a power of p, we try to give effective upper bounds (depending only on q and F) for the number of (pre)periodic points of f in F. Mathematics Subject Classification (2000):primary: 11C08, secondary: 11G09, 37E15     

MODp‐tests,almost independence and small probability spaces

In this paper, we consider approximations to probability distributions over Z . We present an approach to estimate the quality of approximations to probability distributions towards the construction of small probability spaces. These small spaces are used to derandomize algorithms. In contrast to results by Even, Goldreich, Luby, Nisan, and Veličković [EGLNV], the methods which are used here are simple, and we get smaller sample spaces. Our investigations are motivated by recent work of Azar, Motwani, and Naor [AMN]. They considered the problem to construct in time respective space polynomial in n a good approximation to the joint probability distribution of the mutually independent random variables X₁, X₂,…,X_n. Each X_i has values in {0, 1} and satisfies X_i=0 with probability q and X_i=1 with probability 1−q where q∈[0, 1] is arbitrary. Our considerations improve on results in [EGLNV] and [AMN] for q=1/p and p a prime. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 293–313, 2000     

?*-extensions of tori, higher chow groups and applications to incidence equivalence relations for algebraic cycles

LetX be a smooth projective variety over an algebraically closed fieldk. We repeat Bloch's construction of aG _m -biextension (torseur)E over CH _hom ^p (X)×CH _hom ^q (X) forp+q=dim(X)+1. First we show that in characteristic zeroE comes via pullback from the Poincaré biextension over the corresponding product of intermediate Jacobians which has been conjectured by Bloch and Murre. Then the relations betweenE and various equivalence relations for algebraic cycles are studied. In particular we reprove Murre's theorem stating that Griffiths' conjecture holds for codimension 2 cycles, i.e. every 2-codimensional cycle which is algebraically and incidence equivalent to zero has torsion Abel-Jacobi invariant.     

A projection and contraction method for a class of linear complementarity problems and its application in convex quadratic programming

In this paper we propose a new iterative method for solving a class of linear complementarity problems:u 0,Mu + q 0, u^T(Mu + q)=0, where M is a givenl ×l positive semidefinite matrix (not necessarily symmetric) andq is a givenl-vector. The method makes two matrix-vector multiplications and a trivial projection onto the nonnegative orthant at each iteration, and the Euclidean distance of the iterates to the solution set monotonously converges to zero. The main advantages of the method presented are its simplicity, robustness, and ability to handle large problems with any start point. It is pointed out that the method may be used to solve general convex quadratic programming problems. Preliminary numerical experiments indicate that this method may be very efficient for large sparse problems.On leave from the Department of Mathematics, University of Nanjing, Nanjing, People's Republic of China.     

Poisson approximations for 2-dimensional patterns

LetX=(X _ij) _n×n be a random matrix whose elements are independent Bernoulli random variables, taking the values 0 and 1 with probabilityq _ij andp _ij (p _ij+q _ij=1) respectively. Upper and lower bounds for the probabilities ofm non-overlapping occurrences of a square submatrix with all its elements being equal to 1, are obtained. Some Poisson convergence theorems are established forn . Numerical results indicate that the proposed bounds perform very well, even for moderate and small values ofn.This work is supported in part by the Natural Science and Engineering Research Council of Canada under Grant NSERC A-9216.     

Interpolation of Intersections Generated by a Linear Functional

Let (X ₀, X ₁) be a Banach couple such that X ₀ ∩ X ₁ is dense in X ₀ and X ₁. By (X ₀, X ₁)_θ,q , 0 < θ < 1, 1 ⩽ q < ∞, we denote the spaces of the real interpolation method. Let ψ be a nonzero linear functional defined on some linear space M ⊂ X ₀ + X ₁ and such that ψ ∈ (X ₀ ∩ X ₁)*, and let N = Ker ψ. We examine conditions under which the natural formula