On a Test Statistic for Linear Trend

Let {W(s)} _s 0 be a standard Wiener process. The supremum of the squared Euclidian norm Y (t)², of the R²-valued process Y(t)=(1/t W(t), {12/t ³ int_{0 t} s dW (s)– {3/t} W(t)), t [, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {sup _t [, 1] Y(t)² > u as u , of this statistic, for a fixed (0,1), and for a moving = (u) 0 at a suitable rate as u . The statistical interest of our results lie in their use as approximate test levels.     

Existence and construction of Δ1-splines of class Ck on a three-directional mesh

Let ^* be the equilateral triangulation of the plane and let ₁ ^* be the equilateral triangle formed by four triangles of ^*. We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ ^*, having a sufficiently high degree n and which are invariant with respect to the group of symmetries of ₁ ^*. Such splines are called ₁ ^*-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁ ^*-splines of class C ^k and minimal degree, but these splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines.     

On the Integrability of Eigenfunctions of the Laplace-Beltrami Operator in the Unit Ball of C n

Let B denote the unit ball in C ⁿ , n1, and let , , and denote the volume measure, gradient, and Laplacian respectively, with respect to the Bergman metric on B. For R and 0<p<, we denote by L ^p the set of real, or complex-valued measurable functions f on B for which _B (1–|z|²)|f(z)| ^p d(z)<, and by D ^p the Dirichlet space of C ¹ functions f on B for which | f|L ^p . Also, for C, we denote by X the set of C ² real, or complex-valued functions f on B for which f=f. The main result of the paper is as follows: Let 0<p< and suppose R with –n ². Then L ^p X ={0}, and for 0, D ^p X ={0}(a) for all n+ when p1, and(b) for all when 0<p<1.By example it is shown that the result is best possible for all values of p with pn/(n+ .     

Conjugation for polynomial mappings

We consider Keller's functions, namely polynomial functionsf:C ⁿ C ⁿ with detf(x)=1 at allx C ⁿ. Keller conjectured that they are all bijective and have polynomial inverses. The problem is still open.Without loss of generality assumef(0)=0 andf'(0)=I. We study the existence of certain mappingsh , > 1, defined by power series in a ball with center at the origin, such thath(0)=I andh (f(x))=h (x). So eachh conjugates f to its linear part I in a ball where it is injective.We conjecture that for Keller's functionsf of the homogeneous formf(x)=x +g(x),g(sx)=s ^dg(x),g(x)ⁿ=0,xC ⁿ,sC the conjugationh for f is anentire function.     

Sensitivity analysis of the greatest eigenvalue of a symmetric matrix via the ɛ-subdifferential of the associated convex quadratic form

LetA(·) be ann × n symmetric affine matrix-valued function of a parameteruR ^m , and let (u) be the greatest eigenvalue ofA(u). Recently, there has been interest in calculating (u), the subdifferential of atu, which is useful for both the construction of efficient algorithms for the minimization of (u) and the sensitivity analysis of (u), namely, the perturbation theory of (u). In this paper, more generally, we investigate the Legendre-Fenchel conjugate function of (·) and the -subdifferential (u) of atu. Then, we discuss relations between the set (u) and some perturbation bounds for (u).The author is deeply indebted to Professor J. B. Hiriart-Urruty who suggested this study and provided helpful advice and constant encouragement. The author also thanks the referees and the editors for their substantial help in the improvement of this paper.     

A Generalized Double Crossproduct and Drinfeld Double

In this paper we construct a new and more complicated algebra construction of two algebras B and H, a generalized double crossproduct B H. The left generalized smash product, the right generalized smash product, Majids double crossproduct, especially, the smash product, the Drinfeld Double D(H) and Doi-Takeuchi algebra B H are all special cases as our algebra structure. Next, we analyze conditions under which this new algebra B H is a Hopf algebra termed a generalized double crossproduct of Hopf algebra, and describe a coquasitriangular structure over the generalized double crossproduct Hopf algebra B H. Finally, what we do is to construct a new braided monoidal category ^J_JMod^Q_Q obtained from the structure of the generalized double crossproduct, and establish a kind of new quantum Yang-Baxter operators.AMS Subject Classification (1991): 16W20, 16D90, 16S40, 16W30     

C k -B-splines with Square Support on a Three-Direction Mesh of the Plane

Let be the uniform triangulation generated by the usual three-directional mesh of the plane and let ₁ be the unit square consisting of two triangles of . We study the space of piecewise polynomial functions in C ^k (R ²) with support ₁ having a sufficiently high degree n, which are symmetrical with respect to the first diagonal of ₁. Such splines are called ₁-splines. We first compute the dimension of this space in function of n and k. Then, for any fixed k0, we prove the existence of ₁-splines of class C ^k and minimal degree. These splines are not unique. Finally, we describe an algorithm computing the Bernstein–Bézier coefficients of these splines, and we give an example.     

Growth of length of sequential derivation transformed into natural one

We consider the (&, )-fragment of the intuitionistic propositional calculus. It is proved that under the standard transformation of a Gentzen derivation into a natural derivation(), the length of (())2^2·length( ). There is constructed a sequence of Gentzen derivations of length _i, for which the length of (( _i))2^{1/3·length(i}), which shows that the upper bound obtained is not too weak.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 192–196, 1979.     

Fine Analysis of the Quasi-Orderings on the Power Set

We pursue the fine analysis of the quasi-orderings and on the power set of a quasi-ordering (Q,). We set X Y if every xX is majorized in by some yY, and X Y if every yY is minorized in by some xX. We show that both these quasi-orderings are -wqo if and only if the original quasi-ordering is ( )-wqo. For this holds also restricted to finite subsets, thus providing an example of a finitary operation on quasi-orderings which does not preserve wqo but preserves bqo.     

Homogenization of Some Elasticity Problems with Rapidly Oscillating Convex Constraints on Displacements

The problem of homogenization is considered for an elastic body occupying a perforated domain = obtained from a fixed domain and an -contraction of a 1-periodic domain .     

Even diameters of the classes W(r)H ω in the space C2π

For even values of n we find the exact values of the diameters d_n(W^(r)H) of the classes of 2-periodic functions ((t) is an arbitrary convex upwards modulus of continuity) in the space C₂. We find that d_2n(W^(r)H)=d_2n–1(W^(r)H) (n=1, 2, ... r=0, 1, 2, ...).Translated from Matematicheskie Zametki, Vol. 15, No. 3, pp. 387–392, March, 1974.The author expresses his thanks to N. P. Korneichuk for his interest in my work.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

Topological influence of the presence of fully-λ-bases in locally convex spaces

This article reveals the topological impact of fully--bases in locally convex spaces where carries either the traditional normal topology or the fairly generalized-topology of Ruckle. It has been established that the generalized nuclearity of plays a significant role in influencing the topology of the space. Further, the equivalence of normal topology and the topology arising out of the fully--base ( being equipped with normal topology or-topology) has been investigated.We acknowledge with thanks the suggestions of the referee.     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

Quasianapole Interaction of Charged Fermions

We obtain the analytic expression for the total cross section of the reaction e ^– e ⁺l ^– l ⁺ (l=,) taking possible quasianapole interaction effects into account. We find numerical restrictions on the interaction parameter value from data for the reaction e ^– e ^+–+ in the energy domain below the Z ⁰ peak.     

Brownian charges around loops

Summary Let (X _t ⁿ ) be a Poisson sequence of independent Brownian motions in ^d ,d3; Let be a compact oriented submanifold of d, of dimensiond–2 and volume ; let t be the sum of the windings of (X _s ⁿ , 0st) around ; then t/t converges in law towards a Cauchy variable of parameter /2. A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in ^d .     

Stability for Quadratic K1

The general quadratic group GQ _2n and its elementary subgroup EQ _2n are analogs in the theory of quadratic forms of the general linear group GL _n and its elementary subgroup E _n . This article proves that the stabilization map GQ _2n /EQ _2n GQ _2(n+1) /EQ _2(n+1) is an isomorphism whenever n +1 and S denotes the -stable rank of rings with anti-involution. As a corollary, a result is obtained which has been anticipated since the late 1960s: over rings of finite Bass–Serre dimension d, the stabilization map is an isomorphism whenever n d + 2.     

Function Spaces of Posets with Projections

This paper investigates function spaces of structures consisting of a partially ordered set together with some directed family of projections.More precisely, given a fixed directed index set (I,), we consider triples (D,,(p _i ) _iI ) with (D,) a poset and (p _i ) _iI a monotone net of projections of D. We call them (I,)-pop's (posets with projections). Our main purpose is to study structure preserving maps between (I,)-pop's. Such homomorphisms respect both order and projections.Any (I,)-pop is known to induce a uniformity and thus a topology. The set of all homomorphisms between two (I,)-pop's turns out to form an (I,)-pop itself. We show that its uniformity is the uniformity of uniform convergence. This enables us to prove that properties such as completeness and compactness transfer to function pop's.Concerning categorical properties of (I,)-pop's, we will see that we are in a lucky situation from a computer scientist's point of view: we obtain Cartesian closed categories. Moreover, by a D -construction we get (I,)-pop's that are isomorphic to their own exponent. This yields new models for the untyped -calculus.     

Bounding the diameter of a distance regular graph by a function ofk d

Let be a distance regular graph with diameterd, and _d () the set of vertices at distanced from. is said to be thin if the induced subgraph on _d () is a union of cliques for every vertex. We show that the diameterd is bounded above by a function depending only onk _d, which is the cardinality of _d (), if is not thin. We also investigate thin distance regular graphs witha _d 0.     

Constructing one-parameter transformations on white noise functions in terms of equicontinuous generators

Let be a barreled locally convex space. A continuous operator on is called anequicontinuous generator if { ⁿ /n!;n=0,1,2,...} is an equicontinuous family of operators. For each equicontinuous generator a one-parameter group of operators is constructed by means of power series. There is a one-to-one correspondence between the equicontinuous generators and the locally equicontinuous holomorphic one-parameter groups of operators. If two equicontinuous generators ₁, ₂ satisfy [₁,₂]=₂ for some thena₁+b₂ is also an equicontinuous generator for anya, b. These general results are applied to a study of operators on white noise functions. In particular, a linear combination of the number operator and the Gross Laplacian, which are natural infinite dimensional analogues of a finite dimensional Laplacian, is always an equicontinuous generator. This result contributes to the Cauchy problems in white noise (Gaussian) space.Work supported by Alexander von Humboldt-Stiftung and Japan Society for Promotion of Sciences