Note on Fano 3-folds

LetE be an ample vector bundle of rankr on a compact complex manifoldX of dimension 3 with detE=–K _x, andi(X) the index ofX. Then it is proved in this note thati(X)r unless (X,E)(¹ × ²,p*O(1) q*), wherep,q are the projections and is isomorphic toO(2) O(1) or the tangent bundleT of ². This result gives a counterexample to the conjecture formed by T. Peternell.     

A Comparison Theorem on Moment Inequalities Between Negatively Associated and Independent Random Variables

Let {X _i, 1in} be a negatively associated sequence, and let {X* _i , 1in} be a sequence of independent random variables such that X* _i and X _i have the same distribution for each i=1, 2,..., n. It is shown in this paper that Ef( ⁿ _i=1 X _i)Ef( ⁿ _i=1 X* _i ) for any convex function f on R ¹ and that Ef(max_1kn ⁿ _i=k X _i)Ef(max_1kn ^k _i=1 X* _i ) for any increasing convex function. Hence, most of the well-known inequalities, such as the Rosenthal maximal inequality and the Kolmogorov exponential inequality, remain true for negatively associated random variables. In particular, the comparison theorem on moment inequalities between negatively associated and independent random variables extends the Hoeffding inequality on the probability bounds for the sum of a random sample without replacement from a finite population.     

Asymptotic behavior of general M-estimates for regression and scale with random carriers

Summary Let (xini, y _i be a sequence of independent identically distributed random variables, where x _i R ^p and y _i R, and let R ^p be an unknown vector such that y _i =x _i +u _i (^*), where u _i is independent of x _i and has distribution function F(u/), where >0 is an unknown parameter. This paper deals with a general class of M-estimates of regression and scale, ( ^*,^*), defined as solutions of the system: , where r= (y _i –x _i ^1*/)^*, with R ^p ×RR and RR. This class contains estimators of (, ) proposed by Huber, Mallows and Krasker and Welsch. The consistency and asymptotic normality of the general M-estimators are proved assuming general regularity conditions on and and assuming the joint distribution of (x _i , y _i ) to fulfill the model (^*) only approximately.     

Decomposing symmetric exchanges in matroid bases

LetB,B be bases of a matroid, withX B, X B. SetsX,X are asymmetric exchange if(B – X) X and(B – X) X are bases. SetsX,X are astrong serial B-exchange if there is a bijectionf: X X, where for any ordering of the elements ofX, sayx _i ,i = 1, , m, bases are formed by the sets B₀ = B, B_i = (B_i–1 – x_i) f(x _i), fori = 1, , m. Any symmetric exchangeX,X can be decomposed by partitioning X = _i=1 ^m Y_i, X = _i=1 ^m Y_i, X, where (1) bases are formed by the setsB ₀ =B, B _i = (B _i–1 –Y _i ) Y _i ; (2) setsY _i ,Y _i are a strong serialB _i–1 -exchange; (3) properties analogous to (1) and (2) hold for baseB and setsY _i ,Y _i .     

Efficiency of Proximal Bundle Methods

We give efficiency estimates for proximal bundle methods for finding f_*min_Xf, where f and X are convex. We show that, for any accuracy <0, these methods find a point x^kX such that f(x^k)–f^* after at most k=O(1/³) objective and subgradient evaluations.     

On the estimation of ordered means of two exponential populations

Let random samples of equal sizes be drawn from two exponential distributions with ordered means _i . The maximum likelihood estimator _i ^* of _i is shown to have a smaller mean square error than that of the usual estimator X_i, for each i=1,2. The asymptotic efficiency of _i ^* relative to X_i has also been found.     

Locally most powerful rank tests for testing randomness and symmetry

Let X _i, 1 i N, be N independent random variables (i.r.v.) with distribution functions (d.f.) F _i(x,), 1 i N, respectively, where is a real parameter. Assume furthermore that F _i(·,0) = F(·) for 1 i N. Let R = (R ₁,R _N) and R ⁺,...,R _N ⁺ be the rank vectors of X = (X ₁,X _N) and |X|=(|X ₁|,...,|X _N|), respectively, and let V = (V ₁,V _N) be the sign vector of X. The locally most powerful rank tests (LMPRT) S = S(R) and the locally most powerful signed rank tests (LMPSRT) S = S(R ⁺, V) will be found for testing = 0 against > 0 or < 0 with F being arbitrary and with F symmetric, respectively.     

Mathematical expectations of functions of sums of a random number of independent terms

Conditions are found which must be imposed on a function g(x) in order that M g(₁+₂+ + _v < if M g(_i) < and M g(v) < ,, ₁, ₂, , _n, ... being non-negative and independent, being integral, and {_i} being identically distributed. The result is applied to the theory of branching processes.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 387–394, April, 1968.     

On some additive mappings in rings with involution

Summary LetR be a *-ring. We study an additive mappingD: R R satisfyingD(x ²) =D(x)x ^* +xD(x) for allx R.It is shown that, in caseR contains the unit element, the element 1/2, and an invertible skew-hermitian element which lies in the center ofR, then there exists ana R such thatD(x) = ax ^* – xa for allx R. IfR is a noncommutative prime real algebra, thenD is linear. In our main result we prove that a noncommutative prime ring of characteristic different from 2 is normal (i.e.xx ^* =x ^* x for allx R) if and only if there exists a nonzero additive mappingD: R R satisfyingD(x ²) =D(x)x ^* +xD(x) and [D(x), x] = 0 for allx R. This result is in the spirit of the well-known theorem of E. Posner, which states that the existence of a nonzero derivationD on a prime ringR, such that [D(x), x] lies in the center ofR for allx R, forcesR to be commutative.     

OnC *-algebras having linear,polynomial and subexponential growth

Summary Answering a question of Voiculescu [16, Problem 5.9], we show thatC ^*-algebras having filtrations (A _n)_n satisfying the condition lim sup_n ln dimA _n/n=0 (in particular having subexponential growth), are nuclear.For the case of linear growth we obtain the following particular result: letX be a finite dimensional self-adjoint generating system of aC ^*-algebraA such that dim (span (X ⁿ +1))1+dim (span (X ⁿ )), then there exist a finite dimensionalC ^*-algebraC having only irreducible representations of dimension 1 and aC ^*-algebraB, which is generated by a single self-adjoint element, such thatACB.Some other results are given on linear growth and we show that there exist singly generatedC ^*-algebras such that the growth of the filtration (span (X ⁿ ))_n is polynomial, whereX={x,x ^*, 1} is a generating system, and such that in every neighbourhood ofx there exists an invertibley such thatY={1,y,y ^*} is a generating system whose associated filtration (span (Y ⁿ ))_n doesn't satisfy the previous condition of Voiculescu, and in particular does not have subexponential growth.Oblatum 19-X-1991Work partially supported by DFGAllocataire M.R.T., Université Aix-Marseille II (France)     

Decomposition of bipartite multigraphs into matchings

Summary In this paper we show that a bipartite multigraphG in which no vertex is adjacent to more thann edges may be decomposed into any numbern ^*n of matchings in such a way that the number of edges in each of these matchings differ from one another by at most one unit. When the two sets of verticesX and¯X ofG are partitioned into subsetsX _i and¯X _j respectively, we give conditions for the existence of a decomposition ofG inton ^*n matchings such that each matching contains at most _i edges adjacent to vertices inX _i and at most _j edges adjacent to vertices in¯X _j .There are many practical problems of scheduling with resource limitations for which such decompositions of a bipartite graph are required; some examples are given.

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Zusammenfassung Es wird gezeigt, daß ein bipartiter MultigraphG, in dem in keinem Knoten mehr alsn Kanten zusammenstoßen, in eine beliebige Zahln ^*n von matchings dekomponiert werden kann, und zwar derart, daß die Zahl der Kanten in jedem dieser matchings untereinander um höchstens eins differiert. Mit dem Aufteilen der beiden KnotenmengenX und ¯X vonG in UntermengenX _i und¯X _j werden Bedingungen für die Existenz einer Dekomposition vonG inn ^* n matchings angegeben, wobei jedes matching höchstens _i ( _j ) Kanten enthält, die in den KnotenX _i (bzw.¯X _j ) zusammenstoßen.Es gibt viele praktische Planungsprobleme mit beschränkten Ressourcen, für die solche Dekompositionen von bipartiten Graphen verlangt werden; einige Beispiele werden angeführt.

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This work was supported by a grant from the National Research Council of Canada.     

A characterization of plane Lorentz transformations

Denote (x_i,y_i=ct_i), i=1,2, by X_i and (x₂–x₁)²–(y₂–y₁)² by F(X₁,X₂). Then our result is the following: Given a fixed real number 0 and given a bijection of M=IR² such that F(X₁,X₂) = iff F(X _in1 ^su , _in2 ^su ) =p for all X₁, X₂ M. Then must be a Lorentz transformation (time reversal and inhomogeneity included).     

Simultaneous Extensions of Topogenities

Let (X,) be a topological space, X _i X for i I, and < _i be a topogenity on X _i . We look for a topogenity < on X such that is the topology induced by < and <|X _i =< _i for i I.     

Large Deviations of Local Times of Lévy Processes

For X(t) a real-valued symmetric Lévy process, its characteristic function is E(e ^iX(t))=exp(–t()). Assume that is regularly varying at infinity with index 1<2. Let L ^x _t denote the local time of X(t) and L* _t =sup _xR L ^x _t . Estimates are obtained for P(L ⁰ _t y) and P(L* _t y) as y and t fixed.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

A decomposition for the exponential dispersion model generated by the invariant measure on the hyperboloid

For = ₀, ₁, ₂) andx=(x₀, x₁, x₂) in R³, define [,x] = ₀ x ₀ – ₁ x ₁ – ₂ x ₂,C = {x³:x ₀ > 0 and [x, x]>0},R(x)=([x, x]) ^1/2 forx inC andH ₁={xC: x₀>0,R(x)=1}. Define the measure onH ₁ such that if is inC and =R(), then exp (–[,x])(dx = ( exp )^–1. Therefore, is invariant under the action ofSO (1, 2), the connected component ofO(1, 2) containing the identity. We first prove that there exists a positive measure in ³ such that its Laplace transform is ( exp )^– if and only if >1. Finally, for 1 and inC, denotingP(,)(dx) = ( exp )^– exp (–[,x])(dx, we show that ifY ₀,...,Y _n aren+1 independent variables with densityP(,),j=0,...,n and ifS _k =X ₀ + ... +X _k andQ _k =R(S _k) –R(S _k–1) –R(Y _k),k=1,...,n, then then+1 statisticsD _n = [/,S _k ] –R _{k – 1} ),Q ₁,...,Q _n are independent random variables with the exponential () or gamma (1,1/) distribution.This research has been partially funded by NSERC Grant A8947.     

Nonparametric Estimation of a Conditional Quantile for α-Mixing Processes

Let (X_i,Y _i) be a set of observations form a stationary -mixing process and (x) be the conditional -th quantile of Y given X = x. Several authors considered nonparametric estimation of (x) in the i.i.d. setting. Assuming the smoothness of FF(x), we estimate it by local polynomial fitting and prove the asymptotic normality and the uniform convergence.     

p-Frames in Separable Banach Spaces

Let X be a separable Banach space with dual X ^*. A countable family of elements {g _i }X ^* is a p-frame (1 p ) if the norm _X is equivalent to the ^p -norm of the sequence {g _i ()}. Without further assumptions, we prove that a p-frame allows every gX ^* to be represented as an unconditionally convergent series g=d _i g _i for coefficients {d _i } ^q , where 1/p+1/q=1. A p-frame {g _i } is not necessarily linear independent, so {g _i } is some kind of overcomplete basis for X ^*. We prove that a q-Riesz basis for X ^* is a p-frame for X and that the associated coefficient functionals {f _i } constitutes a p-Riesz basis allowing us to expand every fX (respectively gX ^*) as f=g _i (f)f _i (respectively g=g(f _i )g _i ). In the general case of a p-frame such expansions are only possible under extra assumptions.     

A trust region method for implicit orthogonal distance regression

Let a family of curves or surfaces be given in implicit form via the model equationf (x,)=0, wherex ^d and ^m is a parameter vector. We present a trust region algorithm for solving the problem:find a parameter vector ^* such that the contour f(x, ^*)=0is a best fit to given data {z_i} _i ⁿ =1 ^d in a least squares sense. Specifically, we seek ^* and {x _i ^* } _i ⁿ =1 such thatf (x _i ^* , ^*) = 0,i=1,...,n, and _i=1 ⁿ z _i –x _i ^* ₂ ² is minimal. The termorthogonal distance regression is used to describe such constrained nonlinear least squares problems.