A quadratically convergentO((\kappa + 1)\sqrt n L)-iteration algorithm for theP *(κ)-matrix linear complementarity problem

An interior-point predictor-corrector algorithm for theP _*()-matrix linear complementarity problem is proposed. The algorithm is an extension of Mizuno—Todd—Ye's predictor—corrector algorithm for linear programming problem. The extended algorithm is quadratically convergent with iteration complexity . It is the first polynomially and quadratically convergent algorithm for a class of LCPs that are not necessarily monotone.     

A randomized algorithm for fixed-dimensional linear programming

We give a (Las Vegas) randomized algorithm for linear programming in a fixed dimensiond for which the expected computation time is , where lim _{d d} = 0. This improves the corresponding worst-case complexity, . The method is based on a recent idea of Clarkson. Two variations on the algorithm are examined briefly.     

Implicit functions and sensitivity of stationary points

We consider the spaceL(D) consisting of Lipschitz continuous mappings fromD to the Euclideann-space ⁿ ,D being an open bounded subset of ⁿ . LetF belong toL(D) and suppose that solves the equationF(x) = 0. In case that the generalized Jacobian ofF at is nonsingular (in the sense of Clarke, 1983), we show that forG nearF (with respect to a natural norm) the systemG(x) = 0 has a unique solution, sayx(G), in a neighborhood of Moreover, the mapping which sendsG tox(G) is shown to be Lipschitz continuous. The latter result is connected with the sensitivity of strongly stable stationary points in the sense of Kojima (1980); here, the linear independence constraint qualification is assumed to be satisfied.     

On the Korn Inequalities on Thin Periodic Frames

Korn-type inequalities for thin periodic structures of period and width h() with h() 0 are presented. Periodic meshes, three-dimensional road structures, and three-dimensional box structures are considered. A particular attention is paid to structures with the so-called critical width when 0$$ " align="middle" border="0"> .     

On Dirichlet Processes Associated with Second Order Divergence Form Operators

Let be a d - dimensional Markov family corresponding to a uniformly elliptic second order divergence form operator. We show that for any quasi continuous in the Sobolev space the process (X) admits under P ^x a decomposition into a martingale additive functional (AF) M and a continuous AF A of zero quadratic variation for almost every starting point x if q=2, for quasi every x if q>2 and for every if is continuous, d=1 and or d>1 and q>d. Our decomposition enables us to show that in the case of symmetric operator the energy of A equals zero if q=2 and that the decomposition of (X) into the martingale AF M and the AF of zero energy A is strict if for some q>d. Moreover, our decomposition provides a probabilistic representation of A .     

On Some Rings of Arithmetical Functions

In this paper, we consider several constructions which from a given B-product * _B lead to another one We shall be interested in finding what algebraic properties of the ring are shared also by the ring . In particular, for some constructions the rings R _B and will be isomorphic and therefore have the same algebraic properties.     

A remark on H1 mappings

With , we here construct, for each positive integer N, a smooth function of degree zero so that there must be at least N singular points for any map that minimizes the energy in the family . The infimum of over U(g) is strictly smaller than the infimum of over the continuous functions in U(g). There are some generalizations to higher dimensions.Research partially supported by the National Science FoundationResearch supported by an Alfred P. Sloan Graduate Fellowship     

Singular limits in Liouville-type equations

We consider the boundary value problem in a bounded, smooth domain in with homogeneous Dirichlet boundary conditions. Here 0,k(x) $$ " align="middle" border="0"> is a non-negative, not identically zero function. We find conditions under which there exists a solution which blows up at exactly m points as and satisfies . In particular, we find that if , 0 $" align="middle" border="0"> and is not simply connected then such a solution exists for any given Received: 11 February 2004, Accepted: 17 August 2004, Published online: 22 December 2004     

Characterization of Normal Traces on Von Neumann Algebras by Inequalities for the Modulus

It is proved that if a normal semifinite weight on a von Neumann algebra satisfies the inequality for any selfadjoint operators in , then this weight is a trace. Several similar characterizations of traces among the normal semifinite weights are proved. In particular, Gardner's result on the characterization of traces by the inequality is refined and reinforced.     

The radius of a compact set in a Hilbert space

In this note we improve the formular(A) (A) proved by Routledge for Hilbert spaces. We show that if A is a relatively compact set, thenr(A)< (A)Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 157–158, 1988.     

Semilinear hyperbolic systems and equations with singular initial data

We study the weak limits (t, .)of solutions to semilinear strictly hyperbolic systems and wave equations with initial datau (0, .) approximating a distribution, 0 < 1. We propose an optimal link between the singularity of and the growth of the nonlinear term in order that exists. In this way we extend some of the results in [3], [10], [13].     

On the Normal Order of ϕk+1(n)/ϕk(n), where ϕk is the k-Fold Iterate of Euler's Function

Let be the j-fold iterated function of . Let and > 0 be fixed, Q be a prime, and let N _k(Q|x) denote the number of those nx for which Q . We give the asymptotics of N _k(Q|x) in the range .     

Win probabilities and simple majorities in probabilistic voting situations

This paper considers an election between candidatesA andB in which (1) voters may be uncertain about which candidate they will vote for, and (2) the winner is to be determined by a lottery betweenA andB that is based on their vote totals. This lottery is required to treat voters equally, to treat candidates equally, and to respond nonnegatively to increased support for a candidate. The set _n of all such lottery rules based on a total ofn voters is the convex hull of aboutn/2 basic lottery rules which include the simple majority rule. For odd values ofn 3 let , and for even values ofn 4 let . With the average of then voters probabilities of voting forA, it is shown that within _n the simple majority rule maximizes candidateA's overall win probability whenever , and that(n) is the smallest number for which this is true. Similarly, the simple majority rule maximizesB's overall win probability whenever (the average of the voters probabilities of voting forB) is as large as(n). This research was supported by the National Science Foundation, Grant SOC 75-00941.     

Two New Infinite Families of 3-Designs from Kerdock Codes over Z4

In this paper we show that the support of the codewords of each type in the Kerdock code of length 2^m over Z₄ form 3-designs for any odd integer . In particular, twonew infinite families of 3-designs are obtained in this constructionfor any odd integer . In particular, twonew infinite families of 3-designs are obtained in this constructionfor any odd integer , whose parameters are ,and .     

On a General Projection Algorithm for Variational Inequalities

Let H be a real Hilbert space with norm and inner product denoted by and . Let K be a nonempty closed convex set of H, and let f be a linear continuous functional on H. Let A, T, g be nonlinear operators from H into itself, and let be a point-to-set mapping. We deal with the problem of finding uK such that g(u)K(u) and the following relation is satisfied: , where >0 is a constant, which is called a general strong quasi-variational inequality. We give a general and unified iterative algorithm for finding the approximate solution to this problem by exploiting the projection method, and prove the existence of the solution to this problem and the convergence of the iterative sequence generated by this algorithm.     

Weak convergence of an empirical process indexed by the closed convex subsets ofI 2

Summary LetI ² be the unit cube of andX _i be independentI ²-valued random variables that are distributed according to Lebesgue-measure. IfS is the set of closed convex subsets ofI ² we consider the process _n (A) _AS,where .It is proved that this process suitably normalized converges in a suitable weak sense to a Gaussian process.     

The completions of a commutative lattice group with respect to the intrinsic topologies

In this paper we discuss the completions (, ) of a commutativel-groupG with respect to the intrinsic topologies . We give some conditions under which is the intrinsic topology of the same type on as and give the relations between these completions.     

On Bahadur asymptotic efficiency in a semiparametric regression model

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Limiting Laws for Entrance Times of Critical Mappings of a Circle

A renormalization group transformation R ₁ has a single stable point in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number (the golden mean). Let a homeomorphism T be the C ¹-conjugate of . We let denote the sequence of distribution functions of the time of the kth entrance to the nth renormalization interval for the homeomorphism T. We prove that for any , the sequence has a finite limiting distribution function , which is continuous in , and singular on the interval [0,1]. We also study the sequence for k>1.     

Solvability of partial differential equations of infinite order in certain classes of entire functions

In this paper it is shown that under conditions of applicability of the operator to the class [,] =(I,s), ₂ 1, ₂), ₁, ₂< the equation y=f has a particular solution of this class vf[, ]. The general form of a solution of the homogeneous equation y=0 is established. The growth of a solution is investigated by means of a system of conjugate orders and a system of conjugate types. A solvability result is also obtained in the class , where T is a certain set in R ₊ ² depending on the operator .Translated from Matematicheskie Zametki, Vol. 19, No. 2, pp. 225–236, February, 1976.In conclusion, the author would like to express his thanks to his adviser, Yu. F. Korobeinik.