Stability of Jackson Type Network Output

We consider an open Jackson type queueing network N with input epochs sequence I={T _n ⁽⁰⁾,n0}, T ₀ ⁽⁰⁾=0, assume another input ={ _n ⁽⁰⁾} and denote _k =| _k ⁽⁰⁾–T _k ⁽⁰⁾|, ₀=0, _n =max_{1kn k} , n1. Let {T _n } and { _n } be the output points in network N and in modified network, with input , accordingly. We study the long-run stability of the network output, establishing two-sided bounds for output perturbation via input perturbation. In particular, we obtain conditions that imply max _kn |T _k – _k |=o(n ^1/r ) with probability 1 as n for some r>0. This result is also extended to continuous time. We consider successively separate station (service node), tandem and feedforward networks. Then we extend stability analysis to general (feedback) networks and show that in our setting these networks can be reduced to feedforward ones. Similar stability results are also obtained in terms of the number of departures. Application to a tandem network with the overloaded stations is considered.     

Is Gauss quadrature optimal for analytic functions?

Summary We consider the problem of optimal quadratures for integrandsf: [–1,1] which have an analytic extension to an open diskD _r of radiusr about the origin such that 1 on . Ifr=1, we show that the penalty for sampling the integrand at zeros of the Legendre polynomial of degreen rather than at optimal points, tends to infinity withn. In particular there is an infinite penalty for using Gauss quadrature. On the other hand, ifr>1, Gauss quadrature is almost optimal. These results hold for both the worst-case and asymptotic settings.This research was supported in part by the National Science Foundation under Grants MCS-8203271 and MCS-8303111This research was supported in part by the National Science Foundation under Grant MCS-8923676     

Fehlerabschätzungen für Gauß-Quadraturformeln

Summary We consider Gauss quadrature formulaeQ _n ,n, approximating the integral ,w an even weight function. Let be analytic inK _r :={z:|z|<r},r>1, and . The error functionalR _n :=I-Q _n is continuous with respect to |·|_r and the relation , q_2k (x):=x ^2k holds.In this paper estimates for R _n are given. To this end we first derive two new representations of R _n which are essential for our further investigations. The R _n =r ² R _n (), with (x):=1/(r ²-x ²), is estimated in various ways by using the best uniform approximation of in P_2n-1, and also the expansion of with respect to Chebyshe polynomials of the first and second kind. Forw(x)=(1-x ²), =±1/2, R _n is calculated. The asymptotic behaviour, forr1+, of R _n and of the derived error bounds is also discussed. Finally, we compare different error bounds and give numerical examples.

     

Approximations for optimal Laplace transform inversion

A function (p) of the Laplace transform operatorp is approximated by a finite linear combination of functions (p+ _r ), where (p) is a specific function ofp having a known analytic inverse (t), and is chosen in accordance with various considerations. Then parameters _r ,r=1, 2,...,n, and then corresponding coefficientsA _r of the (p + _r ) are determined by a least-square procedure. Then, the corresponding approximation to the inversef(t) of (p) is given by analytic inversion of _r=1 ⁿ A _r (p+ _r ). The method represents a generalization of a method of best rational function approximation due to the author [which corresponds to the particular choice (t)1], but is capable of yielding considerably greater accuracy for givenn.The computations for this paper were carried out on the CDC-6600 computer at the Computation Center of Tel-Aviv University. The author is grateful to Dr. H. Jarosch of the Weizmann Institute of Science Computer Center for use of their Powell minimization subroutine (Ref. 1).     

Nonabelian K-theory: The nilpotent class of K1 and general stability

A functorial filtration GL _n =S^–1L _n S⁰L _n S ⁱ L _n E _n of the general linear group GL _{n, n} 3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S^–1 L _n (A)/S⁰L _n (A) is abelian, that S⁰L _n (A) S¹L _n (A) is a descending central series, and that S ⁱ L _n (A) = E _n(A) whenever i the Bass-Serre dimension of A. In particular, the K-functors k ₁ S ⁱ L _n =S ⁱ L _n /E _n are nilpotent for all i 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism S ⁱ L _n (A)/S ⁱ⁺¹ L _n (A)S ⁱ L _{n+ 1}(A)/S ⁱ⁺¹ L _{n + 1} (A) is injective whenever n i + 3, so that one has stability results without stability conditions, and if A is commutative then S⁰L _n (A) agrees with the special linear group SL _n (A), so that the functor S⁰L _n generalizes the functor SL _n to noncommutative rings. Applying the above to subgroups H of GL _n (A), which are normalized by E _n(A), one obtains that each is contained in a sandwich GL _n (A, ) H E _n(A, ) for a unique two-sided ideal of A and there is a descending S⁰L _n (A)-central series GL _n (A, ) S⁰L _n (A, ) S¹L _n (A, ) S ⁱ L _n (A, ) E _n(A, ) such that S ⁱ L _n (A, )=E _n(A, ) whenever i Bass-Serre dimension of A.Dedicated to Alexander Grothendieck on his sixtieth birthday     

On some properties of multiple conjugate trigonometric series

We have obtained an estimate, in terms of partial and mixed moduli, of the continuity of deviation of the Cesáro (C, ) means ( = (₁,...,_n),_i , ₁ > –1, ) of the sequence of rectangular partial sums ofn-multiple (n>1) conjugate trigonometric series from then-multiple truncated conjugate function. This estimate implies the result on them -convergence (1) of (C, ) means (₁ > 0, ) provided that the essential conditions are imposed on the partial moduli of continuity. Finally, it is shown that them -convergence cannot be replaced by ordinary convergence.     

On the Exact Asymptotics of the Best Relative Approximations of Classes of Periodic Functions by Splines

We obtain the exact asymptotics (as n ) of the best L ₁-approximations of classes of periodic functions by splines s S _{2n, r – 1} and s S _{2n, r + k – 1} (S _{2n, r} is the set of 2-periodic polynomial splines of order r and defect 1 with nodes at the points k/n, k Z) under certain restrictions on their derivatives.     

Polynomial and Regular Maps into Grassmannians

We find a regular deformation retraction _n,r (K): Idem _n,r (K) G _n,r (K) from the manifold Idem _n,r (K) of idempotent n × n matrices with rank r to the Grassmannian manifold G _n,r (K) over K the reals, complex numbers or quaternions. Then we derive an injection from the sets of homotopy classes of complex-valued polynomial to such a set of real-valued regular maps, where denotes the Zariski closure in the affine space ⁿ of a subset ⁿ . Furthermore, we list complex-valued polynomial maps ^{2 2} of any Brouwer degree and deduce that the map ()_2,1: Idem()_2,1 G()_2,1 yields an isomorphism [ ² ] [ ², ²] of cyclic infinite homotopy groups. Finally, we show that every nonzero even Brouwer degree of the spheres ⁿ and ⁿ cannot be realized by a real-valued (resp. complex-valued) homogeneous polynomial map provided that n is even.     

A characterization ofL n (q) by the normalizers' orders of their sylow subgroups

LetG be a finite group. If for every primer, whereR ₁ Syl _r G andR ₂ Syl _r (L _n (q)), thenG L _n (q).     

Divergence of a Random Walk Through Deterministic and Random Subsequences

Let {S _n} _n0 be a random walk on the line. We give criteria for the existence of a nonrandom sequence n _i for which respectively We thereby obtain conditions for to be a strong limit point of {S _n} or {S _n /n}. The first of these properties is shown to be equivalent to for some sequence a _i , where T(a) is the exit time from the interval [–a,a]. We also obtain a general equivalence between and for an increasing function fand suitable sequences n _i and a _i. These sorts of properties are of interest in sequential analysis. Known conditions for and (divergence through the whole sequence n) are also simplified.     

The Minimal Subgroup of a Random Walk

It is proved that for each random walk (S _n ) _n0 on ^d there exists a smallest measurable subgroup of ^d , called minimal subgroup of (S _n ) _n0, such that P(S _n )=1 for all n1. can be defined as the set of all x ^d for which the difference of the time averages n ^{–1 n} _k=1 P(S _k ) and n ^{–1 n} _k=1 P(S _k +x) converges to 0 in total variation norm as n. The related subgroup * consisting of all x ^d for which lim _n P(S _n )–P(S _n +x)=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S _n ) _n0. In the final section we consider quasi-invariance and admissible shifts of probability measures on ^d . The main result shows that, up to regular linear transformations, the only subgroups of ^d admitting a quasi-invariant measure are those of the form ₁×...× _k × ^l–k ×{0} ^d–l , 0kld, with ₁,..., _k being countable subgroups of . The proof is based on a result recently proved by Kharazishvili⁽³⁾ which states no uncountable proper subgroup of admits a quasi-invariant measure.     

Almost everywhere convergence of the spherical partial sums for radial functions

LetfL ^p( ⁿ ),n2, be a radial function and letS _Rf be the spherical partial sums operator. We prove that if thenS _Rf(x)f(x) a.e. asR. The result is false for and \frac{{2n}}{{n + 1}}$$ " align="middle" border="0"> .Partially supported by M.P.I.     

A Signed Generalization of the Bernoulli–Laplace Diffusion Model

We bound the rate of convergence to stationarity for a signed generalization of the Bernoulli–Laplace diffusion model; this signed generalization is a Markov chain on the homogeneous space ( ₂S _n )/(S _r ×S _n–r ). Specifically, for r not too far from n/2, we determine that, to first order in n, n log n steps are both necessary and sufficient for total variation distance to become small. Moreover, for r not too far from n/2, we show that our signed generalization also exhibits the cutoff phenomenon.     

Fourier inversion for multidimensional characteristic functions

A density functionf(x),xR ⁿ is said to bepiecewise smooth if for eachxR ⁿ , the mean value function is piecewiseC with compact support. (d is normalized surface measure on the unit sphere). The Fourier transform is with spherical partial sum . Theorem. For suchf, lim _r f _R (x)=M ₀+f(x) if and only ifrM _r f(x) hask=[(n–3)/2] continuous derivatives. ([]=integer part). Otherwise we have lim where 0 is uniquely determined.     

Some Groups Having Only Elementary Actions on Metric Spaces with Hyperbolic Boundaries

We study isometric actions of certain groups on metric spaces with hyperbolic-type bordifications. The class of groups considered includes SL _n (), Artin braid groups and mapping class groups of surfaces (except the lower rank ones). We prove that in various ways such actions must be elementary. Most of our results hold for non-locally compact spaces and extend what is known for actions on proper CAT(-1) and Gromov hyperbolic spaces. We also show that SL _n () for n 3 cannot act on a visibility space X without fixing a point in . Corollaries concern Floyd's group completion, linear actions on strictly convex cones, and metrics on the moduli spaces of compact Riemann surfaces. Some remarks on bounded generation are also included.     

Eine Klasse nicht einbettbarer reeller Laguerre-Ebenen

Replace in the parabolic model of the classical Laguerre-Plane the parabolas y=a(x–b)²+c, a0, by the curves y=af(x–b)+c with f(x)= if x0, and f(x)=(–x)^r ₂ if x<0. For each pair r₁, r₂>1 we obtain again a Laguerre-Plane (r₁,r₂).(r₁,r₂) can be embedded only if r₁=r₂=2.     

A representation theorem for weak automorphisms of a universal algebra

Given a group G and a descending chainG ₀,G ₁,...,G _n, of normal subgroups ofG, we prove that there exists a universal algebra , such that the chain ...W_n( )...W₁( }) W₀( )W( ) is isomorphic to the chain ...G _n ...G ₁G ₀G, where W( ) is the group of weak automorphisms of , and W_n( ) is the group of weak automorphisms of that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G₀, and G₁=G₂=...=G_n=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.     

Generating Random Elements in SL n (F q ) by Random Transvections

This paper studies a random walk based on random transvections in SL _n(F _q ) and shows that, given > 0, there is a constant c such that after n + c steps the walk is within a distance from uniform and that after n – c steps the walk is a distance at least 1 – from uniform. This paper uses results of Diaconis and Shahshahani to get the upper bound, uses results of Rudvalis to get the lower bound, and briefly considers some other random walks on SL _n(F _q ) to compare them with random transvections.     

Continuous Quotients for Lattice Actions on Compact Spaces

Let < SL _n ( ) be a subgroup of finite index, where n 5. Suppose acts continuously on a manifold M, where ₁(M) = ⁿ , preserving a measure that is positive on open sets. Further assume that the induced action on H ¹(M) is non-trivial. We show there exists a finite index subgroup < and a equivariant continuous map : M ⁿ that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where ₁(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with actions.