Saddlepoint approximations for multivariate <Emphasis Type="Italic">M</Emphasis>-estimates with applications to bootstrap accuracy

We obtain marginal tail area approximations for the one-dimensional test statistic based on the appropriate component of the M-estimate for both standardized and Studentized versions which are needed for tests and confidence intervals. The result is proved under conditions which allow the application to finite sample situations such as the bootstrap and involves a careful discretization with saddlepoints being used for each neighbourhood. These results are used to obtain second-order relative error results on the accuracy of the Studentized and the tilted bootstrap. The tail area approximations are applied to a Poisson regression model and shown to have very good accuracy. An erratum to this article can be found at     

Approximations for the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">GI</Emphasis> type call center

In this paper, we propose approximations to compute the steady-state performance measures of the M/GI/N+GI queue receiving Poisson arrivals with N identical servers, and general service and abandonment-time distributions. The approximations are based on scaling a single server M/GI/1+GI queue. For problems involving deterministic and exponential abandon times distributions, we suggest a practical way to compute the waiting time distributions and their moments using the Laplace transform of the workload density function. Our first contribution is numerically computing the workload density function in the M/GI/1+GI queue when the abandon times follow general distributions different from the deterministic and exponential distributions. Then we compute the waiting time distributions and their moments. Next, we scale-up the M/GI/1+GI queue giving rise to our approximations to capture the behavior of the multi-server system. We conduct extensive numerical experiments to test the speed and performance of the approximations, which prove the accuracy of their predictions.      

Degenerations of <Emphasis Type="Italic">k</Emphasis>[<Emphasis Type="Italic">t</Emphasis>]/(<Emphasis Type="Italic">t</Emphasis><Superscript>r</Superscript>)-Modules Giving Stratification of the Orbits

Let Λ be a finitely generated associative k-algebra where k is an algebraically closed field. For each natural number d, we have the variety of d-dimensional module structures on k^d given by the multiplication of the elements from a generating set of Λ. The general linear group Gl_d(k) acts on this variety by conjugation and the orbits under this action correspond to isomorphism classes of d-dimensional Λ-modules. For two d-dimensional Λ-modules M and N one says that M degenerates to N if the orbit corresponding to N is in the Zariski-closure of the orbit corresponding to M. Now in this situation the stabilizers of the elements in the orbit corresponding to N acts on the orbit corresponding to M. In this paper we characterize degenerations of k[t]/(t^r)-modules with the property that for each y in the orbit corresponding to N, there is an x_y in the orbit corresponding to M such that the orbit corresponding to M is the disjoint union of orbits of the x_y’s under the action of the stabilizer of y where y runs through the orbit corresponding to N. Presented by Idun ReitenMathematics Subject Classifications (2000) 14L30, 16G10.     

Embedding of non-commutative <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-spaces: <Emphasis Type="Italic">p</Emphasis> < 1

If (N,t) ({\cal N},\tau) is a finite von Neumann algebra and if (M,n) ({\cal M},\nu) is an infinite von Neumann algebra, then L_p(M,n) L_{p}({\cal M},\nu) does not Banach embed in L_p(N,t) L_{p}({\cal N},\tau) for all p ? (0,1) p\in (0,1) . We also characterize subspaces of $ L_{p}({\cal N},\tau),\ 0< p <1 $ L_{p}({\cal N},\tau),\ 0< p <1 containing a copy of l_p.     

Analysis of GI<Superscript><Emphasis Type="Italic">X</Emphasis></Superscript>/<Emphasis Type="Italic">M</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)//<Emphasis Type="Italic">N</Emphasis> systems with stochastic customer acceptance policy

We investigate GI ^X /M(n)//N systems with stochastic customer acceptance policy, function of the customer batch size and the number of customers in the system at its arrival. We address the time-dependent and long-run analysis of the number of customers in the system at prearrivals and postarrivals of batches and seen by customers at their arrival to the system, as well as customer blocking probabilities. These results are then used to derive the continuous-time long-run distribution of the number of customers in the system. Our analysis combines Markov chain embedding with uniformization and uses stochastic ordering as a way to bound the errors of the computed performance measures.      

On the existence of <Emphasis Type="Italic">F</Emphasis>-crystals

Let (N,F) be an F-isocrystal, with associated Newton vector ν in . To any lattice M in N (an F-crystal) is associated its Hodge vector in . By Mazur's inequality we have . We show that, conversely, for any with , there exists a lattice M in N such that . We also give variants of this existence theorem for symplectic F-isocrystals, and for periodic lattice chains. Received: March 1, 2002     

On <Emphasis Type="Italic">N</Emphasis>-termed approximations in <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">s</Emphasis></Superscript>-norms with respect to the Haar system

In [9], we proved numerically that spaces generated by linear combinations of some two-dimensional Haar functions exhibit unexpectedly nice orders of approximation for solutions of the single-layer potential equation in a rectangle. This phenomenon is closely related, on the one hand, to the properties of the approximation method of hyperbolic crosses and on the other to the existence of a strong singularity for solutions of such boundary integral equations. In the present paper, we establish several results on the approximation for the hyperbolic crosses and on the best N-term approximations by linear combinations of Haar functions in the H ^s -norms, −1 < s < 1/2; this provides a theoretical base for our numerical research. To the author's best knowledge, the negative smoothness case s < 0 was not studied earlier. __________ Translated from Sovremennaya Matematika. Fundamental'nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 25, Theory of Functions, 2007.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis>,<Emphasis Type="Italic">q</Emphasis></Subscript>-Cohomology and Normal Solvability

We consider some problems concerning the L _p,q -cohomology of Riemannian manifolds. In the first part, we study the question of the normal solvability of the operator of exterior derivation on a surface of revolution M considered as an unbounded linear operator acting from L_p^k (M) into L^k+1_q (M). In the second part, we prove that the first L _p,q-cohomology of the general Heisenberg group is nontrivial, provided that p < q. Received: 17 January 2006 Supported by INTAS (Grant 03–51–3251) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh 311.2003.1, NSh 8526.2006.1).     

The <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 retrial queue with retrials due to server failures

Sherman and Kharoufeh (Oper. Res. Lett. 34:697–705, [2006]) considered an M/M/1 type queueing system with unreliable server and retrials. In this model it is assumed that if the server fails during service of a customer, the customer leaves the server, joins a retrial group and in random intervals repeats attempts to get service. We suggest an alternative method for analysis of the Markov process, which describes the functioning of the system, and find the joint distribution of the server state, the number of customers in the queue and the number of customers in the retrial group in steady state.      

Consecutive customer losses in regular and oscillating <Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript>/<Emphasis Type="Italic">G</Emphasis>/1/<Emphasis Type="Italic">n</Emphasis> systems

We derive fast recursions to compute the probability that k or more consecutive customer losses take place during a busy period of a queue, the so called k-CCL probability, for regular and oscillating M ^X /G/1/n systems.     

The <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/∞ queue in a random environment

We consider the M/M/∞ queueing system with arrival and service rate depending on the state of an auxiliary semi-Markov process (which can be viewed as an external environment) and find the mean number of customers in the system in steady state. In a particular case when the external environment can be only in two states we find the distribution of the number of customers in the system.      

Cc （X） Spaces with X Locally Compact

In this paper, we show, among other results, that if X is a [separable] locally compact space X [satisfying the first countability axiom] then the space Cc （X） has countable tightness [if and only if it has bounding tightness] if and only if it is Frechet-Urysohn, if and only if Cc （X） contains a dense （LM） subspace and if and only if X is a-compact.     

Asymptotic expansions for the conditional sojourn time distribution in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1-PS queue

We consider the M/M/1 queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer’s service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison (SIAM J. Appl. Math. 45:152–167, [1985]) and Flatto (Ann. Appl. Probab. 7:382–409, [1997]). This work was partly supported by NSF grant DMS 05-03745.     

<Emphasis Type="Italic">G</Emphasis>-compactness and groups

Lascar described E _KP as a composition of E _L and the topological closure of E _L (Casanovas et al. in J Math Log 1(2):305–319). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M′ consisting of M and X as two sorts, where X is an affine copy of G and in M′ we have the structure of M and the action of G on X. We prove that the Lascar group of M′ is a semi-direct product of the Lascar group of M and G/G _L . We discuss the relationship between G-compactness of M and M′. This example may yield new examples of non-G-compact theories. The first author is supported by the Polish Goverment grant N N201 384134. The second author is supported by the Polish Goverment grant N201 032 32/2231.     

About the computation of the signature of surface singularities <Emphasis Type="Italic">z</Emphasis><Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> + <Emphasis Type="Italic">g</Emphasis>(<Emphasis Type="Italic">x,y</Emphasis>) = 0

In this article we describe our experiences with a parallel Singular implementation of the signature of a surface singularity defined by z ^N + g(x; y) = 0.     

Asymptotic analysis for loss probability of queues with finite <Emphasis Type="Italic">GI</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/1 type structure

This paper discusses the asymptotic behavior of the loss probability for general queues with finite GI/M/1 type structure such as GI/M/c/K, SM/M/1/K and GI/MSP/1/K queues. We find an explicit expression for the asymptotic behavior of the loss probability as K tends to infinity. With the result, it is shown that the loss probability tends to 0 at a geometric rate. This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology Research Center) support program supervised by the IITA (Institute of Information Technology Assessment).     

On <Emphasis Type="Italic">l</Emphasis><Subscript>2,<Emphasis Type="Italic">p</Emphasis></Subscript>-circle numbers

Circle numbers are defined to reflect the Euclidean area-content and, for p ≠ 2, suitably defined non-Euclidean circumference properties of the l _2,p -circles, p ∈ [1, ∞]. The resulting function is continuous, increasing, and takes all values from [2, 4]. The actually chosen dual l _2,p -geometry for measuring the arc-length is closely connected with a generalization of the method of indivisibles of Cavalieri and Torricelli in the sense that integrating such arc-lengths means measuring area content. Moreover, this approach enables one to look in a new way into the co-area formula of measure theory which says that integrating Euclidean arc-lengths does not yield area content except for p = 2. The new circle numbers play a natural role, e.g., as norming constants in geometric measure representation formulae for p-generalized uniform probability distributions on l _2,p -circles.     

Busy period analysis for <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">PH</Emphasis>/1 queues with workload dependent balking

We consider an M/PH/1 queue with workload-dependent balking. An arriving customer joins the queue and stays until served if and only if the system workload is no more than a fixed level at the time of his arrival. We begin by considering a fluid model where the buffer content changes at a rate determined by an external stochastic process with finite state space. We derive systems of first-order linear differential equations for the mean and LST (Laplace-Stieltjes Transform) of the busy period in this model and solve them explicitly. We obtain the mean and LST of the busy period in the M/PH/1 queue with workload-dependent balking as a special limiting case of this fluid model. We illustrate the results with numerical examples.      

The flatness properties of <Emphasis Type="Italic">S</Emphasis>-poset <Emphasis Type="Italic">A</Emphasis>(<Emphasis Type="Italic">I</Emphasis>) and Rees factor <Emphasis Type="Italic">S</Emphasis>-posets

Let S be a pomonoid and I a proper right ideal of S. In a previous paper, using the amalgamated coproduct A(I) of two copies of _S S over I, we were able to solve one of the problems posed in S. Bulman-Fleming et al. (Commun. Algebra 34:1291–1317, 2006). In the present paper, we investigate further flatness properties of A(I). We also solve another problem stated in the paper cited above. Namely, we determine the condition under which Rees factor S-posets have property (P _w ). Research supported by nwnu-kjcxgc-03-18.