Rate of convergence to stationarity of the system <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">N</Emphasis>/<Emphasis Type="Italic">N</Emphasis>+<Emphasis Type="Italic">R</Emphasis>

We consider the M/M/N/N+R service system, characterized by N servers, R waiting positions, Poisson arrivals and exponential service times. We discuss representations and bounds for the rate of convergence to stationarity of the number of customers in the system, and study its behaviour as a function of R, N and the arrival rate λ, allowing λ to be a function of N.     

Packet loss characteristics for <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1/<Emphasis Type="Italic">N</Emphasis> queueing systems

In this contribution we investigate higher-order loss characteristics for M/G/1/N queueing systems. We focus on the lengths of the loss and non-loss periods as well as on the number of arrivals during these periods. For the analysis, we extend the Markovian state of the queueing system with the time and number of admitted arrivals since the instant where the last loss occurred. By combining transform and matrix techniques, expressions for the various moments of these loss characteristics are found. The approach also yields expressions for the loss probability and the conditional loss probability. Some numerical examples then illustrate our results.     

Three-dimensional <Emphasis Type="Italic">N</Emphasis>=4 supersymmetry in harmonic <Emphasis Type="Italic">N</Emphasis>=3 superspace

We consider the map of three-dimensional N=4 superfields to the N=3 harmonic superspace. The left and right representations of the N=4 superconformal group are constructed on N=3 analytic superfields. These representations are convenient for describing N=4 superconformal couplings of Abelian gauge superfields to hypermultiplets. We investigate the N=4 invariance in the non-Abelian N=3 Yang-Mills theory.     

Different prime <Emphasis Type="Italic">N</Emphasis>-ideals and IFP <Emphasis Type="Italic">N</Emphasis>-Ideals

We provide some characterizations of completely prime (completely semiprime) and 3-prime (3-semiprime) N-groups. The relationship between a 3-prime (completely prime) N-ideal P of an N-group Γ and the ideal (P: Γ) of the near-ring N is investigated. Moreover, the notion of IFP N-ideal is defined. We prove that the concept of IFP N-ideal occurs naturally where N is a left permutable (left self distributive, subcommutative) near-ring and Γ a monogenic N-group. Also, we obtain some relationships between an IFP N-ideal P of an N-group Γ and the ideal (P: Γ) of the near-ring N.     

<Emphasis Type="Italic">N</Emphasis>-commutators

The N-commutator is conjecturally a well-defined nontrivial operation on for x = (x ₁, ... , x _n ) if and only if N = n ² + 2n - 2. This is proved for n = 2 and confirmed by computer experiments for n < 5. Under 2- and 5-commutators the algebra of divergence-free vector fields in two dimensions is an sh-Lie (strong homotopic Lie) algebra in the sense of Stasheff. Similarly, W(2) is an sh-Lie algebra with respect to 2- and 6-commutators.     

<Emphasis Type="Italic">X</Emphasis>-decomposable finite groups for <Emphasis Type="Italic">X</Emphasis>={1, <Emphasis Type="Italic">m</Emphasis>, <Emphasis Type="Italic">m</Emphasis> + 1, <Emphasis Type="Italic">m</Emphasis> + 2}

A normal subgroup N of a finite group G is called n-decomposable in G if N is the union of n distinct G-conjugacy classes. We study the structure of nonperfect groups in which every proper nontrivial normal subgroup is m-decomposable, m+1-decomposable, or m+2-decomposable for some positive integer m. Furthermore, we give classification for the soluble case.     

The <Emphasis Type="Italic">N</Emphasis>-wave equations with <Emphasis Type="Italic">PT</Emphasis> symmetry

We study extensions of N-wave systems with PT symmetry and describe the types of (nonlocal) reductions leading to integrable equations invariant under the P (spatial reflection) and T (time reversal) symmetries. We derive the corresponding constraints on the fundamental analytic solutions and the scattering data. Based on examples of three-wave and four-wave systems (related to the respective algebras sl(3,C) and so(5,C)), we discuss the properties of different types of one- and two-soliton solutions. We show that the PT-symmetric three-wave equations can have regular multisoliton solutions for some specific choices of their parameters.     

The <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1+<Emphasis Type="Italic">G</Emphasis> queue revisited

We consider an M/G/1 queue with the following form of customer impatience: an arriving customer balks or reneges when its virtual waiting time, i.e., the amount of work seen upon arrival, is larger than a certain random patience time. We consider the number of customers in the system, the maximum workload during a busy period, and the length of a busy period. We also briefly treat the analogous model in which any customer enters the system and leaves at the end of his patience time or at the end of his virtual sojourn time, whichever occurs first.     

Further Results for the Queueing System <Emphasis Type="Italic">M</Emphasis><Superscript><Emphasis Type="Italic">x</Emphasis></Superscript>/<Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">c</Emphasis>

For the multi-channel bulk-arrival queue, M ^x /M/c, Abol'nikov and Kabak independently obtained steady state results. In this paper the results of these authors are extended, corrected and simplified. A number of measures of efficiency are calculated for three cases where the arrival group size has: (i) a constant value, (ii) a geometric distribution, or (iii) a positive Poisson distribution. The paper also shows how to calculate fractiles for both the queue length and the waiting time distribution. Examples of extensive numerical results for certain measures of efficiency are presented in tabular and chart form.     

A study of (<Emphasis Type="Italic">x</Emphasis>(<Emphasis Type="Italic">q</Emphasis> + 1), <Emphasis Type="Italic">x</Emphasis>; 2, <Emphasis Type="Italic">q</Emphasis>)-minihypers

In this paper, we study the weighted (x(q + 1), x; 2, q)-minihypers. These are weighted sets of x(q + 1) points in PG(2, q) intersecting every line in at least x points. We investigate the decomposability of these minihypers, and define a switching construction which associates to an (x(q + 1), x; 2, q)-minihyper, with x ≤ q ² − q, not decomposable in the sum of another minihyper and a line, a (j(q + 1), j; 2, q)-minihyper, where j = q ² − q − x, again not decomposable into the sum of another minihyper and a line. We also characterize particular (x(q + 1), x; 2, q)-minihypers, and give new examples. Additionally, we show that (x(q + 1), x; 2, q)-minihypers can be described as rational sums of lines. In this way, this work continues the research on (x(q + 1), x; 2, q)-minihypers by Hill and Ward (Des Codes Cryptogr 44:169–196, 2007), giving further results on these minihypers.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

Bilinear approach to <Emphasis Type="Italic">N</Emphasis> = 2 supersymmetric KdV equations

The N = 2 supersymmetric KdV equations are studied within the framework of Hirota bilinear method. For two such equations, namely N = 2, a = 4 and N = 2, a = 1 supersymmetric KdV equations, we obtain the corresponding bilinear formulations. Using them, we construct particular solutions for both cases. In particular, a bilinear Bäcklund transformation is given for the N = 2, a = 1 supersymmetric KdV equation.     

Investigating the anomaly puzzle in <Emphasis Type="Italic">N</Emphasis>=1 supersymmetric electrodynamics

Using Schwinger-Dyson equations and Ward identities in the N=1 supersymmetric electrodynamics regularized by higher derivatives, we find that we can calculate some contributions to the two-point Greens function of the gauge field and to the -function exactly in all orders of the perturbation theory. We use the results to investigate the anomaly puzzle in the considered theory.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 1, pp. 35–56, January, 2005.     

Integrable equations for the model with <Emphasis Type="Italic">N</Emphasis> sources and <Emphasis Type="Italic">n</Emphasis>-1 modes

We consider simplified models of coupling charged matter to radiation resonance modes generalizing the well-known Jaynes-Cummings and Dicke models. We find that these new models are integrable for arbitrary numbers of dipole sources and resonance modes of the radiation field. We discuss the problem of explicitly diagonalizing the corresponding Hamiltonians.     

Nested regular polygon solutions for planar 2<Emphasis Type="Italic">N</Emphasis>-body problems

In this paper we study the necessary conditions for the masses of the nested regular polygon solutions of the planar2N-body problem.We prove that the masses at the vertices of each regular polygon must be equal to each other     

Cyclotomic Hecke Algebras of <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)

In this note, we find a monomial basis of the cyclotomic Hecke algebra \({\mathcal{H}_{r,p,n}}\) of G(r,p,n) and show that the Ariki-Koike algebra \({\mathcal{H}_{r,n}}\) is a free module over \({\mathcal{H}_{r,p,n}}\), using the Gröbner-Shirshov basis theory. For each irreducible representation of \({\mathcal{H}_{r,p,n}}\), we give a polynomial basis consisting of linear combinations of the monomials corresponding to cozy tableaux of a given shape.     

Availability Analysis of Repairable <Emphasis Type="Italic">N</Emphasis>-Server <Emphasis Type="Italic">n</Emphasis><Subscript><Emphasis Type="Italic">i</Emphasis></Subscript>-Replicated Systems

We develop an exact closed-form expression for the steady-state availability of a repairable, N-server system in which the ith server contains n _i identical, reconfigurable, breakdown-prone units. Our approach, which follows from the Markov chain balance equations and the recursive properties of Hessenberg matrix determinants, is simpler than previously proposed matrix geometric approaches, and can readily be adapted to the availability analysis of more complicated structures. We illustrate this by computing the steady-state availability of a mixed parallel-serial gracefully degrading replicated system.     

(<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">t</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-bi-Koszul algebras

The paper focuses on the 1-generated positively graded algebras with non-pure resolutions and mainly discusses a new kind of algebras called(s,t,d)-bi-Koszul algebras as the generalization of bi-Koszul algebras. An(s,t,d)-bi-Koszul algebra can be obtained from two periodic algebras with pure resolutions. The generation of the Koszul dual of an(s,t,d)-bi-Koszul algebra is discussed. Based on it,the notion of strongly(s,t,d)-bi-Koszul algebras is raised and their homological properties are further discussed.     

A lower bound for the <Emphasis Type="Italic">r</Emphasis>-order of a matrix modulo <Emphasis Type="Italic">N</Emphasis>

For a positive integer N, we define the N-rank of a non singular integer d × d matrix A to be the maximum integer r such that there exists a minor of order r whose determinant is not divisible by N. Given a positive integer r, we study the growth of the minimum integer k, such that A ^k − I has N-rank at most r, as a function of N. We show that this integer k goes to infinity faster than log N if and only if for every eigenvalue λ which is not a root of unity, the sum of the dimensions of the eigenspaces relative to eigenvalues which are multiplicatively dependent with λ and are not roots of unity, plus the dimensions of the eigenspaces relative to eigenvalues which are roots of unity, does not exceed d − r − 1. This result will be applied to recover a recent theorem of Luca and Shparlinski [6] which states that the group of rational points of an ordinary elliptic curve E over a finite field with q ⁿ elements is almost cyclic, in a sense to be defined, when n goes to infinity. We will also extend this result to the product of two elliptic curves over a finite field and show that the orders of the groups of rational points of two non isogenous elliptic curves are almost coprime when n approaches infinity. Author’s address: Dipartimento di Matematica e Informatica, Via Delle Scienze 206, 33100 Udine, Italy     

Nonlinear Preserver Problems on B（H）

Let H be a complex Hilbert space of dimension greater than 2, and B(H) denote the Banach algebra of all bounded linear operators on H. For A, B ∈ B(H), define the binary relation A ≤* B by A*A = A*B and AA* = AB*. Then (B(H), “≤*”) is a partially ordered set and the relation “≤*” is called the star order on B(H). Denote by Bs(H) the set of all self-adjoint operators in B(H). In this paper, we first characterize nonlinear continuous bijective maps on B _s (H) which preserve the star order in both directions. We characterize also additive maps (or linear maps) on B(H) (or nest algebras) which are multiplicative at some invertible operator.