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.     

On a new property of <Emphasis Type="Italic">n</Emphasis>-poised and <Emphasis Type="Italic">G</Emphasis><Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript> sets

In this paper we consider n-poised planar node sets, as well as more special ones, called G C _n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ? is called k-node line for a node set \(\mathcal X\) if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C _n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of \(\mathcal X\) is used by each node in \(\mathcal X\setminus M, \)meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C _n set \(\mathcal {X}\) is used either by exactly \(\binom {n}{2}\) nodes or by exactly \(\binom {n-1}{2}\) nodes. We prove also similar statements concerning n-node or (n ? 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C _n sets. At the end we present a conjecture concerning any k-node line.     

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.     

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.     

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.     

<Emphasis Type="Italic">n</Emphasis>-Abelian and <Emphasis Type="Italic">n</Emphasis>-exact categories

We introduce n-abelian and n-exact categories, these are analogs of abelian and exact categories from the point of view of higher homological algebra. We show that n-cluster-tilting subcategories of abelian (resp. exact) categories are n-abelian (resp. n-exact). These results allow to construct several examples of n-abelian and n-exact categories. Conversely, we prove that n-abelian categories satisfying certain mild assumptions can be realized as n-cluster-tilting subcategories of abelian categories. In analogy with a classical result of Happel, we show that the stable category of a Frobenius n-exact category has a natural \((n+2)\)-angulated structure in the sense of Geiß–Keller–Oppermann. We give several examples of n-abelian and n-exact categories which have appeared in representation theory, commutative algebra, commutative and non-commutative algebraic geometry.     

<Emphasis Type="Italic">n</Emphasis>-Flat and <Emphasis Type="Italic">n</Emphasis>-FP-injective modules

In this paper, we study the existence of the n-flat preenvelope and the n-FP-injective cover. We also characterize n-coherent rings in terms of the n-FP-injective and n-flat modules.     

Some isomorphism results for Thompson-like groups <Emphasis Type="Italic">V</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">G</Emphasis>)

The natural automorphism group of a translation surface is its group of translations. For finite translation surfaces of genus g ≥ 2 the order of this group is naturally bounded in terms of g due to a Riemann–Hurwitz formula argument. In analogy with classical Hurwitz surfaces, we call surfaces which achieve the maximal bound Hurwitz translation surfaces. We study for which g there exist Hurwitz translation surfaces of genus g.     

Extremal <Emphasis Type="Italic">G</Emphasis>-invariant eigenvalues of the Laplacian of <Emphasis Type="Italic">G</Emphasis>-invariant metrics

The study of extremal properties of the spectrum often involves restricting the metrics under consideration. Motivated by the work of Abreu and Freitas in the case of the sphere S ² endowed with S ¹-invariant metrics, we consider the subsequence of the spectrum of a Riemannian manifold M which corresponds to metrics and functions invariant under the action of a compact Lie group G. If G has dimension at least 1, we show that the functional λ _k ^G admits no extremal metric under volume-preserving G-invariant deformations. If, moreover, M has dimension at least three, then the functional is unbounded when restricted to any conformal class of G-invariant metrics of fixed volume. As a special case of this, we can consider the standard O(n)-action on S ⁿ ; however, if we also require the metric to be induced by an embedding of S ⁿ in , we get an optimal upper bound on .      

Analyzing the Steady-State Queue <Emphasis Type="Italic">GI</Emphasis><Superscript><Emphasis Type="Italic">X</Emphasis></Superscript><Emphasis Type="Italic">/G/</Emphasis>1

It is proved in this note that the delay in the queue GI ^X /G/1 can be expressed as the sum of two independent components, such that known results of the queue GI/G/1 (e.g. approximations) can be readily applied. Based on this result, closed-form expressions are also derived for other performance measures of interest.     

Groups with less than <Emphasis Type="Italic">n</Emphasis> subgroups of index <Emphasis Type="Italic">n</Emphasis>

We characterise (residually-finite) groups which possess less than n subgroups of index n for almost all n ∈ ℕ.     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where

2-Connected factor-critical graphs <Emphasis Type="Italic">G</Emphasis> with exactly |<Emphasis Type="Italic">E</Emphasis>(<Emphasis Type="Italic">G</Emphasis>)| + 1 maximum matchings

A connected graph G is said to be a factor-critical graph if G ?v has a perfect matching for every vertex v of G. In this paper, the 2-connected factor-critical graph G which has exactly |E(G)| + 1 maximum matchings is characterized.     

Mappings of preserving <Emphasis Type="Italic">n</Emphasis>-distance one in <Emphasis Type="Italic">n</Emphasis>-normed spaces

We give a positive answer to the Aleksandrov problem in n-normed spaces under the surjectivity assumption. Namely, we show that every surjective mapping preserving n-distance one is affine, and thus is an n-isometry. This is the first time the Aleksandrov problem is solved in n-normed spaces with only the surjectivity assumption even in the usual case \(n=2\). Finally, when the target space is n-strictly convex, we prove that every mapping preserving two n-distances with an integer ratio is an affine n-isometry.     

<Emphasis Type="Italic">C</Emphasis>*-simplicity of <Emphasis Type="Italic">n</Emphasis>-periodic products

The C*-simplicity of n-periodic products is proved for a large class of groups. In particular, the n-periodic products of any finite or cyclic groups (including the free Burnside groups) are C*-simple. Continuum-many nonisomorphic 3-generated nonsimple C*-simple groups are constructed in each of which the identity xⁿ = 1 holds, where n ≥ 1003 is any odd number. The problem of the existence of C*-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.     

The Combinatorial Quantum Cohomology Ring of <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">B</Emphasis>

A purely combinatorial construction of the quantum cohomology ring of the generalized flag manifold is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of our previous papers [12, 13], we obtain a presentation of this ring in terms of generators and relations, and formulas for quantum Giambelli polynomials. We show that these polynomials satisfy a certain orthogonality property, which—for G = SL_n( )—was proved previously in the paper [5].     

Exponentially Many Monochromatic <Emphasis Type="Italic">n</Emphasis>-Matchings in <Emphasis Type="Italic">K</Emphasis><Subscript>3<Emphasis Type="Italic">n</Emphasis>-1</Subscript>

Erdös et al and Gerencsér et al had shown that in any 2-edge-coloring of K _3n-1, there is a n-matching containing edges with the same color(we call such matching monochromatic matching). In this paper we show that for any 2-edge-coloring of K _3n-1 there exists a monochromatic subgraph H of K _3n-1 which contains exponentially many monochromatic n-matchings.     

On holomorphic curves in a complex Grassmann manifold <Emphasis Type="Italic">G</Emphasis>(2, <Emphasis Type="Italic">n</Emphasis>)

Let s : S ² → G(2, n) be a linearly full totally unramified non-degenerate holomorphic curve in a complex Grassmann manifold G(2, n), and let K(s) be its Gaussian curvature. It is proved that K(s) = \frac4n-2{K(s) = \frac{4}{n-2}} if K(s) satisfies K(s) 3 \frac4n-2{K(s) \geq \frac{4}{n-2}} or K(s) ￡ \frac4n-2 {K(s) \leq \frac{4}{n-2} } everywhere on S ². In particular, K(s) = \frac4n-2{K(s) = \frac{4}{n-2}} if K(s) is constant.     

Colored-descent representations of complex reflection groups <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r,p, n</Emphasis>)

We study the complex reflection groups G(r, p, n). By considering these groups as subgroups of the wreath products , and by using Clifford theory, we define combinatorial parameters and descent representations of G(r, p, n), previously known for classical Weyl groups. One of these parameters is the flag major index, which also has an important role in the decomposition of these representations into irreducibles. A Carlitz type identity relating the combinatorial parameters with the degrees of the group, is presented.     

A new class of lattice paths and partitions with <Emphasis Type="Italic">n</Emphasis> copies of <Emphasis Type="Italic">n</Emphasis>

Agarwal and Bressoud (Pacific J. Math. 136(2) (1989) 209–228) defined a class of weighted lattice paths and interpreted several q-series combinatorially. Using the same class of lattice paths, Agarwal (Utilitas Math. 53 (1998) 71–80; ARS Combinatoria 76 (2005) 151–160) provided combinatorial interpretations for several more q-series. In this paper, a new class of weighted lattice paths, which we call associated lattice paths is introduced. It is shown that these new lattice paths can also be used for giving combinatorial meaning to certain q-series. However, the main advantage of our associated lattice paths is that they provide a graphical representation for partitions with n + t copies of n introduced and studied by Agarwal (Partitions with n copies of n, Lecture Notes in Math., No. 1234 (Berlin/New York: Springer-Verlag) (1985) 1–4) and Agarwal and Andrews (J. Combin. Theory A45(1) (1987) 40–49).