On Geodesic <Emphasis Type="Italic">E</Emphasis>-Convex Sets,Geodesic <Emphasis Type="Italic">E</Emphasis>-Convex Functions and <Emphasis Type="Italic">E</Emphasis>-Epigraphs

In this paper, we introduce a new class of sets and a new class of functions called geodesic E-convex sets and geodesic E-convex functions on a Riemannian manifold. The concept of E-quasiconvex functions on R ⁿ is extended to geodesic E-quasiconvex functions on Riemannian manifold and some of its properties are investigated. Afterwards, we generalize the notion of epigraph called E-epigraph and discuss a characterization of geodesic E-convex functions in terms of its E-epigraph. Some properties of geodesic E-convex sets are also studied.     

On semi-<Emphasis Type="Italic">I</Emphasis>-open sets and semi<Emphasis Type="Italic">-I-</Emphasis>continuous functions

Summary We investigate further properties of semi-I-open sets and semi-I-continuous functions introduced in [4] and give the notions of semi-I-open and semi-I-closed functions.     

<Emphasis Type="Italic">E</Emphasis><Superscript>*</Superscript>-dense <Emphasis Type="Italic">E</Emphasis>-semigroups

We investigate E^*-dense semi\-groups as analogues of E-densesemigroupsfor semigroups with zero. We give a characterisation theorem forE^*-dense semigroups whose idempotents form a *-rectangularband. The construction methods of generalised Rees matrix semigroupsare employed to provide examples and illustrations. Our results areanalogous to those of Weipoltshammer for E-dense semigroups.     

Integrals of <Emphasis Type="Italic">K</Emphasis> and <Emphasis Type="Italic">E</Emphasis> from lattice sums

We give closed form evaluations for many families of integrals, whose integrands contain algebraic functions of the complete elliptic integrals K and E. Our methods exploit the rich structures connecting complete elliptic integrals, Jacobi theta functions, lattice sums, and Eisenstein series. Various examples are given, and along the way new (including 10-dimensional) lattice sum evaluations are produced.     

<Emphasis Type="Italic">E</Emphasis>-inversive Dubreil-Jacotin semigroups

Using group congruences, we obtain necessary and sufficient conditions for an ordered E-inversive semigroup to be a Dubreil-Jacotin semigroup. We also determine when such a semigroup is naturally ordered. In particular, when the subset of regular elements is a subsemigroup it contains a multiplicative inverse transversal.     

Structure theorems of <Emphasis Type="Italic">E</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)-Azumaya algebras

Let k be a field and E(n) be the 2 ⁿ⁺¹-dimensional pointed Hopf algebra over k constructed by Beattie, Dăscălescu and Grünenfelder [J. Algebra, 2000, 225: 743–770]. E(n) is a triangular Hopf algebra with a family of triangular structures R _M parameterized by symmetric matrices M in M _n (k). In this paper, we study the Azumaya algebras in the braided monoidal category $ E_{(n)} \mathcal{M}^{R_M } $ E_{(n)} \mathcal{M}^{R_M } and obtain the structure theorems for Azumaya algebras in the category $ E_{(n)} \mathcal{M}^{R_M } $ E_{(n)} \mathcal{M}^{R_M } , where M is any symmetric n×n matrix over k.     

Generalised domain and <Emphasis Type="Italic">E</Emphasis>-inverse semigroups

A generalised D-semigroup is here defined to be a left E-semiabundant semigroup S in which the \(\overline{\mathcal R}_E\)-class of every \(x\in S\) contains a unique element D(x) of E, made into a unary semigroup. Two-sided versions are defined in the obvious way in terms of \(\overline{\mathcal R}_E\) and \(\overline{\mathcal L}_E\). The resulting class of unary (bi-unary) semigroups is shown to be a finitely based variety, properly containing the variety of D-semigroups (defined in an order-theoretic way in Communications in Algebra, 3979–4007, 2014). Important subclasses associated with the regularity and abundance properties are considered. The full transformation semigroup \(T_X\) can be made into a generalised D-semigroup in many natural ways, and an embedding theorem is given. A generalisation of inverse semigroups in which inverses are defined relative to a set of idempotents arises as a special case, and a finite equational axiomatisation of the resulting unary semigroups is given.     

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.     

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.     

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.     

Generalized convex spaces, <Emphasis Type="Italic">L</Emphasis>-spaces,and <Emphasis Type="Italic">FC</Emphasis>-spaces

We show that FC-spaces due to Ding are particular types of L-spaces due to Ben-El-Mechaiekh et al., and hence particular types of G-convex spaces. Some counter-examples are given and related matters are also discussed.     

Integral varieties of the canonical cone structure on <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">P</Emphasis>

The canonical cone structure on a compact Hermitian symmetric space G/P is the fiber bundle where is the cone of the highest weight vectors under the action of the reductive part of P. It is known that the cone coincides with the cone of the vectors tangent to the lines in G/P passing through x, when we consider G/P as a projective variety under its homogeneous embedding into the projective space of the irreducible representation space V of G with highest weight associated to P. A subvariety X of G/P is said to be an integral variety of at all smooth points xG/P. Equivalently, an integral variety of is a subvariety of G/P whose embedded projective tangent space at each smooth point is a linear space We prove a kind of rigidity of the integral varieties under some dimension condition. After making a uniform setting to study the problem, we apply the theory of Lie algebra cohomology as a main tool. Finally we show that the dimension condition is necessary by constructing counterexamples.     

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">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.     

The Center of <Emphasis Type="Italic">Dist</Emphasis> (<Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">m</Emphasis>|<Emphasis Type="Italic">n</Emphasis>)) in Positive Characteristic

The purpose of this paper is to investigate central elements in distribution algebras D i s t(G) of general linear supergroups G = G L(m|n). As an application, we compute explicitly the center of D i s t(G L(1|1)) and its image under Harish-Chandra homomorphism.     

Approximating the Distribution of Customers in <Emphasis Type="Italic">M/E</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript><Emphasis Type="Italic">/s</Emphasis> Queues

Approximation formulae are suggested for the mean and variance of customers in M/E _n /s queues. It is shown that the distributions can be approximated by using the mean and variance to fit Gamma functions. A brief comment on the more general E _m /E _n /s case is given.     

<Emphasis Type="Italic">C</Emphasis>-distribution semigroups and <Emphasis Type="Italic">C</Emphasis>-ultradistribution semigroups in locally convex spaces

The main purpose of this paper is to study C-distribution semigroups and C-ultradistribution semigroups in the setting of sequentially complete locally convex spaces. We provide a few important theoretical novelties in this field and some interesting examples. Under consideration are stationary dense operators in a sequentially complete locally convex space.