EpGs with minimal μ: II

There are two known lower bounds for (P, Q) in an EpG, called ₁ and ₂, see for example [3]. In [4], =₁ was studied for the case of triangular EGQs and, in [3], =₂ was considered for EpGs in general. Here we extend this to the case =₁ for EpGs in general, including non-triangular EGQs, and we give a number of characterizations. For instance a triangular EpG with =₁ locally is an EGQ, an extended dual net or a semibiplane; if t>2–1, then an EpG(s, t) with =₁ locally is an EGQ. In general we have only partial results for t2–1.     

Miklós–Manickam–Singhi conjectures on partial geometries

In this paper we give a proof of the Miklós–Manickam–Singhi (MMS) conjecture for some partial geometries. Specifically, we give a condition on partial geometries which implies that the MMS conjecture holds. Further, several specific partial geometries that are counterexamples to the conjecture are described.     

Integrable Extended Blaszak–Marciniak Lattice and Another Extended Lattice with Their Lax Pairs

Two new lattices are given. One is an extended system of the Blaszak–Marciniak lattice, and the other is the extended system of the lattice given by Wu and Hu. Lax pairs for these two lattices are derived from their corresponding bilinear Bäcklund transformation.     

A ℤ-Simplex Algorithm with partial updates

The Simplex primal and dual methods, for the solution of $$\max \left\{ {c^T x:Ax = b, x \geqslant 0} \right\},$$ were presented previously in terms of certain bases ? and \(\mathbb{Y}\) ofN(A) andR(A ^T ) respectively. In these implementations, called the ?-Simplex Algorithm and the \(\mathbb{Y}\) -Dual Method, the bases ? and \(\mathbb{Y}\) (giving the edges of the polyhedron in question at the given basic feasible solution) are updated at each iteration. In this paper we show that only partial updates of ? are needed in the ?-Simplex Algorithm, analogously to the partial updates in the Revised Simplex Algorithm. Similar results can be given for the \(\mathbb{Y}\) -Dual Method.     

Small embeddings of partial directed triple systems and partial triple systems with even λ

In 1976, Lindner and Rosa (Ars Combin. 1 (1976), 159–166) showed that a partial triple system with λ > 1 can be embedded in a finite triple system with the same λ. This result is improved in the case when λ is even by embedding a partial triple system on υ symbols in a triple system on t symbols, t ≡ 0,1 (mod 3), for all t >/ 3(λυ² + υ(2 ? λ) + 1). In the process, it is shown that for any λ >/ 1, a partial directed triple system on υ symbols can be embedded in a directed triple system on t symbols, t ≡ 0, 1 (mod 3), for all t ? 6λv² + 6v(1 ? λ) + 3, thus generalizing a result of Hamm (Proceedings, 14th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, Boca Raton, Florida, 1983).     

On the equivalence of μ-invariant measures for the minimal process and its q-matrix

In this paper we obtain necessary and sufficient conditions for a measure or vector that is μ-invariant for a q-matrix, Q, to be μ-invariant for the family of transition matrices, {P(t)}, of the minimal process it generates. Sufficient conditions are provided in the case when Q is regular and these are shown not to be necessary. When μ-invariant measures and vectors can be identified, they may be used, in certain cases, to determine quasistationary distributions for the process.     

On the minimal Keller–Segel system with logistic growth

This paper applies a delicate method which is inspired by Deuring (1987) and is different from those of Winkler (2010) and Yang et al. (2015) to show the known conclusion: The weak chemotactic effect can ensure the global existence and boundedness of the solutions of the minimal Keller–Segel model with logistic growth in any dimensional cases. Moreover, we obtain the explicit uniform-in-time upper bound for the global solution. It is noted that the method used in the paper may be employed to study other chemotaxis systems.     

Orthogonal geometries over finite fields with characteristic ≠2 and block designs

By taking as blocks certain subspace-pairs of an orthogonal geometry over a finite field with characteristic 2 we construct some new types of BIB designs and PBIB designs whose parameters are also given.     

1-Homogeneous Graphs with Cocktail Party μ-Graphs

Let be a graph with diameter d 2. Recall is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of the distance partition{{z V() | (z, y) = i, (x, z) = j} | 0 i, j d}is equitable and its parameters do not depend on the edge xy. Let be 1-homogeneous. Then is distance-regular and also locally strongly regular with parameters (v,k,,), where v = k, k = a ₁, (v – k – 1) = k(k – 1 – ) and c ₂ + 1, since a -graph is a regular graph with valency . If c ₂ = + 1 and c ₂ 1, then is a Terwilliger graph, i.e., all the -graphs of are complete. In [11] we classified the Terwilliger 1-homogeneous graphs with c ₂ 2 and obtained that there are only three such examples. In this article we consider the case c ₂ = + 2 3, i.e., the case when the -graphs of are the Cocktail Party graphs, and obtain that either = 0, = 2 or is one of the following graphs: (i) a Johnson graph J(2m, m) with m 2, (ii) a folded Johnson graph J¯(4m, 2m) with m 3, (iii) a halved m-cube with m 4, (iv) a folded halved (2m)-cube with m 5, (v) a Cocktail Party graph K _{m × 2} with m 3, (vi) the Schläfli graph, (vii) the Gosset graph.     

Remarks on minimal hypersur faces of sphere with constant scalar curvature

Let M be a n-dimension closed minimally immersed hypersurface in the unit sphere S~(n 1),and let h denote the second fundamental form of M.We denote the square of the length of h by S.Then we have S=n(n-1)-R,where R is the scalar curvature of M, which shows that S is intrinsic.In particular,S is constant if and only if M has constant scalar curvature.     

Extended Gerber–Shiu functions in a risk model with interest

We consider a compound Poisson risk model with interest. The Gerber–Shiu discounted penalty function is modified with an additional penalty for reaching a level above the initial capital. We show that the problem can be split into two independent problems; an original Gerber–Shiu function and a first passage problem. We also consider the case of negative interest. Finally, we apply the results to a model considered by Embrechts and Schmidli (1994).     

On Crown-Free Graphs with Regular μ-Subgraphs,II

The crown is the complete multipartite graph K _1,1,3. Terwilliger graphs without crowns and graphs without 3-cocliques with regular -subgraphs of given positive degree are studied. As a corollary, the local structure of graphs in which the neighborhoods of vertices are regular Terwilliger graphs of diameter 2 and some of these neighborhoods contain no 7-paws is determined. Connected crown-free graphs in which -subgraphs are edge regular graphs of diameter not exceeding 2 with given parameters are described.     

A note on spaces with a countable μ-base

In topological spaces every normal space with a countable base is metacompact. We show that this is not necessarily true in generalized topological spaces; more exactly we give an example of a μ-normal space with a countable μ-base which has a μ-open cover with no μ-open point-finite refinement.     

Finite p-groups with a minimal non-abelian subgroup of index p(Ⅲ)

This paper finishes the classification of three-generator finite p-groups G such that Φ(G) Z(G).This paper is a part of classification of finite p-groups with a minimal non-abelian subgroup of index p, and partly solves a problem proposed by Berkovich(2008).     

On the Morse index of minimal surfaces in ℝp with polygonal boundaries

The minimal surfaces spanning a polygon in ^p (p2) correspond to the critical points of an analytic function in finitely many variables, namely Shiffman's function. We shall prove that the Morse index of the minimal surface coincides with the Morse index of at the corresponding critical point. Alternatively expressed, the Schwarz operator of the minimal surface and the Hessian of have the same number of negative eigenvalues. Finally we control the degeneration of the critical points.     

Finite p-groups with a minimal non-abelian subgroup of index p(Ⅱ)

We classify completely three-generator finite p-groups G such that Ф(G)≤Z(G)and|G′|≤p2.This paper is a part of the classification of finite p-groups with a minimal non-abelian subgroup of index p,and solve partly a problem proposed by Berkovich.     

Hölder continuous periodic solution of Boussinesq equation with partial viscosity

We show the existence of Hölder continuous periodic solution with compact support in time of the Boussinesq equations with partial viscosity. The Hölder regularity of the solution we constructed is anisotropic which is compatible with partial viscosity of the equations.     

Stability of hyperbolic–parabolic mixed type equations with partial boundary condition

In this paper, we are concerned with the hyperbolic–parabolic mixed type equations with the non-homogeneous boundary condition. If it is degenerate on the boundary, the part of the boundary whose boundary value should be imposed, is determined by the entropy condition from the convection term. If there is no convection term in the equation, we show that the stability of solutions can be proved without any boundary condition. If the equation is completely degenerate, we show that the stability of solutions can be established just based on the partial boundary condition.