Extended partial geometries with minimal μ

Recently Fisher and Hobart extended the original work of Buekenhout and Hubaut on extended generalized quadrangles with a certain minimal value (minimal ) for the nonempty residue intersections. We show that there is another minimal which is sometimes better than the earlier one, and we extend all this to the case of extended partial geometries. We examine especially the triangular case, and find only a few new possibilities, which we are unable to settle, although some of these look extremely interesting.     

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.     

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.     

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.     

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.     

Evolution equations with Gel’fond-Leont’ev generalized differentiation operators: II

We establish the solvability of the Cauchy problem for evolution equations with Gel’fond-Leont’ev generalized differentiation operators in spaces of the type W as well as in spaces of generalized functions (analytic functionals) of the type W′.     

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).     

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.     

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.     

Performance evaluation of demodulation with diversity—a combinatorial approach II: bijective methods

This paper is devoted to the presentation of a combinatorial approach for analyzing the performance of a generic family of demodulation methods used in mobile telecommunications. We show that a fundamental formula in this context is in fact highly connected with a slight modification of a very classical bijection of Knuth between pairs of Young tableaux of conjugate shapes and $\{0,1\}$-matrices. These considerations allowed us to obtain the first explicit expressions for several important specializations of the performance evaluation formula that we studied.     

Existence result for minimal hypersurfaces¶with a prescribed finite number of planar ends

Paralleling what has been done for minimal surfaces in ℝ³, we develop a gluing procedure to produce, for any k≥ 2 and any n≥ 3 complete immersed minimal hypersurfaces of ℝ ^{n +1} which have k planar ends. These surfaces are of the topological type of a sphere with k punctures and they all have finite total curvature. Received: 1 July 1999 / Revised version: 31 May 2000     

Completeness for μ-calculi: A coalgebraic approach

We set up a generic framework for proving completeness results for variants of the modal mu-calculus, using tools from coalgebraic modal logic. We illustrate the method by proving two new completeness results: for the graded mu-calculus (which is equivalent to monadic second-order logic on the class of unranked tree models), and for the monotone modal mu-calculus.Besides these main applications, our result covers the Kozen–Walukiewicz completeness theorem for the standard modal mu-calculus, as well as the linear-time mu-calculus and modal fixpoint logics on ranked trees. Completeness of the linear-time mu-calculus is known, but the proof we obtain here is different and places the result under a common roof with Walukiewicz' result.Our approach combines insights from the theory of automata operating on potentially infinite objects, with methods from the categorical framework of coalgebra as a general theory of state-based evolving systems. At the interface of these theories lies the notion of a coalgebraic modal one-step language. One of our main contributions here is the introduction of the novel concept of a disjunctive basis for a modal one-step language. Generalizing earlier work, our main general result states that in case a coalgebraic modal logic admits such a disjunctive basis, then soundness and completeness at the one-step level transfer to the level of the full coalgebraic modal mu-calculus.     

On the bicriterion – minimal cost/minimal label – spanning tree problem

We address a bicriterion spanning tree problem relevant in some application fields such as telecommunication networks or transportation networks. Each edge is assigned with a cost value and a label (such as a color). The first criterion intends to minimize the total cost of the spanning tree (the summation of its edge costs), while the second intends to get the solution with a minimal number of different labels. Since these criteria, in general, are conflicting criteria we developed an algorithm to generate the set of non-dominated spanning trees. Computational experiments are presented and results discussed.     

The Duffin–Schaeffer conjecture with extra divergence II

In 2012 the authors set out a programme to prove the Duffin–Schaeffer conjecture for measures arbitrarily close to Lebesgue measure. In this paper we take a new step in this direction. Given a non-negative function $\psi : \mathbb N \rightarrow \mathbb R $ , let $W(\psi )$ denote the set of real numbers $x$ such that $|nx -a| < \psi (n) $ for infinitely many reduced rationals $a/n \ (n>0) $ . Our main result is that $W(\psi )$ is of full Lebesgue measure if there exists a $c > 0 $ such that $$\begin{aligned} \sum _{n\ge 16} \, \frac{\varphi (n) \psi (n)}{n \exp (c(\log \log n)(\log \log \log n))} \, = \, \infty \, . \end{aligned}$$      

On the practical μ-stability of solutions of standard systems with delay

We study the problem of μ-stability of a dynamical system with delay. Conditions of the practical μ-stability are established for the general case and for a quasilinear system. The conditions suggested are illustrated by an example. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 204–213, February, 1999.