The type set for some measures on \mathbb{R}^{2n} with n-dimensional support

Let be realhomogeneous functions in ofdegree and let bethe Borel measure on given by

Almost periodic compactifications of group extensions

Let and be groups and let be an extension of by . Given a property of group compactifications, one can ask whether there exist compactifications and of N and K such that the universal -compactification of G is canonically isomorphic to an extension of by . We prove a theorem which gives necessary and sufficient conditions for this to occur for general properties and then apply this result to the almost periodic and weakly almost periodic compactifications of G.     

On a Problem Concerning k-Subdomination Numbers of Graphs

One of numerical invariants concerning domination in graphs is the k-subdomination number of a graph G. A conjecture concerning it was expressed by J.H. Hattingh, namely that for any connected graph G with n vertices and any k with the inequality holds. This paper presents a simple counterexample which disproves this conjecture. This counterexample is the graph of the three-dimensional cube and k = 5.     

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

The All-Paths Transit Function of a Graph

A transit function R on a set V is a function satisfying the axioms and , for all . The all-paths transit function of a connected graph is characterized by transit axioms.     

Another Perron type integration in n dimensions as an extension of integration of stepfunctions

For a new Perron-type integral a concept of convergence is introduced such that the limit f of a sequence of integrable functions is integrable and any integrable f is the limit of a sequence of stepfunctions .     

A Full Characterization of Multipliers for the Strong ϱ-Integral in the Euclidean Space

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong -integral, introduced by Jarník and Kurzweil. Let be the space of all strongly -integrable functions on a multidimensional compact interval E, equipped with the Alexiewicz norm We show that each element in the dual space of can be represented as a strong -integral. Consequently, we prove that fg is strongly -integrable on E for each strongly -integrable function f if and only if g is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on E.     

Interpolation theorems for a family of spanning subgraphs

Let G be a graph with order p, size q and component number . For each i between p – and q, let be the family of spanning i-edge subgraphs of G with exactly components. For an integer-valued graphical invariant if H H is an adjacent edge transformation (AET) implies |(H)-(H^')|1 then is said to be continuous with respect to AET. Similarly define the continuity of with respect to simple edge transformation (SET). Let M _j() and m _j() be the invariants defined by . It is proved that both M _p–() and m _p–(;) interpolate over , if is continuous with respect to AET, and that M _j() and m _j() interpolate over , if is continuous with respect to SET. In this way a lot of known interpolation results, including a theorem due to Schuster etc., are generalized.     

Quotient Groups of Groups of Certain Classes

For an arbitrary variety of groups and an arbitrary class of groups that is closed on quotient groups, we prove that a quotient group G/N of the group G possesses an invariant system with - and -factors (respectively, is a residually -group) if G possesses an invariant system with - and -factors (respectively, is a residually -group) and N (respectively, N is a maximal invariant -subgroup of the group G).     

On the Best Ranges for A p + and RH r +

In this paper we study the relationship between one-sided reverse Holder classes and the classes. We find the best possible range of to which an weight belongs, in terms of the constant. Conversely, we also find the best range of to which a weight belongs, in terms of the constant. Similar problems for , and are solved using factorization.     

Commutator Subgroups of the Extended Hecke Groups \bar H(\lambda _q )

Hecke groups H(_q) are the discrete subgroups of generated by S(z) = –(z+ _q)^–1and T(z) = –1/z. The commutator subgroup of H(_q), denoted by H(_q), is studied in [2]. It was shown that H(_q) is a free group of rank q– 1.Here the extended Hecke groups obtained by adjoining to the generators of H(_q) are considered. The commutator subgroup of is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the H(_q) case, the index of H(_q) is changed by q, in the case of this number is either 4 for qodd or 8 for qeven.     

On signed majority total domination in graphs

We initiate the study of signed majority total domination in graphs. Let G = (V, E) be a simple graph. For any real valued function f: V and S V, let . A signed majority total dominating function is a function f: V {–1, 1} such that f(N(v)) 1 for at least a half of the vertices v V. The signed majority total domination number of a graph G is = min{f(V): f is a signed majority total dominating function on G}. We research some properties of the signed majority total domination number of a graph G and obtain a few lower bounds of .This research was supported by National Natural Science Foundation of China.     

Light paths with an odd number of vertices in polyhedral maps

Let P _k be a path on k vertices. In an earlier paper we have proved that each polyhedral map G on any compact 2-manifold with Euler characteristic contains a path P _k such that each vertex of this path has, in G, degree . Moreover, this bound is attained for k = 1 or k 2, k even. In this paper we prove that for each odd , this bound is the best possible on infinitely many compact 2-manifolds, but on infinitely many other compact 2-manifolds the upper bound can be lowered to .     

Rings of maps: sequential convergence and completion

The ring B(R) of all real-valued measurable functions, carrying the pointwise convergence, is a sequential ring completion of the subring C(R) of all continuous functions and, similarly, the ring of all Borel measurable subsets of R is a sequential ring completion of the subring of all finite unions of half-open intervals; the two completions are not categorical. We study -rings of maps and develop a completion theory covering the two examples. In particular, the -fields of sets form an epireflective subcategory of the category of fields of sets and, for each field of sets , the generated -field yields its epireflection. Via zero-rings the theory can be applied to completions of special commutative -groups.     

On posets with isomorphic interval posets

Let be a partially ordered set, Int the system of all (nonempty) intervals of partially ordered by the set-theoretical inclusion . We are interested in partially ordered sets with Int isomorphic to Int . We are going to show that they correspond to couples of binary relations on A satisfying some conditions. If is a directed partially ordered set, the only with Int isomorphic to Int are corresponding to direct decompositions of ( denotes the dual of . The present results include those presented in the paper [11] by V. Slavík. Systems of intervals, particularly of lattices, have been investigated by many authors, cf. [1]–[11].     

Asymptotic behavior of solutions of A 2nth order nonlinear differential equation

In this paper we prove two results. The first is an extension of the result of G. D. Jones [4[:Every nontrivial solution for

The completions of a commutative lattice group with respect to the intrinsic topologies

In this paper we discuss the completions (, ) of a commutativel-groupG with respect to the intrinsic topologies . We give some conditions under which is the intrinsic topology of the same type on as and give the relations between these completions.     

Strictly Cyclic Algebra of Operators Acting on Banach Spaces H p(β)

Let be a sequence of positive numbers and 1 p< . We consider the space H ^p() of all power series such that . We investigate strict cyclicity of the weakly closed algebra generated by the operator of multiplication by zacting on H ^p(), and determine the maximal ideal space, the dual space and the reflexivity of the algebra . We also give a necessary condition for a composition operator to be bounded on H ^p() when is strictly cyclic.     

Estimation of the Coefficient of Multiple Determination

Assume that we have iid observations on the random vector X = (X ,...,X ) following a multivariate normal distribution N (,) where both R and (p.d.) are unknown. Let denote the multiple correlation coefficient between X and (X ,...,X ). The parameter = , called the multiple coefficient of determination, indicates the proportion of variability in X explained by its best linear fit based on (X ,..., X ). In this paper we consider the point estimation of under the ordinary squared error loss function. The usual estimators (MLE, UMVUE) have complicated risk expressions and hence it is quite difficult to get exact decision-theoretic results. We therefore follow the asymptotic decision theoretic approach (as done by Ghosh and Sinha (1981, Ann. Statist., 9, 1334-1338)) and study Second Order Admissibility of various estimators including the usual ones.     

Higher Degrees of Distributivity in MV-Algebras

In this paper we deal with the (, )-distributivity of an MV-algebra , where and are nonzero cardinals. It is proved that if is singular and (, 2)-distributive, then it is (, )-distributive. We show that if is complete then it can be represented as a direct product of MV-algebras which are homogeneous with respect to higher degrees of distributivity.