A note on harmonious families

The concept of a harmonious family of sets, which generalizes, in a particular setting, the concept of a balanced collection, is characterized in set-like termsThis work was supported, in part, by the Army Research Office under Contract DA-31-124-ARO(D)-366; and, in part, was done while the author was an IBM World Trade Corporation Fellow at IBM France.     

A note on linear Sperner families

In an earlier work we described Gröbner bases of the ideal of polynomials over a field, which vanish on the set of characteristic vectors \(\mathbf {v}\in \{0,1\}^n\) of the complete d uniform set family over the ground set [n]. In particular, it turns out that the standard monomials of the above ideal are ballot monomials. We give here a partial extension of this fact. A set family is a linear Sperner system if the characteristic vectors satisfy a linear equation \(a_1v_1+\cdots +a_nv_n=k\), where the \(a_i\) and k are positive integers. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials, provided that \(0<a_1\le a_2\le \cdots \le a_n\). As an application, we confirm a conjecture of Frankl in the special case of linear Sperner systems.     

A note on families of hyperelliptic curves

We give a stack-theoretic proof for some results on families of hyperelliptic curves. Received: 5 February 2008     

A note on the optimality of the N- and D-policies for the M/G/1 queue

This note considers the N- and D-policies for the M/G/1 queue. We concentrate on the true relationship between the optimal N- and D-policies when the cost function is based on the expected number of customers in the system.     

A note on three-parameter families and generalized convex functions

The concept of generalized convex functions introduced by Beckenbach [E.F. Beckenbach, Generalized convex functions, Bull. Amer. Math. Soc. 43 (1937) 363–371] is extended to the two-dimensional case. Using three-parameter families, we define generalized convex (midconvex, M-convex) functions and show some continuity properties of them.     

A note on confidence sequences in multiparameter exponential families

In a paper by Lai [(1976) Ann. Statist.4] a construction of sequences of confidence intervals for one-parameter exponential families is given, where the intersection of these intervals contains the true parameter value θ₀ with a prescribed level of confidence and the respective interval bounds tend a.s. to θ₀. In this paper an analogous result for multiparameter exponential families is proved. More presicely, sequences of convex confidence sets are obtained which shrink to the actual parameter value.     

A note on equal unions in families of sets

A family of sets has the equal union property if and only if there exist two nonempty disjoint subfamilies having the same union. We prove that any n nonempty subsets of an n-element set have the equal union property if the sum of their cardinalities exceeds n(n+1)/2. This bound is tight. Among families in which the sum of the cardinalities equals n(n+1)/2, we characterize those having the equal union property.     

Continuity in G

For a discrete group G, we consider βG, the Stone– ech compactification of G, as a right topological semigroup, and G^*=βGG as a subsemigroup of βG. We study the mappings λ_p^* :G^*→G^*and μ^* :G^*→G^*, the restrictions to G^* of the mappings λ_p :βG→βG and μ :βG→βG, defined by the rules λ_p(q)=pq, μ(q)=qq. Under some assumptions, we prove that the continuity of λ_p^* or μ^* at some point of G^* implies the existence of a P-point in ω^*.     

G-Loops and Permutation Groups

A G-loop is a loop which is isomorphic to all its loop isotopes. We apply some theorems about permutation groups to get information about G-loops. In particular, we study G-loops of order pq, where p < q are primes and p  (q − 1). In the case p = 3, the only G-loop of order 3q is the group of order 3q. The notion “G-loop” splits naturally into “left G-loop” plus “right G-loop.” There exist non-group right G-loops and left G-loops of order n iff n is composite and n > 5.     

The Levi problem on strongly pseudoconvex G-bundles

Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle G → M → X so that M is also a strongly pseudoconvex complex manifold. In this study, we show that if G acts by holomorphic transformations satisfying a local property, then the space of square-integrable holomorphic functions on M is infinite-dimensional.     

A solution of the Steenrod problem for G-Moore spaces

The purpose of this paper is to prove the following theorem: If G is a finite group, then every G-module is isomorphic to the homology of a G-Moore space if and only if all Sylow subgroups of G are cyclic.     

A note on the analytic families of compact submanifolds of complex manifolds

In this paper, we prove a result of the deformation of the complex structure of a submanifold. Our result is a modification of the result of Kodaira (Ann. Math 75(1), 146-162, 1962).

     

A note on the norm-continuity for evolution families arising from non-autonomous forms

We consider evolution equations of the form $$\begin{aligned} \dot{u}(t)+{\mathcal {A}}(t)u(t)=0,\ \ t\in [0,T],\ \ u(0)=u_0, \end{aligned}$$where $${\mathcal {A}}(t),\ t\in [0,T],$$ are associated with a non-autonomous sesquilinear form $${\mathfrak {a}}(t,\cdot ,\cdot )$$ on a Hilbert space H with constant domain $$V\subset H.$$ In this note we continue the study of fundamental operator theoretical properties of the solutions. We give a sufficient condition for norm-continuity of evolution families on each spaces V, H and on the dual space $$V'$$ of V. The abstract results are applied to a class of equations governed by time dependent Robin boundary conditions on exterior domains and by Schrödinger operator with time dependent potentials.     

G-pre-invex functions in mathematical programming

In the present paper, we introduce the concept of G-pre-invex functions with respect to η defined on an invex set with respect to η. These function unify the concepts of nondifferentiable convexity, pre-invexity and r-pre-invexity. Furthermore, relationships of G-pre-invex functions to various introduced earlier pre-invexity concepts are also discussed. Some (geometric) properties of this class of functions are also derived. Finally, optimality results are established for optimization problems under appropriate G-pre-invexity conditions.