For each value of r ≥ 4, r even, we construct infinitely many r-regular, r-connected graphs whose cyclability is not greater than 6r − 4 if r = 0 (mod 4) and 8r − 5 if r ≡ 2 (mod 4).

Let A = {a1, a2, …} be an increasing sequence of positive integers whose upper density, θ, is less than 1, Then the counting function of the subsequence does not exceed .

A homomorphism f: R → T of (commutative) rings is said to be universally incomparable in case each base change R → S induces an incomparable map S → S⊗RT. The most natural examples of universally incomparable homomorphisms are the integral maps and radiciel maps. It is proved that a homomorphism f: R → T is universally incomparable if and only if f is an incomparable map which induces algebraic field extensions of fibres, k(f-1(Q))→k(Q), for each prime ideal Q of T. In several cases (f algebra-finite, T generated as R-algebra by primitive elements, T an overring of a one-dimensional Noetherian domain R), each universally incomparable map is shown to factor as a composite of an integral map and a special kind of radiciel.

Let F be a field. If A is an F-algebra with involution that is generated (as a space) by symmetric idempotents, then A is a subdirect product of copies of F if and only if every idempotent in A is symmetric.

Recently, Varshney and Singh [Rend. Mat. (6) 2 (1982), 219–225] have given sharper quantitative estimates of convergence for Bernstein polynomials, Szasz and Meyer-Konig-Zeller operators. We have achieved improvement over these estimates by taking moments of higher order. For example, in case of the Meyer-Konig-Zeller operator, they gave the following estimate

wherein ∥·∥ stands for sup norm. We have improved this result to

We may remark here that for this modulus of continuity ) our result cannot be sharpened further by taking higher order moments.

An empire map is a map where the set of countries is divided into disjoint subsets which we call empires. If one empire consists of a single country then the remaining map is called an empire island map, the single country being referred to as the ocean. If every empire consists of at most M countries then it is called an M-pire map.

It is proved that an M-pire map is colorable by 6M – 2 colors. Further there exists such a map which has 6M – 2 mutually adjacent empires.

If an M-pire island map has every country adjacent to the ocean, then it is colorable in 4M colors. Such a map exists with 4M mutually adjacent empires where every country is adjacent to the ocean.

If A is a Dedekind domain and f generates a prime ideal of A[X] which is not maximal, then the domain A[X]/(f) is Dedekind if and only if f is not contained in the square of any maximal ideal of A[X]. This criterion is used find the ring of integers of a cyclotomic field, and to determine when a plane curve is normal.

Every group is the automorphism group of an orthomodular lattice.

In this paper a new translation plane of order 81 is constructed. Its collineation group is solvable and acts on the line at infinity as a permutation group K which is the product of a group of order 5 belonging to the center of K with a group of order 48. A 2-Sylow subgroup of K is the direct product of a dihedral group of order 8 with a group of order 2. K admits six orbits. They have lengths 4, 6, 12, 12, 24, 24.

We construct a counterexample, for p = 1, to the conjecture posed by Milaszevitch in 1970: is the space of functions which are analytic in the upper half plane and uniformly almost periodic in its closure dense in the Hardy space Hp (0 < p ∞) of analytic almost periodic functions?

For each odd prime p this paper gives an example of a non-abelian p-group with a non-abelian automorphism group in which each automorphism is central.

An investigation is made of the stability of block LU-decomposition of matrices A arising from boundary value problems of differential equations, in particular of ordinary differential equations with separated boundary conditions. It is shown that for such matrices the pivotal growth can be bounded by constants of the order of ‖A‖ and, if solution space is dichotomic, often by constants of order one. Furthermore a method to estimate the growth of the pivotal blocks is given. A number of examples support the analysis.

We give a definition of any discrete dynamical system in compact ultrametric space we study the behaviour of its trajectories.

Let be a nondecreasing sequence of positive numbers. We consider Gelfond-Leontev derivative Df(z), of a function , defined by for univalence and growth properties, and extend some results of Shah and Trimble. Set en = {d1d2 … dn), n≥l, e0 = 1, . Let r be the radius of convergence of p(z). We state parts of Theorem 1 and Corollaries. Let f and all Dkf, k = 1, 2,…, be analytic and univalent in the unit disk U. Then

(iii) if p is entire and of growth (ρ, T) then f must be entire and of growth not exceeding (ρ, 2d2T),

(iv) if D corresponds to the shift operator (dn ≡ l), then .

Another class of functions is defined by a condition of the form |an+1/an| ≤ bn+1/dn+1, where is a sequence of positive numbers satisfying and inequality, and it is shown that all functions in this class along with all their Gelfond–Leontev successive derivatives are regular and univalent in U. An extension of the definition of a linear invariant family is given and results analogous to (i) and (ii) are stated.

A formula is given for the Möbius function of the poset Hom(P, Q) of all order-preserving maps between two finite posets P and Q. Two applications of the formula are presented.

In this paper we show that the countable sum theorem for locally-closed sets, for sheaf theoretic cohomological dimension over a given ring L, holds in all perfectly normal spaces as well as in all locally compact spaces.

In J. Math. Anal. Appl. 61 (1977), 466–474, Leader has given a fixed point principle for an operator f:X → X, where X is a metric-space, based on conditional uniform equivalence of orbits. We generalize this principle for two mappings f1 and f2 to give common fixed point results in two different ways. Further we derive an f-generalized fixed point theorem for two commuting mappings.

The classic HNN-embedding theorem for groups does not transfer to associative rings or algebras. In its first part this paper presents constructions which provide such a theorem if an additional condition is put on the isomorphic subalgebras or if one restricts to algebras over fields and drops the associativity. The main part of the paper deals with applications of these results. For example, it is known that every existentially closed group is ω-homogeneous. It is shown that the corresponding is false for existentially closed associative Δ-algebras but true for existentially universal nonassociative K-algebras. Further-more, orthogonal sequences of idempotents in existentially closed associative Δ-algebras over a regular ring Δ are investigated. It is shown that the conjugacy class of such a sequence depends only on a corresponding order sequence. In particular, in every existentially closed K-algebra all idempotents different from 0 and 1 are conjugated.

We show that recent integral versions of the classic Jordan-Von Neumann characterization of Euclidean space may he viewed as special cases of a general averaging principle over sets of isometries.

A system of parabolic equations is considered: Lui = uit − uix = fi (x, t, u, uix) on Q, Biui(j, t) = ωij(t), t ∈ (−∞, ∞), j = 0, 1, i = 1, 2, …, n, where Bi is one of the boundary operators Biui = ui or Biui = ∂ui/∂v + βi(x, t)ui, x = 0, 1, Ω = (0, 1), Q = Ω × R, u(=(u1, …, un)): Q → Rn, v(x) is the outward normal to the boundary ∂Ω, f, u, ω0, ω1 are n-valued functions and f, ω0, ω1 are periodic in t with period T and Bi is a positive function.

The paper is classified into two parts. The first part deals with the existence and uniqueness of periodic solutions of the above system of parabolic equations. The second part deals with a monotone iterative method which develops a monotone iterative scheme for the solution of the above system of equations. In this paper we establish the existence of coupled quasi-solutions of the above equation. Also we give a monotone iterative scheme for the construction of such a solution.