The paper establishes an effective bound for the solutions of the equation , under some conditions.

Subject classification (Amer. Math. Soc. (MOS) 1970): 10 B 15.

Let p be a prime, let R be a finite p-ring with identity and suppose that the radical of R has pm elements. If R is indecomposable as a ring then there are at most m+1 minimal ideals in R/Rad R.

1980 Mathematics subject classification (Amer. Math. Soc): 16 A 44.

The coincidence degree for the pair (L, N) developed by Mawhin (1972) provides a method for proving the existence of solutions of the equation Lx = Nx where L: dom L ⊂ X → Z is a linear Fredholm mapping of index zero and is a (possiblv nonlinear) mapping and Ω is a bounded open subset of X, X and Z being normed linear spaces over the reals. In this paper we have extended the coincidence degree for the pair (L, N) to solve the equation , where L: dom L ⊂ X → Z is a linear Fredholm mapping of index zero, and X, Z and Ω are as above, CK(Z) being the set of compact convex subsets of Z.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 47 H 15, 47 A 50; secondary 47 H 10, 47 A 55.

We show that, for any Tychonoff space X with base point θ, the infinite symmetric product SP∞ X of X is a subspace of an abelian group A(X) generated by X. (This clarifies the continuity of the multiplication in SP∞ X.) Furthermore, SP∞ X is a retract of A(X). Analogous results hold for reduced product spaces, with respect to non-abelian groups.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 22 A 99; secondary 54 B 15.

Characterizations of collectively precompact and collectively semi-precompact sets of operators on topological vector spaces are obtained. These lead to the characterization of totally bounded sets of semi-precompact operators on locally convex spaces.

1980 Mathematics subject classification (Amer. Math. Soc): primary 47 B 05, 47 D 15; secondary 46 A 05, 46 A 15.

It is proved that there exists a universal good inner function in the open unit polydisc Un, that is its non Euclidean translates are dense in the closed unit ball of H∞ (Un) and that there exists a universal function in the open unit ball Bn of Cn. These generalize Heins' result on universal Blaschke products.

1980 Mathematics subject classification (Amer. Math. Soc.): primary 32 A 10.

For certain problems in the weak convergence theory of probability measures on non-separable metric spaces it is convenient to consider measures defined on a σ-field B0 smaller than the Borel σ-field. A suitable theory has been developed by Dudley and Wichura. In this paper it is shown that some of the key results in that theory can be deduced directly from the better known weak convergence theory for Borel measures. This is achieved by a remetrization of the underlying space to make it separable. The σ-field B0 contains the new Borel σ-field and weak convergence in Dudley's sense becomes equivalent to convergence of restrictions of the measures to the latter σ-field.

1980 Mathematics Subject Classification (Amer. Math. Soc.): primary 60 B 10, 60 B 05.

The concept of very weak local uniform rotundity (very WLUR) is introduced. It is shown that if a Banach space E is WLUR, very WLUR, or LUR then its dual E* is smooth, very smooth, or Fréchet differentiable, respectively, on a norm dense subset. This leads to examples of nonreflexive spaces which are Fréchet differentiable at every nonzero point of a (relatively) norm dense subset of the embedding of E** in the fourth dual. When E is reflexive, necessary and sufficient conditions for E and E* to be WLUR and LUR are given.

Subject classification (Amer. Math. Soc. (MOS) 1970): 46 B 99.

Let Zn be the factor ring of the integers mod n and t be a positive integer. In this paper some results are given on the structure of the semigroup of all mappings from Zn into Zn and on the structure of the group of all permutations on Zn, which, for any t elements, can be represented by a polynomial function. If n = ab and a, b are relatively prime, then this (semi)group is isomorphic to the direct product of the respective (semi)groups for a and b. Thus it is sufficient to consider only the case where n = pe, p being a prime. In this case it is proved, that the (semi)group is isomorphic to the wreath product of a certain sub(semi)group of the symmetric (semi)group on Zpe−1 by the symmetric (semi)group on Zp. Some remarks on these sub(semi)groups are given.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 B 99, 13 B 25.

The paper examines the classes K1 and Γ1 of Hausdorff uniform spaces which are Gδ-closed in their Samuel compactifications, or completions. It is shown that the classes are epi-reflective, the reflections K1 and Γ are described, K1 and Γ1 are represented as epi-reflective hulls, membership in the classes is described by fixation of certain zero-set ultrafilters, and it is shown that k1 = Γ1 exactly on spaces without discrete sets of measurable power. The results include familiar facts about realcompact and topologically complete topological spaces and are closely connected with the theory of metric-fine uniform spaces.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 54 C 50, 54 E 15, 18 A 40; secondary 54 B 05, 54 B 10, 54 C 10, 54 C 30.

Generalizations are proved for theorems of Carathéodory (1907), Kirchberger (1903) and Watson (1973), the theme of these results being how thickly the convex hull of a family of points is covered by simplexes whose vertices are chosen from the points of the family.

Subject classification (Amer. Math. Soc. (MOS) 1970): 52A35.

For a class of primes p ≡ 1 (mod 20) for which 2 is a quintic nonresidue, unambiguous evaluations of parameters of bidecic Jacobi and Jacobsthal sums (mod p) are presented, in terms of the partition p = a2+5b2+5c2+5d2, ab = d2–c2–cd. Similar results for sums of others orders have been obtained by E. Lehmer and by K. S. Williams.

Subject classification (Amer. Math. Soc. (MOS) 1970): 10 G 05.

The extended diagram of a two-bridged knot is introduced, and it is shown how the coefficients of the Alexander polynomial of the knot may be read straight from this diagram. Using this result, it is shown by diagram manipulation that a conjecture of Fox about the coefficients of the Alexander polynomial of an alternating knot is true at least for two-bridged knots (which are all alternating).

1980 Mathematics subject classification (Amer. Math. Soc.): 57 M 25.

A class of cyclically presented groups with n generators and n Fibonacci type relations is discussed. Conditions are given for these groups to be finite and metacyclic. With these conditions the presentations are reduced to the standard form for metacyclic groups with trivial Schur multiplicator. This enables certain ismorphisms between the groups to be found.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 F 05.