The extension of the Rellich-Kondrachov theorem on the complete continuity of Sobolev space imbeddings of the sort

1

to unbounded domains G has recently been under study [1–5] and this study has yielded [4] a condition on G which is necessary and sufficient for the compactness of (1). Similar compactness theorems for the imbeddings

2

are well known for bounded domains G with suitably regular boundaries, and the question naturally arises whether any extensions to unbounded domains can be made in this case.

Let be two foliations on a Cr manifold M. We say and are Ck-conjugate if there exists a Ck diffeomorphism h:M→M such that h maps the leaves of onto the leaves of .

Let A be a finite dimensional commutative and associative algebra with identity, over a field K. We assume also that A is generated by one element and consequently, isomorphic to a quotient algebra of the polynomial algebra K[X]. If A=K[a] and bi=fi(A), fi(X) ∊ K[X], 1≤i≤r we find necessary and sufficient conditions which should be satisfied by fi(X) in order that A = K[b1, …, br].

The result can be stated as a theorem about matrices. As a special case we obtain a recent result of Thompson [4].

In fact this last result was established earlier by Mirsky and Rado [3]. I am grateful to the referee for supplying this reference.

Let H be a cyclic group, K⊂H a subgroup and x, y generators of H, K. We shall say that x, y are related if y=xa where a is the index of K in H, in other words, y is the smallest positive power of x in K. The main purpose of this note is to show that for any group G one may, by means of the axiom of choice, choose for each cyclic group H⊂G a generator xH such that when K⊂H then xK, xH are related.

Consider a one-dimensional random walk model where a particle starting at the origin at any instant either takes a jump through a unit distance to the right with probability p1, or stays at the same position with probability p0, or else takes a jump through either of 1, 2, … μ, units of distance to the left with probabilities p-1, p-2, …, p-μ respectively.

Let A be a ring (associative) with unity, and let denote the category of unital left A-modules. If is a strongly complete Serre class in then (see [3], and also [6]) there is an exact functor S: , where is the quotient category , and is an abelian category.

In connection with a statistical problem concerning the Galtontest Cśaki and Vincze [1] gave for an equivalent Bernoullian symmetric random walk the joint distribution of g and k, denoting respectively the number of positive steps and the number of times the particle crosses the origin, given that it returns there on the last step.

In [3, p. 149], J. Lambek gives a proof of a theorem, essentially due to Grothendieck and Dieudonne, that if R is a commutative ring with 1 then R is isomorphic to the ring of global sections of a sheaf over the prime ideal space of R where a stalk of the sheaf is of the form R/0P, for each prime ideal P, and . In this note we will show, this type of representation of a noncommutative ring is possible if the ring contains no nonzero nilpotent elements.

This paper deals with the convergence of iterations of the equation

1.1

with u0 ∊ Rn, both as a root and a point of repulsion. Here u and ϕ are real n-vectors and ϕ is continuously differentiate with respect to u. By taking u0 as an initial approximation to u0 in a neighbourhood of u0 we define the sequence of iterates {uk+1} by

1.1

For the sake of convenience we shall, sometimes, call the equation (1.1) an iteration.

Throughout this paper we consider associative rings with unity elements. In §1 various results on the representation of rings by rings of sections of special rings are compared. In particular, it is shown that results enunciated by Dauns and Hofmann, Koh, and the present author may all be deduced from one statement, the proof of which appears in §3.

In [ 2 ] P. Fletcher proved that a finite topological space has a unique compatible quasi-uniformity; C. Barnhill and P. Fletcher showed in [1] that a topological space (X, ), with finite, has a unique compatible quasiuniformity. In this note we give some necessary conditions for unique quasiuniformizability.

Dans cette note on donne une démonstration, fondée sur des propriétés élémentaires des opérateurs linéaires dans des espaces munis de deux normes, d'un théorème d'interpolation établi par J. Lions et J. Peetre [2] dans le cadre de leur théorie des espaces de moyenne.

Raghavarao and Chandrasekhararao [3] introduced a family of PBIB designs having three associate classes known as cubic designs. In this paper a detailed analysis of the case of PBIB designs having four associate classes, which are called quartic designs, is given. Results are obtained pertaining to construction and existence of quartic designs. Moreover, using methods similar to those used by Shah [5], [6], [7], the block structure of certain quartic designs is studied.

Since 1963 several authors ([13], [2], [6], [14], [10], [11], [12], [9]) have considered integral equations each one of which is contained as a special case in one of the two equations

1.1

1.2

for Re b > 0 and x ∊ [α, β].

Let R be a Dedekind domain with quotient field K and ∧ an R-order in the finite-dimensional separable K-algebra A. If K is an algebraic number field with ring of integers R, then the Jordan-Zassenhaus theorem states that for every left A-module L, the set SL(M)={M: M=∧-lattice, KM≅L} splits into a finite number of nonisomorphic ∧-lattices (cf. Zassenhaus [5]).

Let the random variables X, Y, Z have respectively the extreme value densities as

1

2

3

where θ>0, λ>0 and μ are real numbers.

In [5], DeMeyer extended one consequence of Wedderburn's theorem; that is, if R is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents except 0 and 1 or if R is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective module over a central separable R-algebra A.

In this paper, we study the oscillatory behavior of the solutions of the linear differential equation

(1.1)

where

(1.2)

and all functions are assumed to be continuous on a bounded interval [a, b). An «th-order linear equation is said to be disconjugate on an interval I provided it has no nontrivial solution with more than n — 1 zeros, counting multiplicities, in I.

Let GF(q) denote the finite field with q = pn elements and let

(1)

where each ai ∊ GF(q) and 1 < d <p. For r=2, 3, …, d we let nr denote the number of solutions (x1, …, xr) over GF(q) of

(2)

for which x1, …, xr are all different. Birch and Swinnerton-Dyer [1] have shown that

(3)

where each vr is a nonnegative integer depending on f and q and the constant implied by the O-symbol depends here, and throughout the paper, only on d.