In 1876, Klein published the following result: “If a crunode of a real, irreducible, plane, algebraic curve changes into an acnode via the intermediary stage of a real cusp, two real inflexions are introduced in a neighborhood of the double points” [2].

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]).

Paul Halmos expressed [3, p. 110] the general dissatisfaction with the usual proofs of this famous and important theorem. They all make it seem like an accidental product of a computation. A more conceptual proof was devised by N. P. Dekker. In spite of the elegance of his proof, the one offered below may have some claim to be regarded as "the reason the theorem is true".

The theory of integration in division spaces introduced by Henstock ([3], [4]) serves to unite and simplify much of the classical material on nonabsolute integration as well as to provide a new approach to Lebesgue integration. In this paper we sketch a simplified approach to the division space theory and show how it can lead rapidly to the standard Lebesgue-type theory without a substantial departure from the usual methods; some applications to integration in locally compact spaces are briefly developed.

Let C be an operator on the subsets of a set X with values among the subsets of X. We assume that C is a closure operator in X, i.e. a monotone, idempotent and extensive operator in X (cf., e.g., Birkhoff [3, p. 39], Schmidt [1], [2]). If A ⊆ X and B ⊆ X, we say that A and B are C-equivalent if C(A) = C(B) (Bleicher- Marczewski [4, p. 210]).

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.

We shall be concerned with the following problem. Let a11, …, ann be complex numbers and λ1, …, λn nonnegative real numbers. Under what conditions does there exist an n × n complex matrix A with a11, …, ann as principal elements and λ1, …, λn as singular values? This problem has been suggested in [3] but, to our knowledge, has not yet been solved.

Let (a1, a2, …, am), ai ∊ {l, 2, …, n}, be an m-permutation of n (repetitions allowed) with exactly kj of the a's equal to j, j = 1, 2, …, n, m = k1+ …+kn, fixed nonnegative integers. An inversion is a pair i, j such that i < j, ai > aj. Denote by N(r; k, …, kn) the number of such permutations with exactly r inversions.

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.

The purpose of this note is to show that there is one and only one coloring of the sides of a dodecahedron with four colors, under the condition that no two adjacent sides share the same color.

In [1] Busacker and Saaty give a graphical solution to the game known as “Instant Insanity”.

A Banach space X is said to be subspace homogeneous if for every two isomorphic closed subspaces Y and Z of X, both of infinite codimension, there is an automorphism of X (i.e. a bounded linear bijection of X) which carries Y onto Z. In [1] Lindenstrauss and Rosenthal showed that c0 is subspace homogeneous, a property also shared by l2, and conjectured that c0 and l2 are the only subspace homogeneous Banach spaces. In that paper no mention was made of subspaces of c0.

L'emploi de comme estimateur de la moyenne est très fréquent en statistiques pour plusieurs bonnes raisons. Ainsi pour une densité régulière de la classe exponentielle, il est l'estimateur sans biais pour la moyenne, à variance minimale. Mais il perd cette importante propriété pour la classe des densités tronquées étudiées ici: cet article démontre que lorsque la taille de l'échantillon est suffisamment grande, il faut préférer Δ = [(mini=1, …, nXi + maxi=1, …, nXi/2] à .

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.

In this note we answer in the affirmative the following question.

If S is a finite family of sets such that each subset of a member of S also belongs to S, is it possible to arrange the sets in pairs so that the first member of the pair is contained in the complement of the second member?

By compactness, the unit ball Bn in Rn has a finite covering by translates of rBn, for any r > 0. The main theorem of this note shows that a weaker covering property does not hold in any infinite-dimensional normed space.

Let Q be a finite partially ordered (by ≤) set with universal bounds O, I. The height function h of Q is defined by the rule: h(x) is the maximum length of a chain from O to x. Let h(I)=n. Suppose that for each k≥0, there exist positive integers a(k) and b(k) such that all elements of height k

1. (i) are covered by a(k) elements of height k+1;

2. (ii) cover b(k) elements of height k—1.

Then we call Q a U-poset. Call a subset S of a partially ordered set an antichain if no two elements of S are comparable.

We wish to establish the existence of a periodic solution to

1

where x, g and p are n-vectors, A is an n × n constant matrix, and ∊ is a small scalar parameter. We assume that g and p are locally Lipschitz in x and continuous and T-periodic in t, and that the origin is a point of asymptotically stable equilibrium, when ∊ = 0.

In this note we give a very elementary proof of a result of Meyer's [1]. Let r(A) be the rank of the matrix A.

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.

In a recent publication [2], D. J. Lewis stated that the Diophantine equation x3+117y3 = 5 has at most 18 integer solutions, but the exact number is unknown. In this paper we shall solve this problem by proving the following

Theorem. The equationx3+117y3 = 5 has no integer solutions.