Edelblute [1] has given a method of finding an inverse of a nonsingular matrix by rank annihilation. The purpose of this paper is to show that the method can be extended in the case of a singular matrix. This method will produce a singular inverse satisfying condition (3) of Penrose [3].

Several authors have recently considered the problem of obtaining sufficient conditions for the oscillation of the quasilinear matrix differential equation

(1)

and the associated inequality VTLV ≤ 0 (as a form). Here A, B, and V are m x m matrix functions, A(x) is symmetric, positive semidefinite and continuous in an interval [a, ∞) and B(x,V, V') is symmetric and continuous in a≤ x < ∞ for all V and V'.

Let denote the automorphism group of the tournament T. Let g(n) be the maximum of taken over all tournaments of order n. It was noted in [3] that g(n) is also the order of the subgroups of Sn of maximum odd order where Sn denotes the symmetric group of degree n.

In Meir and Keeler [3], the authors proved a fixed point theorem in a complete metric space (X, d) for a mapping f that satisfies the following condition of weakly uniformly strict contraction:

Given ∊ > 0, there exists δ > 0 such that

(A)

The basic solvability theorems of Farkas [2] and Levinson [4] were recently extended in different directions by Ben-Israel [1] and Kaul [3].

The theorem stated in this note generalizes both results of Ben-Israel and Kaul and is applicable to nonlinear programming over complex cones.

Dans [2] nous avons introduit, en nous inspirant des notions équivalentes pour les anneaux [3], les notions de monoïde maximal Q(S) et de monoïde complet B(S) des quotients d'un monoïde donné S, la premiére construction n'étant valide que lorsque S est commutatif, et nous avons montré que dans ce cas, on a toujours les inclusions (en général strictes) de monoïdes suivantes: S ⊂ Q(S) ⊂ B(S).

The study of group types was completed by Meldrum [1]. The concept of ring type described here is based on analogous definitions.

The series R = R0 ⊃ R1 ⊃ … ⊃ Rα = Rα+1is the lower central series for the ring R if Rγ+1=RRγ + RγR for ordinal number γ and Rγ∩δ<γ Rδ if γ is a limit ordinal. The upper central series for R is the series 0=J0 ⊂ J1 ⊂ … ⊂jβ = Jβ+1 where Jγ+1={x ∊ R:xR + Rx ⊆ Jγ} for every ordinal number γ and Jγ = ∪ ∩δ<γ Jδ if γ is a limit ordinal. The length of the upper central series is the smallest ordinal number γ for which Jβ = Jβ+1.The length of the lower central series is defined similarily. We shall say the ring has type (β,α) if the length of the upper central series is β and the length of the lower central series is α.

Throughout this note, we adopt the graph-theoretical terminology and notation of Harary [3]. A graph G is hypohamiltonianif G is not hamiltonian but the deletion of any point u from G results in a hamiltonian graph G-u. Gaudin, Herz, and Rossi [2] proved that the smallest hypohamiltonian graph is the Petersen graph. Using a computer for a systematic search, Herz, Duby, and Vigué [4] found that there is no hypohamiltonian graph with 11 or 12 points. However, they found one with 13 and one with 15 points. Sousselier [4] and Lindgren [5] constructed independently the same sequence of hypohamiltonian graphs with 6k+10 points. Moreover, Sousselier found a cubic hypohamiltonian graph with 18 points. This graph and the Petersen graph were the only examples of cubic hypohamiltonian graphs until Bondy [1] constructed an infinite sequence of cubic hypohamiltonian graphs with 12k+10 points. Bondy also proved that the Coxeter graph [6], which is cubic with 28 points, is hypohamiltonian.

Letf:S1→S1 be an orientation preserving C1-diffeomorphism. Denote by the flow on S1 x S1 which is the suspension of f(see Smale [5]).

We consider the problem of approximating by a smoother foliation.

It follows from [1, p. 377, Lemma 1] that a noncommutative subdirectly irreducible ring, with no nonzero nilpotent elements, cannot possess any proper zero-divisors. From [2, p. 193, Corollary 1] a subdirectly irreducible distributive lattice, with more than one element, is isomorphic to the chain with two elements. Hence we can say that a subdirectly irreducible distributive lattice with 0 possesses no proper zero-divisors.

It is the purpose of this paper to establish oscillation criteria for second order nonlinear differential equations with retarded argument. Specifically, we consider the equation

1.1

where f ∊ C[0, + ∞) x R2, g ∊ C[0, + ∞), and

1.2

We shall restrict attention to solutions of (1.1) which exist on some ray [T, + ∞). A solution of (1.1) is called oscillatory if it has no largest zero.

Recall that a function f:X→ Yis called connected if f(C) is connected for each connected subset C of X. These functions have been extensively studied. (See Sanderson [6].) A function f:X → Y is monotone if for each y ∊ Y, f-1(y) is connected. We shall use the techniques of multivalued functions to prove that if f: X→ Y is open and monotone onto Y, then f-1(C) is connected for each connected subset C of Y. This result is used to prove that the product of semilocally connected spaces is semilocally connected and that the image of a maximally connected space under an open, connected, monotone function is maximally connected.

If In, I∞ denote the intervals [0, n), [0, ∞) respectively, we propose to examine the properties of an upper and lower density

(1)

(1)

of a set S belonging to the power set of I∞ where m*,m* denote the outer and inner (linear) Lebesgue measures.

The central limit theorem was originally proved for independent random variables. The independence is a very strong notion and hard to check. There are various efforts to prove different theorems on independent variables (e.g. strong law of large numbers, central limit theorem, the law of iterated logarithm, convergence theorem of Kolmogorov) under weaker conditions, like mixing, martingale-difference, orthogonality. Among these concepts the weakest one is orthogonality, but this ensures only the validity of law of large numbers.

Consider a life testing experiment in which (Xl, X2,…,Xn) are such that (n—1) of them are distributed as f(x, σ)=(l/σ)e-x/σ, x ≥ σ, > 0 and one of them is distributed as f(x, σ/α), 0 < α< l. A priori each Xi has probability 1/n of being a spurious observation distributed as f (x,σ/α). For such an experiment Kale and Sinha [2] showed that if ur denotes the probability that X(r), the rth component of the order statistic, corresponds to the spurious observation, then u1 < u2 < … < un.

Let f be a real valued function on an open subset of R2. It is assumed that f satisfies Carathéodory's conditions: f (t,x) is continuous in x for each t, Lebesgue measurable in t for each x and there is a locally integrable function m(t) such that |f(t, x)| ≤ m(t) uniformly in x. A proof will be given of the following theorem.

If (Xl, X2,…, Xn), is a set of n independent random variables, such that EXi=0, Var and if t is a real positive number and , then Bernstein [2] has given an upper bound for Pr when the X's are bounded. The best English language discussion of Bernstein's work is probably by Bennett [1].

In the series of simple groups 2F4(q),q =2 2m+1, discovered by Ree, Tits [4] showed that the group 2F4(2) was not simple but contained a simple subgroup of index 2. In this note we extend the characterization of obtained by the author in [3].

Let f be an L integrable real valued function of period 2π and let

(1)

be its Fourier series. It is known that if f is of bounded variation then all nan and nbn(n=1,2,3,…) lie in the interval [-V(F)/π, V(F)/π;] where V(f) is the total variation of f. M. Izumi and S. Izumi [3] have recently asserted the following theorem A about the density of the positive and negative Fourier sine coefficients of a function of bounded variation.

Strong oscillation criteria will be obtained for the linear elliptic partial differential equation

(1)

in unbounded domains R of general type in n-dimensional Euclidean space En. It will be assumed throughout that B and each Aij are real-valued continuous functions in R, and that the matrix (Aij(x)) is symmetric and positive definite in R.