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I regret to announce that there is a serious error in my paper in these Transactions, volume 84, pp. 430-443. The error was discovered by Louis Kokoris who found that on line 8 of page 434 the expression given as 4[g(bz)](az) should have been 4[g(az)](bz). As a consequence the computation of P(z, g, az, b) yields nothing, the proof of formula (30) is not valid, and the important Lemma 9 is not proved. Thus the paper does not give a proof of its major result stated as Theorem 1. Nevertheless, the theorems of the paper are all correct and we shall provide a revision of the proof here. This revised proof has been checked by Louis Kokoris to whom the author wishes to express his great thanks. We observe first that the equation

With respect to the printability of paper, several problems of color were investigated.Summary was as follows.(1) Max., min., and average qualities of each grade of printing Papers in Japan.(2) Inter relation of each other properties of paper testing figures of the middle class printing Paper.(3) Spectral reflect curve of the typical papers and pulps.(4) Effect of dyes on the paper color.(5) Effect of the kinds of color inks and paper qualities on the printing coloring effect.(6) Relation between paper testing values and theprinting coloring effect.Coloring effect was measured by means of comparing the spectral reHect curve of test sample with that of standard sheet printed under the standard condition.Calculation of the relation between A and B was done as follows.pA→B=SA (B) -a-1/b-1SC (B) /ST (B) -a (r-1) /b (r-1) SM (B)

The reading of Ayres' interesting paper suggested to me the following remarks: 1. The order of a subset of a set S in a point p4 cannot surpass the order of S in p. Hence if S2 denotes the set of all points of S of order 2, then S2 has in each point of S the order 2, the order 1, or the order 0, where the terms "order 0" and "0-dimensional" are used synonymously. SI(M), S2(1), S" may denote the set of all points of S in which S2 has the order 0, 1, 2, respectively. The points of order 2 of S are also called the ordinary points of S, and the set S2 of all ordinarv points of S may be called the ordinary part of S. The set S" of all ordinary points of the ordinary part of S may be designated the ordinary kernel of S. WVe have

The purpose of this note is to present a new proof of a generalized form of Day's fixed point theorem. The proof we give is suggested by the work of T. Mitchell in his paper, Function algebras, means, and fixed points, [2]. The version of Day's theorem which we present here has not appeared explicitly in the literature before, and seems especially well suited for application to questions concerning fixed point properties of topological semigroups. 1. Preliminaries. We adopt the terminology and notation of [2] except where otherwise specified. New terminology will be introduced as needed. Let y be a convex compactum (compact convex set in a real locally convex linear topological space E), and let A( Y) denote the Banach space of all (real) continuous affine functions on Y under the supremum norm. Observe that A(Y) contains every function of the form h=f\Y + r where fe E* and r is real; thus A(Y) separates points of Y.

1. Although the decision problem of the first order predicate calculus has been proved by Church to be unsolvable by any (general) recursive process, perhaps it is not superfluous to investigate the possible reductions of the general problem to simple special cases of it. Indeed, the situation after Church's discovery seems to be analogous to that in algebra after the Ruffini-Abel theorem; and investigations on the reduction of the decision problem might prepare the way for a theory in logic, analogous to that of Galois.It has been proved by Ackermann that any first order formula is equivalent to another having a prefix of the form(1) (Ex1)(x2)(Ex3)(x4)…(xm).On the other hand, I have proved that any first order formula is equivalent to some first order formula containing a single, binary, predicate variable. In the present paper, I shall show that both results can be combined; more explicitly, I shall prove theTheorem. To any given first order formula it is possible to construct an equivalent one with a prefix of the form (1) and a matrix containing no other predicate variable than a single binary one.2. Of course, this theorem cannot be proved by a mere application of the Ackermann reduction method and mine, one after the other. Indeed, Ackermann's method requires the introduction of three auxiliary predicate variables, two of them being ternary variables; on the other hand, my reduction process leads to a more complicated prefix, viz.,(2) (Ex1)…(Exm)(xm+1)(xm+2)(Exm+3)(Exm+4).

In this note the authors call for a change of the optimality criteria given by Theorem 3 in section 5 of the paper of W. Szwarc “On Some Sequencing Problems” in NRLQ Vol. 15, No. 2 [2]. Further, two cases of the three machine problem, namely, (i) ≦ and (ii) ≦ are considered, and procedures for obtaining optimal sequences in these cases are given. In these cases the three-machine problem is solved by solving n (the number of jobs) two-machine problems.

We illustrate a technique for computing certain integrals that arise in probability theory by giving a new derivation of a formula of Kingman. This formula contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t) = inf{s: X(t - s) = b} where X is a time homogeneous, continuous parameter, Markov process and b is a fixed point in its state space. We then extend this formula to the situation in which b is replaced by a finite set {b 1, …, b n }.

The aim of this short note is to improve some recent results of B. Grunbaum by some remarks. We use Grunbaum's notations.

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It is shown that a transversal matroid of rank r on a set of cardinality m is representable over any field of order greater than m + ( inr −1 m ).

For every separable Banach spaceX there is a Banach spaceY with a separable dual such thatY ⊕X* ≈Y**. There is also a separable spaceZ so thatZ**/JZ is isomorphic toX.

Let H be a real or complex Hilbert space and let L p(1≦p

In H. K. Weiss' letter to the Editor (“The Fiske Model of Warfare”, Opns Res. 10, 569–571 (1962).), Weiss pointed out that Rear Admiral Fiske, USN, seemed to have foreshadowed by 11 years the developments that have come to be known as Lanchester's Square Law. Weiss stated that Fiske appears to have been the first to observe publicly that, when the damage each of two forces can inflict on the other is proportional to its own size, the change of size of each force was geometric rather than arithmetic.

1. In a recent paper [2] Reich has made the observation that an example of Koebe given in 1918 [1] does not fulfill its asserted purpose. This example was to show that vanishing measure of the complement did not assure that a slit domain was minimal. Reich proceeded to fill the gap by carrying out the following somewhat more general construction. Let A be a compact perfect nowhere dense set on the x-axis in the z-plane (z=x+iy). Then there exists a compact set S in the z-plane with the properties: (i) A is the projection of S on the x-axis, (ii) S is composed of segments symmetric with respect to the x-axis and points on the x-axis, at least one segment being present, (iii) any point in S, not on the x-axis and not at the end of a segment in S, is the limit, both from the left and right, of points of S. Once this construction is performed the desired examples are easily given [2, ?4]. The object of the present paper is to give an alternative construction which is very explicit and direct.

It is shown that the method of Chernoff developed in the preceding paper can be modified to prove the essential self-adjointness on C 0 ∞ ( R m ) of all positive powers of the Schrodinger operator T = − Δ + q if q real and in C ∞ ( R m ) and if T ⩾ −a − b ¦ x ¦ 2 on C 0 ∞ (R m ) .

It is shown that one of the results of Schaefer can be obtained in a more elementary way and can be improved. This is an immediate corollary of a paper of the author Duke Math. J.19, 75---91 (1952).

In the paper of the title [1], a number of problems are posed. Ne­ gative solutions of two of them (Problems 2 and 3) are derived in a straightforward way from a paper of L. Gillman and the present author [2]. Motivation will not be supplied since it is given amply in [1], but enough definitions are given to keep the presentation reasonably self­ contained. 1. A Hausdorff space X is said to satisfy (Qm), where m is an in finite cardinal, if, whenever U and V are disjoint open subsets of X such that each is a union of the closures of less than m open subsets of X, then U and V have disjoint closures. In particular, a normal (Hausdorff) space X satisfies (Q*,) if and only if disjoint open F.-SUbsets of X have disjoint closures. (For, an open set that is the union of less than ~, closed sets is a fortiori an F.. Conversely if U is the union of countably many closed subsets Fo> then since X is normal, for each n there is an open set Un containing Fn whose closure is contained in U. Thus U is the union of the closures of the open sets Un.) In Prob­ lem 3 of [1], it is asked if every compact (HaUSdorff) space satisfying (Qm) for some m>~o is necessarily totally disconnected, and it is re­ marked that this is the case if the first axiom of countability is also as­ sumed. If X is a completely regular space, let C(X) denote the ring of all continuous real-valued functions on X, and let Z(f)= {x EX: f(x)=O}, let P(f)={xEX:f(x»O}, and let N(f)=P(-f). As usual, let (IX denote the Stone-Cech compactification of X. If every finitely generated ideal of C(X) is a principal ideal, then X is called an F-space. The fol­ lowing are equivalent. ( i) X is an F-space. ( ii) If f E C(X), then P(f) and NU) are completely separated [2, Theorem 2.3]. (iii) If f E C(X), then every bounded g E C(X-ZU» has an ex­ tension g E C(X) [2, Theorem 2.6]. A good supply of compact F-spaces is provided by the fact that if X is locally compact and ,,-compact, then {lX-X is an F-space [2, Theo­ rem 2.7].

The purpose of this note is to present a lemma which will settle a question of completeness left open in Section 6 of the above mentioned paper [5]. We give two applications of the lemma, (i) by proving that, in addition to Herbach's results, also the standard $F$-test for $\sigma^2_{ab} = 0$ is a uniformly most powerful similar test, (ii) by pointing out that the standard form introduced in [5] together with our lemma provide convenient tools to prove that in a balanced model II design (with the usual normality assumptions) the standard estimates of variance components are minimum variance unbiased. This result is well known ([2], [3]) and it has in fact been pointed out by Graybill and Wortham [3] that a completeness argument may be used to demonstrate the minimum variance property of the usual estimators for the variance components. The present lemma shows that the estimators do indeed have the necessary completeness property. We will follow Herbach's notation throughout.

WE WISH to approximate a simple root a' of the equation f (x) = 0. A sequence of approximants [xi] is generated by an iteration function

This chapter discusses that C r structural stability implies Kupka–Smale. It proves that if a diffeomorphism is C r structurally stable, r ≥ 1, then the stable and unstable manifolds of periodic points meet transversally. This result is well known for r = 1, but the chapter gives a proof that works for any r , 1 ≤ r ≤ ∞. There is a corollary that if f is C r structurally stable and satisfies Axiom Aa of Smale, then f satisfies the strong transversality condition.

The class of simple syntactic operators is defined. If a full AFL ℒ is not closed under a simple syntactic operator ⊗, then repeated applications of ⊗ to ℒ produces an infinite hierarchy of full AFLs and the closure of ℒ under ⊗ is not full principal. If ℒ1 and ℒ2 are incomparable full AFLs, then the least full AFL containing ℒ1 and ℒ2 is not closed under any simple syntactic operator. If L is any generator of a full AFL ℒ closed under any simple syntactic operator, then all of ℒ may be expressed as finite state translations of L (without applying concatenation or star). It is shown that substitution, insertion, intercalation and homomorphic replication are all simple syntactic operators.