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Since the Iast papers the research has been continued at intervals through the nine years. Lome numerical errors on pp. 325 and 327 in the last paper have been detested by the author, but nothing which in the least decree alters the main results of that paper, viz., that the number of revolutions per second which a disc having a white sector w , and a black sector 360° — w , must make in order that the flicker may just vanish, under an illumination I, is given by an equation of the form n = a+b log w (360- w ), a being a constant, and b of the form c+d log I, where c and d are constants. If the disc is considered to be under unit illumination when lighted by a standard stearins candle, burning 8·273 grm. of stearins per hour, at a distance of 4 metres; that is, if I is taken to be 1 under these conditions, then, from the combination of the series of observations described in papers I and II, with man others made since, it follows that the numerical value of n is determined with a very fair decree of accuracy to within at most threes revolutions per second, and generally within one, from the equations n = 10·7+12·57 log I, for discs half white, half black, and n = -27·83+(8·57+20·79 log I) log w (360 - w ), when tbs angle of the white sector is w °.

In a paper which appeared a few years ago the authors investigated the exponential representation of functions analytic in the upper half-plane with positive imaginary part there [1]. We refer to that paper in the sequel as A-D. One of the principal results of A-D, there called Theorem A, can be extended to a considerably more general result, the proof of which is perhaps simpler than that given in A-D. We give the extended version of Theorem A here. We will use the notations and results of A-D without further explanations. Before we present the extension and its proof we would like to add some information that by oversight was omitted from the list of fundamental properties of the functions in the class P given in Section 1 of A-D. In such a comprehensive review one should mention that the classical theorem on representation of a positive harmonic function in a circle by a Poisson-Stieltjes integral is due to G. Herglotz [2]. The following results of L. H. Loomis [3] were not given: XVII. For all ~ for which /t[~] = O, the limits lira Im[~b(~ + iq)] ~l~ O and lira hOg(A) exist and are f ini te simultaneously and are equal. Their ~ 0 common value is the symmetric derivative of #(A) at 2 = ~ multiplied by ~r. XVIII. I f for two values of 0 in the interval 0 < 0 < ~

Introduction. In this paper a proof is given of some results obtained by Sanov (see [2]). Some identities are used which are of interest in themselves. Let F= { u, v} be a free group generated by u, v. Let (X, y) = x-ly-lxy for x, yF. If S, T are subgroups of F, let (S, T) = {(s, ), s C S, t C T}. Let F(k) = {Xk, x CF}, F1 = F, Fl = (Fk11,F). Let (u, v, 1)=(u, v); (u, v, n) =((u,v, n-1), v). Then Sanov [2] proved that (1) (U, V, apa 1)PO-"a E F(pA)Fapa+j, ,B, a = 1, 2, where p is an arbitrary prime. In this paper a proof simpler than Sanov's is given for the case p=2 and p=3, with a =1, and /3 arbitrary. It also gives a result if instead of p being a prime, p is divisible by 2 or 3. Identities (4), (6), (9) and (10) below are of interest in themselves. THEOREM 1. Let F= {u, v } be a free group generated by u, v. Let k=2n. Then (2) (u, v) 112 E F(k)F3 k = 2, 4, 6, * and, in particular, if k=20, then (3) (u, V)21-1 G F(21)F3 A = 1, 2, . Comment. (3) is (1) for a = 1, p=2. PROOF. By induction on k d = (u, V)(VZ, u2) . . . (v-1', u')(ui, vi) . . . (uk-1 vk-1) (4) = (U-1V-1) k(VUV-1) 1V1 C F(k), k = 2, 3, * * Received by the editors December 5, 1956.

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.

R. F. Fox considered in his paper [2] a method for calculating the ordinary irreducible characters of the symmetric group S n from the so-called Frobenius compound characters, a method which is based on the fact that the matrix of the multiplicities of irreducible constitutents of the compound characters is a triangular one for a certain order of rows and columns. This matrix is discussed here. That this matrix is a triangular one Fox derives from relations between idempotents. However this can be seen directly by an easy application of the so-called Littlewood-Richardson rule, which also permits evaluation of the entries of this matrix. Further remarks can be made which come from this rule, too, and which facilitate and shorten the computation. They also answer some questions of Fox concerning the general form of this matrix. First column and main diagonal consist of 1's, and the last row is the row of the degrees of the irreducible representations of S n . An additional remark concerns rows which can be used for those matrices for higher degrees n again. For every degree n 0 a number m(n 0 ) is derived so that the non-vanishing part of the i -th row for every i≤m(n 0 ) is the same in every i -th row for all n ≥ n 0 . Fox observed that the first five rows of this matrix for S 5 are the first five rows of the partition number table. An example is given that this cannot be generalized to higher degrees n .

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).

About thirty years ago T. BARANOV developed a theory of exploitation of fish stock, entitled “On the question of the biological basis of fisheries”*. Recently Prof. W. E. RICKER translated this paper into English. Denoting by the letter n the abundance of any group of fish and by the letter t time, he assumed that the decrease dn in the number of fish in the small time dt is proportional to the abundance of that groun and put dn/dt=-k1n…… (1). Following to the results of Heincke he assumed that the increase of length of fish is proportional to time (t) and put t=rl…… (3). From equations (1) and (3), after integrating, we have the relation n=n0e-kl…… (4), where the coefficient k=k1r is called the coefficient of decrease. In Japan, Spring Herring are caught in the Hokkaido District from 3 age group to 8 age group. Since 1910 Hokkaido Fishery Experiment Station has published the statistcs of the catch of herring for each year and the age composition of herring which calculated from the scales. The growth of herring of each age was determined by KAWAT using the scales of age 8 herrings, Tsuda had published the wean body weight of herring of each age. From these data I have made Table 1, from which we know that the increase of body length in each age is not constant and growth curve will be represented more correctly by the Logistic curve. From Table 1, as the number of fish is decreasing exponentially after 5 age, we can obtain the following values of constant. M=33.33, a=0.246, b=0.448, k=1.12, αk=0.307…(*). From equation. (13) we obtain the decreasing rate ψ=1-ak…… (16), where α=(M-l2)/(M-l1) and l1 and l2 are body length for age t1 and t2. Table 1 show that body length of age 3 is l1 ?? 24cm. and that of age 4 is l2 ?? 27cm, hence α=0.675 and ?? =0.355. Similarly from age 2 to age 3, ?? =0.33. The number R of fish of commercial size (that is, fish whose length is greater than L ) is given by the integral (17) and (18), where L/M=β, 0

Let Em and En be orthogonal Euclidean spaces of dimensions m and n respectively and with the origin of each as their only common point. In a previous paper (3) I gave what was intended to be a proof of the relationwhere the dimension of A, dim A, is the Besicovitch dimension, i.e. the number s such that the Hausdorff measure in any dimension greater than s is zero whilst that in any dimension less than s is infinite, where A and B are subsets of En and Em respectively and where A × B is the Cartesian product of A with B.

The circumstances explained in the footnote on p. 61 of the former paper with this title might well have necessitated a re-writing of the whole. Fortunately it appears that only a few short comments are required. For example it may be noted that the first question in §4 can be answered by counting the degrees of freedom in the two configurations. Eight points of a twisted cubic have freedom 20; four pairs of planes drawn at random through four lines of a regulus have freedom 21; therefore the eight planes of the second paragraph of §4 cannot always lead back to the eight points of §2. This is corroborated by the corresponding numbers in [4], which are 31 and 34.

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.

If (a1, b1, c1), (a2, b2, c2), (a3, b3, c3) be three sets of numbers subject to the six relations such as and with a positive determinant, then the four tetrahedra, M, M1, M2, M3, whose angular points have coordinates given by the rows of the four matricesare such that every two are mutually inscribed. The sixteen points so arising are the nodes of a Kummer surface having for one form of its equation

On each trial Ss viewed one of the patterns of six connected squares that result when the faces of a cube are unfolded onto a flat surface. The Ss tried, as rapidly as possible, to decide whether two arrows, each marked on an edge of a (different) square, would or would not meet if the squares were folded back up into the cube. The time required to make such decisions increased linearly (from 2 to about 15 sec) with the sum of the number of squares that would be involved in each fold, if those folds were actually performed physically.

Let M0 be the set of measures μ on the real line such that open sets are μ*-immeasurable. While attempting to find out whether a set μ*-measurable for all μ in Mo is mapped into a similar set by a continuous function of bounded variation, Maurice Sion develops a theory for what he calls variational measure (4). As an application of the theory, he gets conditions on a function f and a set of measures M in order that f map a set, which is μ*-measurable for all μ ∈ M, into a set of the same kind. In particular he proves for his class M2 (def. 2.5), the following theorem (4, § 8.11).

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.

This note contains some results on the asymptotic distribution of the random vector $(\nu _1 ,\nu _2 , \cdots ,\nu _{k - 1} ,\nu _k )$, where $\nu _1 ,\nu _2 , \cdots ,\nu _{k - 1} ,\nu _k $ are the numbers of A-series of lengths $1,2, \cdots ,k - 1$ greater or equal to k, respectively, in the simple homogeneous Markov chain with two states A and B. The asymptotic distribution of the above-mentioned vector (when appropriately formed) is shown to be multivariate normal with the parameters of the distribution calculated.Possible extensions for a number of states greater than two are also discussed.

It is proved that \[ \mathop {\lim }\limits_{n \to \infty } \mathop {\inf }\limits_{\mathop { - \infty < a < \infty }\limits_{ < 0 < \sigma < \infty } } \mathop {\sup }\limits_x \sqrt n \left| {F_n (x) - \Phi \left( {\frac{{x - a}}{\sigma }} \right)} \right| \leqq \frac{1}{{\sqrt {2\pi } }}\rho _3 , \] where $\Phi (x)$ is a normal distribution function and $F_n (x)$ is a distribution function of a normed sum of independent identically distributed random variables. The constant $( 2\pi )^{ - 1/ 2} $ cannot be improved.

Dans cet article nous attirons l'attention sur un resultat du aGaunt et que voici: Si pour resoudre l'equation differetielle $$\dot x = f\left( {t,x} \right), x\left( 0 \right) = x_0 $$ on utilise la methode des differences $$\mu \delta \xi _i = h f\left( {t,\xi } \right)_i , \xi _0 = x_0 , \xi _1 $$ la valeur de ξ1 doit etre calculee d'une facon determinee, distincte dex(h). L'erreur varie alors uniformement au sens decrit ci-dessus et on peut appliquer la metamethode deRichardson. Nous donnons un enonce precis et une demonstration elegante que nous etendons ensuite au cas d'une equation differentielle de second ordre. Nous indiquons d'autres generalisations possibles et signalons une methode pratique de calcul pour ξ1.

About the filterpaper electrophoresis of proteins, we come to a conclusion as the following, from the result of experiment, ussing serum (plasma) proteins as trial material after studing the provision of experimentation, material of experimentation according to the method of the experimentation.1. We find inportant condition in the position on which we put the trial material, how much to have an overdose of it, and how fresh to have it.2. The quality of the filterpaper has a close relation with the result of electrophoresis of proteins. We find the good result in the filterpaper of Whatman No. 1, Schleicher & Schull No. 2043 a, b or Toyo filterpaper corp. No. 51, 51a.3. About to the condition of the electrophoresis, the voltage, the electric current and the time of the phoresis are concerned each other. As to decide strictly on these point, we find some deficult problem, but we know that the condition of the phoresis had decided naturally, if we have the fixed filterpaper and buffer.4. We find the best conditioh in about 7cm in phoresis distance and in 100-110°C, 20 minutes in the method of the desiccate as the same as we find from the result of experiment as to the phoretic distance and the desiccate method.5. 20 minutes are the best in the time of dyeing so for as Amido black 10B is concermed.We think as if We will find some deference between Amido black 10B and B. P. B. dyeingin in the result of the some amount, but it is not only clear above mentioned writting in this point of view but it is also necessary to study in future.

We take a measurable set E on the positive η-axis and denote by μ(r) the linear measure of the part of E in the interval 0 < η < r. The lower density of E at η = 0 is defined byTheorem by Kawakami [1] asserts that if λ is positive, if a function f(ζ) = f(ξ + iη) is bounded analytic in ξ > 0 and continuous at E, and if f(ζ) → A as ζ → 0 along E, then f(ζ) → A as ζ → 0 in ∣η∣ ≦ kξ for any k > 0.

In sections 2 and 3 two methods for proving the non existence of certain universal Banach spaces, are presented. In section 4 it is proved that every infinite-dimensional conjugate Banach space has a two-dimensional subspace whose unit cell is not a polygon.

(1973). An Addendum to the Paper “A Characterization of the n × n Matrices Over a Finite Field”. The American Mathematical Monthly: Vol. 80, No. 9, pp. 1041-1043.

I in part reproduce this very interesting paper for the sake of a remark which appears to me important. I write (a, b, c, f, g, h) in place of Muir's (A, B, C, A′, B′, C′), and take as usual (A, B, C, F, G, H) and K to denote (bc − f2, ca − g2, ab − h2, gh − af, hf − bg, fg − ch) and the discriminant abc − af2 − bg2 − ch2 + 2fgh.

In [4], Phelps showed that in certain function algebras the unit ball is the closed convex hull of its extreme points. The algebra, C(X), of complex valued continuous functions on a compact Hausdorff space, will always have this property. The class of logmodular algebras which have a Gleason part which is total also was shown to have the property. In this paper we give an elementary proof of the first result (a proof which is, in theory at least, constructive). The simplest nontrivial example of a logmodular algebra with a total part is the disk algebra (i.e. the functions continuous on the closed disk and analytic in the interior). For this algebra we show that the extreme points (in fact the exposed points) of the unit ball form a dense subset of the boundary of the unit ball. Let U be the unit ball of C(X). It is well known that q in U is an extreme point of U if I q(x) I = 1 for all x in X. Now if f in U never vanishes, then f is in the closed convex hull of the extreme points; in factf is between two uniquely determined extreme points. We need only observe that for each x in X,f(x) is halfway between two uniquely determined extreme points of the disk and that these points vary continuously with x. Now it is not necessarily the case that each function in U can be approximated by a nonvanishing function. For a counterexample we need only look at h(z) = z in the algebra of continuous functions on the disk. (If If(z)-zj 1/e. Letfi, i = 1, *, N be the retract maps obtained from these rotations. Then I 1/N Efi(z)-z

In [5] S. Holm proposed teststatistics for testing simple hypotheses by means of the probability paper for distribution functions (d.f.) of the form F 0(x) = Φ[(x - μ0)/σ0], where μ0 is location parameter, σ0 scale parameter, and Φ is an absolutely continuous distribution function with Φ(0) = 1/2. If μ0 and (σ0 are known, the hypothesis H 0 is: H 0: H(x) = F 0(x) = Φ[(x−μ0)/σ0], while the three possible alternatives are H 1: H(x) > F 0(x) H 2: H(x) < F 0(x) H 3: H(x) ≠ F 0(x).

In what follows the word matrix denotes a square matrix of order n with complex elements, and the word vector denotes an n-dimensional vector with complex components. The norm of a vector and its subordinate operator norm of matrices, are fixed; this operator norm of matrices is associated, as in [21, with the logarithmic norm. The symbols A, B, 2 denote matrices; x, y, 4, 7 denote vectors. The symbols \]A\\, [lx\\ and y(A) denote, respectively, the norm of A, the norm of x and the logarithmic norm of A.

Romesh Saigal presents a zero-one linear program and a dynamic programming algorithm for finding the shortest route containing exactly q arcs from node 1 to node n in a network (N, A) with distances c(i,j). This note shows that the linear programming formulation and his extension based on it are defective, and that the dynamic programming algorithm can lead to suboptimal solutions, but a minor change in the dynamic programming formulation relieves the difficulty.

Within the class of offspring distributions with given meanm>1 and probability of no offspringp o, the probabilityq of ultimate extinction in a Galton-Watson branching process starting from one individual satisfiesp 0

A system of five points is completely determined by the vectors joining one of them with the other four. If a, β, γ be three of these, the fourth is necessarily δ = xα + yβ + zγ. Hence any property characteristic of a group of five points will remain when x, y, z are eliminated.

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

Publisher Summary Thorn [1] has .described seven differentiable catastrophes, and these have been elaborated by Zeeman [2] and by both Thorn and Zeeman in this symposium on dynamic systems. In this paper, we present a beginning of the theory of polyhedral catastrophes and, in particular, we examine analogues of each of the four basic differentiable catastrophes which map the line to the line. The unfoldings of these polyhedral singularities are topologically equivalent to those of the differentiable category, so this alternative view of catastrophe theory can give some additional insight into the structure of differentiable unfoldings. The main advantage of the polyhedral theory is that there is a precise algorithm for determining the singular behavior as opposed to the progressively complicated algebraic calculations required in the differentiable theory.

Publisher Summary This chapter discusses the combinatorial problems of arrays and applications to design of experiments. The combinatorial arrangements are known as hypercube, and more generally as orthogonal arrays. They are applied in the construction of confounded symmetrical and asymmetrical factorial designs, and multifactorial designs. The chapter discusses some arrays with a number of other combinatorial structures slightly weaker than that of orthogonal arrays and explains their use in the design of experiments. It also reviews the use of orthogonal arrays with a variable number of symbols in the rows. The chapter also discusses the concepts of semi-orthogonal arrays, balanced arrays, semi-balanced arrays, and partially balanced arrays.

In a recent paperf entitled A note on the linear fractional group, Todd obtained an abstract definition for the group LF(2, 2) in terms of n+2 generators. Apparently he gave no consideration to the question of the independence of the defining relations, for they can be considerably simplified. First, in view of the condition RSi = Si+iR (which is the same as i£S0^~* = Si), the three generators U, R and S0 are sufficient to generate the entire group. If we give a definition in teims of these three generators alone, the relations