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1. Main theorem. Let x = (xo, xi, • • • ) be a sequence of complex numbers and let | x \ he the norm | x | = (^Z | x„ |2)1/2 2:0. Let an asterisk denote complex conjugation. If \x\ <

IN part I of this paper1 methods for the synthesis of the optimum nonlinear bang-bang control system with complex roots was presented. In this, the scope and utility of this synthesis are verified by analytical studies of the dynamic response capabilities of this systems.

The aim of the present paper is to extend Daniel Bernoulli's method of approximating to the numerically greatest root of an algebraic equation. On the basis of the extension here given it now becomes possible to make Bernoulli's method a means of evaluating not merely the greatest root, but all the roots of an equation, whether real, complex, or repeated, by an arithmetical process well adapted to mechanical computation, and without any preliminary determination of the nature or position of the roots. In particular, the evaluation of complex roots is extremely simple, whatever the number of pairs of such roots. There is also a way of deriving from a sequence of approximations to a root successive sequences of ever-increasing rapidity of convergence.

The closure, W, of the set of values (Cx, x) when ||x|| = 1 is a closed convex set (Hausdorff, cf. [8, p. 34]). A complex number z will be said to belong to the interior of W if z is in W and if one of the following conditions holds: (i) If W is two-dimensional, then z does not lie on the boundary of W; (ii) If W is a line segment, then z is not an end point; (iii) W consists of z alone. It was shown in [4] that if A (or B) is normal, or even semi-normal, so that AA* -A *A is semi-definite, then 0 belongs to W, but is not necessarily in the interior of W. (That, for arbitrary A and B, in general 0 need not even belong to W was shown in [2].) In fact, if A =(aij) is defined by aj,+=1,aij=0 ifj$i+1, then C=AA*-A*A = (cij) is the self-adjoint matrix all elements of which are zero except cl =1. Consequently, C>0 with a spectrum consisting of X=0, 1; hence W is the segment 0

Here, we present various new forms of the Bernstein-Vazirani algorithm beyond qubit systems. First, we review the Bernstein-Vazirani algorithm for determining a bit string. Second, we discuss the generalized Bernstein-Vazirani algorithm for determining a natural number string. The result is the most notable generalization. Thirdly, we discuss the generalized Bernstein-Vazirani algorithm for determining an integer string. Finally, we discuss the generalized Bernstein-Vazirani algorithm for determining a complex number string. The speed of determining the strings is shown to outperform the best classical case by a factor of the number of the systems in every cases. Additionally, we propose a method for calculating many different matrices simultaneously. The speed of solving the problem is shown to outperform the classical case by a factor of the number of the elements of them. We hope our discussions will give a first step to the quantum simulation problem.

THEOREM. Suppose that A is a complex sequence, t> 0, m is a nonnegative integer, API t(p + 1 )m for each nonnegative integer p, and f is a function such that, for each complex number z with IzI

1. The usual theory relates to systems of numbers =,a,e, in which the co6rdinates ai range independently over all real nulmbers or else over all ordinary complex numbers; for example, the real quaternion system, or the complex quaternion system. As an obvious generalization,t the co6rdinates may range independently over all the marks of any field F; for example, the rational quaternion system. As a further generalization, the sets of co6rdinates al, *, a in the various numbers of a system may include only a part of the sets bl, -, b , each b ranging independently over F; for example, the integral quaternion system. The various coordinates a1, , a. need not have the same range; for example, the numbers (a + 2b V/2) el + (c + 4d 12 ) e2 (a, b, c, d arbitrary integers)

Zusammenfassung In the theory of vector and matrix norms, one poses problems of the following type: New norms are generated by simple operations on existing norms, and one asks which properties of the old norms carry over to the new ones. Further, certain classes of norms, obtained by special constructions, are characterized by algebraic properties. An example of the latter is the characterization of operator norms in the set of all submultiplicative matrix norms (Schneider, Strang, Stoer). Putting aside the question of the direct application to numerical problems, many of the concepts used in these investigations do not depend on the special character of matrix algebra. In this paper, one considers an arbitrary (not necessarily associative) algebra A over the field C of complex numbers. If p is a norm on A then, under appropriate assumptions, (1)—defined below—is again a norm. Properties of this mapping from the old norm to the new norm are considered. For example, it is shown that, on any finite dimensional algebra with unit element, there exists norms satisfying (3) and (4).

The problem to be studied in this paper concerns the closure properties on an interval of a set of characters {e~nx}~, where A = {2n}~ is a given set of real or complex numbers without finite point of accumulation. This problem is for obvious reasons depending on the distribution of zeros of certain entire functions of exponential type. The main problem of the paper is to determine the closure radius Q = Q(A)defined as the upper bound of numbers r such that (ei~x)~EA span the space L 2 ( r , r ) . The value of r does not change if a finite number of points are removed from or adjoined to A. Nor does Q(A) change if the metric in the previous definition is replaced by any other LV-metric, or by a variety of other topologies. I f A contains complex numbers we shall always assume (1)< 6~t ~ (0.1) 9 ~eA ~

It is shown that the Volterra integral Equation (2) can be explicitely solved for all complex number λ under the assumption that the parameter α is less than 1 and rational. The solution is given by Formulas (3), (4).

The determination of the coefficients is quite complicated (and, for the coefficient of n-4, incorrect). In the present paper we obtain Copson's series by a simpler method which yields an asymptotic expansion for Sn(z) valid for every complex number z except z= 1. For our principle result we require the following three lemmas concerning the function Sn(z) defined by (1) and the function Tn(z) defined by

Let, as usual, x = { xG } E (c) mean that x is a convergent sequence. We write l|Xii=iiXiiA=suPni Zn=1 an7kXki =SUPn iAn(x)l, where A = (ark) is a matrix of complex numbers. By the ordinary norm of x we shall understand I x I = supn I Xn I . PROBLEM 1. What conditions on A are necessary and sufficient that (c) be a Banach space with this norm? The first result is that |lxii n, while ann #O (A remark on reversibility will be appended.) Finally, let (A) be the class of sequences x such that Ax = {An(X) } is convergent.

Let g be the magnitude of the gradient and c0 be the sound speed at the observer. The apparent distance (c0t) and direction (θ0) of a sound source determine the true distance (R) and direction (θ), which are given by R = (2c0/g)[cos2θ0+ (sinθ0+coth12gt)2]−12 and θ = θ0+Tan−1[cosθ0/(sinθ0+coth12gt)]. A compact graph for generating angular correction is based on the fact that θ−θ0 is the argument of the complex number 1+ie−iθ0 tanh12gt. Taking c0/g to be unit distance and g−1 to be unit time, only one ray‐wavefront diagram is needed to represent all unbounded constant‐gradient media. Scaled distance (Rg/c0) and true direction (θ) can be read directly from this diagram. Intensity (I) in an unbounded constant‐gradient medium, measured in decibels, differs from 20 logR by a term that depends only on travel time: −10 logI = 20 logR + 20 log cosh12gt.

In this note the secular equation for Rayleigh surface waves in classical elasticity theory is examined in detail. It is shown, in particular, that when the equation has complex roots the associated displacement components are inadmissible — they do not satisfy the condition that they tend to zero at infinity. Rayleigh [1]3) demonstrated this result numerically for an incompressible body.

The nature of the eigenvalues of a square quaternion matrix had been considered by Lee [1] and Brenner [2]. In this paper the author gives another elementary proof of the theorems on the eigenvalues of a square quaternion matrix by considering the equation Gy = μȳ, where G is an n x n complex matrix, y is a non-zero vector in Cn, μ is a complex number, and ȳ is the conjugate of y. The author wishes to thank Professor Y. C. Wong for his supervision during the preparation of this paper.

For the group of real numbers R, an exponential monomial is defined as a function of the form xr(−ixz), for some non-negative integer r and some complex number z. Similarly, an exponential polynomial is a function P(x) exp (−ixz), for a polynomial P. In a now famous paper ((15)), Schwartz proved that every closed translation invariant subspace (variety) of the space of continuous functions on R is determined by the exponential monomials it contains. His techniques do not generalize to groups other than R as they use the theory of functions of one complex variable. A shorter proof of this result, using the Carleman transform of a function, was given by Kahane in his thesis ((9)). Ehrenpreis ((5)) proved results similar to those of Schwartz for certain varieties in the space of analytic functions of n complex variables, and Malgrange ((13)) proved the related result that any solution in ℰ(Rn) (for the notation see (16)) of the homogeneous convolution equation μ*f = 0, for some μ∈ℰ′, belongs to the closure of the exponential polynomial solutions of the equation.

The following nonlinear eigenvalue problem is studied : Let $T(\lambda )$ be an $n \times n$ matrix, whose elements are analytical functions of the complex number $\lambda $. We seek $\lambda $ and vectors x and y, such that $T(\lambda )x = 0$, and $y^H T(\lambda ) = 0$.Several algorithms for the numerical solution of this problem are studied. These algorithms are extensions of algorithms for the linear eigenvalue problem such as inverse iteration and the $QR$ algorithm, and algorithms that reduce the nonlinear problem into a sequence of linear problems. It is found that this latter method can be extended into a global strategy, finding a complete basis of eigenvectors in the cases where it is proved that such a basis exists.Numerical tests, performed in order to compare the different algorithms, are reported, and a few numerical examples illustrating their behavior are given.

Two programs based on Wilkinson's method [6] have been coded for the UNIVAC Scientific computer, Model 1103A for computing all of the eigenvalues and eigenvectors of complex non-Hermitian matrices. These programs make use of floating point, single precision, interpretive complex arithmetic, and take advantage of the built-in floating point feature of the 1 103A. The first program, which is described in Section 2, handles matrices of orderup to70. The second,which is described in Section3, handles matrices of order up to 25. The latter program is currently being reprogrammed to make possible the handling of matrices of order 75. In case the Jordan canonical form of the matrix is nondiagonal, the programs supply principal vectors for the missing eigenvectors, although, in this case, convergence is slow. Both programs employ 62-acceleration [6], which does not appear to be entirely satisfactory for complex matrices. A substitute acceleration is discussed in Section 2.5. The second program, employing the Wielandt "inverse power method" (see [1] pp. 293-294), in conjunction with the power method, has proved to be quite successful. In this program 62-acceleration is used with the power iterations and not with the Wielandt iterations. Matrix deflation is used in both programs and has yielded good results thus far. This is in contrast to experience reported by others [2]. For this reason some conclusions concerning matrix deflation are stated at the end of this paper.

The story of the development of the concept and uses of complex numbers provides a fine example of how the history of mathematics may shed light on the meaning of terminology, the relative roles of “practical” needs and intellectual curiosity in the motivation of mathematicians, the utility of pure mathematics, and the development not only of mathematics itself but also of the concepts of rigor and proof. The story also involves intrigue and illustrates the international nature of mathematical scholarship.

The Cauchy-Ostrowski theorem on convergence of Newton iterates for an analytic function in one variable is extended to include computational errors using complex interval arithmetic. Several numerical examples are given for polynomials with real and complex roots and one example for the Bessel function of the first kind.

Let K K be a complex valued measurable function on ( 0 , 1 ] (0,1] such that ∫ 0 1 t − 1 / p | K ( t ) | d t \int _0^1 {{t^{ - 1/p}}|K(t)|dt} is finite for some p > 1 p > 1 . Let H H be the Hausdorff operator on l p {l^p} determined by the moments μ n = ∫ 0 1 t n K ( t ) d t {\mu _n} = \int _0^1 {{t^n}K(t)} dt . Define Ψ ( z ) = ∫ 0 1 t z K ( t ) d t \Psi (z) = \int _0^1 {{t^z}K(t)} dt . Then for each z z with Re Re ⁡ z > − 1 / p , Ψ ( z ) \operatorname {Re} z > - 1/p,\Psi (z) is an eigenvalue of H ∗ {H^\ast } . The spectrum of H H is the union of { 0 } \{ 0\} with the range of Ψ \Psi on the half-plane Re Re ⁡ z ≧ − 1 / p \operatorname {Re} z \geqq - 1/p .

Suppose throughout that l, a n ( n = 0, 1, …) are arbitrary complex numbers, that α is a fixed positive number and that x is a variable in the interval [0,µ]. Let

The matrix spaces under consideration are the four main types of irreducible bounded symmetric domains given by Cartan (5). Let z = (zjk) be a matrix of complex numbers, z' its transpose, z* its conjugate transpose and I = I(n) the identity matrix of order n. Then the first three types are defined by(1)where z is an n by m matrix (n ≤ m), a symmetric or a skew-symmetric matrix of order n (16). The fourth type is the set of complex spheres satisfying(2)where z is an n by 1 matrix. It is known that each of these domains possesses a distinguished boundary B which in the first three cases is given by(3)(In the case of skew symmetric matrices the distinguished boundary is given by (2) only if n is even.)

THE following dialogue ought to take place one of these days. Well-Trained Teacher: As you all know, a linear equation has one root, a quadratic equation has two, and in general an alge braic equation of the nth degree with com plex coefficients has exactly complex roots. Typical Insolent Student: I know that a linear equation always has just one root, but we've solved quadratic equations that turned out to have just one root. WTT: Can you give an example? TIS: Sure, 2 = 0 has just one root, zero.

where the ~v are distinct complex numbers, and the coefficients A,v are arbitrary complex constants. In the present paper I continue his investigations a little further and prove a general theorem which may have some interest in itself. In order to make the paper self-contained, I have repeated some o f Gelfond's proofs. 1. Let ~o, ~ . . . . . ~,-1 be finitely many distinct complex numbers; let mo, ml . . . . . m,_a be an equal number of positive integers; and let

The objective of this paper is to give some concrete distinctions between positive linear maps and completely positive linear maps on C*-algebras of operators.Herein, C*-algebras possess an identity and are written in German type . Capital letters A, B, C stand for operators, script letters for vector spaces, small letters x, y, z for vectors. Capital Greek letters Φ, Ψ stand for linear maps on C*-algebras, small Greek letters α, β, γ for complex numbers.We denote by the collection of all n × n complex matrices. () = ⊗ is the C*-algebra of n × n matrices over .

This theorem answers in the negative the question: Is there a "universal scrambling sequence" {I ek}I Ei,k = ?1, turning every power series EakZk with positive coefficients into a power series >.-kakZk having the circle of convergence as natural boundary? This problem was raised by Mrs. Turan, and I am indebted to Dr. P. Erdos for communicating it to me. An example (?4) shows that the semi-circle in the statement of the theorem can not be replaced by a larger arc. A question which remains open is to find a corresponding theorem for the case that {11} is a given sequence of complex numbers of absolute value one.

( = 0 for p 7± q, (1.2) S'{Dp{z)Dq{z))\^^ \ \ 7* 0 for p = q, hold relative to the operator 5 ' and the sequence {cp\. The polynomials Dp(z) are given recurrently by the formulas D0(z) = l, Dp(z) = (dp+z)Dp„1(z)-bp-1Dp-2(z), £ = 1, 2, • • • (£>_!(*) =0) . In this paper orthogonality relations similar to (1.2) are developed for the polynomials Bp(z) which are derived from the denominators Bp(z) of the successive approximants of a C-fraction a,\zi a2Z2 azz* (1.3) 1 + 1 + 1 + 1 +• • • , where the ap denote complex numbers and the ap positive integers (cf. [3]). In fact, conditions (1.2) for a certain J-fraction are shown to be a specialization of the orthogonality relations for a C-fraction. Furthermore, necessary and sufficient conditions are obtained for the unique existence of the polynomials B*(z) in terms of the sequence {cp} (Theorem 2.2).

The familiar Taylor’s series expansion of the function , $f(z)$ has for its general term $D^n f(z_0 ){{(z - z_0 )^n } / {n!}}$. A new generalization of Taylor’s series in which the general term is $D^{an + \gamma } f(z_0 ){{(z - z_0 )^{an + \gamma } } / {\Gamma (an + \gamma + 1)}}$, where $a > 0$ and $\gamma $ is an arbitrary complex number, is examined. This new series is extended further to a form which includes the familiar Lagrange’s expansion as a special case. The derivatives appearing in this series are of order $an + \gamma $ and are called “fractional derivatives.” Examples of the use of this new series for discovering generating functions are given.

THE applicability of the method of differentiation with respect to a parameter for calculating the complex roots of a system of non-linear equations with analytic functions is studied.

(1917). The Graph of F(X) for Complex Numbers. The American Mathematical Monthly: Vol. 24, No. 9, pp. 409-420.

quantum field theory. Although few nonspecialists have had opportunity to become familiar with the language of modern pure mathematics, quantum theory seems to have reached a point where it must use that language if it is to find a genuine escape from the divergence difficulties. Divergence can not be properly coped with when convergence itself has never been rigorously defined. In the classical analysis of real and complex numbers, results, even correct results, can be obtained by algebraic manipulation of formal power series; but these numbers are not just algebras, they are topological algebras, and only with Cauchy's introduction of the epsilon-delta treatment was mathematics provided an explicit method of separating sense from nonsense. Similarly, in the modern analysis of infinite-dimensional algebras results can be obtained by algebraic manipulation of formal expressions, but these results often require topological justification. One standard way of introducing a topology into the algebra of observables is to make them operators on a Hilbert space. This method, which does not seem to be extensively employed in quantum electrodynamics, can be used to construct a mathematically rigorous formalism the manipulation of which is directly followable by one's physical intuition. This construction requires the exercise of two dissimilar disciplines, mathematics and physics, so the exposition is divided into two parts upon which relative emphasis can be adjusted to suit individual tastes. In particular, physicists can greatly simplify the mathematics by ignoring: (1) operator-domain considerations (as is done here in the derivation of the Yukawa-potential); (2) discussions involving the group algebra of the symmetric group (since only the FermiDirac and Bose-Einstein cases have ever actually occurred); (3) material depending on the simply-connected covering group of the Lorentz group (since it is not needed to derive Maxwell's equations). However, Part I is empty, unmotivated mathematics without Part II; and Part II does not exist without Part I. The two are designed to be read, not consecutively, but in parallel. Sections are numbered accordingly. I would like to thank Professor I. E. Segal for liberal use of his time and advice in the preparation of this paper. It is to be submitted to the Depart

The first author wishes to acknowledge with gratitude many instructive consultations with J. A. Schatz, and correspondence with M. S. MacPhail. Let A = (ank), n, k = 1, 2, , be a matrix of complex numbers. Let D be the set (linear sequence space) of sequences x = { x, } such that y =Ax is defined; y being the sequence {yn }I, where Yn = EkankXk for each n. Let R be the set of all Ax, xeD. We call D and R the domain and range of A. They are linear subspaces of (s), the space of all sequences. To emphasize the distinction between inverse matrix and inverse transformation, we denote Ax by T(x), thus defining T:D>R, and investigate, under various hypotheses: (a) the existence of right, left, and two-sided inverses for A, denoted by A', 'A, A-', (b) the same for T, denoted by T', 'T, T-1, (c) connections between (a) and (b). By A' we mean any matrix satisfying AA' = I, the identity matrix. By T' we mean any function T':R->D satisfying T(T'(x)) =x for all xCR. The other symbols are interpreted similarly. By "T' exists" we mean "there exists at least one T'." Similarly for the others. Our main results concern row-finite matrices, i.e. such that almost all the elements in each row are zero; column-finite matrices, i.e. matrices whose transpose is row-finite; and reversible matrices, i.e. matrices A such that for each convergent sequence y, the equation y =Ax has a unique solution (we shall see that if A is row-finite, reversibility is equivalent to the existence of a unique solution for all y). A discussion is given of the constants Cn of Banach [1, p. 50 ] which appear in the inverse transformation of a reversible matrix. Let E be the (countably infinite-dimensional) set of sequences x such that xn=0 for almost all n, (c) the set of convergent sequences. Clearly, DDE for all A; A is row-finite if and only if D = (s), column-finite if and only if AxCE whenever xCE, reversible if and only if RD(c), and T is 1-1 (i.e. to each y R corresponds exactly one xCD; A is 1-1 will mean that the associated T is 1-1).

The consequences of different symmetry groups for strong interactions are studied in terms of relations between polarizations. Some experimental tests concerning antihyperon-hyperon production by means of {ie2}p and {ie3}d annihilations are proposed.

A complex vector space X will be called an F 2 n {F_{2n}} space if and only if there is a mapping ⟨ ⋅ , ⋯ , ⋅ ⟩ \langle \cdot , \cdots , \cdot \rangle from X 2 n {X^{2n}} into the complex numbers such that: ⟨ x , ⋯ , x ⟩ > 0 \langle x, \cdots ,x\rangle > 0 if x ≠ 0 ; x k → ⟨ x 1 , ⋯ , x 2 n ⟩ x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle is linear for k = 1 , ⋯ , n ; ⟨ x 1 , ⋯ , x 2 n ⟩ = ⟨ x 2 n , ⋯ , x 1 ⟩ − k = 1, \cdots ,n;\langle {x_1}, \cdots ,{x_{2n}}\rangle = {\langle {x_{2n}}, \cdots ,{x_1}\rangle ^ - } where denotes complex conjugate; ⟨ x σ ( 1 ) , ⋯ , x σ ( n ) , y τ ( 1 ) , ⋯ , y τ ( n ) ⟩ = ⟨ x 1 , ⋯ , x n , y 1 , ⋯ , y n ⟩ \langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (n)}},{y_{\tau (1)}}, \cdots ,{y_{\tau (n)}}\rangle = \langle {x_1}, \cdots ,{x_n},{y_1}, \cdots ,{y_n}\rangle for all permutations σ , τ \sigma ,\tau of { 1 , ⋯ , n } \{ 1, \cdots ,n\} . In the case of a real vector space the mapping is assumed to be into the reals such that: ⟨ x , ⋯ , x ⟩ > 0 \langle x, \cdots ,x\rangle > 0 if x ≠ 0 ; x k → ⟨ x 1 , ⋯ , x 2 n ⟩ x \ne 0;{x_k} \to \langle {x_1}, \cdots ,{x_{2n}}\rangle is linear for k = 1 , ⋯ , 2 n ; ⟨ x σ ( 1 ) , ⋯ , x σ ( 2 n ) ⟩ = ⟨ x 1 , ⋯ , x 2 n ⟩ k = 1, \cdots ,2n;\langle {x_{\sigma (1)}}, \cdots ,{x_{\sigma (2n)}}\rangle = \langle {x_1}, \cdots ,{x_{2n}}\rangle for all permutations σ \sigma of { 1 , ⋯ , 2 n } \{ 1, \cdots ,2n\} . In either case, if ‖ x ‖ = ⟨ x , ⋯ , x ⟩ 1 / 2 n \left \| x \right \| = {\langle x, \cdots ,x\rangle ^{1/2n}} defines a norm, X is called a G 2 n {G_{2n}} space (Trans. Amer. Math. Soc. 150 (1970), 507-518). It is shown that an F 2 n {F_{2n}} space is a G 2 n {G_{2n}} space if and only if | ⟨ x , y , ⋯ , y ⟩ | 2 n ≦ ⟨ x , ⋯ , x ⟩ ⟨ y , ⋯ , y ⟩ 2 n − 1 |\langle x,y, \cdots ,y\rangle {|^{2n}} \leqq \langle x, \cdots ,x\rangle {\langle y, \cdots ,y\rangle ^{2n - 1}} and that G 2 n {G_{2n}} spaces are examples of uniform semi-inner-product spaces studied by Giles (Trans. Amer. Math. Soc. 129 (1967), 436-446).

The present article discusses means for the systematic description of programming methods based on the use of control words, different from those employed in [1], and the structure of a digital machine (DM) which would enable these programming methods to be used. The references cited contain ideas which may be regarded as in some degree precursors of the ideas proposed in the present article. A problem is solved on a DM “in steps”. The steps may be coarse or fine, but each of them is determined by what is called the command. Accordingly, one command can determine the operation of the machine over a longer time interval (we shall say that this command is of higher order), and another over an elementary cycle only of the machine operation, i.e. a so-called microcommand. There are correspondingly two different codes: a code of operations and a number code. We use the term “number code” intentionally here, understanding by this that, in some form acceptable to the machine, an indication is given of the numbers on which a given operation is to be performed. In line with the code of an operation we can write directly the number on which the operation is to be performed, i.e. the number can be specified by a number address. We can go even further, and instead of the address of the number, indicate the address where the address of the required number will be found. Similarly, along with the code of the operation we can indicate the address where this code is found. In modern machines the second and third methods are used. This enables the program writing to be shortened, since for the most part the same operation is performed on a whole set of numbers, and not merely on one. In order to change the numbers on which the operation is performed, the address of these numbers can be changed. In this case the address of the “changed” address is indicated in the command. An example of this can be found in so-called systems of index registers: in line with the code of the operation the number of the index register, the content of which is the number address, is indicated. This system is used in the Larc and Stretch machines. However it leads to a need for special commands which transform the contents of the index registers. In this case the program is written in the form of a consecutive set of commands of the index arithmetic (called in the literature commands on the contents of the index registers) and commands of the computational arithmetic (see [1]). In the present work an attempt is made to standardize the formation of the addresses, by having a limited number of standard subprograms, in accordance with which the addresses will be formed in accordance with the initial data introduced. As a result the total subprograms are substantially shortened. Three standard methods of formation of the addresses of a number set are introduced: points, straight lines, planes. The initial data for these subprograms form standard words, which will in future be called control words. We shall consider methods of constructing codes of number and command sets in the form of control words. Development of these methods leads to a new system of programming. In this system the introduction into the machine of a fairly wide class of problems does not require the writing of an ordinary program, consisting of an ordered set of elementary commands. Instead, we have to write the control words, and if necessary, certain pieces of programs for the operations on the numbers. The realization of such a system requires a machine possessing certain special structural features. We can regard the machine as consisting of individual units (“submachines”). The first of these submachines is used for operating on numbers, the second for operating on addresses. To match the operation of the arithmetic and address submachines, a third “control submachine” is employed. In essence, each submachine has its arithmetic (transforming organ), memory and control with program. The arithmetic submachine has a relatively complex arithmetic, a large memory and programs for operation on number codes. The address submachine has a primitive arithmetic, a smallish memory (index registers) and programs for address transformation which will in future be termed address development laws. The control submachine has as its own program control words. This machine contains an arithmetic in a strongly reduced form and a small memory. Generally speaking, constructional separation of the submachines is not essential. In existing machines of the Larc and Stretch type it is not carried out in this clear-cut way. However, constructional separation of the submachines would be an advantage. This functional structure of the computational complex can simplify programming and greatly facilitate the exchange of information between machine and man. The formation of more complex computational complexes can be carried out with the aid of these triads. The possible functional arrangement of a computer is shown in the figure. The black squares denote the local control units, and the arrows the connections between the functional blocks. The notation is as follows: OM is the working memory, Pr.OM is the working memory for the program, Ar.M is the arithmetic submachine, Ad.M is the address submachine, Con.M is the control submachine. The figures indicate the transmitted information, namely: 1, 2 are numbers; 3 are predicates (ω-signs), 4 are initial data for changing the content of the index registers, 5 are ω-signs, 6 are the codes of the subprograms to vary the contents of the index registers, 7 are the number codes, 8 are the command codes, 9 are the symbols of the operations.

Using mathematical tools developed by Hermann Weyl, the Wigner classification of group‐representations and co‐representations is clarified and extended. The three types of representation, and the three types of co‐representation, are shown to be directly related to the three types of division algebra with real coefficients, namely, the real numbers, complex numbers, and quaternions. The author's theory of matrix ensembles, in which again three possible types were found, is shown to be in exact correspondence with the Wigner classification of co‐representations. In particular, it is proved that the most general kind of matrix ensemble, defined with a symmetry group which may be completely arbitrary, reduces to a direct product of independent irreducible ensembles each of which belongs to one of the three known types.

Let V be an n-dimensional vector space over complex numbers C. Let W be the mth tensor product of V. If T C Homc(V, V), let ?m T EHomC(W, W) be the mth tensor product of T. The homomorphism T -4 (m T is a representa- tion of the full linear group GL (C). If H is a subgroup of the symmetric group SMI and X a linear character on H, let VX(G) be the subspace of W consisting of all tensors symmetric with respect to H and x. Then Vm(H) is invariant under gm T. Let K(T) be the restriction of (m T to Vxm(H). For n large compared with m and for H transitive, we determine all cases when the representation T - K(T) is irreducible.

A numerical procedure is described for processing the data of a microwave measurement in which the measured points are distributed in a form of a circle in a complex plane. Instead of plotting the measured data on a Smith chart and analyzing them by graphical methods, the data are analyzed by the method of least squares. The result of this analysis consists of three complex numbers K, L, and M, which define the bilinear transformation in question. The procedure is illustrated on the example of impedance versus bias measurements on a varactor diode which was recently described by E. W. Sard. The necessary formulas are derived for computation of elements of the equivalent circuit from the above constants K, L, and M. The procedure is well-suited for programming a digital computer.

The parameter p in (1) is required to be a positive odd integer and z is any complex number with 9T(z) >0. The parameters ae and f3 are real numbers restricted as follows: When p = 1 we must have 0 ? ce ? 1 and 0 1, Equation (1) is valid for 0 1, it yields a transformation formula for the Lambert series

This paper contains a complete system of ALGOL procedures which enable arithmetic operations to be carried out upon complex numbers. Further procedures for carrying out the evaluation of certain elementary functions (e.g. ln, exp, sin, ...) of a complex variable are given. Application of these procedures is then illustrated by their use in the computation of the confluent hypergeometric function and the Weber parabolic cylinder function. Procedures relating to the application of the e-algorithm to series of complex terms are described. Two integrated series of procedures, relating to Stieltjes typeS-fractions and to corresponding continued fractions respectively, are given. Complete programmes, which illustrate the use of these procedures, may be used for the computation of the incompleteβ-function, the incompleteΓ-function (of arguments of large and small modulus) and the Weber function.

Mit Hilfe elementarer Satze aus der Geometrie der ebenen Massenpunkthaufen werden bekannte Einschliesungssatze von Kryloff-Bogoliubov, Temple, Collatz und anderen unter gemeinsamem Gesichtspunkt hergeleitet und aufs Komplexe erweitert. Auch einige neue Ergebnisse werden dabei gewonnen. By means of basic principles of the plane geometry of particle clusters the well known theorems by Kryloff-Bogoliubov, Temple, Collatz and others are developed from a viewpoint which is common to all. They are extended to complex numbers, also some new results are obtained by this procedure.