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If D is a set of complex numbers and a and b are given numbers then by aD+b we mean the set of numbers ad+b where dczD. Translations of D will be denoted by D +b, and the number |b will be called the length of D+b. Let D denote the image of I z I < 1 under an analytic function f(z). The first theorem proved in this paper is the following: if f(z) = a0 +Zn+an+?zn+l+ * * * then each translation of D of length less than 7r/2 meets D. In the case where f(z) is univalent (and therefore n = 1) the existence of such a nonzero constant follows from the fact that D covers the circle lw-aol

The paper describes a machine for determining the real and complex roots of higher-degree algebraic equations. The principle of operation is found in the correspondence between sine wave quantities and complex numbers. The particular machine is designed for equations of the eighth degree, and finds all the roots with engineering accuracy in the space of a few minutes. Though designed primarily with a view to determining the indicial admittance of electric networks, the machine should find utility in other fields of applied mathematics as well.

Let A be the set of all functions from N, the natural numbers, to C the field of complex numbers. The Dirichlet product of elements f, g of A is given bywhere the summation condition means sum over all positive integers d which divide n.

This paper presents a clear interpretation of the pole migration on the real axis where many root loci starting from different poles lie together or conflict with one another, by introducing the concept of a pair of real roots. Except in the case of a single root locus, the root loci on the real axis appear always as a pair. Just like the complex number, a pair of real roots is expressed in two parts and the whole root loci for both real and complex roots are plotted in the conventional rectangular coordinates as continuous curves, by which the migration of each pole on the real axis is shown very clearly. Three examples are given.

By programming pairs of amplifiers to represent the real and imaginary parts of complex variables on an analog computer, techniques are developed which 1) permit the mechanization of the method of steepest descents, for finding roots of complex functions, in Cartesian coordinates; 2) allow the generation of complex functions utilizing the Cauchy integral theorem; 3) enable a programmer to scan the z plane efficiently when electronic mode control and track-and-hold circuitry are available. Examples are given which show that these techniques frequently provide more efficient programs, in terms of equipment, than previous methods.

1. For a square matrix A the so-called field of values G[A] is defined as the following set of complex numbers:' (1.1) G[A] := {xHAx|xHx = 1}. It is well known that this set contains the spectrum A(A) = {Xj(A) } of A. Moreover, Toeplitz [11] proved in 1918 that G[A] also contains the convex hull 5C( A(A)) of A(A). Hausdorff [5] generalized this result by proving that G[A] is convex. The concept of a field of values was generalized by Bauer [1] in 1962. Starting with an arbitrary norm || v 1l in Cn (or R') and its dual norm (defined on the dual space of Cn (or Rn) of row vectors yH) IIyH D =max Re y X

A generalized Barker sequence is a finite sequence \{a_{r}\} of complex numbers having absolute value 1 , and possessing a correlation function C(\tau) satisfying the constraint |C(\tau)| \leq 1, \tau \neq 0 . Classes of transformations leaving |C(\tau)| invariant are exhibited. Constructions for generalized Barker sequences of various lengths and alphabet sizes are given. Sextic Barker sequences are investigated and examples are given for all lengths through thirteen. No theoretical limit to the length of sextic sequences has been found.

Someone once said that mathematics consists of finding new solutions to old problems and old solutions to new problems. Sometimes, however, it becomes necessary to revive old solutions for old problems. Approximately a hundred years ago C. A. Laisant and other mathematicians of that time used vector methods, with coordinates in the complex plane, to solve certain types of problems in geometry involving polygons having the same centroid, called concentric polygons. This method has been neglected, and it is almost forgotten today. It is the purpose of this article to attempt to revive this method by illustrating its effectiveness in solving certain types of problems in elementary geometry.

When the Newton-Raphson method for complex roots is used, numerical solutions of both the Morse and Scott equations for attenuation in a lined duct can readily be obtained. The method of calculation and some comparative results using both theories are presented in this note.

1. A summability matrix is called conservative if it attaches a limit to, that is, sums, every convergent sequence. If moreover this limit is a fixed multiple m of the ordinary limit of the sequence the matrix is called multiplicative m. If a matrix A is multiplicative m then the matrix kA, where k $0 is a number, is multiplicative km and sums exactly the same sequences as A. Thus it is immaterial to specify m except to say whether or not it is zero. The dichotomy of matrices into those for which m =0, m S 0 is well known to be significant. A single example is the theorem of Steinhaus [1 ] (1) that if m #0 a multiplicative m matrix cannot sum all bounded sequences, the result being false if m = 0. The principal object of this paper is to extend this classification into the whole set of conservative matrices. This will be separated into the subclasses of co-regular and co-null matrices; the division of multiplicative matrices induced will be that mentioned above. It will then be shown that a class of theorems which have been proved about multiplicative matrices can be so generalized as to apply to conservative matrices in general. The value of the classification will appear in that certain results in which the condition m 0 plays an essential role will hold for co-regular but not for co-null matrices. Some results are new even for multiplicative matrices, for example, the specialization of Theorem 2.0.3. 1.1. Preliminary. Let A = (ac,k) be a matrix of complex numbers and x = {xn4 a complex sequence, then y =Ax is called the transformed sequence where in the multiplication x and y are treated as column vectors, thus Y= {yn} where yn= Z_o ankXk=An(x). Then, if it exists, lim yn=lim An(x) is called A(x). Finally the domain of the functional A(x), that is, the set of sequences x such that Ax is convergent, is called the summability field of A and written (A). We shall denote the identity matrix by I so that (I) is the set of convergent sequences. Setting, after Brudno (1), jAj =supn Ek= ank and denoting by F the set of sequences i= {I, 1, 1, . . . }, Io= {1, 0, 0, . . . }, 8'= {0, 1, 0, . . *} . . , Sk, * * * , we have the classical result: the matrix A is conservative if and only if (i) |jA|| < Xo and (ii) FC(A). Denoting A(Sk) = limn,O0 ank by ak we can easily show that

(1929). Some Geometrical Applications of Complex Numbers. The American Mathematical Monthly: Vol. 36, No. 10, pp. 504-511.

Because complex arithmetic takes only a little more time than real arithmetic on modern computers it is advisable to take a closer look at some of the known mathematical tools and compare the advantages in complex computations.

If {λ; n }, { b n } are sequences of complex numbers, and we consider the series ∑ b n exp (−λ n x ), given as convergent in (0, 1) (i.e. the open invertal (0,1)) to f(x) ∈ L , then, writing (if λ n = 0 the corresponding term is ½ b n x 2 ) where the series is supposed is to be uniformly convergent in (0, 1), we have for 0 h h(x) .If we know that the second member of (2) tends to f(x) as h → +0, it will follow that F(x) is a repeated integral of f(x) ((1), 671). If there is a sequence { φ v (x) } of integrable functions with the property that then, on multiplying (1) by φ v (x) and integrating over (0,1), we obtain a formula for b v in terms of F(x) . On integrating by parts twice, b v will be expressed in terms of f(x) , and this will constitute a uniqueness theorem for the series ∑ b n exp (− λ n x ).

This paper is concerned with the theory of ideals in the algebra L1 of integrable functions on a locally compact abelian group. After some preliminaries an analytical proof is given of the known theorem that an analytic function of a Fourier transform represents again a Fourier transform (p. 406). Then, in part I, the continuous homomorphisms of closed ideals I of L1 upon C, the field of complex numbers, are studied. Any such homomorphism is given by a Fourier transform and, if I o is its kernel, the quotient-algebra I/Io, normed in the usual way, is not only algebraically isomorphic, but also isometric with C (Theorem 1.2). Another result states that homomorphic groups have homomorphic L'-algebras and that a corresponding property of isometry holds (Theorem 1.3). In part II, which may be read independently of part I, a theorem of S. Mandelbrojt and S. Agmon, which generalizes Wiener's theorem on the translates of a function in Ll, is extended to groups (Theorem 2.2). Several generalizations of Wiener's classical theorem have been published in the past few years; references to the literature are given on p. 422. The rest of part II is devoted to some applications (pp. 422-425). In conclusion it should be said that the work is carried out in abstract generality, with the methods, and in the spirit, of analysis, which is then applied to algebra. To Professors S. Mandelbrojt, B. L. van der Waerden, and A. Weil I owe my mathematical education. The inspiration which I have received in their lectures, in letters, and above all in personal contact, is at the base of this work ; may I here express my gratitude.

The vector method is just as useful in solving problems involving transient conditions in electric circuits as it has proved to be when the currents and potentials are steady sinusoids. As far as the writer is aware, the vector method for determining transients in rotating electric machines was first used by L. Dreyfus. Previously the method had been applied to fixed combinations of resistances, inductances and capacitances by Kennelly and others. By making certain assumptions that are, however, quite reasonable in many cases, the transient currents in nearly all of the common types of electric machinery are damped sinusoids. Fortunately the damping is exponential and is thus readily accounted for. It is interesting to trace the development of the method. In the solution of all problems in direct currents the potentials, currents, and circuit constants are real numbers. In the corresponding problem in which the applied potentials are steady sinusoids, these quantities are all represented by complex numbers. In all other respects the working out of the solution is identical with that followed in the direct-current case. When the currents are damped sinusoids, they and the potentials and the circuit constants can still be represented by complex numbers. There is this difference, however; the vectors which represent the currents and potentials shrink exponentially as they rotate and the values of the circuit constants depend not only upon the frequency of the current, but also upon its rate of shrinking. Again the solution of any problem follows the same procedure that would the corresponding one in which the currents are steady sinusoids. In both the steady and damped sinusoidal cases the circuit constants depend upon the angular velocity of the vectors which represent the currents. In the former, the angular velocity is purely imaginary while in the latter it is complex, the real part being the rate at which the current vector shrinks and the imaginary portion, its angular velocity. In electric machinery in which rotating magnetic fields are produced, these fields shrink exponontially as they rotate when the currents are damped sinusoids. If these rotating magnetic fields are represented by vectors, the vectors will have a complex angular velocity just as do the currents. The e. m. f. which is produced by a steady sinusoidal variation of flux lags the flux by 90 degrees, whereas if the flux variation is a damped sinusoid, the angle of lag is less than 90 degrees, depending upon the damping. The mathematical relation, however, is the same, vis., the e. m. f. is proportional to the negative of the product of the flux and its angular velocity. It is then readily appreciated that the form of the solution for the transient state is the same as that which is used for the steady state. Before the method can be expected to give as accurate results as are obtained when predicting the steady operation, considerable experimental data must be obtained in order to determine the best methods of measuring the necessary constants, for these may be somewhat different during the transient period than during steady operation.

An approximate linear form of the Peierls-McManus equations of motion for an electron is shown to yield runaway solutions for a particular influence function, which otherwise satisfies all the conditions imposed by the theory. The general problem is to discuss the existence of complex roots of a transcendental equation for an arbitrary form of the influence function. It is shown that this equation always has roots. These, however, may be such that the corresponding motion of the free electron is of a damped nature. It has not been found possible to construct a function which avoids runaway solutions or to prove whether or not such a function exists when all conditions are satisfied.

Let G be an arbitrary discrete group and let F = C [G] be its group algebra over the complex numbers C. If 9 is an irreducible representation of the algebra then 9T(F)=P is primitive and hence isomorphic to a dense set of linear transformations over D, the commuting ring of St [4, p. 28]. Let L be the center of D. If dimL P < 00 then we say that St is finite and since P is central simple over L [4, p. 122 ] we have dimL P =m2. We set m = deg St, the degree of St. If G is finite then C is always the commuting ring of St so this agrees with the usual definition of degree. Again let P =T9(F). Then by a theorem of Amitsur [1] deg _n if and only if for every 2n elements xi, * *, x2n in P we have

In low latitude the spatial distribution functions of the source field over the surface and the dimensions of the source, are important in any theory of electromagnetic induction developed for studying the conductivity structure of the Earth. The author has built up a mathematical structure for a theory of electromagnetic induction in anyn-layered earth model in low latitude. No simple solution is assumed for the horizontal distribution function of the source field and no assumption is made about the horizontal gradients of the source. The mathematical structure involves the concept of downward continuity of the field equations inside then-layered earth model. The resulting mathematical functions derived for anyn-layered earth model are complex. Hence a new matrix algebra of complex numbers is introduced by the author and this is built into the theory. From the upward continuity of the field equations, an inequality equation is derived in order to determine the heighth0 at which the induction field of the earth becomes negligible compared with the source field. The comparison of such heights at two or more stations under the same influence of the source field can be used for the resolution of the lateral distribution of the earch conductivity structure at these stations. The application of the theory will follow in a subsequent paper.

THE REDUCTION OF THE EXPERIMENTAL DATA obtained from an oscillating concentric cylinder rheometer to values of the moduli of rigidity and viscosity of a viscoelastic material involves considerable difficulty and labor. A method is here proposed by which such computations can be performed by a digital computer.The differential equation applicable to the apparatus and sample contains the moduli as a complex number whose value is not known. An approximate value of this parameter is used in the numerical integration of the equation subject to its initial boundary conditions at the radius of the inner cylinder. The deviation of the solution from a boundary condition at the outer cylinder is used in the Newton-Raphson equation to compute a better approximation to the desired parameter and the numerical integration is then repeated. The mathematical basis of this iterative procedure and the programming of it on an IBM 742 is discussed.

A network containing switches is equivalent to a number of networks that differ in the values of their components, in the arrangement of the components, or in both respects. When analyzing or synthesizing such a network, one may treat each different network by itself, and then combine the results. This paper describes a method by which the different aspects of a switchable network may be treated simultaneously. The mathematics by which the network is treated is a combination of ordinary field algebra (complex numbers) and Boolean algebra. The mathematical foundation is first laid out, then interpreted in terms of switchable network elements. The paper is concluded with some examples of analysis and synthesis of switchable networks.

Alternating hydrocarbons have a number of characteristic physical and chemical properties which were originally discussed in terms of the molecular orbital theory by Coulson and Rushbrooke, and later given an algebraic formulation by Ruedenberg and Scherr in terms of anoperator J which anticommutes with their Hamiltonian, H . Although in a certain sense, the alternating compounds comprise the most general class having these specialized properties, nevertheless the relation JH = − HJ may be generalized to an “exchange relation” of the type JH = ωHJ , ω being a complex number. While a Hamiltonian may not satisfy this relationship unless ω = ±1, it is still possible that it is a function of such an operator, or that it may be decomposed into non-Hermitian submatrices which satisfy the more gereral rule. In this paper, a canonical form is given to characterize finite operators satisfying an exchange rule, and it is thereby shown how these rules may be used to simplify the secular equation of a Hamiltonian operator. When one has a group G of exchange operators, there is an identity between possible exchange factors ω ( J ) and one-dimensional representations of G . When H is not a Hamiltonian, but itself a representation matrix of a group, the theory of the paper becomes a tool for constructing group representations, closely related however to the theories of Schur, Clifford, and Mackey. A simple example, the perinapthenyl free radical, is analyzed to illustrate the methods described.

Introduction. The theories of hypercomplex numbers and of continuous groups were first explicitly connected by POINCAR4t in 1884 with the statement that the problem of complex numbers was reduced to that of finding all the linear continuous group of substitutions in n variables of which the coefficients are linear functions of n arbitrary parameters, and since that time the advance in the theory of hypercomplex numbers has been largely suggested by the theory of continuous groups. In 1889 STUDYf and SCHEFFERS? developed the relation between these theories to a considerable extent, and the latter in 1891 arrived at a complete enumeration of systems in less than six units which are inequivalent (of different "' Typus "), non-reciprocal, irreducible, and which possess moduli. Previously (1889) STUDY?f had enumerated all inequivalent systems with moddli in less than five units without direct use of the theory of continuous groups, and in 1890 ROHR** continued the work through systems in five units. The problem of enumerating hypercomplex number systems had been attacked by BENJAMIN PEIRCE about twenty years before the investigations of STUDY and SCHEFFERS. His results were not printed, however, until after his death.tt With the methods of the European investigators in mind, I have else

Let A be a commutative semisimple F-algebra with identity, and let D 0 , D 1 , ⋯ {D_0},{D_1}, \cdots be a system of derivations from A into the algebra of all continuous functions on the spectrum of A. It is shown that the transformations D 0 , D 1 , ⋯ {D_0},{D_1}, \cdots are necessarily continuous. This result is used to obtain a characterization of derivations on Hol ( Ω ) {\text {Hol}}(\Omega ) where Ω \Omega is an open polynomially convex subset of C n {C^n} .

A joint paper by Tranter and the present author concerning the heat flow in an infinite medium heated by a sphere has necessitated a proof that no complex zeros of the equation

A vectorial pseudonorm (norm) of order k k on the vector space C n {C^n} of all n n -tuples of complex numbers is a mapping from C n {C^n} into the positive cone of R k {R^k} which satisfies the usual axioms of a pseudonorm (norm). The vector space R k {R^k} of the k k -tupies of real numbers is partially ordered componentwise. Vectorial norms have been introduced by Kantorovitch. Recently they have been studied by Robert and Stoer. In the present paper different properties of vectorial pseudonorms are investigated. They deal mainly with the following topics: regularity of pseudonorms, the dual of a vectorial norm, inequality between vectorial pseudonorms and the G G -transform of a vectorial pseudonorm.

The density matrix for a many-electron system has been examined. A formalism has been arrived at which facilitates its evaluation in terms of a basis corresponding to a discrete spectrum of eigenvalues. This allows a representation to be employed associated with a suitably chosen approximation to the Hamiltonian. Thereby, reasonably accurate estimates of the properties of many-electron systems may be anticipated for low orders of approximation.In developing the formalism, attention was focused upon a means for approximating the quantity $\mathrm{exp}(z\mathrm{H})$, where H is the many-particle Hamiltonian and $z$ is a complex number. It was found possible to represent this quantity exactly as an infinite product of exponential factors, each of which depends upon a portion of the Hamiltonian alone. The relation of this result to statistical mechanical applications is indicated. An approximation procedure is described in terms of which various finite products of exponential factors approach the desired quantity in the limit of indefinitely increasing numbers of factors; this limit is approached in a manner that permits any given approximation to contain all the terms of the previous approximation.The first order of approximation was employed to approximate the many-particle density matrix. The resulting theory was applied to the helium-like systems: He, ${\mathrm{Li}}^{+}$, ${\mathrm{Be}}^{++}$. The energies of the ground $^{1}S$ state of helium and the three lowest states of ${\mathrm{Li}}^{+}$ and ${\mathrm{Be}}^{++}$ were calculated. The agreement with experimental term values is reasonably satisfactory, the discrepancy between calculations and experiments not exceeding one percent. The triplet-singlet splitting for ${\mathrm{Li}}^{+}$ was in the order observed; the magnitude calculated was about one-half that observed.The relation between the present theory and the Thomas-Fermi theory is discussed. It is pointed out that the former is the analog of the latter when the density matrix is expressed in other than eigenfunctions of momentum. The quasi-classical character of the approximation is discussed.

The partition functions of the two- and three-dimensional finite Ising models in the presence of a magnetic field have been calculated with use of a high-speed computer. Periodic boundary conditions were imposed. All the numbers of configurations for the 3×3×3, 5×5, and 4×6 lattices are tabulated. The distribution of zeros of the partition functions in the complex magnetic-field plane has been obtained both for the ferro- and antiferromagnetic cases. The zeros in the ferromagnetic cases are distributed on the unit circle according to the Yang-Lee theorem, and the dependence on temperature of the distribution functions has been studied. In the antiferromagnetic case most of zeros are on the negative real axis, but a certain number of complex roots appear in the left half of the plane. The magnetization and the magnetic susceptibility have been calculated as functions of temperature.

Solutions to Laplace’s equation in three dimensions are obtained by the introduction of a suitable commutative and associative algebra over the field of the complex numbers. In one formulation these solutions turn out to be the coefficients of powers of $\sigma $ in the expression of a function of $w = z + ( i\rho /2 )( {\sigma e^{i\phi } + \sigma ^{ - 1} e^{ - i\phi } } )$ as a formal series. A variety of relations from the subject of spherical harmonics are shown to follow from the consideration of simple functions of w. While there is a condition on the gradients of the coefficients of powers of $\sigma $ that is analogous to the orthogonality of the gradients of the real and imaginary parts of an analytic function of $x + iy$, since here the number of coefficients may be infinite, the relation is much more complicated than for the two-dimensional case.

Given an Nth degree polynomial $P(z)$ one may use Laguerre’s method to generate a sequence of complex numbers $x_0 ,x_1 ,x_2 , \ldots $ which usually converges to a, zero of $P(z)$. This note shows that each circle $\left| {z - x_n } \right| \leqq \sqrt N \left| {x_{n + 1} - x_n } \right|$ contains at least one zero of $P(z)$. If N is not a perfect square, then the constant $\sqrt N $ can be replaced by a slightly smaller constant $R_N $.

Let A = (ank) be an infinite matrix of complex numbers ank (n, k = 1, 2,…) and X, Y two subsets of the space s of complex sequences. We say that the matrix A defines a (matrix) transformation from X into Y, and we denote it by writing A: X → Y, if for every sequence x = (xk)∈X the sequence Ax = (An(x)) is in Y, where An(x) = Σankxk and the sum without limits is always taken from k = 1 to k = ∞. The sequence Ax is called the transformation of x by the matrix A. By (X, Y) we denote the class of matrices A such that A: X → Y.

Our main object in this note is to establish (Theorem 1) a necessary and sufficient condition to be satisfied by a sequence {en} so that a series Σ an enmay be summable | A |whenever the series Σanis summable (C, — 1). We suppose that an and en are complex numbers. The condition is unchanged if the an are restricted to be real, but our proof is adapted to the case where they may be complex. Theorem 1 has been quoted by Bosanquet and Chow [12] in order to fill a gap in the theory of summability factors. We also obtain some related results, which are discussed in the Appendix.

where { xi } is a sequence of complex elements and the Kn,i are complex numbers, has been widely studied, and the conditions which must be fulfilled by the Kn,i in order that the property of convergence of the sequence may remain invariant were given by Schur [I ].t In recent studies by Hurwitz [2, 3] and Knopp [4] modes of measuring the divergence of bounded sequences were given, and the conditions on the Kn,i were found under which the divergence of the sequence { yn } is no greater than that of {Xn}. In this paper the effects of the transformations will be investigated with fewer restrictions on the Kn,i than those imposed by earlier writers. The problem will be approached by means of the new concept of the limit circle defined as follows: The limit circle of a bounded sequence of complex elements is the (unique) circle of least radius which contains within or on its boundary the limit points of the sequence. The limit circle of a bounded function F(y) of the complex variable y as y->t (finite or infinite) is analogously defined in terms of the limit points of F(y) as y->t; this concept will be used in the study of transformations of sequences and functions into functions. 2. Sequence to function transformations. Instead of the transformation mentioned in the introduction we shall study the following more general transformation S. Let T be a set of points in the complex plane having a limit point to (finite or infinite) not belonging to T. We shall speak of a point t in T as being sufficiently advanced if for some a >0, jt toI < when to is finite, or j i/tI < 5 when to is infinite. Then let Ki(t) be a set of complex numbers defined for i= 1, 2, ... , and each t in T, and such that

Expressions are derived for computing the mean and variance of the number of renewals in observation times of any length and for any values of the scaling parameter of the censored Poisson process. These results depend on certain identities involving the complex roots of unity. Computations show the oscillatory behavior to be expected when the average interval between renewals of the censored Poisson process is of the order of the observation time. Both ordinary and equilibrium renewal processes are considered.

1. Introduction .-Part I and II of the present paper give a brief description of two methods, which are believed to be substantially novel, for the numerical resolution of equations with complex roots. As regards rapidity of computation, the methods will probably be found inferior to others such as Aitken's extension of Bernoulli's method, or the root-squaring method as improved by Brodetsky and Smeal. On the other hand, a feature of the alternatives proposed is that arithmetical errors need not be cumulative. Certain further merits will be suggested in due course. Part III contains a résumé of Bairstow's variant of the root-squaring method, and Part IV deals with successive approximation to a quadratic factor. The substance of these parts has already appeared in aeronautical pulications, but may not be familiar to the general scientific public.

A set of principles and a systematic procedure are presented to establish the exact solutions of very large and complicated physical systems, without solving a large number of simultaneous equations and without finding the inverse of large matrices. The procedure consists of tearing the system apart into several smaller component systems. After establishing and solving the equations of the component systems, the component solutions themselves are interconnected to obtain outright, by a set of transformations, the exact solution of the original system. The only work remaining is the elimination or solution of the comparatively few superfluous constraints appearing at the points of interconnection.The component and resultant solutions may be either exact or approximate and may represent either linear or, with certain precautions, nonlinear physical systems. The component solutions may be expressed in numerical form or in terms of matrices having as their elements real or complex numbers, functions of time, ...

Let f be a meromorphic function in [z I 0. It is implicit in the method of proof that for any B > 1 there is a corresponding A for which the desired representation holds for all f . We show that in general B cannot be chosen to be 1 by giving an example of a meromorphicfsuch that if f = fl]f2 where f l and f2 are entire then T(r,f2) # O(T(r,f)). Rubel and Taylor have obtained the above theorem for special classes of meromorphic functions. In particular it is shown in I-5] that such a representation exists for any meromorphic f such that either sup T(2r,f)]T(r,f) _l or such that log T(r,f) is a convex function of log r. Although the representation is not obtained in [-5] for all meromorphic functions, the general result is shown to be equivalent to a seemingly more elementary proposition concerning sequences of complex numbers. The contribution of this paper is to prove the result concerning sequences of complex numbers and to provide an example showing the theorem is sharp. Results in this direction for functions of several complex variables appear

It is shown that if a normed division ring has a norm which is "multiplicati on monotone" in the sense that N(x) 0 for x φ 0, (ii) N(-x) = N(x) for all x, (iii) N(x + y) ^ N(x) + N(y) for all x and y, (iv) N(xy) < N(x)N(y) for all x and y.) Ostrowski's results may be regarded as the treatment of the special case of this problem in which the norm N satisfies the additional condition N(xy) = N(x)N(y) for all x and y. This extra requirement is replaced here by the weaker condition that N be multiplication monotone in the sense that whenever N(x) < N(x') and N(y) < N(y f) then N(xy) ^ N(x'y'). Specifically, it is shown in the corollary of Theorem 3 that if a commutative connected normed ring with unity has a multiplication monotone norm then that ring is (algebraicall y and topologically isomorphic to) a subring of the field of complex numbers. (The version of this statement which appears below actually includes the noncommutative case as well.) The basic device employed in obtaining this result is Theorem 2, which asserts that if a normed division ring has a multiplication monotone norm N such that

A method of implementing two single-bit adders is discussed. These adders can be used individually to realize the conventional functions of serial addition and serial multiplication on a pair of operands, or they can be cascaded to allow the serial addition of three operands for forming the product of complex numbers. In either case, the circuits will detect the occurrence of an overflow or the generation of the number minus one, and they will allow an addition to be rescaled by outputting the correct bits during the additional shifts, whether the addition overflowed or not.

The formulae enabling in a simple way to determine the explicit form of factors of non-equivalent extensions of the point groups by lattice groups and multiplicative group of complex numbers are derived. It is shown that the number of all non-equivalent extensions of the groups C 2 h , D 2 h , D 4 h by lattice groups Γ m , Γ o , Γ q is equal 88, an and the explicit form of the factors of these extensions is determined. Also the explicit form of the factors of all non-equivalent extensions of the groups C nh , D n , D nh by the multiplicative group of complex numbers is derived.

ENGINEERING, and particularly electrical engineering, has become a science whose future progress depends largely on the extensive use of mathematics and physics. In the writer's early experience it was not unusual to hear prominent engineers state that the place for an integral sign is the violin. The use of complex numbers in circuit analysis, the introduction of operational and transform methods, and the application of tensor analysis in electrical engineering have brought to our profession the realization that not only the integral sign, but also other mathematical hieroglyphics, are indeed just as essential to the engineer as they are to the mathematician.

This paper synthesizes the five-link mechanism for the maximum possible number of associated relative positions[1] when the transmission functions of the zero, first and second orders are given[2]. The complex number approach is used together with Newton's method of successive approximations. It is shown that when two driving elements have the same angular speed or when a driving and a driven element have the same angular speed in prescribed positions, the five-link mechanism is equivalent to a four-bar linkage of the type shown in Figs. 2 and 3. The method developed in this paper permits the design of a five-link mechanism for a greater number of associated relative positions than the five possible through the use of the four-bar linkage mechanism. The results are applicable i in the automotive and automation industries.