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Using approximation by a micropolar continuum, the extensional buckling of a multi-story, multibay rectangular frame with rectangular boundary is solved by formulating the problem in terms of a system of six linear ordinary differential equations. The buckling loads must be computed by a trial-and-error procedure because all coefficients of the 6 × 6 determinant to vanish depend on explicitly inexpressible complex roots and eigenvectors of another 6 × 6 determinant which in turn depends nonlinearly on the initial stress. It is shown that for tall frames consideration of long-wave buckling loads is important. Also, the continuum approximation is found to give very accurate results, as compared with the exact solutions of large frames.

We give a variation of a theorem of Gelfond. One of the corollaries is Schneider's conjecture that either exp( e ) or exp( e 2 ) is transcendental. It also follows that at least one of the numbers exp(α), exp( α 2 ), exp( α 3 ) is transcendental for any nonzero complex number α.

In a study [6] of the equivalence of forms, Jordan derived the finiteness of the orthogonal group of a form defined over the complex number field whose degree is at least three and whose discriminant is not zero. Herein we demonstrate the validity of Jordan’s result for any field, with the usual restriction that the characteristic of the field, if not zero, dominate the degree of the form. Jordan’s proof rests on an ingenious inequality argument applied to the coefficients of the form. Our argument begins with a proposition of Bott and Tate [3] which focuses attention on unipotent isometries. The proof provided here (due to Harrison [Sj) is regardless of the characteristic of the underlying field, as is the subsequent elimination of unipotent isometries. The argument is completed by showing the group under consideration to be periodic of bounded period with exponent relatively prime to the characteristic of the field, if it is nonzero. Finiteness then follows from a theorem of Burnside. Rather than a theorem of forms we adopt throughout the equivalent setting of symmetric multilinear maps on a vector space in the hope of achieving greater clarity of exposition and with the risk of obscuring the elementary nature of the argument. Let 0 denote a symmetric multilinear map of degree Y 2 3 on a vector space V of dimension n over a field F whose characteristic p, if nonzero, is greater than r. Let 0 = o(8) denote the orthogonal group of 0 in GL(n,F). Employing the standard extension of the scalar field to its algebraic closure, we may assume that F is algebraically closed. Recall that B is singular if and only if for some 0 # z E V, 0(X,..., z, v) = 0 for all 0 E V.

Wolter's multilayer reflection and transmission formulae are extended for the case, that the multilayer system is composed of orthorhombic media. These are oriented in such a way, that two of their principal dielectric axes lie in the surface of a layer. The third axis forms with a second one the plane of incidence. An analysis of the complex roots in the formulae, always present when dealing with absorbing media, is given to choose the correct solutions. The extended formulae are well suited for a computer program. They allow e.g. the computation of Attenuated Total Reflection (ATR)-spectra of anisotropic crystals.

A Dirichlet series associated with a positive definite form of degree δ in n variables is defined by D F (s,p,α) = ∑ α∈Z n −{0} F(α) −s e(ρF(α)+〈α, α〉) where ϱ ∈ Q , α ∈ Q n, 〈x, y〉 = x1y1 + ⋯ + xnyn, e(a) = exp (2πia) for a ∈ R , and s = σ + ti is a complex number. The author proves that: (1) DF(s, ϱ, α) has analytic continuation into the whole s-plane, (2) DF(s, ϱ, α), ϱ ≠ 0, is a meromorphic function with at most a simple pole at s = n δ . The residue at s = n δ is given explicitly. (3) ϱ = 0, α ∉ Z n, DF(s, 0, α) is analytic for α>, n (δ − 1) .

A mechanical synthesizer with thirty harmonic elements (fifteen sine components and fifteen cosine components) may be used to graph a polynomial in a complex plane. The sum of the sine components is recorded by a tracing point which moves vertically. The sum of the cosine components is recorded by horizontal motion of a drawing board. Expansion of the polynomial by De Moivre's theorem expresses the function as a sum of sine terms and a sum of cosine terms. The polynomial may then be graphed by the machine, and thereby the complex roots determined. A number of typical curves are given to illustrate the machine method of determining real, imaginary, and complex roots. An auxiliary method is described which gives all the real roots of the polynomial from a single graph.

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 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.

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.

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.

The difficulty in using the classical general formulae of Appleton , for determining the refractive and absorption indices and the polarization of each of the two wave-modes, is primarily due to the presence of the radical term ρ = √γT4δ2 + γL2 where δ is the reciprocal of a given complex number. This difficulty has been overcome by means of each of four methods of calculation developed in the School of Physics of the University of Sydney. The first three, although previously published, are here presented again but briefly. The fourth method makes use of the following approximations: √1 + s2 equals 4 + 3s 2 4 + s 2 or s· 4s 2 + 3 4s 2 + 1 according as ¦s¦ ≶ 1 (except when s 2 ≑ -1) . On applying them to Appleton's formulae, by taking s = 2γLγT−2(1 + α + iβ), we obtain the good approximate formulae (32) and (33) for M2 (square of the complex refractive index) which, in effect, cover all possible situations with an error not exceeding about 1 per cent. Other applications of these approximations to √ + s2 can be made: for example to the polarization numbers R, for, with the same value of s, R = is−1{1 ∓ √1 + s2}.

In this paper a generalized inner product 〈 x | y 〉 is defined as a binary function with complex values which satisfies the following: (i) for any nonzero vector y and any complex number ζ there exists a vector x such that 〈 x | y | = ζ , ( ii ) 〈 x 1 + x 2 | y 1 + y 2 〉 = 〈 x 1 | y 1 〉 + 〈 x 2 | y 1 〉 + 〈 x 1 | y 2 〉 + 〈 x 2 | y 2 〉, ( iii ) 〈 y | x 〉 = f [〈 x | y 〉], where f is a continuous function and, (iv) 〈 x | μy 〉 = g [ μ , 〈 x | y 〉], where g is a continuous function. These conditions induce several functional equations which are then solved. By making a linear combination of 〈 x | y 〉 with its complex conjugate a new function ( x | y ) is obtained which is either symmetric, antisymmetric, or Hermitian. The functions 〈 x | y 〉 and ( x | y ) have the same orthogonal vectors.

THE numerical inversion of the Laplace transform and other problems of applied analysis Involve a study of the nature of the distribution of the zeros of the exponential polynomial ʋ(s) = ∑ i =0 m ∑ j =0 n a i j s j exp (−ω i s) , where α ij are complex numbers, 0 = / GW 0 GW 1 GW m . If the indices / GW i are essential in (1), i.e. v ( s ) does not degenerate into a polynomial, then v ( s ) has an Infinite set of zeros, the unique limit point of which is infinity. If a 0 n ≠ 0 all the zeros of (1) lie in the s-plane to the left of some straight line Re( s ) = h [1]. This line can be found from a direct estimate. We write v ( s ) in the form ʋ(s) = s N A 0 n + ∑ j = 0 n −1 a 0 j s j−n + ∑ i=1 m ∑ j=1 n a 0 j s j−n exp (−ω i s) . Now, if h is sufficiently large, on any straight line Re( s ) = x ≥ h we have |ʋ (s) | ≶ |s| n |a 0 n | − ∑ j=0 n−1 |a oj |H j−j − ∑ i=1 m ∑ j=0 n |a ij |h j−h exp (−ω i h) > 0 . In the majority of cases this method gives a rough estimate for h . Other possible ways of finding h are of interest, and we give one of these below. We consider the cases of commensurate and incommensurate Indices / GW i ( i = 1, …, m ) separately.

Sinusoidal currents in time, arising from impressed sinusoidal voltages in time, need no explanation; but circular and hyperbolic functions of unobvious circular and hyperbolic space angles arising apparently spontaneously in electrical circuits call for some explanation. Accordingly, the concept of an angle, adequate both for circular and hyperbolic functions, is first developed geometrically, pointing out that an angle is nothing but the percentage value (more precisely, percentage/100 or per-unit value) of an arc recognized as the vector increment of the radius vector. Naturally, the radius vector itself is used as the per-unit base, hence the name radians (meaning, radii) for this kind of measure numbers. The idea naturally leads to the compound-interest like calculation of the per-unit value of a gross change in the radius vector. The vector concept recognizes as angles those incremental arcs that are parallel to the radius vector just as well as those that are normal. The former are called hyperbolic angles, the latter circular. Thus an arc that represents a circular angle and one that represents a hyperbolic angle are j with respect to each other. Obviously, in the first case, only the direction of the radius vector changes; in the latter, only its magnitude. Applied to a physical variable, this concept of an angle means that a change in the variable expressed in per-unit 2 is the angular value of that change in radians. This concept of angles leads to logarithms and exponentials, yielding a simple interpretation of the logarithms of negative, imaginary and complex numbers so that, in its light, by inspection log (− 1) is recognized as πj, log j is recognized as ( π 2 )j , etc. The cosine of an angle θ (whether θ is circular, or hyperbolic, or mixed) is interpreted as the resultant of two variables, initially alike, but one having undergone the vector per-unit change (+ θ), and the other (− θ). A sine is interpreted as the resultant of two variables, initially equal and opposite, and one having undergone the vector per unit change (+ θ), the other (− θ). The origin of such pairing of vectors is found in reflection. In transmission lines, discontinuities like open circuits and short circuits act as perfect reflectors, and provide reflected variables which match the original variable in the indicated manner to yield with the original a resultant corresponding to the sine or cosine (or sinh or cosh) of the per-unit change involved.

When the polarization states of light are represented by complex numbers, the chart formed by the two orthogonal families of circles representing the equi-azimuth and equiellipticity (EA-EE) contours in the complex plane provides the medium for convenient graphical solutions of problems in ellipsometry. Thus, in conventional ellipsometry, the specimen ψ and Δ are found to be easily related to any set of nulling angles P, C and A via this chart. In generalized ellipsometry, the coefficients of the polarization transfer function of an optical system (or the reflection coefficients of an anomalous surface) are determined by the circle locus of the incident polarization states, χ , which are mapped into linear vibrations, ξ , at the output of the system. The EA-EE chart provides, again, a direct method to locate the points χ , determine the circle and hence the polarization transfer function.

The parametric (or impact parameter) method has been used to derive the equations describing proton-hydrogen collisions in the energy range of 0.5–50 keV. Only the 1s, 2s, 2p0, 2p±1 states of the target and the (neutralised) projectile have been accounted for. First the Schroedinger equation is replaced by a set of 10 simultaneous differential equations of the first order in the time. The coefficients form a (complex) matrix and are functions of the time; they are parametrically dependent on the impact parameter and the impact velocity. The evaluations of these coefficients are considered for nearadiabatic as well as for near-impulsive treatments. In case these coefficients combine electrons states defined on different nuclei, a (single) numerical quadrature finally remains; often complex arithmetic is required. The differential equations possess several properties which may serve as criteria to test the correctness of the solution. They are: the normalisation of the probability amplitudes (the conservation of probability), the invariance with respect to an initial phase factor (connected with the irrelevance of the internuclear potential energy), the invariance with respect to the sign of the magnetic quantum number and the symmetry because of the identity of the nuclei. The two last properties are also used to reduce the number of equations.

Let A 0 , A 1 be n × n matrices of complex numbers and let E n be the vector space of n × 1 matrices of complex numbers. Let N 1 = \s{ x ∈ E n ¦ A 1 x = 0\s}, N 0,-1 = \s{0\s} ⊂ E n , and for k ⩾− 1 define R 1 k = A 1 N 0 k and N 0 k +1 ={ x ∈ E n ∣ A 0 x ∈ R 1 k }. In any case μ = min{ k ⩾ − 1 ∣ N 0, k +1 = N 0, k exists and μ ⩾ 0 or μ = − 1 according as A 0 is singular or not. The main result presented is the following: There exists δ> 0 such that the matrix A 0 + zA 1 is invertible for all complex numbers z such that 0 z ¦ N 1 ⋂ N 0 k = \s{0\S} for all ⩾ 0. Moreover, if this condition holds, then there exist n × n matrices Q k such that , the series converging for 0 z ¦ 0, and Q -μ-1 ≠ 0.

If H is a subgroup of the symmetric group of degree n and χ is a complex character on H of degree 1, then the Schur function for H and χ is defined by d χ H (Y) = ∑ σϵH χ(σ) ∏ i=1 n y iσ(i) for any n -square matrix Y = ( y ij ). It is shown that, if A 1 is a positive definite matrix, A 2 a positive semidefinite nonzero matrix, and μ 1 , μ 2 complex numbers, then |d χ H (μ 1 A 1 + μ 2 A 2 )| ⩽ d χ H (|μ 1 |A 1 + |μ 2 |A 2 ) .Other inequalities relating functions of μ 1 A 1 + μ 2 A 2 and of | μ 1 | A 1 + | μ 2 | A 2 are also obtained.

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.

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.

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.

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.

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.

Let x = (x1, xs , 0-e) x,) be an r-dimensional vector, where q are real or complex numbers, and let 9 be the domain defined by I xj I < l (j = 1,2, *.., r). (l-1) Consider a family 3 of functions of x which are defined on 9, and consider an n by TZ matrix A(x) whose elements belong to this family 9. Let p(h; x) be the characteristic polynomial of A(x): p(h; x) = det (A(x) Al,), (1.2) where 1, is the TZ by TZ identity matrix, h is a complex variable, and det means the determinant. We assume that the polynomial p(X; X) is factored as PC& 4 = P&t x) PA& 4 for every x in 9, (1.3) where

The study of the algebraic theory of characters of commutative semigroups was initiated by Schwarz [8, 9, IO] and Hewitt and Zuckerman [#, 51. The theory has been further developed and expanded by Clifford and Preston [I] and several other authors [3,6, 7, II]. A character of a semigroup S is simply a homomorphism of S into the multiplicative semigroup of complex numbers. The collection of all characters of a semigroup S is a semigroup with respect to pointwise multiplication. We denote the semigroup of characters of S by V(S) and we define

A problem that confronts the engineer desiring to use time optimal discontinuous control in a system is the determination of the loci of points in state space at which the steering functions should be switched to the opposite direction. The equations which define these loci are required for synthesis of the system controller. This paper is an exposition of an approach of R. Oldenburger and a solution of R. Bellman applied to seven examples to illustrate a method for finding these loci and to indicate the problems involved. The method works without complication for normal systems described by equations with real roots. By the approach of LaSalle, the method is extended to cover normal systems described by equations with complex roots. A final section shows how non-symmetry of steering functions can be treated in practice. The steering functions for the non-symmetric case are treated by replacing the non-symmetric steering function by the sum of a symmetric steering function and a constant which describes the degree of asymmetry.

Let T(n) denote the number of arithmetic operations needed to evaluate the normalized derivatives P ^ ( t ) / i ! , i = 0,...,n, for an nth degree polynomial p(t) over the field of complex numbers. We show 1 2 T(n) ^ — c n log n + lower order terms where c may be taken as 12.

This note shows that the multiplicative group of non-zero complex numbers and the multiplicative group of complex numbers of absolute value 1 are isomorphic as groups.