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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.

T. N. E. Greville has shown that the equations $BB^ + A^ * AB = A^ * AB$ and $A^ + ABB^ * A^ * = BB^ * A^ * $ are necessary and sufficient in order that the reverse order law $(AB)^ + = B^ + A^ + $ hold for pseudoinverses of matrices over the field of complex numbers. E. Arghiriade’s results give the more concise formulation that $(AB)^ + = B^ + A^ + $ if and only if $A^ * ABB^ * $ is an $EP_r $ matrix. In this paper, we analyze modifications of the reverse order law which are obtained by replacing one of the pseudo-inverses in the reverse order law by either a normalized generalized inverse or a weak generalized inverse. The results obtained are related to those of Greville.

The theory of classification of homogeneous bounded domains in the complex number space Cn has been developed mainly in the recent papers [10], [6], [3] and [7]. As a result, the classification is reduced to that of S-algebras due to Takeuchi [7] which correspond to irreducible Siegel domains of type I or type II (For the definition of irreducibility see § 1). On the other hand Pjateckii-Sapiro [5] found large classes of homogeneous Siegel domains obtained from classical self-dual cones. Even in lower-dimensional cases, however, there are still homogeneous Siegel domains which do not appear in his results.

Let F and G be $n \times n$ and $n \times m$ matrices of complex numbers, respectively, and let I be the $n \times n$ identity matrix. For any complex number $\lambda $, the index of $( {\lambda I - F} )$ is the least nonnegative integer p such that $\operatorname{rank} ( {\lambda I - F} )^p = \operatorname{rank} ( {\lambda I - F} )^{p + 1} $. By associating the linear differential system $\mathop x\limits^ \circ = Fx + Gu$ with this pair of matrices, controllability results can be used to obtain conditions involving the index of $( {\lambda I - F} )$ which guarantee the existence of limits such as $\lim _{s \to \lambda } ( {sI - F} )^{ - 1} ,s$ being a complex variable. An expression for the value of such limits is given.

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.

Lets n be a given sequence, $$U = \left[ {\mu _{i,j} } \right]_{i,j = 0}^\infty $$ an infinite array of complex numbers $$\mu _{i,j} = 0,i< j,\sum\limits_{k = 0}^i {\mu _{i,k} = 1} $$ and $$s_{n,m} = \sum\limits_{k = 0}^m {\mu _{m,k} s_{n + k} } $$ . We develop and discuss linear sequence transformations defined by $$s_{n,m + 1} = a_m s_{n + 1,m} + b_m s_{n,m} $$ or $$s_{n,m + 1} = a_m s_{n,m} + b_m s_{n + 1,m} + c_m s_{n,m - 1} ,$$ n, m=0, 1, 2,... We ask, ifs n ?? asn??, doess n,m ?? asm??? This convergence and the rapidity of it are seen to depend on the location of the zeros of the polynomial $$P_m \left( \lambda \right) = \sum\limits_{k = 0}^m {\mu _{m,k} \lambda ^k } $$ . When theP m are chosen to be certain polynomials encountered in special function theory, such as Bessel polynomials or Legendre polynomials, the result is an elegant and powerful set of algorithms for improving the convergence of large classes of sequences.

Let the sequences {λn} and {αn} of complex numbers satisfy the conditions: 1) sup ¦Im λn¦=h

Let ƒ(z) be a meromorphic function of finite non-zero orderρ. Assuming certain growth estimates on ƒ by comparing it withr p L (r) whereL (r) is a slowly changing function we have obtained the bounds on the zeros of ψ (z) —w where ψ (z) is a differential polynomial. These bounds are satisfied except for special values ofw. Examples are given to show that the exceptional values exist.

Let B be a closed dissipative operator in a Hilbert space H with an arbitrary domain of definition. We will give a description of all closed (and, in particular, closed maximal) dissipative extensions\(\tilde B\) of B in terms of extensions\(\tilde W\) of a nonexpanding operator W associated with B. We construct a family {Bz} of maximal closed dissipative extensions of B, where z is a complex number in the lower half-plane. We present an example which illustrates the above concepts.

A complete solution of the neutron slowing-down problem is developed for isotropic elastic scattering in an infinite homogeneous medium. An exact formula for obtaining all the complex roots of the transcendental equation is given. This, together with the exact nontrivial real root determined elsewhere, constitutes the exact solution of the problem. A general misconception about the application of these roots in the first collision interval is pointed out. Numerical results obtained for different magnitudes of absorption rates show agreement with more conventional solutions, like that of Teichmann. The number of terms in the infinite sum necessary for convergence depends on the relative magnitude of lethargy and the pre-assigned permissible error. Good results are obtained with a modest number of terms for small values of relative lethargy, but only with a few for larger values of u/.

We define a class of operations on formal power series that includes practically all operations of interest. The class is closed under composition. We prove that an operation in this class vanishes for complex numbers if and only if it vanishes for functions analytic at the origin. A variety of applications of this “lifting theorem” are discussed.

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.

This paper studies the linear programming problem in which all coefficients (even those of the stipulations matrix) are rational functions of a single parameter t called “time,” and provides an algorithm that can solve problems of the following two types: (1) Steady-state behavior [the algorithm can be used to determine the functional form x(t) of the optimal solution as a function of t, this form being valid for all “sufficiently large” values of t], and (2) sensitivity analysis [if a value t0 of “time” is given, the algorithm can be used to determine the two possible functional forms of the optimal solution for all values of t “sufficiently close” to t0 (the first functional form valid for t < t0, the second for t < t0)]. In addition, the paper gives certain qualitative information regarding steady-state behavior, including the following result: If for some one of the properties of consistency, boundedness, or bounded constraint set, there exists a sequence tn ↗ +∞ such that the linear program at n has this property for all n, then the program has this property for all “sufficiently large” values of t.

We present specific polynomials in $\mathbb{C}[x]$ with algebraic or rational coefficients which are hard to compute (even though arbitrary complex numbers are allowed as inputs for the computation). Examples are: $\sum_{\delta = 0}^d e^{2\pi i/2^\delta } x^\delta $, $\sum_{\delta = 0}^d 2^{2^\delta } x^\delta $. We also show that the minimum number of arithmetic operations to compute polynomials in $\mathbb{C}[x]$ is itself computable. Finally, we study computational complexity in finite-dimensional algebras over an algebraically closed field.

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.

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) .

In this paper the simultaneous uncertainties of the generalized cosine and sine operators are studied. We obtain states which minimize (ΔC)2 (ΔS)2/ 2; these states are characterized by a complex number λ and form an over-complete set within an ellipse in the λ-plane.

A very important example of almost strongly minimal theories are the algebraically closed fields. A. Macintyre has shown [3] that every ω1-categorical field is algebraically closed. Therefore every ω1-categorical field is almost strongly minimal. It will be shown that not every ω1-categorical ring is almost strongly minimal.Let R0 be the factor ring C[y/(y2), where C[y] is the ring of polynomials in the indeterminate y over the field of complex numbers and (y2) the ideal generated by y2 in C[y].It is straightforward to prove that R0 has the following properties:1. R0 is a commutative ring with identity.2. R0 is of characteristic 0.3. For every polynomial p(x) = ∑ a1x1 ∈ R0[x] with of ai2 ≠ 0 for some i > 0 there is an a ∈ R0 such that p(a) · p(a) = 0.4. For all x, y ∈ R0 such that x2 = 0 and y ≠ 0 there exists a z ∈ R0 with y · z = x.5. There is an x ≠ 0 such that x2 = 0.These properties can be ∀∃-axiomatised in a countable first order logic (see [4]). Let T be the set of these sentences. With Theorem 7 we get that T is model-complete.If R is a model of T then I shall denote {a ∈ R ∣ a2 = 0}.

Let , where pk are complex numbers, have 0 < ρ ≤ ∞ for radius of convergence and assume that P(x) ≠ 0 for α ≤ x < ρ (α < ρ is some real constant). Assuming that is convergent for all (x ∈ [0, ρ), we define the P-limit of the sequence s = {sk} byThis, so called, power method of limitation (see (3), Definition 9 and (1) Definition 6) will be denoted by P. The best known power methods are Abel's (P(x) = 1/(1 – x), α = 0, ρ = 1) and Borel's (P(x) = ex, α = 0, ρ = ∞). By Cp we denote the set of all sequences, P-limitable to a finite limit and by the set of all sequences, P-limitable to zero.

In this paper three theorems dealing with the properties of replaceablel−l matrix methods of summation are proved. A typical result is given below: Theorem. LetA be anl−l matrix method of summation. IfA is replaceable by an absolutely regular matrix methodB, then there exists anl−l matrix methodD depending on some sequencet satisfying the propertyP such thatl A⊆lD and thatD(x)=0 for allx∈l A. Here, the sequencet={t n} is said to satisfy the propertyP ift∈m and if $$\sum\limits_{k = 1}^\infty {} \sum\limits_{k = 1}^\infty {t_n } a_{nk} x_k $$ converges for any givenl−l methodA and every sequencex={x k}∈lA.

A result of Singh and Sreenivasulu is proved under less restrictive hypothesis. It is also shown that if a condition on the coefficients is not satisfied, the theorem will not hold.

Complex analysis is used to derive exact solutions of the transcendental equation $\alpha \sin \zeta = \zeta $, where $\alpha $is an arbitrarily assigned complex number. The method involves the use of canonical solutions to suitably posed Riemann problems, and the explicit results are expressed in terms of elementary quadratures.

A recurrence relation is presented which connects the general (n' not=n) hydrogenic radial matrix element of rbeta , where beta is a complex number to matrix elements involving exponents beta +2, beta -1, and beta -2 but the same sets of quantum numbers. This identity is found solely from the boundary conditions on the wavefunctions and the radial differential equation and should be of use in simplifying rk forms given elsewhere. For n'=n, the recurrence relation reveals a proportionality between the - beta -2, beta -1 matrix elements that includes the Pasternack-Sternheimer identity as a special case. Further results may be obtained by beta -differentiation, and new identities for matrix elements of rk log (r) are given. The n'=n matrix elements of rk are studied for k>or=-1; closed-product expressions are found for k-values near the sum and difference of the angular momenta, so that simple prescriptions can be given for the iterative calculation of n-dependent explicit forms. It is suggested that complex-variable theory be applied to a study of these results.

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.

The convergence field of a Toeplitz matrix is a monotonic function of the set of rows that compose the matix, in the sense that the deletion of some of the rows of the matrix (followed by appropriate renumbering of the rows that remain) can never decrease the convergence field. In the case of certain matrices, the deletion of infinitely many rows always increases the convergence field; but there exist matrices that do not have this property. This dichotomy was considered with special reference to the space of bounded sequences and certain classical families of matrices.

The object of this article is to show how the problems of transformation geometry, and in particular the determination of the composition of two or more successive transformations, can be solved far more easily by Argand diagram methods leading to algebraic equations involving complex numbers, rather than by purely geometric considerations. It must be conceded, of course, that the methods advocated here cannot be extended to three dimensions.

An alternative method for deriving the generalized inverse of an arbitrary rectangular matrix with real or complex coefficients is described. The method applies a classical minimization procedure without iteration processes and is readily applicable in digital computer programming.

Letf (z) be an entire function λn(n=0,1,2,...) complex numbers, such that the system f(λn n=0 ∞ is not complete in the circle ¦z¦

In the electromagnetic theory of harmonic waves in linear systems, the various field quantities are often represented by complex numbers which contain the time t through a factor eiωt. Then the electric and magnetic field vectors are complex vectors, and in general may be thought of as having complex lengths and directions. If the instantaneous field associated with a complex vector E is drawn as a real vector with a fixed origin, its end point traverses an ellipse whose features can be expressed in terms of the scalar and vector products of E with itself or with its complex conjugate. Some applications of this to the design of aerials for space vehicles, and to the relations between the electric and magnetic fields in magnetoionic theory are discussed.