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Let X(t) be a continuous n×n symmetric matrix function of t for 0≤t≤1, monotone in the sense that X(t)−X(s) is non-negative definite for 1≥t≥s≥0. Denote by [X(t)−X(s)]1/2 the unique non-negative definite square root of X(t)−X(s) for t≥s. Take 0≤t1≤t2≤…≤tN=1 to be a sub-division of [0, 1] and consider the sum $$S_N = \sum\limits_{i = 1}^{N - 1} {\left[ {X(t_{i + 1} ) - X(t_i )} \right]} ^{1/2} F(t_i )\left[ {X(t_{i + 1} ) - X(t_i )} \right]^{1/2} $$ , where F(t) is a given continuous matrix function of t in [0, 1].

1. Throughout we shall denote by X a Banach space and by S(X) the algebra of all linear bounded operators acting in X. N resp. A will be the set of all natural resp. complex numbers. The field of all Borel subsets of A will be denoted by (B. For nEN, ao(E6 we shall write a1/n for the set { X: XCA, Xnea } . For any 0 < r we shall denote Ar ={ I :XEA, r < I xi LEMMA 1.1. Let a, rCG and n, meN be two relatively prime numbers. If we have J arg X-arg sJI < 2wr/mn for any X, ,ukaJUr, XAu FO, then the equality (aym) i/ml) (rn) 1/n = orflr holds.

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

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.

Let P , Q P,Q be compact selfadjoint operators in a Hilbert space. It is proven that the characteristic and associated vectors of the quadratic eigenvalue problem, x = λ P x + ( 1 / λ ) Q x x = \lambda Px + (1/\lambda )Qx , form a Riesz basis for the cartesian product of the closure of the range of P P and the closure of the range of Q Q .

The approach to plane mechanics by vector methods is often considered to be an unnecessarily sophisticated one and it is true that the preliminary work in vector algebra is lengthy and out of keeping with the elegance achieved. The natural tool, it has been said, is the complex number, and I shall obtain some of the fundamental results by using only complex number theory and vector addition.

The general solution to the slightly damped network is expressed in terms of the undamped solution by means of series expansions. The first part of the paper gives a method for evaluating the complex roots of the determinantal equation, and the second part shows how the expansions of the first part may be correlated with the Heaviside formula to form the complete approximate solution. An example illustrates the application to a simple network.

1. When a heat source travels along the surface of a semi-infinite solid with a constant velocity ν, the temperature θ in the semi-infinite solid is given by [numerical formula] where w=x-vt, β=ν/α, and the temperature distribution of the heat source can be expressed as θ(w). 2. Though the eigen-values of this partial differential equation are complex numbers, the solution can be obtained in the form of a real Fourier's integral. Thus the temperature distribution in a semi-infinite solid and heat transmission through the surface are calculated. 3. This Fourier's integral was numerically calculable by a digital computer. 4. It was much easier to solve the above partial differential equation by the accelerated Liebmann's method using a digital computer than to solve it by the Fourier's integral method, and the result obtained by one method coincided with the result obtained by the other.

THERE appeared in your pages some three years ago (vol. xv. p. 515) an article of mine “On the Simplest Continuous Manifoldness of Two Dimensions and of Finite Extent.” In a succeeding number a correspondent (Mr. Monro, of Barnet) propounded a query which may be shortly stated as follows:— “How does it happen that the perpendicular on a geodesic from a point moving along another geodesic changes sign without passing through either the value zero (0) or the value infinity (∝)?” The problem here suggested is a peculiarly knotty one. In the case of the Euclidian plane the perpendicular of course changes sign by passing through the value ∝, while in the case of a spherical surface it is equally obvious that the perpendicular passes through zero, since the two geodesies intersect twice. But what are we to say of the strange hybrid surface which formed the subject-matter of my paper? Your correspondent appeared to insinuate that the problem was insoluble, and that the definition of the surface must therefore involve a logical contradiction. For a while I was greatly puzzled by this unforeseen difficulty, but after a little thought came to the conclusion that the perpendicular changes sign by passing through the value l/2√−1, where l is positive and represents the absolute length of a complete geodesic. In other words, I conceived that the sign of the perpendicular changed from + to − by a continuous variation of the real numbers a and b in the complex number a + b√- 1. I conceived a to diminish continuously till, passing through o, it became − -a, while b at the same time increased with simple harmonic motion from o to a maximum, and then decreased from a maximum to o.

Dirichlet-Taylor series are considered, where is a sequence of complex numbers, is a nondecreasing sequence of positive numbers, and is the number of times ?j occurs in the segment {?1,?,?j}. An adherence principle is established for these series. As applications of this principle, uniqueness theorems are proved for analytic functions which are asymptotically representable by partial sums of Dirichlet-Taylor series in strips. Bibliography: 16 items.

The paper contains 2 sections. In section 1 the spectral theory for rectangular complex matrices, as given by Penrose and Hestenes is reformulated so as to include the spectral theory for normal matrices as a special case. These results, which are mostly well known, are used in section 2 to study matric functions of rectangular complex matrices. These functions are shown in theorem 4 to satisfy an appropriate version of the Fantappie requirements. Theorems 5 and 6 are the analogs of Cauchy's integral theorem and Taylor's theorem respectively. (Author)

A complex number λ will be said to be free if the multiplicative group Fλ generated by the two matricesis a free group, and non-free, otherwise. Very little is known about the distribution of free and non-free numbers [1]. It is, for instance, unknown whether the domaincontains an open set which consists of only free points.

Introduction. The title refers to analytic functions s(x) which behave like nonconstant logarithmic monomials M(x) = cxmo(log x)ml ... (log, x)mr (where c is a complex number 0, the mi are real, and logN is the N-fold iterate of the principal determination of log) in the sense that s(x)/M(x)-*1 as x-* oo in the complex plane. DEFINITION. An analytic function E is said to -3O rapidly enough for M if E- O and (M/M')E'- O as x- oo. Theorem 2 states that if s = M(1 +E) where E-*O rapidly enough for M, then s is 1-1 in some neighborhood of infinity. The neighborhood bases for oo with which we shall be concerned are families F(a, ,B) whose elements are sector-like regions V(ax, ,B, t) defined as follows: Let -r?a<

is called an infinite radical. The numbers aj are the elements of the infinite radical, and the uj are called partial radicals. Throughout this paper a'12 will mean that square root whose real part is positive or zero. If a is a negative real number, a"/2 will be defined to be on the positive imaginary axis. A set which is open and connected is called an open region, and an open region with part or all of its boundary is called a region. A region A is called a convergence region for an infinite radical {u2 n} if the assumption that all the elements aj lie in A is sufficient for the convergence of { u2 } . A region A is called a conditional convergence region if the assumption that all the elements aj lie in A, together with another condition which restricts the rapidity of growth of J a, J,insures the convergence of { u2. It is not difficult to prove that a necessary and sufficient condition for a sequence { Sn} of real numbers to be an infinite radical { un } is that { Sn } be a monotonically nondecreasing sequence of non-negative numbers. However, if {s.} is a sequence of complex numbers, the situation is not so simple. If Pn is the region from which Sn may be chosen, Pn CPn_lx for every n, but each Pn depends on the entire set of numbers si, S2, * * *, Sn-1, and therefore necessary and sufficient conditions are rather involved and not very meaningful geometrically. Vieta was probably the first to conceive of the idea of an infinite radical [1, p. 595]. The famous infinite product

1. Examples have been given [5, p. 185] of functions f(z), analytic in the unit-circle K: j z j

Fur die biquadratische Gleichung erhalt man fur komplexe Wurzeln durch Trennung von Real- und Imaginarteil zwei reelle Gleichungen, deren Resultante die Realteile als Wurzeln hat. Es wird ein praktisches Rechenverfahren angegeben zur iterativen Ermittlung von Naherungswerten der reellen Wurzeln dieser Resultante, bei dem man nur das vollstandige, reelle Horner-Schema der gegebenen Gleichung benotigt. Die zugehorigen Imaginarteile sind einfach auszurechnen. For the biquadratic equation one obtains for complex roots by separation of real and imaginary part two real equations the resultant of which has the real parts as roots. A practical iterative method to is indicated approximate the real roots of this resultant at which only the complete, real Horner schema of the given equation is needed. The correspondend imaginary parts can be calculated in a simple way.

We prove here an unpublished conjecture of Milnor which gives a complete set of multiplicative relations between the numbers e′(ζ) = 1−ζ,where ranges over complex roots of unity. Information of this type is useful in certain areas of topology as well as in number theory.

AMONG the various solutions to quad ratic and cubic equations, some possess an intuitive appeal arising from direct identification with geometric analogs. One analog, discussed here, shows that the quadratic solution known as "completing the square" actually conforms to the pic torial situation of fitting rectangles to gether in the shape of a square figure. A second analog likewise deals with the cubic equation and demonstrates the cor respondence between graphical and analy tical versions of a solution that might be appropriately called "completing the cube." Geometric analogs usually lack the ele gance that abstract symbolism can give to analytic solutions. Examples include the matrix formulation of vectors in theoreti cal mechanics and the complex-number representation of phasors in electrical engineering. Historically, however, pic torial solutions were the first to come within reach of man's intuitive compre hension. One historian speaks of a graphical solution to the cubic equation as early as the eleventh century, which he credits to the celebrated poet Omar Khayyam (Lamb 1936). Analytic solu tions that appeared in the sixteenth cen tury were due to the work of Scipio Ferro and Niccol? Fontana, also known as Tartaglia (Newman 1956, pp. 118-19). The most elegant of all analytic solutions to the cubic equation, due to Niels Abel, appeared only a little over one hundred years ago (Encyclopaedia Britannica 1960, p. 33). The present article begins with a geometric construction for obtaining one solution of a quadratic equation when both roots are real. Progressive generali zations then demonstrate constructions for the second real root, for both real roots simultaneously, and finally for the case of two complex roots. Geometric construc tions for the cubic equation demonstrate (a) the removal of the quadratic term and (b) a means for associating one solu tion with the edge-length of a "completed cube" under certain conditions. These constructions nevertheless serve as a de tailed guide to an analytic solution, free of all restrictive conditions.

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.

for some vector x, where (x, x)1z=1x1+ * ** x,x. It is known that F,(A) is a closed bounded convex region containing the characteristic values X1, * * *, Xn of A and with these points as foci of the algebraic curve bounding F,(A). A summary of results and a bibliography has recently been given by Parker [1]1 who observes that several authors have given bounds for the roots which are actually bounds for F,(A). Since F,(A) is invariant when A is replaced by B = UA U-1 for U unitary but may be altered if B = TA T-1 for arbitrary nonsingular T, information about F,(A) cannot be expected to give precise information on the Xi. Evidently the difficulty is that F,(A) depends not only on the linear mapping described by A but also involves a choice of metric which is unrelated to the linear mapping. If, however, we define for each positive definite hermitian matrix H, a field of values FH(A) consisting of all complex numbers

We first consider the construction of a norm on a direct sum of normed linear spaces and call a norm absolute if it depends only on the norms of the component spaces. Several clharacterizations are given of absolute norms. Absolute norms are then used to construct norms on tensor products of normed linear spaces and on tensor products of operators on normed linear spaces. 1. Introduction. In this paper, we consider the construction of norms on composite linear spaces formed from direct sums and tensor products of normed linear spaces and we consider properties of norms of operators on these spaces. The notion of an absolute norm is introduced as a natural generalization of the relatively familiar idea of an absolute vector norm on the space Cn of ordered n-tuples of complex numbers. Such norms on Cn correspond to the "coordinatewise symmetric" gauge functions as described by Ostrowski (3), and it is shown that our absolute norms on composite spaces correspond in a one-to-one fashion with the absolute vector norms on Cn. We are particularly interested in operator norms for which, in an appropriate sense to be detailed later

A point x in the real or complex space lpis an infinite sequence,(x1, x2, x3,…) of real or complex numbers such that is convergent. Here p ≥ 1 and we writeThe unit sphere S consists of all points x ε lp for which ¶ x ¶ ≤ 1. The sphere of radius a≥ ≤ 0 and centre y is denoted by Sa(y) and consists of all points x ε lp such that ¶ x - y ¶ ≤ a. The sphere Sa(y) is contained in S if and only if ¶ y ¶≤1 - a, and the two spheres Sa(y) and Sa(z) do not overlap if and only if¶ y- z ¶≥ 2aThe statement that a finite or infinite number of spheres Sa (y) of fixed radius a can be packed in S means that each sphere Sa (y) is contained in S and that no two such spheres overlap.

A method is presented for finding the zeros of any nth degree real or complex polynomial as its coefficients (or their parameters) are varied. Several new theorems are presented. The method is based on new theorems, some concerning determinants, and on known theorems. It is applied to characteristic equations of linear physical systems with or without feedback. Root trajectories in the complex plane can be visualized without plotting them from ``root trajectory diagrams'' which show curve families for real or complex roots of a polynomial equation in a plane having two coefficients of the polynomial equation as coordinates. The diagrams can be used in the converse manner to determine the coefficient values corresponding to a desired dominant complex pair or real root.

1. It is trivial thatfor all integers n ≥ 0 implies sn = 0 (n = 0, 1, 2, …). This conclusion is no longer true if it is only known that (1·1) is true for an infinity of n. But we shall show that the truth of (1·1) for, roughly speaking, one-half of all positive integers, together with an order condition on the magnitude of sn, ensures sn = 0 for all n. This follows fromTheorem 1. Let a1 < a2 < … be positive integers, n(R) the number of a's not exceeding R, sn a sequence of complex numbers. If(ii) sn = ο(nK)as n tends to infinitythen sn = P(n), where P(x) is a polynomial of degree less than K.

The condition z, : 1 insures that not all of the numbers Sk vanish. In some instances it is convenient to have inequalities which assert that, in some sense or other, not all of the numbers lSkl are small. P. TURAN [7] has shown that such estimates have significant application in other areas, notably analytic number theory and the numerical solution of algebraic equations. The example z 2 = z a . . . . . z , 0 shows that, in general, the largest of the numbers ISkr need not exceed 1. The problem of finding a positive lower bound for max [Ski was proposed by TURIN [7], and settled in 1961 by F. V. ATKINSON [1], who proved that 1 (1.1) max [ski > ~ l~_k~_n

The (γ, r) summation method, 0 < r < 1, is the "circle method" employed by G. H. Hardy and J. E. Littlewood. It is also known as the Taylor method. Its Lebesgue constants, say L(Tr, n), n = 1, 2, …, were studied by K. Ishiguro [1] in the notation L*(n;1-r). He noted that1where Im{z} denotes the imaginary part of the complex number z, and proved that2Here3

Let Ω be the set of the analytic functions F(z), regular in some neighbourhood of the origin with the expansion There may exist a function F(s, z) arndytic in s and satisfying the following conditions (s and s′ are any complex numbers): and the ƒ k(s) are polynomials in s.

Given a sequence j tf8, • • ••• of complex numbers, we denote by § («„) the set of all limit points of the sequence, i.e., the set of all such that lim = l p —>■ oo for some increasing sequence of positive integers If is an infinite matrix, with complex elements, and xx a sequence of complex numbers, then, formally at any rate, we can define another sequence jyK by oo y k ^ A = 1 The object of this paper is to investigate relations between $ (*A) and $ (jvK). In particular we shall be concerned with the following question. Given a class $ of matrices and a set 9ft of complex numbers, what sets £) (jv*) can be generated by sequences (#A) with *x

If σμ is a reference tetrad at an eventP of space-time, then the bivectors\(V_{\mu \nu } = \sigma _{[\mu } \bar \sigma _{\nu ]} \) span a three-dimensional complex vector space, and any infinitesimal circuit in the neighbourhood ofP can be correlated with a vector dz of this space. If a tetrad is parallely propagated around the circuit, it suffers an infinitesimal Lorentz transformation, which may be described by an antiselfadjoint quaternion dq. In a general space-time, dq is the sum of a term linear and a term antilinear in dz. The necessary and sufficient condition for the space-time to be Einstein is that the antilinear term should vanish, so that dq =kdz, wherek is a complex number. We apply this result to show how, in principle, one can distinguish Einstein from non-Einstein spacetime by means of a simple experiment.

A(x) = Σ x**Pμ(log x) + O {x

In 1849, Cayley and Salmon discovered that a general cubic surface in projective space of three dimensions over the complex numbers has twenty-seven lines on it. They remarked that all the properties of the twenty-seven lines would not become apparent until a better notation than they had given was found. This notation was discovered by Schläfli in 1858 in the double-six theorem (henceforth referred to as given five skew linesa1, …, a5with a single transversal b6such that no four of the ai lie in a regulus, the four ai excluding aj have a second transversal bj and the five lines b1, …, b5thus obtained have a transversal a6—the completing line of the double-six. The other fifteen lines of the cubic surface are then , where ai bj is the plane containing ai and bj.

This problem was originally attacked by Stokes, and the formula obtained by Stokes for the case of steady motion (given at the end of § 1 below) is now generally used in experimental work. However, this formula throws no light on the way in which the velocity of the sphere approaches its steady value: and the only known discussion of this problem was given by Boggio (his results will be found in § 2 below). Unfortunately, when the specific gravity of the sphere exceeds ⅝ (or the constant σ, defined later, is less than 4) the solution so obtained cannot be converted into numerical results, because the constants α, β become complex numbers; and even when α, β are real, the formula is not very convenient for large values of t (as the velocity approaches the steady value).

Leaking-mode dispersion and attenuation is computed for two single-layer models. Roots of the dispersion relation are obtained in the complex wave-number plane, using real frequency as the independent variable. At low frequencies the root loci on the (+, +) sheet determine the fundamental propagating mode plus a series of attenuated standing-wave modes in the vicinity of the source. As frequency increases, the complex roots on all four sheets migrate through the wave-number plane, producing (successively) organ-pipe modes, PL modes, and normal modes. For higher frequencies, the modes exhibit a tendency to couple with pure P- (and S-) type motion in the wave guide. These effects are related to similar phenomena which occur for the case of the free elastic plate. Results are also calculated for a six-layer continental model. Here the pseudo-modes on the (+, −) sheet noted by Cochran et al. (using real wave-number analysis) are found to be decoupled P- and S-related modes whose phase-velocity curves freely intersect one another when frequency is assumed real.

The problem of the influence of nuclear structure on the calculation of conversion coefficients is discussed and a distinction is made between those effects (static) which are essentially calculable without a detailed nuclear model and the effects (dynamic) for which such a model is mandatory. The theory of the conversion process is developed in such a way as to provide formulas wherein both types of effects are exhibited. No attempt is made to provide specific numerical results for the dynamic effects but instead a framework is provided within which the internal conversion predictions of any nuclear model can be readily tested. In order to do this it is necessary to know not merely the conversion coefficient but the matrix elements for the separate final states as complex numbers. This information, which is available at present for only the $K$ shell, is utilized to provide numerical results for the $K$ shell and for several values of transition energy and atomic number. Electric and magnetic ${2}^{L}$-pole transitions with $1l~Ll~5$ are considered.

αφ6 = P(α,6), aQb = Q(a,b), for all a and b in F yield a field (F, 0, 0)? It turns out that the answer is different for infinite fields than for finite fields, as shown in §§2 and 3. Next let R be the field of real numbers. For what quadruples Pu Pz> Qu Qz of real polynomials in four variables is (R x R, 0, 0) a field, when we set (α, 6) 0 (c, d) - (Pτ(a9 b, c, d), P2(a, b, c, d)) , (α, b) 0 (c, d) = (Q^α, 6, c, d), Q2(α, 6, c, d)) , where (x, y) denotes an ordered pair of real numbers? This question is partially answered in §§ 4 and 5, and in § 6 it is shown that the polynomials may be of arbitrarily high degree. In §7 it is proved that if definitions (II) do give a field, it must be isomorphic to the field of complex numbers.

Multiparameter systems with dominant complex roots constrained by degrees of freedom to construct reduced characteristic equation

where c is a scalar matrix. It is well knownf that a relation of this type can not be satisfied by finite matrices. However, in the calculation of commutation formulas for polynomials in p and q no use is made of the fact that c is a scalar but merely that it is commutative with both p and q.% And there do exist pairs of finite matrices x, y of the same order such that xy—yx is not zero and is commutative with both x and y. Such matrices will be called quasi-commutative matrices and either may be said to be quasi-commutative with the other. In a certain sense the algebra of polynomials in a pair of quasi-commutative matrices is homeomorphic to the algebra arising in quantum mechanics. It is hoped to discuss such algebras in some detail in a later paper. In the present paper we shall make a brief study of quasi-commutative matrices whose elements belong to the complex number field. The concept of quasi-commutativity is an extension or generalization of commutativity, and as would be expected, some of the results obtained are generalizations of known theorems concerning commutative matrices. The problem of determining quasi-commutative matrices is that of finding matrices x, y, z (^0) which satisfy the equations

The mathematician Thomas Harriot has hitherto been evaluated almost exclusively from his Artis analyticœ praxis, which in 1631 was published posthumously from an imperfect manuscript. In this praxis algebraic equations and their solutions are discussed, but negative and complex roots are almost entirely neglected. However, Harriot's manuscripts (Add 6782–9 Br. Mus. and H.M.C. 240 and 241, Petworth House) reveal that Harriot in his own working papers listed negative roots and in many cases also complex roots. Also in other respects Harriot had a better understanding of algebraic equations than appears from the praxis. According to its title page, this work was composed as a textbook for the instruction of the Earl of Northumberland. Harriot may not have thought it advisable to introduce in such a book numbers which were more difficult to visualize than natural numbers, fractions, and positive roots.

Let $\{X_t, t \geq 1\}$ be a stochastic process which satisfies the following set of assumptions: ASSUMPTION 1: For every $t, X_t$ satisfies \begin{equation*}\tag{1}X_t = \alpha_1X_{t - 1} + \alpha_2X_{t - 2} + \cdots + \alpha_kX_{t - k} + u_t,\end{equation*} where $\alpha_1, \cdots, \alpha_k$ are $k$ finite real numbers (unknown parameters) and $u_t, t$ positive, are independent, identically distributed random variables with mean zero and a finite positive variance $\sigma^2$. ASSUMPTION 2: The distribution of $u_t$ is continuous. (Actually $\mathrm{Pr}\{u_t = 0\} = 0$ suffices.) ASSUMPTION 3: The roots $m_1, \cdots, m_k$ of the characteristic equation \begin{equation*}\tag{2}m^k - \alpha_1m^{k - 1} - \alpha_2m^{k - 2} - \cdots - \alpha_k = 0,\end{equation*} of (1), are distinct. ASSUMPTION 4: There is a unique root $\rho$ of (2) such that $|\rho| > 1$, and $|\rho| > \max_{j = 2, \cdots, k} |m_j|$. Here $\rho$ is identified with $m_1$ for convenience. Since complex roots enter in pairs, it follows from this assumption that $\rho$ is real. Note that there can be $m_j, j > 1$, such that $|m_j| > 1$. ASSUMPTION 5: For $t$ non-positive, $u_t = 0$. If Assumption 4 holds, the process $\{X_t, t \geqq 1\}$ is said to be (strongly) explosive, and the corresponding difference equation (1) is called an explosive (linear homogeneous) stochastic difference equation; this is the subject of the present paper. Under the assumptions above, it follows (cf., C. Jordan [5], p. 564, Mann and Wald [8], p. 178, and also the footnote on p. 22 of [10]) that $X_t = \sum^t_{r = 1}\sum^k_{q = 1} \lambda_qm^{t - r}_qu_r,$ $t$ positive, and that $\lambda_q$ satisfy the relations \begin{equation*}\tag{3}\delta_{1t} = \sum^k_{q = 1} \lambda_qm^{t - 1}_q,\quad t = 1, 0, -1, \cdots, - (k - 2),\end{equation*} where $\delta_{1t} = 1$ if $t = 1$ and 0 otherwise. (Note that $\sum^k_{q = 1}\lambda_q = 1$.) For convenience, define the random variables \begin{equation*}\tag{4}X_{i,t} = \sum^t_{r = 1} m^{t - r}_iu_r,\quad i = 1, 2, \cdots, k, (m_1 = \rho),\end{equation*} so that $X_{i,t} = 0$ for $t$ non-positive. Thus one may write $X_t$ as follows: \begin{equation*}\tag{5}X_t = \lambda_1X_{1,t} + \lambda_2X_{2,t} + \cdots + \lambda_kX_{k,t}.\end{equation*} The first part of this paper is devoted to finding a consistent estimator of $\rho$ and its limit distribution. Consequently, in Section 3 some lemmas will be proved for use in the consistency proof (Theorem I). Similarly, in Section 5, some lemmas leading to the proof of the limit distribution of the estimator (Theorem II) will be given. In the second part, the consistency of the Least Squares (L.S.) or Maximum Likelihood (M.L.) estimators of the "structural parameters" $\alpha_i$ of (1) will be considered (Theorem III). The procedure becomes much more involved because the direct application of the usual limit theorems is not possible, since the process under consideration is explosive. It is noteworthy that Lemmas 9, 10, 14-16, and Theorem I are rather general, in that they hold under the only global Assumptions 1-5 above, and the further requirement $|m_j| < 1, j = 2, \cdots, k$, so essential for the rest of the analysis of this paper, is unnecessary for them. The corresponding problem, in the case $|\rho| < 1$, has been completely solved by Mann and Wald [8]. If $k = 1$ in (1), the results of this paper reduce to those obtained by Rubin [13], White [14], and T. W. Anderson [1]. The vector case has also been treated by Anderson in [1], but a comparison of the results in this case with those of the present paper shows that they do not imply each other except in the first order. In the latter case, however, both reduce to Rubin's [13] result. The available results on stochastic difference equations are summarized in a table at the end of the paper. Some of the details and computations omitted in this paper may be found in [10]. In the following section, some known lemmas related to stochastic convergence are collected and stated in a convenient form, as they will be constantly referred to in both parts of the paper. (For proofs, see [2], [3], [4], [6] and [9].)

The necessary procedure involved in the transition from the Laplace transformation to the solution of linear differential equations is summarized, and a particular form of partial-fraction expansion which may be advantageous in special cases is noted. The remainder theorem with regard to algebraic polynomials is restated, and it is shown how this theorem may be applied to the evaluation of high-degree polynomials for real and complex numbers. A numerical example is treated to illustrate application to a typical transfer function. Finally, the method is shown to be useful in the evaluation of the frequency response of such a transfer function.

Let A be a nonzero matrix of complex numbers and let I be the corresponding identity. The index of A (the least nonnegative integer k such that rank $A^k = {\operatorname{rank}} A^{k + 1}$ is related to the least integer m such that $\lim _{z \to 0} z^{m - 1} ( {A + zI} )^{ - 1} A$ exists (z being a complex variable). It is shown that the two are equal if the index of A is at least two; the latter is one if the index of A is zero or one. It is also shown that if b is a nonzero $n \times 1$ matrix of complex numbers and the index of A equals $\nu $, then b is in the range of $A^\nu $ if and only if $\lim _{z \to 0} ( {A + zI} )^{ - 1} b$ exists.

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.