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In what follows, a polycyclic series, for the group G, is any finite seriesG = G0 ≧ G1 ≧ . . . ≧ Gl = 1of subgroups of G, such that Gi+1 ⊲ Gi and Gi/Gi+1 is cyclic, for all i = 0, . . ., l — 1. A group that has a polycyclic series is called a polycyclic group, and if G is a polycyclic group, then the polycyclic length of G, which we denote by ρ(G), is the number of non-trivial factors of a polycyclic series for G of shortest length.

In this paper, we show that if Kn∗ denotes the symmetric oriented complete graph on n vertices, with n ≠ 6, then we can partition the arcs of Kn∗ into oriented 3-circuits if and only if n(n−1) ≡ 0 (mod 3). We use the solution of Kirkman's schoolgirl problem given by Ray-Chaudhuri and Wilson [1].

The accurate space derivative (ASD) method for the numerical treatment of nonlinear partial differential equations has been applied to the solution of Fisher's equation, a nonlinear diffusion equation describing the rate of advance of a new advantageous gene, and which is also related to certain water waves and plasma shock waves described by the Korteweg-de-Vries-Burgers equation. The numerical experiments performed indicate how from a variety of initial conditions, (including a step function, and a wave with local perturbation) the concentration of advantageous gene evolves into the travelling wave of minimal speed. For an initial superspeed wave this evolution depends on the cutting off of the right-hand tail of the wave, which is physically plausible; this condition is necessary for the convergence of the ASD method. Detailed comparisons with an analytic solution for the travelling waves illustrate the striking accuracy of the ASD method for other than very small values of the concentration. NONLINEAR DIFFUSION; EPIDEMIC WAVES; NUMERICAL SIMULATION

Galton-Watson processes where the reproduction of individuals in different generations can have different distributions retain many characteristic features of classical processes.

In this article, we are concerned with the following inequality(1)where 0

Let {X N (t)} be a sequence of continuous time Markov population processes on an n-dimensional integer lattice, such that X N has initial state Nx(0) and has a finite number of possible transitions J from any state X: let the transition X → X + J have rate NgJ (N –1 X), and let gJ (x) and x (0) be fixed as N varies. The rate of convergence of √N(N –1 X N (t) — ζ (t)) to a Gaussian diffusion is investigated, where ζ(t) is the deterministic approximation to N –1 XN (t), and a method of deriving higher order asymptotic expansions for its distribution is justified. The methods are applied to two birth and death processes, and to the closed stochastic epidemic.

If X(t) is a Wiener process with EX(t) = vt and var X(t) = σ 2 t (where v > 0) and if M(t) = max0≦τ≦t X(τ) and Ta is the time of first passage through level a (where a > 0) we show that where is the coefficient of variation of Ta . Applications of the result to the “maturity-time” representation of cell-growth and to queues with heavy traffic are discussed.

OFTEN, more than one probability distribution is theoretically feasible when considering statistical models for an experiment. The problem of determination of the more plausible distribution using likelihood procedures (see, for example, Sprott and Kalbfleisch, 1969) will be discussed for the simple case where all observations are made under the same response conditions. (Lindsey, 1974, will consider this problem when independent variables are present.) To do this using likelihood inference, a base statistical model must be introduced with which all other distributions under consideration may be compared. The derivation which follows yields the multinomial model as the base model. Several approaches have been suggested in the literature to the problem of determining which of a number of possible models best describes a set of data. Cox (1961, 1962) develops asymptotic Neyman-Pearson likelihood ratio tests and suggests an alternative approach involving a combination, either additive or multiplicative, of the density functions, with estimation of additional parameters. This approach is further developed by Atkinson (1970). When prior probabilities, both for each model and for the parameters within the models, are available, Lindley (1961, p. 456) gives a posterior odds ratio of the two models using Bayes's theorem. When applicable (i.e. when prior probabilities are available), this approach may be used with the methods developed below.

Dans toute la suite X désignera un groupe abélien localement compact et dénombrable à l’infini; ξ sera sa mesure de Haar. Un noyau de convolution N sur X signifie une mesure de Radon positive dans X, et, pour une mesure de Radon réelle μ dans X, N * μ s’appelle le N-potentiel de μ dès que cette convolution a un sens.

We have several definitions of the positivity of a vector bundle, differentiate definitions, an algebro-geometric definition, a topological definition etc. In § 1 we review the definitions and the relations between them. For a line bundle all the definitions are equivalent and every one agrees that they are reasonable. For a vector bundle, however, the definitions are not necessarily equivalent. One of the main results of this paper is the equivalence of the definitions over a complete non-singular curve. The proof is given in §2. We proved this over an elliptic curve in Umemura [18]. In this case the proof was based on Atiyah’s classification. To prove the equivalence over a curve of genus ≥ 2, the fundamental lemma is; A stable bundle of positive degree is positive in the sense of Nakano. The tool used to prove this lemma is the theory of stable bundles due to Narasimhan and Seshadri [11] —they establish a correspondence between stable bundles and certain types of irreducible unitary representations of a Fuchsian group.

Cohen (1969) has studied the transient and stationary queue length distributions for the M/G/1, K queue, with a fixed maximum number of customers, K, in the system at any time. The present note applies Cohen's method to generalize his results to the MX/GY/1, K queue.

The well-known connection between the Poisson process and empirical c.d.f.'s is exploited from a new point of view. Distributions of functions of empirical c.d.f.'s for finite sample size n are explicitly described in some new examples, and new qualitative information is obtained for some classical examples.

In recent years the problem of concretely representing a class of nonlinear functionals on Banach spaces has received considerable attention. Suppose B is a Banach space equipped with an orthogonality relation ⊥ ⊂ B X B.

Many physical problems are described by stochastic processes with stationary increments. We present a general description of such processes. In particular we give an expression of a process in terms of its increments and we show that there are two classes of processes: diffusion and asymptotically stationary. Moreover, we show that the nth increments are given by a linear filtering of an arbitrary stationary process. STOCHASTIC PROCESSES; STATIONARY; DIFFUSION; LINEAR FILTERING; CORRELATION THEORY Stochastic processes (s.p.) are very often used as statistical models to represent random physical phenomena. In most cases the assumption of stationarity is introduced, either for mathematical simplifications or for physical reasons. Nevertheless there are some situations in which the observed phenomenon is nonstationary, even if it is driven by a stationary source. The best example is the phase ((t) of an oscillator the instantaneous frequency F(t) of which is stationary. Thus we have

We name a functional equation with restricted argument one in which at least one of the variables is restricted to a certain discrete subset of the domain of the other variable(s). In particular, the subset may consist of a single element.The purpose of this paper is to present a functional equation satisfied only by cosine functions.

Luria and Delbrück (1943) have observed that, in old cultures of bacteria that have mutated at random, the distribution of the number of mutants is extremely long-tailed. In this note, this distribution will be derived (for the first time) exactly and explicitly. The rates of mutation will be allowed to be either positive or infinitesimal, and the rate of growth for mutants will be allowed to be either equal, greater or smaller than for non-mutants. Under the realistic limit condition of a very low mutation rate, the number of mutants is shown to be a stable-Lévy (sometimes called “Pareto Lévy”) random variable, of maximum skewnessß, whose exponentαis essentially the ratio of the growth rates of non-mutants and of mutants. Thus, the probability of the number of mutants exceeding the very large valuemis proportional tom–α–1(a behavior sometimes referred to as “asymptotically Paretian” or “hyperbolic”). The unequal growth rate casesα≠ 1 are solved for the first time. In theα= 1 case, a result of Lea and Coulson is rederived, interpreted, and generalized. Various paradoxes involving divergent moments that were encountered in earlier approaches are either absent or fully explainable.The mathematical techniques used being standard, they will not be described in detail, so this note will be primarily a collection of results. However, the justification for deriving them lies in their use in biology, and the mathematically unexperienced biologists may be unfamiliar with the tools used. They may wish for more details of calculations, more explanations and Figures. To satisfy their needs, a report available from the author upon request has been prepared. It will be referred to as Part II.

Let C be an elliptic curve (an abelian variety of dimension one) defined over the field Q of rational numbers. A minimal Weierstrass model for C at all primes p in the sense of Néron [3] is given by a plane cubic equation of the formwhere aj belongs to the ring Z of integers of Q, the zero of C being the point of infinity.

The intrinsic random functions (IRF) are a particular case of the Guelfand generalized processes with stationary increments. They constitute a much wider class than the stationary RF, and are used in practical applications for representing non-stationary phenomena. The most important topics are: existence of a generalized covariance (GC) for which statistical inference is possible from a unique realization; theory of the best linear intrinsic estimator (BLIE) used for contouring and estimating problems; the turning bands method for simulating IRF; and the models with polynomial GC, for which statistical inference may be performed by automatic procedures.

This paper deals with the notion of equational compactness and related concepts in the special case of G-sets for an arbitrary group G. It provides characterizations of pure extensions, pure-essential extensions, and equational compactness in terms of the stability groups of a G-set, proves the general existence of equationally compact hulls, and gives an explicit description of these. Further, it establishes, among other results, that all G-sets are equationally compact iff all subgroups of the group G are finitely generated, that every equationally compact G-set is a retract of a topologically compact one, and that for free groups G with infinite basis there are homogeneous G-sets which are not equationally compact.

The classical secretary problem is generalized to admit stochastically successful procurement of previous interviewees, but each has a certain probability of refusing the offer. A general formula for solving this problem is obtained. Two special cases: constant probability of refusing and geometric probability of refusing are discussed in detail. The optimal stopping rules in these two cases turn out to be simple.

This note is concerned specifically with links of two (disjoint) n-spheres in an (n + 2)-manifold M, i.e. embeddings L:Sn + Sn —> M. The links L0 and L1 are isotopic if they are t he ends of a continuous family Li:Sn + Sn —> M (0 ≦ t ≦ 1) of links. They are ambient isotopic or equivalent if there is a continuous family of self-homeomorphisms ht: M—> M (0 ≦ t ≦ 1) such that h0 = identity and h1◯L0 = L1. Ambient isotopic links are isotopic, but not conversely. For example, an isotopy can tie and untie little knots (as in Figure 3) in the components of a link, thus changing the original link into one which is inequivalent to the original.

(1974). More on Involutions of a Circle. The American Mathematical Monthly: Vol. 81, No. 6, pp. 613-616.

It has been shown (Mallory and Sion [6]) that the problem of finding "limit" measures for inverse systems of measure spaces (Xi,μi)i∊Ican be successfully attacked by establishing the existence of a “limit” measureon the product space, then considering the restriction to the inverse limit set.In this paper we use a similar point of view to establish conditions under which a system of Radón measures has a “limit” measure which is also Radón.

The structure theory of hereditary noetherian prime (hnp) rings—in particular of Dedekind prime rings—has been recently developed by many authors including Eisenbud, Griffith, Michler and Robson; this theory extends some of the well-known results concerning commutative Dedekind domains. In this paper we study quasi-injective modules and quasi-projective modules over those (hnp) rings which are not right primitive and establish some results which extend the corresponding well-known results concerning commutative Dedekind domains. LetRbe an (hnp) ring, which is not right primitive.

The results obtained by Lehoczky on the first emptiness of a dam subject to random outputs are extended to a more general model.

SynopsisThis paper establishes the existence of an infinite set of zeros for the solution of a certain functional differential equation. The primary condition assuring this oscillatory behavior is expressed in terms of the magnitude of the delay.

This paper gives several monotonicity properties of all oscillatory solutions of equations with separable and nonseparable nonlinearities which are more general than the Emden- Fowler equations*Principally, if x(t) is an oscillatory solution, conditions are given such that; if a(t)↑ ∞ as t → ∞, then x(t) → 0; and, if a(t) ↓ 0 as t → ∞, then lim sup | x(t) | = ∞.

The geometries considered here are the Möbius plane M() (W. Benz [1]), the Laguerre plane L() (W. Benz and H. Mäurer [7]) and the Minkowski plane A() (W. Benz [5], G. Kaerlein [18]) over a field . All of them are geometries of an algebra with identity over a field. The characterization of the projective plane over a field by the proposition of Pappus first gave a close relation between algebraic and geometric structures. B. L. v. d. Waedern and L. J. Smid [28] presented a further example by characterizing the Möbius and Laguerre plane with incidence axioms and the "complete" proposition of Miquel.

Ce travail donne des conditions sous lesquelles un sousmodule pur de M est une sommante directe, non nécessairement de M luimême, mais d'un sous-module “proche” de celui-ci. (M est supposé sans torsion, dans un certain sens.) Ces conditions sont exprimées à partir d'un module H d'endomorphismes du mondule des fractions de M, étroitement lié à la notion d'“endomorphisme partiel” d'un module sans torsion, introduite par Fuchs [3] et que nous avons étudiée dans [5] et [6].

Particles enter a state space at random times. Each particle travels in the space independent of the other particles until its death. Functionals of the particle system are studied with strong laws and central limit theorems being obtained. PARTICLE SYSTEMS; INDEPENDENT MOTIONS; FUNCTIONALS; COMPOUND STOCHASTIC PROCESSES 0. Summary Consider a countable collection of particles. The nth particle enters the state space (E, &) at time Tn; it then travels in E according to a stochastic process Xn = {Xn(s); s > O} until its death. Associated with its motion there arises a secondary process An such that (a) for fixed t > 0, B -+ An(B, t) is a a-finite measure on (E, 8); and (b) for each B e d, t -> An(B, t) is right-continuous and nondecreasing. The processes (Xn,A,), (n = 1,2,...), are assumed to be independent of each other and to have the same probability law independent of {T1}. For B E , define Dt(B) = z An(B, t-Tn). Tn S t For fixed t > 0, the mapping B -4 Dt(B) is a random measure on (E, od). The joint probability law of Dt(B), "', Dt(Bn) is completely determined for any finite sequence of sets B1, -,Bn in g. Strong laws and central limit theorems are obtained for Dt as t -+ oo.

Let GF(q) denote a finite field of order q = py, p a prime. Let A and C be symmetric matrices of order n, rank m and order s, rank k, respectively, over GF(q). Carlitz [6] has determined the number N(A, C, n, s) of solutions X over GF(q), for p an odd prime, to the matrix equation1.1where n = m. Furthermore, Hodges [9] has determined the number N(A, C, n, s, r) of s × n matrices X of rank r over GF(q), p an odd prime, which satisfy (1.1). Perkin [10] has enumerated the s × n matrices of given rank r over GF(q), q = 2y, such that XXT = 0. Finally, the author [3] has determined the number of solutions to (1.1) in case C = 0, where q = 2y.

We illustrate a technique for computing certain integrals that arise in probability theory by giving a new derivation of a formula of Kingman. This formula contains the joint distribution of the processes F(t) = inf {s: X(t + s) = b} and B(t) = inf{s: X(t - s) = b} where X is a time homogeneous, continuous parameter, Markov process and b is a fixed point in its state space. We then extend this formula to the situation in which b is replaced by a finite set {b 1, …, b n }.

In 1963, P. L. Butzer [4, p. 180] asked whether it was possible to prove Jackson's theorem by means of an operator which is "almost" interpolatory in the sense that it is based on the values of the approximee at a finite number of nodes. In answer to this question, G. Freud introduced [4] a sequence of operators which led to an independent proof of Jackson's theorem. Strictly speaking these operators are not interpolatory but they are "almost" interpolatory in the above sense.

The Lie algebra gr of all infinitesimal automorphisms of a Siegel domain in terms of polynomial vector fields was investigated by Kaup, Matsushima and Ochiai [6]. It was proved in [6] that gr is a graded Lie algebra; gr = g-1 + g-1/2 + g0 + g1/2 + g1 and the Lie subalgebra ga of all infinitesimal affine automorphisms is given by the graded subalgebra; ga = g-1 + g-1/2 + g0. Nakajima [9] proved without the assumption of homogeneity that the non-affine parts g1/2 and g1 can be determined from the affine part ga.

Given a first-order theory T, welet be the category of models of T and homomorphisms between them. We shall show that a morphism A→B of is an epimorphism if and only if every element of B is definable from elements of A in a certain precise manner (see Theorem 1), and from this derive the best possible Cowell- power Theorem for .

The product moment equations previously derived are first discussed for the infinite one- and two-dimensional Ising (or autologistic) model in the case of non-zero mean, as a prelude to an examination of the probability structure in the higher-dimensional (and nominally zero mean) case below the ‘critical temperature’. Of two simple possible models, A and B, both consistent with the division of the product moment p into ergodic, and long-range non-ergodic, components, such that ρ = r (1 – m 2) + m 2, where r is the intrinsic correlation coefficient, it is shown that the second model B appears appropriate to the three-dimensional ‘spherical model’, but the first model A to the Ising model. Model A is defined by xi = yi + M, where M = +m or –m, and E{yi } = 0; and Model B by

It is shown that two different population models are equivalent to each other when a certain marginal stationary distribution is considered.

(1974). Strong Derivatives and Inverse Mappings. The American Mathematical Monthly: Vol. 81, No. 9, pp. 969-980.

A transversal of a family of non-empty sets is a 1-1 mapsuch that φ(v) ∊ Fv (v ∊ I) . A number of problems in combinatorial mathematics reduce to the question of whether or not a certain family of sets has a transversal. An up-to-date account of this theory is to be found in the book by Mirsky [9]. The best known result of this kind is the following theorem.

giving a way to calculate as many of the B's as desired. Formula (1.2), stated in tha way or in the form (1.3) or in various other forms, has been known a long time. Thus, in 1748 Euler gave (1.3) (obscurely expressed) in section 76 of his Introductio [4], [5]. Adams and Hippisley [1] gave the formula in the form (1.2) (their formula 6.361). Formula (1.3) is given as formula (0.314) in any of the several editions (Russian original, German, or English) of Ryshik and Gradstein [16], where, however, p is restricted to be a natural number. Thinking to remove this restriction of Ryshik and Gradstein, von Holdt [17] published still another derivation using properties of double sums to establish that (1.2) holds true for rational real p. His derivation avoids use of differentiation of the series in (1.1). We mention this because differentiation of formal power series affords a quick proof of (1.2) and has been used before. The basic recurrence relation is not as widely known as it should be, and has been rediscovered repeatedly. Thus Barrucand [2] found (1.2) again and made applications of it. Cappellucci [18] found it in the form (1.3) but attributed it to Hansted [19]. Hindenburg [12], in his treatise on the multinomial theorem (p. 291) gave (1.3) in a perfectly obscure notation. Many other references could be cited. Actually, the recurrence is implicit in still another class of formulas widespread in the literature, stemming from the early work of Hindenburg's student Rothe [15]. I myself have written a number of papers, e.g., [6]-[11] having to do with a formula of Rothe and its consequences for combinatorics, special functions and number theory. What we shall do here is to derive the formulas again and put them in a

The model considered here consists of n operating units which are subject to stochastic failure according to an exponential failure time distribution. Failures can be of two types. With probability p(q) a failure is of type 1(2) and is sent to repair facility 1(2) for repair. Repair facility 1(2) operates as a -server queue with exponential repair times having parameter μ 1 (μ 2). The number of units waiting for or undergoing repair at each of the two facilities is a continuous-parameter Markov chain with finite state space. This paper derives limit theorems for the stationary distribution of this Markov chain as n becomes large under the assumption that both and grow linearly with n. These limit theorems give very useful approximations, in terms of the six parameters characterizing the model, to a distribution that would be difficult to use in practice.

Consider a nonnegative Hölder continuous 2-form P(z)dxdy on a hyperbolic Riemann surface R (z = x + iy). We denote by PB(R) the Banach space of solutions of the equation Δu = Pu on R with finite supremum norms. We are interested in the question how the Banach space structure of PB(R) depends on P. Precisely we consider two such 2-forms P and Q on R and compare PB(R) and QB(R). If there exists a bijective linear isometry T of PB(R) to QB(R), then we say that PB(R) and QB(R) are isomorphic.

In this article, we consider maximum likelihood estimators of the initial probabilities and the mean of a supercritical Galton-Watson process; we find for these estimators, as well as for the Lotka-Nagaev estimator of the mean, the asymptotic distributions and deduce confidence intervals. As these results hold even if the independence between individuals of the same generation is not satisfied, application to living populations may be considered.

The concept of a heat polynomial defined earlier by Rosenbloom and Widder is extended and the polynomials are characterized using integrals of Poisson type. One particular class of polynomials is examined in detail and leads to expansions of solutions of the problem \[u_{xx} = u_t ,\qquad u(x,0) = f(x)\] which are different from those obtained by Rosenbloom and Widder. In the last section some applications of polynomials to heat equation problems are given. In most results, much use is made of the close relationship between these polynomials and Hermite polynomials.

This paper is concerned with a class of dynamic network flow problems in which the amount of flow leaving node i in one time period for node j is the fraction pij of the total amount of flow which arrived at node i during the previous time period. The fraction pij whose sum over j equals unity may be interpreted as the transition probability of a finite Markov chain in that the unit flow in state i will move to state j with probability pij during the next period of time. The conservation equations for this class of flows are derived, and the limiting behavior of the flows in the network as related to the properties of the fractions Pij are discussed.

Let denote the family of all functions of the formthat are analytic in the unit disc U, f′(z) ≠ 0 in U and f maps U onto a domain of boundary rotation at most . Recently Brannan, Clunie and Kirwan [2] and Aharonov and Friedland [1] have solved the problem of estimating |amp+1| for all , provided m = 1.

The theory of the linear model is incomplete in that it fails to deal with variables possessing infinite variance. To fill an important part of this gap, we give an unbiased estimate, the “screened ratio estimate”, for λ in the regression E(X|Z) = λX; X and Z are linear combinations of independent, identically distributed symmetric random variables that are either stable or asymptotically Pareto distributed of index α ≤ 2. By way of comparison, the usual least squares estimate of λ is shown not to converge in general to any constant when α < 2. However, in the autoregression X n = a 1 X n-1 + … + a k X n-k + U n , the least squares estimates are shown to be consistent as long as the roots of 1 - a 1 x 2 - a 2 x 2 - … - a k x k = 0 are outside the complex unit circle, X n is independent of U n+j ,j ≥ 1, and the U n are independent and identically distributed and in the domain of attraction of a stable law of index a ≤ 2. Finally, the consistency of least squares estimates for finite moving averages is established.