55 results
Search Results
2. A semi-markov model for clinical trials
- Author
-
George H. Weiss and Marvin Zelen
- Subjects
Statistics and Probability ,Markov chain ,Stochastic modelling ,General Mathematics ,Variable-order Markov model ,010102 general mathematics ,Stochastic matrix ,Markov process ,Markov model ,01 natural sciences ,Combinatorics ,010104 statistics & probability ,symbols.namesake ,symbols ,Applied mathematics ,Probability distribution ,Markov property ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper applies the theory of semi-Markov processes to the construction of a stochastic model for interpreting data obtained from clinical trials. The model characterizes the patient as being in one of a finite number of states at any given time with an arbitrary probability distribution to describe the length of stay in a state. Transitions between states are assumed to be chosen according to a stationary finite Markov chain.Other attempts have been made to develop stochastic models of clinical trials. However, these have all been essentially Markovian with constant transition probabilities which implies that the distribution of time spent during a visit to a state is exponential (or geometric for discrete Markov chains). Markov models need also to assume that the transitions in the state of a patient depend only on absolute time whereas the semi-Markov model assumes that transitions depend on time relative to a patient. Thus the models are applicable to degenerative diseases (cancer, acute leukemia), while Markov models with time dependent transition probabilities are applicable to colds and epidemic diseases. In this paper the Laplace transforms are obtained for (i) probability of being in a state at timet, (ii) probability distribution to reach absorption state and (iii) the probability distribution of the first passage times to go from initial states to transient or absorbing states, transient to transient, and transient to absorbing. The model is applied to a clinical study of acute leukemia in which patients have been treated with methotrexate and 6-mercaptopurine. The agreement between the data and the model is very good.
- Published
- 1965
3. The minimum of a stationary Markov process superimposed on a U-shaped trend
- Author
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H.E. Daniels
- Subjects
Statistics and Probability ,Stationary process ,Gaussian ,General Mathematics ,010102 general mathematics ,Boundary (topology) ,Markov process ,01 natural sciences ,symbols.namesake ,010104 statistics & probability ,Distribution (mathematics) ,Simple (abstract algebra) ,symbols ,Applied mathematics ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Gaussian process ,Mathematics - Abstract
1. This paper was motivated by some questions of Barnett and Lewis (1967) concerning extreme winter temperatures. The temperature during the winter can be hopefully regarded as generated by a stationary Gaussian process superimposed on a locally U-shaped trend. One is interested in statistical properties of the minimum of sample paths from such a process, and of their excursions below a given level. Equivalently one can consider paths from a stationary process crossing a curved boundary of the same form. Problems of this type are discussed by Cramer and Leadbetter (1967), extensively in the trend-free case and in less detail when a trend is present, following the method initiated by Rice (1945). While results on moments are easy to obtain, explicit results for the actual probability distributions are not usually available. However, in the important case when the level of values of interest is far below the mean, the asymptotic independence of up-crossing times makes it possible to derive simple approximate distributions. (See Cramer and Leadbetter (1967) page 256, Keilson (1966).) There is a dearth of particular examples of processes and trends for which the distributions of interest are known exactly. Such examples could give useful experience of the form of distribution to be expected in typical cases, and could serve as material on which to test out approximate methods. The object of the present paper is to provide an example of this kind. One process for which exact results are available in the trend-free case is the Ornstein-Uhlenbeck process, i.e., the stationary Gaussian Markov process X(t) generated by
- Published
- 1969
4. Green’s Functions For Generalized Schroedinger Equations
- Author
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John A. Beekman
- Subjects
010308 nuclear & particles physics ,General Mathematics ,Gaussian ,010102 general mathematics ,Markov process ,Sample (statistics) ,01 natural sciences ,Green S ,symbols.namesake ,chemistry.chemical_compound ,chemistry ,0103 physical sciences ,symbols ,Applied mathematics ,Boundary value problem ,0101 mathematics ,Schrödinger's cat ,Mathematics - Abstract
I. Introduction. The purpose of this paper is to discuss functions defined on the continuous sample paths of Gaussian Markov processes which serve as Green’s functions for pairs of generalized Schroedinger equations. The results extend the author’s earlier paper [2] to a forward time version, and consider different boundary conditions.
- Published
- 1969
5. A generalized bivariate exponential distribution
- Author
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Albert W. Marshall and Ingram Olkin
- Subjects
Statistics and Probability ,Exponential distribution ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Laplace distribution ,Univariate distribution ,010104 statistics & probability ,Compound Poisson distribution ,Exponential family ,Statistics ,Gamma distribution ,Applied mathematics ,Phase-type distribution ,Natural exponential family ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In a previous paper (Marshall and Olkin (1966)) the authors have derived a multivariate exponential distribution from points of view designed to indicate the applicability of the distribution. Two of these derivations are based on “shock models” and one is based on the requirement that residual life is independent of age. The practical importance of the univariate exponential distribution is partially due to the fact that it governs waiting times in a Poisson process. In this paper, the distribution of joint waiting times in a bivariate Poisson process is investigated. There are several ways to define “joint waiting time”. Some of these lead to the bivariate exponential distribution previously obtained by the authors, but others lead to a generalization of it. This generalized bivariate exponential distribution is also derived from shock models. The moment generating function and other properties of the distribution are investigated.
- Published
- 1967
6. On large sample sequential analysis with applications to survivorship data
- Author
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Norman Breslow
- Subjects
Statistics and Probability ,Class (set theory) ,Sequence ,General Mathematics ,010102 general mathematics ,Random walk ,01 natural sciences ,Exponential function ,Metric space ,010104 statistics & probability ,Convergence of random variables ,Sequential analysis ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
Although his work on the application of invariance concepts to the sequential testing of composite hypotheses is better known, Cox (1963) has also outlined a large sample approach to the same problem. His method is based on Bartlett's (1946) recognition that the sequence of maximum likelihood estimates (MLE) of the parameter of interest, calculated from an increasing number of observations, resembles asymptotically a random walk of normally distributed variables. However, the large sample theory needed to justify this approach rigorously is left largely implicit. At the end of his paper, Cox suggests that these msthods may be extended to yield a sequential comparison of survival curves (Armitage (1959)), a suggestion which has been reiterated as a research problem in the monograph of Wetherill (1966). In this paper we first present a general theoretical framework in which the asymptotic validity of a wide class of large sample sequential tests may be examined, thus making explicit the justification for Cox's approach. The results of this section are fairly straightforward consequences of the increasingly well known theory of convergence in distribution for random variables which take values in separable metric spaces. Next we illustrate the theory by re-examining Cox's results on the comparison of two binomial parameters. Finally, and of greater consequence from the practical point of view, we present a large sample solution to the problem of the sequential comparison of exponential survival
- Published
- 1969
7. Prediction of a noise-distorted, multivariate, non-stationary signal
- Author
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Eugene Sobel
- Subjects
Statistics and Probability ,Polynomial ,Stationary process ,Conjecture ,Series (mathematics) ,Differential equation ,General Mathematics ,Mathematical analysis ,010102 general mathematics ,Generating function ,Hilbert space ,01 natural sciences ,symbols.namesake ,010104 statistics & probability ,symbols ,Applied mathematics ,Elementary divisors ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
The paper represents a generalization of one of the main theoretical results of my Ph.D. thesis. The work is an outgrowth of work first begun by E. J. Hannan and a correct 'conjecture' by P. Whittle. The main theorem of this paper proves the existence of, and gives an explicit formula for, the asymptotic best linear predictor of a certain type of non-stationary multivariate time series from noise distorted observations. The non-stationarity arises from the fact that the signal satisfies a difference equation, which when considered as a polynomial, has only elementary divisors. The proof is accomplished by showing, through Hilbert space and harmonic analysis methods, that the generating function is a limit of the generating functions of the stationary analogue; that is, where the difference function has elementary divisors. Finally, it is shown that this asymptotic generating function exactly predicts null solutions to the difference equation. The proof is direct and due to E. J. Hannan.
- Published
- 1967
8. Mixed Problems for Hyperbolic Equations of General Order
- Author
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G. F. D. Duff
- Subjects
Flux-corrected transport ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Hyperbolic function ,Applied mathematics ,Order (group theory) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Hyperbolic partial differential equation ,Mathematics - Abstract
The object of this paper is the extension to linear partial differential equations of order m in N independent variables, of the existence theorems for mixed initial and boundary value problems which have been established for systems of first order equations in (3). In such mixed problems an initial surface S and a boundary surface T are the carriers of the two types of data, and the number of datum functions to be assigned on T depends on the configuration of the characteristic surfaces relative to S and T.For the first part of the paper (§§ 1-5) the coefficients in the differential equation, the initial and boundary surfaces, and the data prescribed are all taken to be real analytic in the variables x1 … xN. In this “analytic” case an existence theorem is established for boundary conditions of considerable generality. We assume that the differential equation is regularly hyperbolic with respect to 5 and T, a notion which is stated precisely in § 1, and is weaker than the usual regular hyperbolic condition.
- Published
- 1959
9. On strong Rieszian summability
- Author
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Martin Glatfeld
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,16. Peace & justice ,01 natural sciences ,Mathematics - Abstract
Recently H.-E. Richert [10] introduced a new method of summability, for which he completely solved the “summability problem” for Dirichlet series, and which led also to an extension of our knowledge of the relations between the abscissae of ordinary and absolute Rieszian summability. This non-linear method, which may best be characterized by the notion “strong Rieszian summability” †, depends on three parameters, on the order k;, the type λ, and the index p;. While Richert's paper deals almost exclusively with the application of that method of summability in a specialized form (namely the case p = 2, λn=log n) to Dirichlet series, it is the object of the present paper, to consider the general theory of strong Rieszian summability.
- Published
- 1957
10. Absolute Continuity of Some Vector Functions and Measures
- Author
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William J. Knight
- Subjects
Contiguity (probability theory) ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,Absolute continuity ,01 natural sciences ,Vector-valued function ,Mathematics - Abstract
In the theory of vector valued functions there is a theorem which states that if a function from a compact interval I into a normed linear space X is of weak bounded variation, then it is of bounded variation. The proof uses in a straightforward way the Uniform Boundedness Principle (see [2, p. 60]). The present paper grew from the question of whether an analogous theorem holds for absolutely continuous functions. The answer is in the negative, and an example will be given (Theorem 7). But it will also be shown that if X is weakly sequentially complete (e.g. an Lp space, 1 ≦ p < ∞ ), then a weakly absolutely continuous point function from / into X is absolutely continuous. The method of proof involves the construction of a countably additive set function in the standard Lebesgue-Stieltjes fashion.The paper is divided into three parts. In Section 1 extensions of finitely additive, absolutely continuous set functions are carried out in an abstract setting. Section 2 applies this to vector valued (point) functions on the real line.
- Published
- 1972
11. Ergodic Theory and Averaging Iterations
- Author
-
J. J. Koliha
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,Applied mathematics ,Ergodic theory ,010307 mathematical physics ,0101 mathematics ,Stationary ergodic process ,01 natural sciences ,Mathematics - Abstract
Suppose X is a Banach space and T a continuous linear operator on X. The significance of the asymptotic convergence of T for the approximate solution of the equation (I - T)x = f by means of the Picard iterations was clearly shown in Browder's and Petryshyn's paper [1], The results of [1] have stimulated further investigation of the Picard, and more generally, averaging iterations for the solution of linear and nonlinear functional equations [2; 3; 4; 8; 9]. Kwon and Redheffer [8] analyzed the Picard iteration under the mildest possible condition on T, namely that T be continuous and linear on a normed (not necessarily complete) space X. The results of [8] (still waiting to be extended for the averaging iterations) seem to give the most complete story of the Picard iterations for the linear case. Only when T is subject to some further restrictions, such as asymptotic 4-boundedness and asymptotic A -regularity, one can agree with Dotson [4] that the iterative solution of linear functional equations is a special case of mean ergodic theory for affine operators. This thesis is rather convincingly demonstrated by results of De Figueiredo and Karlovitz [2], and Dotson [3], and most of all by Dotson's recent paper [4], in which the results of [1; 2; 3] are elegantly subsumed under the afrine mean ergodic theorem of Eberlein-Dotson.
- Published
- 1973
12. A Cosine Functional Equation with Restricted Argument
- Author
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L. B. Etigson
- Subjects
Discrete mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,010102 general mathematics ,Characteristic equation ,010103 numerical & computational mathematics ,01 natural sciences ,Algebra ,Integro-differential equation ,Argument ,Functional equation ,Riccati equation ,Discrete Mathematics and Combinatorics ,Trigonometric functions ,0101 mathematics ,Mathematics - Abstract
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.
- Published
- 1974
13. The Stability of Solutions of Generalized Emden-Fowler Equations
- Author
-
Hugo Teufel
- Subjects
Independent equation ,Simultaneous equations ,General Mathematics ,010102 general mathematics ,Applied mathematics ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Stability (probability) ,Mathematics - Abstract
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) | = ∞.
- Published
- 1974
14. Some limit theorems for a class of network problems as related to finite Markov chains
- Author
-
Masao Nakamura
- Subjects
Statistics and Probability ,Discrete mathematics ,Markov kernel ,Markov chain ,General Mathematics ,Variable-order Markov model ,Node (networking) ,010102 general mathematics ,01 natural sciences ,010104 statistics & probability ,Flow (mathematics) ,Applied mathematics ,Irreducibility ,Examples of Markov chains ,Limit (mathematics) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
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.
- Published
- 1974
15. Lyapunov Inequalities and Bounds on Solutions of Certain Second Order Equations*
- Author
-
Stanley B. Eliason
- Subjects
Lyapunov function ,Inequality ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Second order equation ,Lyapunov exponent ,01 natural sciences ,symbols.namesake ,0103 physical sciences ,symbols ,Applied mathematics ,Lyapunov equation ,010307 mathematical physics ,0101 mathematics ,media_common ,Mathematics - Abstract
In this paper we consider the equation(1.1) (r(t)y′(t))′+p(t)f(y(t)) = 0under the conditions((H0): the real valued functions r, r′ and p are continuous on a non-trivial interval J of reals, and r(t)>0 for t∈J;and(H1):f:R→R is continuously differentiable and odd with f'(y)>0 for all real y. We also consider the equation(1.2) y″(t)+m(t)y′(t)+n(t)f(y(t)) = 0under the conditions (H1) and(H2): the real valued functions m and n are continuous on a non-trivial interval J of reals.
- Published
- 1974
16. Deformations of group actions
- Author
-
Allan L. Edmonds
- Subjects
Finite group ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_GENERAL ,57E10 ,58D99 ,16. Peace & justice ,01 natural sciences ,Action (physics) ,law.invention ,010101 applied mathematics ,Piecewise linear function ,010104 statistics & probability ,Group action ,law ,0101 mathematics ,Manifold (fluid mechanics) ,Mathematics - Abstract
Let G G be a finite group and M M be a compact piecewise linear (PL) manifold. Define a PL G G -isotopy to be a level-preserving PL action of G G on M × [ 0 , 1 ] M \times [0,1] . In this paper PL G G -isotopies are studied and PL G G -isotopic actions (which need not be equivalent) are characterized.
- Published
- 1973
17. Almost Convergence, Summability And Ergodicity
- Author
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J. Peter Duran
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,Ergodicity ,Applied mathematics ,010307 mathematical physics ,Convergence (relationship) ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
The notion of almost convergence introduced by Lorentz [15] has been generalized in several directions (see, for example [1; 8; 11 ; 14; 17]). I t is the purpose of this paper to give a generalization based on the original definition in terms of invariant means. This is effected by replacing the shift transformation by an "ergodic" semigroupof positive regular matrices in the definition of invariant mean. The resulting "- invariant means" give rise to a summability method which we dub-almost convergence.
- Published
- 1974
18. Bounds for coverage probabilities with applications to sequential coverage problems
- Author
-
Peter J. Cooke
- Subjects
Statistics and Probability ,010104 statistics & probability ,General Mathematics ,010102 general mathematics ,Stopping rule ,Stopping rules ,Applied mathematics ,Stirling number ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences ,Mathematics - Abstract
This paper discusses general bounds for coverage probabilities and moments of stopping rules for sequential coverage problems in geometrical probability. An approach to the study of the asymptotic behaviour of these moments is also presented. STOPPING RULE; SEQUENTIAL COVERAGE; STIRLING NUMBERS; ASYMPTOTIC BEHAVIOUR
- Published
- 1974
19. On infinite dams with inputs forming a stationary process
- Author
-
R. M. Phatarfod and Pyke Tin
- Subjects
Statistics and Probability ,Stationary distribution ,Stationary process ,Markov chain ,General Mathematics ,Computer Science::Neural and Evolutionary Computation ,010102 general mathematics ,Stationary case ,Process (computing) ,Bivariate analysis ,Expected value ,Computer Science::Numerical Analysis ,01 natural sciences ,Physics::Geophysics ,010104 statistics & probability ,Computer Science::Computational Engineering, Finance, and Science ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Unit (ring theory) ,Computer Science::Distributed, Parallel, and Cluster Computing ,Mathematics - Abstract
This paper considers a dam of infinite capacity with a discrete-valued stationary input process and a unit release whenever possible. It is shown how, by suitable manipulations of the equation governing the dam content process, the stationary distribution of the dam being empty can be obtained, as also can (with a few additional assumptions) the expected value of the dam content in the stationary case. Results obtained are applied to particular cases of input independent and identical, Markov, Bivariate Markov and moving-average. INFINITE DAMS; STATIONARY INPUT; UNIT RELEASE; EMPTINESS PROBABILITY; EXPECTED DAM CONTENT
- Published
- 1974
20. A finite dam with exponential release
- Author
-
G. F. Yeo
- Subjects
Independent and identically distributed random variables ,Statistics and Probability ,Exponential distribution ,Recurrence relation ,Series (mathematics) ,Differential equation ,General Mathematics ,010102 general mathematics ,Poisson distribution ,01 natural sciences ,Exponential function ,Numerical integration ,symbols.namesake ,010104 statistics & probability ,symbols ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
This paper considers a finite dam with independently and identically distributed (i.i.d.) inputs occurring in a Poisson process; the special cases where the inputs are (i) deterministic and (ii) negative exponentially distributed are considered in detail. The instantaneous release trate is proportional to the content, i.e., there is an exponential fall in conten except when inputs occur. This model may arise in several other situations such as a geiger counter or integrated shot noise. The distribution of the number of inputs, and of the time, to first overflowing is obtained in terms of generating functions; in Case (i) the solution is obtained through recurrence relations involving iterated integrals which can be evaluated numerically, and in Case (ii) using a series solution of a second order differential equation. Numerical results, in particular for the first two moments, are obtained for various values of the parameters of the model, and compared with a large number of simulations. Some remarks are also made about the infinite dam. FINITE DAMS; POISSON INPUTS; EXPONENTIAL RELEASE; FIRST PASSAGE TIMES; RECURRENCE RELATIONS, NUMERICAL INTEGRATION; SERIES SOLUTION; SIMULATION
- Published
- 1974
21. Orthogonal Decompositions of Multivariate Weakly Stationary Stochastic Processes
- Author
-
James B. Robertson
- Subjects
Combinatorics ,Multivariate statistics ,Stochastic process ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Decomposition (computer science) ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,Characterization (mathematics) ,01 natural sciences ,Mathematics - Abstract
In this paper we shall study the relations between the ranks of g-variate, discrete-parameter, weakly stationary stochastic processes x, y, and z satisfying the condition1.1and derive from them a characterization for the Wold decomposition and conditions for the concordance of the Wold and the Lebesgue-Cramér decompositions.
- Published
- 1968
22. Solutions of some two-sided boundary problems for sums of variables with alternating distributions
- Author
-
G. Yeo and J. Chover
- Subjects
Statistics and Probability ,Queueing theory ,General Mathematics ,010102 general mathematics ,Boundary (topology) ,01 natural sciences ,010104 statistics & probability ,Simple (abstract algebra) ,Applied mathematics ,First-hitting-time model ,0101 mathematics ,Statistics, Probability and Uncertainty ,Random variable ,Mathematics - Abstract
In this paper we present a method for obtaining explicit results for some two-sided boundary problems involving sums of independent random variables with alternating distributions. We apply the method to finding the first passage time to either one of two finite barriers, and to some situations arising in queueing and dam theory. The results can be expressed in terms of a finite sum of simple repeated integrals (or sums) of known functions (cf. formulae (3.6)– (3.11)).
- Published
- 1965
23. The cost of a general stochastic epidemic
- Author
-
J. Gani and D. Jerwood
- Subjects
Statistics and Probability ,010104 statistics & probability ,General Mathematics ,010102 general mathematics ,Applied mathematics ,Kleene's recursion theorem ,0101 mathematics ,Statistics, Probability and Uncertainty ,Duration (project management) ,01 natural sciences ,Mathematics - Abstract
This paper is concerned with the cost Cis = aWis + bTis (a, b > 0) of a general stochastic epidemic starting with i infectives and s susceptibles; Tis denotes the duration of the epidemic, and Wis the area under the infective curve. The joint Laplace-Stieltjes transform of (Wis, Tis ) is studied, and a recursive equation derived for it. The duration Tis and its mean Nis are considered in some detail, as are also Wis and its mean Mis . Using the results obtained, bounds are found for the mean cost of the epidemic.
- Published
- 1972
24. Generalized de la Vallée Poussin Disconjugacy Tests for Linear Differential Equations(1)
- Author
-
D. Willett
- Subjects
Linear differential equation ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we study the oscillatory behavior of the solutions of the linear differential equation(1.1)where(1.2)and all functions are assumed to be continuous on a bounded interval [a, b). An «th-order linear equation is said to be disconjugate on an interval I provided it has no nontrivial solution with more than n — 1 zeros, counting multiplicities, in I.
- Published
- 1971
25. On Some Non-Linear Problems
- Author
-
K. Srinivasacharyulu
- Subjects
Nonlinear system ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Non-linear problems have been studied by Krasnoselski, Browder, and others; in fact Browder and independently Kirk (cf., 1; 5) have proved the following remarkable theorem: let X be a uniformly convex Banach space, U a non-expansive mapping of a bounded closed convex subset C of X into C, i.e., ||Ux — Uy|| ⩽ ||x — y|| for x, y ∊ C; then U has a fixed point in C. The aim of this paper is to give some existence theorems for non-linear functional equations in uniformly convex Banach spaces. Similar results may be found in (3 ; 6).
- Published
- 1968
26. Erlang's formula and some results on the departure process for a loss system
- Author
-
D. N. Shanbhag and D. G. Tambouratzis
- Subjects
Statistics and Probability ,Exponential distribution ,General Mathematics ,010102 general mathematics ,Asymptotic distribution ,Limiting ,Poisson distribution ,01 natural sciences ,Erlang (unit) ,010104 statistics & probability ,symbols.namesake ,symbols ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Mathematics - Abstract
The present paper investigates the limiting distribution of the number of busy channels (queue size) and the remaining lengths of holding times at an epoch of departure for a loss system with general holding times and exponentially distributed interarrival times. Further, it is established that for this loss system in the limit an interdeparture interval length is independent of the queue size at the end of the interval and is distributed according to an exponential distribution with mean X-1. It is also seen that in the limit interdeparture times are mutually independent. LOSS SYSTEM WITH POISSON ARRIVALS, ERLANG'S FORMULA, LIMITING DEPARTURE PROCESS; REMAINING HOLDING TIMES; BUSY CHANNELS
- Published
- 1973
27. On Some New Generalizations of the Functional Equation of Cauchy
- Author
-
Gy. Muszély and P. Fischer
- Subjects
Cauchy problem ,General Mathematics ,010102 general mathematics ,Functional equation ,Mathematical analysis ,Applied mathematics ,Cauchy distribution ,Cauchy principal value ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Examining certain problems in physics M. Hosszu [l] obtained the functional equation(1)where x, y, f are real.In another paper M. Hosszu [2] proved that the equation (1) is equivalent to the functional equation of Cauchy; i. e., to the equation(1)under the assumption that x is real and f is real and continuous.
- Published
- 1967
28. Some Refinements of Lyapunov's Second Method
- Author
-
Fred Brauer
- Subjects
Lyapunov function ,symbols.namesake ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,symbols ,Applied mathematics ,Lyapunov equation ,010307 mathematical physics ,0101 mathematics ,Lyapunov redesign ,01 natural sciences ,Mathematics - Abstract
Lyapunov's second method is a well-known and powerful tool for studying the behaviour of solutions of a system of differential equations. One approach to the theory is the comparison method developed by Corduneanu (4). This approach has the advantage that it also leads to other results on asymptotic behaviour which originally appeared to be unrelated to Lyapunov's method. Some of these results have been obtained by the author in (2). The purpose of this paper is to make use of the comparison method to obtain some refinements of Lyapunov's theory.
- Published
- 1965
29. A Generalization of an Inversion Formula for the Gauss Transformation
- Author
-
P. G. Rooney
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,Gauss ,Mathematical analysis ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,Quadratic Gauss sum ,01 natural sciences ,Inversion (discrete mathematics) ,Mathematics - Abstract
In an earlier paper [3] we considered an inversion formula for the Gauss transformation G defined by1.1We noted there that formally G is inverted by,1.2and we showed that if e-D2 is interpreted via the power series for the exponential function, that is if1.3then under certain conditions on φ,1.4
- Published
- 1963
30. Interconnected population processes
- Author
-
E. Renshaw
- Subjects
Statistics and Probability ,Birth and death process ,education.field_of_study ,General Mathematics ,010102 general mathematics ,Population ,01 natural sciences ,010104 statistics & probability ,Simple (abstract algebra) ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Mathematics - Abstract
This paper investigates the effect of migration between two colonies each of which undergoes a simple birth and death process. Expressions are obtained for the first two moments and approximate solutions are developed for the probability generating function of the colony sizes.
- Published
- 1973
31. Disconjugacy Criteria for Nonselfadjoint Differential Equations of Even Order
- Author
-
Kurt Kreith
- Subjects
Order (business) ,Differential equation ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
Disconjugacy criteria have been established for linear selfadjoint differential equations of order 2n by Sternberg [4] and Ahlbrandt [1]. Such differential equations can be written in the form1.1where it is assumed that the coefficients are real and that Pn(x) ≠ 0. We shall be interested in nontrivial solutions v(x) of (1.1), which satisfy1.2for distinct points α and β. The smallest β> α such that (1.2) is satisfied nontrivially by a solution of (1.1), is denoted by μ1(α) and called the first conjugate point of x = α with respect to (1.1). If no such conjugate point exists we write μ1(α) = ∞, and say that (1.1) is disconjugate on [α, ∞).The principal purpose of this paper is to generalize these disconjugacy criteria to the general linear nonselfadjoint differential equation of the form1.3
- Published
- 1971
32. Traffic light queues with dependent arrivals as a generalization to queueing theory
- Author
-
Hisashi Mine and Katsuhisa Ohno
- Subjects
Statistics and Probability ,Independent and identically distributed random variables ,Queueing theory ,Queue management system ,Generalization ,General Mathematics ,010102 general mathematics ,Fork–join queue ,01 natural sciences ,Computer Science::Performance ,010104 statistics & probability ,Traffic signal ,Light control ,Computer Science::Networking and Internet Architecture ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Queue ,Mathematics - Abstract
This paper considers a fixed-cycle and a semi vehicular-actuated traffic light queue with strictly stationary arrivals and independent and identically distributed departure headways and lost times. These queues are reduced to a generalized model of Loynes and sufficient conditions are derived under which these queues have stationary distributions. Two typical examples of semi vehicular-actuated traffic light queues are discussed.
- Published
- 1972
33. Numerical Integration of Functions of Several Variables
- Author
-
G. W. Tyler
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Variable (mathematics) ,Quadrature (mathematics) ,Numerical integration ,Mathematics - Abstract
Methods of mechanical quadrature of functions of more than one variable apparently have received little systematic investigation and the few available results are widely scattered in the literature. In this paper a systematic approach to this problem is given and a number of formulae are derived which may prove to be useful.
- Published
- 1953
34. Some results for dams with Markovian inputs
- Author
-
R. M. Phatarfod and K. V. Mardia
- Subjects
Statistics and Probability ,Stationary distribution ,General Mathematics ,010102 general mathematics ,Duality (mathematics) ,Autocorrelation ,Process (computing) ,Mathematics::General Topology ,Markov process ,01 natural sciences ,Identity (music) ,010104 statistics & probability ,symbols.namesake ,Content (measure theory) ,symbols ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
The paper considers the dam problem with Markovian inputs, with special reference to the serial correlation coefficient of the input process. An input model is proposed which by giving particular values to the parameters makes the stationary distribution of the inputs one of the standard discrete distributions. The probabilities of first emptiness before overflow are first obtained by using the Markovian analogue of Wald's Identity. From these, the stationary distributions of the dam content are obtained by a duality argument. Both, finite and infinite dams are considered. MARKOVIAN INPUT OF LINEAR REGRESSIVE KIND; MARKOVIAN ANALOGUE OF WALD'S IDENTITY; PROBABILITIES OF EMPTINESS; STATIONARY DISTRIBUTIONS OF DAM CONTENT; DEPENDENCE ON SERIAL CORRELATION COEFFICIENT
- Published
- 1973
35. The equivalence of some overlapping and non-overlapping generation models for the study of genetic drift
- Author
-
C. Cannings
- Subjects
Statistics and Probability ,010104 statistics & probability ,Genetic drift ,General Mathematics ,010102 general mathematics ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences ,Equivalence (measure theory) ,Mathematics - Abstract
The paper discusses models for genetic drift in haploid models; non-overlapping models following Wright, and overlapping models following Moran. It is shown that these models, and their extensions by Chia and Watterson, can all be restated as non-overlapping models. This equivalence between the two sets of models greatly facilitates the specification of latent roots and vectors.
- Published
- 1973
36. Combinatorial methods in the theory of dams
- Author
-
Lajos Takács
- Subjects
Statistics and Probability ,Algebra ,Combinatorial analysis ,010104 statistics & probability ,Mathematical model ,General Mathematics ,010102 general mathematics ,Applied mathematics ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences ,Mathematics - Abstract
In this paper we shall be concerned with two mathematical models of infinite dams. In the first model independent random inputs occur at regular time intervals and in the second model independent random inputs occur in accordance with a Poisson process. The first model has already been studied by Gani, Yeo and others, and the second model by Gani and Prabhu, Gani and Pyke, Kendall, and others. For both models we shall find explicit formulas for the distribution of the content of the dam and that of the lengths of the wet periods and dry periods. The proofs are elementary and based on two generalizations of the classical ballot theorem.
- Published
- 1964
37. On probability properties of measures of random sets and the asymptotic behavior of empirical distribution functions
- Author
-
Gedalia Ailam
- Subjects
Discrete mathematics ,Statistics and Probability ,Asymptotic analysis ,General Mathematics ,010102 general mathematics ,Asymptotic distribution ,V-statistic ,Moment-generating function ,Empirical distribution function ,01 natural sciences ,010104 statistics & probability ,Joint probability distribution ,Applied mathematics ,Probability distribution ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
Probability properties of the measure of the union of random sets have theoretical as well as practical importance (David (1950), Garwood (1947), Hemmer (1959)). In the present paper we derive asymptotic properties of the distributions of these measures and apply the derived properties to the investigation of the asymptotic behavior of empirical distribution functions. Thus, an asymptotic distribution function for the relative lengths of steps in the empirical distribution function is obtained.
- Published
- 1968
38. Weak Containment and Induced Representations of Groups
- Author
-
J. M. G. Fell
- Subjects
Pure mathematics ,Dual space ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Group algebra ,Locally compact group ,01 natural sciences ,Unitary state ,Set (abstract data type) ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Special case ,Topology (chemistry) ,Mathematics - Abstract
Let G be a locally compact group and G† its dual space, that is, the set of all unitary equivalence classes of irreducible unitary representations of G. An important tool for investigating the group algebra of G is the so-called hull-kernel topology of G†, which is discussed in (3) as a special case of the relation of weak containment. The question arises: Given a group G, how do we determine G† and its topology? For many groups G, Mackey's theory of induced representations permits us to catalogue all the elements of G†. One suspects that by suitably supplementing this theory it should be possible to obtain the topology of G† at the same time. It is the purpose of this paper to explore this possibility. Unfortunately, we are not able to complete the programme at present.
- Published
- 1962
39. Mixed Problems for Linear Systems of first Order Equations
- Author
-
G. F. D. Duff
- Subjects
Partial differential equation ,Dynamical systems theory ,Independent equation ,Differential equation ,General Mathematics ,010102 general mathematics ,Relaxation (iterative method) ,System of linear equations ,01 natural sciences ,Simultaneous equations ,0103 physical sciences ,Applied mathematics ,010307 mathematical physics ,Boundary value problem ,0101 mathematics ,Mathematics - Abstract
A mixed problem in the theory of partial differential equations is an auxiliary data problem wherein conditions are assigned on two distinct surfaces having an intersection of lower dimension. Such problems have usually been formulated in connection with hyperbolic differential equations, with initial and boundary conditions prescribed. In this paper a study is made of the conditions appropriate to a system of R linear partial differential equations of first order, in R dependent and N independent variables.
- Published
- 1958
40. On Wald's equations in continuous time
- Author
-
W. J. Hall
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Sobel operator ,Variance (accounting) ,Wald test ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Identity (mathematics) ,Discrete time and continuous time ,Wiener process ,Stopping time ,symbols ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,Variety (universal algebra) ,Mathematics - Abstract
Various formulas of Wald relating to randomly stopped sums have well known continuous-time analogs, holding in particular for Wiener processes. However, sufficiently general forms of most of these do not appear explicitly in the literature. Recent papers by Robbins and Samuel (1966) and by Brown (1969) provide general results on Wald's equations in discrete time and these are here extended (Theorems 2 and 3) to Wiener processes and other homogeneous additive processes, that is, continuous-time processes with stationary independent increments. We also give an inequality (Theorem 1) related to Wald's identity in continuous time, and we derive, as corollaries of Wald's equations, bounds on the variance of an arbitrary stopping time. The Wiener process versions of these results find application in a variety of stopping problems. Specifically, all are used in Hall ((1968), (1969)); see also Bechhofer, Kiefer, and Sobel (1968), Root (1969), and Shepp (1967).
- Published
- 1970
41. On a stochastic integral equation of the Volterra type in telephone traffic theory
- Author
-
W. J. Padgett and Chris P. Tsokos
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Type (model theory) ,01 natural sciences ,Volterra integral equation ,Stochastic integral equation ,010104 statistics & probability ,symbols.namesake ,symbols ,Applied mathematics ,Three-phase traffic theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Abstract
In mathematical models of phenomena occurring in the general areas of the engineering, biological, and physical sciences, random or stochastic equations appear frequently. In this paper we shall formulate a problem in telephone traffic theory which leads to a stochastic integral equation which is a special case of the Volterra type of the form where: (i) ω∊Ω, where Ω is the supporting set of the probability measure space (Ω,B,P); (ii) x(t; ω) is the unknown random variable for t ∊ R +, where R + = [0, ∞); (iii) y(t; ω) is the stochastic free term or free random variable for t ∊ R +; (iv) k(t, τ; ω) is the stochastic kernel, defined for 0 ≦ τ ≦ t < ∞; and (v) f(t, x) is a scalar function defined for t ∊ R + and x ∊ R, where R is the real line.
- Published
- 1971
42. Time dependence of queues with semi-Markovian services
- Author
-
Erhan Çinlar
- Subjects
Statistics and Probability ,Service time ,General Mathematics ,010102 general mathematics ,Process (computing) ,Markov process ,Queueing system ,Poisson distribution ,01 natural sciences ,symbols.namesake ,010104 statistics & probability ,symbols ,Applied mathematics ,Transient (computer programming) ,0101 mathematics ,Statistics, Probability and Uncertainty ,Finite set ,Queue ,Mathematics - Abstract
A single server queueing system with Poisson input is considered. There are a finite number of types of customers and the service time of the nth customers depends on the types of the nth and the (n – l)th customers. The time dependence of the queue size process will be studied, (it will be clear how the methods of the paper can be applied to other processes of interest,) and limiting as well as transient results will be given.
- Published
- 1967
43. Laplace Transformations of Distributions
- Author
-
J. L. B. Cooper
- Subjects
Laplace transform ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In a previous paper (2), I discussed conditions in order that a holomorphic function f(w) be a Laplace transform of a function F(t) such that, for some a, F(t)e-at ∈ Lp(0, ∞). These conditions involved (a) the behaviour of integral transforms of the values of the function on a vertical line w = const, and (b) conditions involving the order of magnitude of the function at infinity. The present article discusses the corresponding question for Laplace transforms of distributions.
- Published
- 1966
44. Selective interaction of a poisson and renewal process: the dependency structure of the intervals between responses
- Author
-
A. J. Lawrance
- Subjects
Statistics and Probability ,General Mathematics ,010102 general mathematics ,Process (computing) ,Type (model theory) ,Poisson distribution ,Stationary point ,01 natural sciences ,symbols.namesake ,010104 statistics & probability ,Joint probability distribution ,Asynchronous communication ,Statistics ,symbols ,Applied mathematics ,Renewal theory ,0101 mathematics ,Statistics, Probability and Uncertainty ,Mathematics ,Event (probability theory) - Abstract
This paper studies the dependency structure of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b)). A new approach to the intervals between events in a stationary point process, based on the idea of an average event, is introduced. Average event initial conditions (as opposed to equilibrium initial conditions previously determined) for the renewal inhibited Poisson process are obtained and event stationarity of the resulting response process is established. The joint distribution and correlation between pairs of contiguous synchronous intervals is obtained; further, the joint distribution of non-contiguous pairs of synchronous intervals is derived. Finally, the joint distributions of pairs of contiguous synchronous and asynchronous intervals are related, and a similar but more general stationary point result is conjectured.
- Published
- 1971
45. A General Perron Integral
- Author
-
P. S. Bullen
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,Mathematical analysis ,Applied mathematics ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Perron's formula ,Mathematics - Abstract
In this paper integrals are considered from the point of view of inverting differential operators. In order to do this it is necessary to introduce integrals more general than the Lebesgue integral and these integrals turn out to have other interesting properties (6, 7, 12). The integral introduced here is defined in the setting of axiomatic potential theory (2, 4). By defining it as generally as possible it not only includes the James P2-integral but inverts many of the standard second-order differential operators.
- Published
- 1965
46. Approximation of Piecewise Continuous Functions by Quotients of Bounded Analytic Functions
- Author
-
Donald Sarason
- Subjects
General Mathematics ,010102 general mathematics ,Mathematical analysis ,01 natural sciences ,Bounded function ,0103 physical sciences ,Analytic capacity ,Piecewise ,Applied mathematics ,Uniform boundedness ,010307 mathematical physics ,0101 mathematics ,Quotient ,Analytic function ,Mathematics - Abstract
This paper concerns a certain subalgebra of the Banach algebra of complex valued, essentially bounded, Lebesgue measurable functions on the unit circle in the complex plane (denoted here by L∞). My interest in this subalgebra was prompted by a question of R. G. Douglas. Let H∞ denote the space of functions in L∞ whose Fourier coefficients with negative indices vanish (equivalently, the space of boundary functions for bounded analytic functions in the unit disk). Douglas [5] has asked whether every closed subalgebra of L∞ containing H∞ is determined by the functions in H∞ that it makes invertible. More precisely, is such an algebra generated by H∞ and the inverses of the functions in H∞ that are invertible in the algebra? An affirmative answer is known for L∞ itself and for certain subalgebras of L∞ recently studied by Davie, Gamelin, and Garnett [3]. At the time of this writing, no algebra is known for which the above question can be answered negatively.
- Published
- 1972
47. Some Distribution Problems of Order Statistics From Exponential and Power Function Distributions1
- Author
-
D. G. Kabe
- Subjects
Exponential-logarithmic distribution ,General Mathematics ,010102 general mathematics ,Order statistic ,010103 numerical & computational mathematics ,01 natural sciences ,Laplace distribution ,Exponentially modified Gaussian distribution ,Exponential family ,Statistics ,Generalized beta distribution ,Gamma distribution ,Applied mathematics ,0101 mathematics ,Natural exponential family ,Mathematics - Abstract
This paper gives alternative straightforward and simpler proofs of some of the results of Laurent's [10], and Likes' [11], [13]. The derivation of the results is simplified by using the theory of Dirichlet's multiple integral and the transformation used to derive this multiple integral. Some applications of Dirichlet's transformation to order statistic theory from gamma, and normal populations, have been already given by Kabe [7].
- Published
- 1968
48. Physical nearest-neighbour models and non-linear time-series. II Further discussion of approximate solutions and exact equations
- Author
-
M. S. Bartlett
- Subjects
Statistics and Probability ,010104 statistics & probability ,Nonlinear system ,Series (mathematics) ,General Mathematics ,010102 general mathematics ,Nearest neighbour ,Applied mathematics ,Exact differential equation ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences ,Mathematics - Abstract
The approximate two- and three-dimensional solutions for spatial correlations, using the non-linear time-series approach for nearest-neighbour systems developed in my previous paper, are further discussed. Orthogonal expansions for the correlation functions are also developed which determine with this approach, though so far only in principle, the exact solutions.
- Published
- 1972
49. A stochastic model for two interacting populations
- Author
-
Niels G. Becker
- Subjects
Statistics and Probability ,Birth and death process ,education.field_of_study ,Component (thermodynamics) ,Stochastic modelling ,Differential equation ,General Mathematics ,010102 general mathematics ,Population ,Type (model theory) ,01 natural sciences ,Birth–death process ,010104 statistics & probability ,Nonlinear system ,Applied mathematics ,0101 mathematics ,Statistics, Probability and Uncertainty ,education ,Mathematics - Abstract
To explain the growth of interacting populations, non-linear models need to be proposed and it is this non-linearity which proves to be most awkward in attempts at solving the resulting differential equations. A model with a particular nonlinear component, initially proposed by Weiss (1965) for the spread of a carrierborne epidemic, was solved completely by different methods by Dietz (1966) and Downton (1967). Immigration parameters were added to the model of Weiss and the resulting model was made the subject of a paper by Dietz and Downton (1968). It is the aim here to further generalize the model by introducing birth and death parameters so that the result is a linear birth and death process with immigration for each population plus the non-linear interaction component. Consider two populations which we refer to as type 1 and type 2. We suppose that at time t there are Xi(t) individuals of type i present in the habitat and X;(0) = min is the initial number. Further let the birth, death and immigration rates for population i be i, ~i and vi respectively. It is then supposed that the probability of increasing the type i population by one individual during the time
- Published
- 1970
50. On dams with Markovian inputs
- Author
-
A. G. Pakes
- Subjects
Statistics and Probability ,010104 statistics & probability ,symbols.namesake ,General Mathematics ,010102 general mathematics ,symbols ,Applied mathematics ,Markov process ,0101 mathematics ,Statistics, Probability and Uncertainty ,01 natural sciences ,Mathematics - Abstract
Some recent work on discrete time dam models has been concerned with special cases in which the input process is a Markov chain whose transition probabilities, p ij , are given by where A(·) and B(·) are probability generating functions (p.g.f.'s). In this paper we obtain some results for the general situation. The convergence norm of the matrix [p ij xj] is found and the results are used to obtain the p.g.f. of the first emptiness time. Distributions of the dam content are obtained and conditions are found for the existence of their limits. The p.g.f. of this distribution is so complicated that its identification in any special case is extremely difficult, or even impossible. Thus useful approximations are needed; we obtain a ‘heavy traffic’ limit theorem which suggests that under certain circumstances the limiting distribution can be approximated by an exponential distribution.
- Published
- 1973
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