Possible Rate Of Growth Research Articles

Near equilibrium fluctuations for supermarket models with growing choices

We consider the supermarket model in the usual Markovian setting where jobs arrive at rate nλn for some λn>0, with n parallel servers each processing jobs in its queue at rate 1. An arriving job joins the shortest among dn≤n randomly selected service queues. We show that when dn→∞ and λn→λ∈(0,∞), under natural conditions on the initial queues, the state occupancy process converges in probability, in a suitable path space, to the unique solution of an infinite system of constrained ordinary differential equations parametrized by λ. Our main interest is in the study of fluctuations of the state process about its near equilibrium state in the critical regime, namely when λn→1. Previous papers, for example, (Stoch. Syst. 8 (2018) 265–292) have considered the regime dn nlogn→∞ while the objective of the current work is to develop diffusion approximations for the state occupancy process that allow for all possible rates of growth of dn. In particular, we consider the three canonical regimes (a) dn/n→0; (b) d n/n→c∈(0,∞) and, (c) d n/n→∞. In all three regimes, we show, by establishing suitable functional limit theorems, that (under conditions on λ n) fluctuations of the state process about its near equilibrium are of order n−1/2 and are governed asymptotically by a one-dimensional Brownian motion. The forms of the limit processes in the three regimes are quite different; in the first case, we get a linear diffusion; in the second case, we get a diffusion with an exponential drift; and in the third case we obtain a reflected diffusion in a half space. In the special case dn/(nlogn)→∞, our work gives alternative proofs for the universality results established in (Stoch. Syst. 8 (2018) 265–292).

Order of growth of distributionally irregular entire functions for the differentiation operator

We study the rate of growth of entire functions that are distributionally irregular for the differentiation operator D. More specifically, given and , where , we prove that there exists a distributionally irregular entire function f for the operator D such that its p-integral mean function grows not more rapidly than . This completes related known results about the possible rates of growth of such means for D-hypercyclic entire functions. It is also obtained, the existence of dense linear submanifolds of all whose ***nonzero vectors are D-distributionally irregular and present the same kind of growth. Furthermore, the D-distributional irregularity in weighted Banach spaces of entire functions is analysed.

Modeling the optimum conditions for the formation of defect-free CVD graphene on copper melt

The nucleation and growth of nuclei of graphene (graphene islets) on the surfaces of copper melts during catalytic CVD, i.e., the catalytic decomposition of a gas-phase carbon support, is considered. It is shown that on a copper melt the optimum combination of conditions for the preservation of islets with almost perfect hexagonal shape and the necessary conditions of the CVD-process are reached at the same time. The average distance between the islets and the dimensionless parameter that determines changes in the shape of islets is calculated. The maximum rate of decomposition of the carbon support at which this parameter simultaneously promotes the growth of defect-free islets and the maximum possible rate of growth of the graphene monolayer is determined.

On a devil’s staircase associated to the joint spectral radii of a family of pairs of matrices

The joint spectral radius of a finite set of real $d \times d$ matrices is defined to be the maximum possible exponential rate of growth of products of matrices drawn from that set. In previous work with K.\~G.\~Hare and J.\~Theys we showed that for a certain one-parameter family of pairs of matrices, this maximum possible rate of growth is attained along Sturmian sequences with a certain characteristic ratio which depends continuously upon the parameter. In this note we answer some open questions from that paper by showing that the dependence of the ratio function upon the parameter takes the form of a Devil's staircase. We show in particular that this Devil's staircase attains every rational value strictly between $0$ and $1$ on some interval, and attains irrational values only in a set of Hausdorff dimension zero. This result generalises to include certain one-parameter families considered by other authors. We also give explicit formulas for the preimages of both rational and irrational numbers under the ratio function, thereby establishing a large family of pairs of matrices for which the joint spectral radius may be calculated exactly.

Chaos of the Differentiation Operator on Weighted Banach Spaces of Entire Functions

Motivated by recent work on the rate of growth of frequently hypercyclic entire functions due to Blasco, Grosse-Erdmann and Bonilla, we investigate conditions to ensure that the differentiation operator is chaotic or frequently hypercyclic on generalized weighted Bergman spaces of entire functions studied by Lusky, whenever the differentiation operator is continuous. As a consequence we partially complete the knowledge of possible rates of growth of frequently hypercyclic entire functions for the differentiation operator.

The Rates of Growth of the Galton–Watson Process in Varying Environments

Let {Zn } be a supercritical Galton–Watson process in varying environments, and W be the limit of the non-negative martingale {Zn /EZ n }. Under a condition which ensures that W is not identically equal to zero we give an upper bound on the possible rates of growth of the process on the set {W = 0}, and find a sufficient condition for the process to have only one rate of growth. We also give an example of a process whose offspring distributions have bounded pth moments, for some p > 1, and which has an infinite number of rates of growth.

The Rates of Growth of the Galton–Watson Process in Varying Environments

Let {Zn} be a supercritical Galton–Watson process in varying environments, and W be the limit of the non-negative martingale {Zn/EZn}. Under a condition which ensures that W is not identically equal to zero we give an upper bound on the possible rates of growth of the process on the set {W = 0}, and find a sufficient condition for the process to have only one rate of growth. We also give an example of a process whose offspring distributions have bounded pth moments, for some p > 1, and which has an infinite number of rates of growth.

The supercritical Galton-Watson process in varying environments

Let { Z n } be a supercritical Galton-Watson process in varying environments. It is known that Z n when normed by its mean EZ n converges almost surely to a finite random variable W. It is possible, however, for such a process to exhibit more than one rate of growth so that in particular { W > 0} need not coincide with { Z n → ∞}. Here a natural sufficient condition is given which ensures that this cannot happen. Under a weaker condition it is shown that the possible rates of growth cannot differ very much in that { Z n EZ n } 1 n → 1 on { Z n → ∞}.

On possible rates of growth of age-dependent branching processes with immigration

It is shown that if ϕ is a given function out of a large class satisfying a certain regularity condition, then a supercritical age-dependent branching process {Z(t)} exists with deterministic immigration and given life-length and family-size distributions such that Z(t)/(eat ϕ(t)) converges in probability to a non-zero constant, a being the appropriate Malthusian parameter.As an easy corollary one discovers the asymptotic behaviour of some processes with random immigration.

On possible rates of growth of age-dependent branching processes with immigration

It is shown that if ϕ is a given function out of a large class satisfying a certain regularity condition, then a supercritical age-dependent branching process {Z(t)} exists with deterministic immigration and given life-length and family-size distributions such that Z(t)/(eat ϕ(t)) converges in probability to a non-zero constant, a being the appropriate Malthusian parameter. As an easy corollary one discovers the asymptotic behaviour of some processes with random immigration.

Investment Policy for Economic Development: Some Lessons of the Communist Experience

Communist experience in the field of investment policy for economic development is particularly instructive in connection with such problems as the determination of the rate of investment which results in the highest possible rate of growth; balanced versus unbalanced development; and the role of the rate of interest in the efficient allocation of investment funds. Our purpose is to examine this experience in relation to the first of the above problems. Since the three problems overlap, however, the paper relates to some extent to the other two.It is suggested that the investment experience of Communist countries is relevant for developing countries in general. Although the results of various policies have been conditioned by the specific institutional framework of the Communist countries, they seem to depend also on the logic of the process of economic growth and of the basic economic problem of allocating scarce resources. Communist economists have been forced to admit that some aspects of economic development are as difficult under socialism as they are under capitalism, although until recently this fact was consistently denied. It is also possible, as has been suggested by Professor T. Haavelmo, that the process of economic growth can be seen most clearly when it is studied in a centrally planned economy, where major complications of a market economy are absent. Moreover, some features of the Communist system reflect the leaders' desire to achieve fast economic growth, and such measures as a high rate of forced saving and a large volume of public investment may be found in any country trying to accelerate its development.

Nutrition of the bacon pig XVII. The nutritive value of condensed fish solubles

Investigations have been carried out in this department in recent years with the object of establishing the minimum protein requirements of the bacon pig. The results were recently summarized in thisJournal(Woodman & Evans, 1948). The minimum protein standards recorded in that communication are consistent with the attainment of the maximum rate of growth compatible with the net energy content of the diet. The basal foods employed consisted throughout of 2 parts of barley meal and 1 part of middlings (fine bran), together with a little lucerne meal and minerals. Briefly, it was found that such a diet, when supplemented with 7% of white-fish meal, supplied the minimum amount of digestible protein required for the quickest possible rate of growth between weaning and 90 lb. live weight, when the level of feeding was such as is shown in the feeding chart (see Table 7). It was also found unnecessary to include any protein concentrate in the diet beyond 90 lb. live weight to ensure the maximum rate of growth over this later period. The requirements were based on the results of statistically designed growth trials, and were confirmed by nitrogen-balance determinations carried out in metabolism crates. It was found that nitrogen retention was just as favourable on the basal diet supplemented with 7% of white-fish meal as when the protein level was increased by feeding higher amounts offish meal.

Nutrition of the bacon pig XIV. The determination of the relative supplemental values of vegetable protein (extracted, decorticated ground-nut meal) and animal protein (white-fish meal)

Nutrition of the bacon pig XIV. The determination of the relative supplemental values of vegetable protein (extracted, decorticated ground-nut meal) and animal protein (white-fish meal)

Organic productivity of an Atoll

The production of organic matter by phytoplankton in the waters in and around the northern Marshall Islands is extremely low. The amount of plankton swept across the reefs of these islands by the wind‐driven currents is grossly inadequate to support the animals living on the reefs. The reef community, consisting of attached algae and animals, is self‐supporting; all the algae produce at least as much organic matter as all the animals consume. The rate of production of organic matter by an atoll as a whole ‐ water, reef, and bottom ‐ is several times as high as that of the surrounding sea, and permits maintenance of a marine population considerably denser than that of the ocean. The maximum possible rate of growth of the reef is found to be 1.4 cm per year.

On the Possible Rate of Growth of an Analytic Function

On the Possible Rate of Growth of an Analytic Function

On the possible rate of growth of an analytic function

Introduction('). The present paper deals with a number of diverse topics, ranging from purely topological considerations, through a general theory of possible distributions of values of an analytic function, to more special theorems on simultaneous expansions of an infinity of analytic functions. No unifying principle is presented as an excuse for treating such a variety of subjects; but there is a slight sequence of argument running throughout. The parts of the paper were actually written in reverse order. The initial investigation (Part III) was started as an attempt to generalize, from n to infinity, a known theorem(2) on simultaneous expansions of n analytic functions. This generalization was found to depend on an affirmative answer to the following question on level curves of an analytic function: Given any sequence of points a1, a2, . , in the complex plane, which has the point at infinity as its only limit point, does there exist an analytic function with a level curve C such that C contains a distinct branch about each given point which separates that point from all the other points? This was, in turn, made to depend on a certain problem relative to the possible rate of growth of an integral function. It was shown long, ago by Poincare(3), Borel(4), and others that an integral function may be made to grow arbitrarily fast along the real axis or along other lines or curves extending to infinity. Our problem was to obtain an affirmative answer to the following related question: Does there exist a sequence of regions Si, S2, . , with ai interior to Si, such that, no matter how fast the sequence of numbers m1, M2, increases there will be an integraL function f(z) for which