Branched Regular Curve Families and Finite Asymptotic Values

1964 ◽  
Vol 86 (3) ◽  
pp. 493
Author(s):  
David V. V. Wend
2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Hynek Bednář ◽  
Aleš Raidl ◽  
Jiří Mikšovský

Initial errors in weather prediction grow in time and, as they become larger, their growth slows down and then stops at an asymptotic value. Time of reaching this saturation point represents the limit of predictability. This paper studies the asymptotic values and time limits in a chaotic atmospheric model for five initial errors, using ensemble prediction method (model’s data) as well as error approximation by quadratic and logarithmic hypothesis and their modifications. We show that modified hypotheses approximate the model’s time limits better, but not without serious disadvantages. We demonstrate how hypotheses can be further improved to achieve better match of time limits with the model. We also show that quadratic hypothesis approximates the model’s asymptotic value best and that, after improvement, it also approximates the model’s time limits better for almost all initial errors and time lengths.


1969 ◽  
Vol 9 (04) ◽  
pp. 403-411 ◽  
Author(s):  
B.K. Sinha ◽  
Harvey T. Kennedy

Abstract Recommendations are made for obtaining consistent and reproducible test data on drilling fluids having identical composition. Previously, such a procedure has been difficult to accomplish even when the fluids were mixed in similar equipment. A survey of work in this area indicates that previous methods have been unsatisfactory because previous methods have been unsatisfactory because (1) the muds are extremely sensitive to the duration and violence of agitation during a normal mixing routine, and (2) gelling of the muds occurs before the properties can reach constant values. This gelling is caused by water evaporation resulting from the increase in temperature associated with the agitation. The work shows that these problems largely can be overcome by (1) agitating the constituents of the drilling fluid more vigorously, (2) maintaining a fairly constant temperature, and(3) Protecting the fluid from evaporation. When these steps are followed, the fluid properties approach asymptotic values that do not change by prolonged or accelerated agitation or by aging for a month. The time required to reach asymptotic values or a stabilized state is from 2 to 6 hours and is a function of the mud composition. Introduction Preparation of drilling fluids in the laboratory to determine their suitability to meet specific drilling requirements or to serve as a base fluid to evaluate the effectiveness of thinners, dispersants or other additives normally begins with combining measured quantities of the constituents and stirring them for a short time in a low-speed mixer. This is done to obtain a uniform mixture and to hydrate clays. Then the fluid is further agitated in a higher-speed device (Hamilton Beach mixer or Waring blender) to disperse more thoroughly and clay particles The biggest obstacle in the laboratory investigation of drilling fluids has been the lack of a method of producing a mixture by which reproducible results of the measured properties could be obtained. Numerous investigators have encountered this difficulty. Prior to 1929, density was the only property of mud that customarily was measured. The use of Wyoming bentonite on a large scale after 1929 was mainly responsible for the development of more elaborate testing procedures and for the application of the principles of colloid chemistry to the drilling fluids. Ambrose and Loomis in 1931 were among the first to recognize the plastic flow characteristics of drilling fluids, although Bingham in 1916 had observed The same phenomenon with dilute clay suspensions. Marsh introduced the Marsh funnel for field testing in 1931. By this time, non-Newtonian characteristics of drilling fluids were established. The Stormer and MacMichael viscometers were used to study the rheological properties of the fluids. In the 1930's and early 1940's, the work conducted by several investigators contributed toward a better understanding of drilling fluids. In the mid 1930's, fluid-loss and the associated mud-cake-forming properties of drilling fluids were recognized as important to the behavior of these fluids. The other properties of drilling fluids, including gel strength, pH, and sand content soon were recognized. In 1937, API published its first recommended procedure for test methods. Since that time, these procedures have been revised periodically. The latest edition, RP-13B, was published in 1961 However, in spite of the recognized need for a method of mixing that provides drilling fluids with stabilized properties, no such method previously has been described. SPEJ P. 403


1996 ◽  
Vol 20 (1) ◽  
pp. 67-77 ◽  
Author(s):  
Hideyuki Narumi ◽  
Hideaki Kita ◽  
Haruo Hosoya

Fluids ◽  
2020 ◽  
Vol 5 (4) ◽  
pp. 232
Author(s):  
Panos N. Papanicolaou ◽  
George C. Stamoulis

Turbulent round buoyant jets and fountains issuing vertically into a linearly density-stratified calm ambient have been investigated in a series of laboratory experiments. The terminal (steady-state) height of rise and the mean elevation of subsequent horizontal spreading have been measured in positively buoyant jets (at source level), including pure momentum jets and plumes, as well in momentum-driven negatively buoyant jets (fountains). The results from experiments confirmed the asymptotic analysis that was based on dimensional arguments. The normalized terminal height and spreading elevation with respect to the elevation of injection of momentum-driven (positively) buoyant jets and fountains attained the same asymptotic values. The numerical results from the solution of entrainment equations, using an improved entrainment coefficient function, confirmed the results related to buoyancy dominant flows (plumes), while their predictions in momentum-driven flows were quite low if compared to measurements.


1976 ◽  
Vol 28 (6) ◽  
pp. 1210-1215
Author(s):  
P. M. Gauthier ◽  
J. S. Hwang

Let ƒ be a function meromorphic in the finite complex plane C. If for some number θ, 0 ≦ θ < 2 π, the family, fr(z) = f(reθz), is not normal at z = 1, then the ray arg z = θ is called a Julia ray. Such a ray has the property that in every sector containing it, F assumes every value infinitely often with at most two exceptions. Many authors have taken this property as the definition of a Julia ray.


1973 ◽  
Vol 25 (4) ◽  
pp. 812-819
Author(s):  
David C. Haddad

A function denned in a domain D is n-valent in D if f(z) — w0 has at most n zeros in D for each complex number w0. Let denote the class of nonconstant, holomorphic functions f in the unit disc that are n-valent in each component of the set . MacLane's class is the class of nonconstant, holomorphic functions in the unit disc that have asymptotic values at a dense subset of |z| = 1.


1967 ◽  
Vol 29 ◽  
pp. 7-18 ◽  
Author(s):  
F. Bagemihl

Let Γ be the unit circle and D be the open unit disk in the complex plane, and denote the Riemann sphere by Ω. By an arc at a point ζ∈Γ we mean a continuous curve such that |z(t)| < 1 for 0 ≦ t < 1 and . A terminal subarc of an arc Λ at ζ is a subarc of the form z = z (t) (t0 ≦ t < 1), where 0 ≦ t0<1. Suppose that f(z) is a meromorphic function in D. Then A(f) denotes the set of asymptotic values of f; and if ζ∈Γ, then C(f, ζ) means the cluster set of f at ζ and is the outer angular cluster set of f at ζ (see [13]).


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