Showing papers on "Bernoulli's principle published in 1976"

NewK-automorphisms and a problem of Kakutani

Abstract: A property is introduced, for 1-1 measure-preserving transformations of probability spaces, calledloose Bernoulliness (LB), which is invariant under taking factors, inducing, and tower-building. It amounts to replacing, in Ornstein’s definition ofvery weak Bernoulli, the Hamming distance on strings by a coarser metric. The main result is the construction of a transformationT0 which is ergodic and of entropy 0 butnot LB. On the other hand, any irrational rotationis LB. Consequently, the equivalence relation generated by inducing and tower-building (which I callKakutani equivalence, and the Russians callmonotone equivalence) has at least two distinct equivalence classes among the ergodic entropy zero transformations. A similar situation exists for ergodic positive-entropy transformations: on the one hand, any Bernoulli shift is LB, while on the other hand a non LBK-automorphism\(\hat T_0 \) can be made by skewingT0 over a Bernoulli base.

Application of Spencer’s Ideal Soil Model to Granular Materials Flow

Abstract: In 1964, Spencer proposed a model for plane deformation of soils based upon the concept that the strain at any point may be considered as the resultant of simple shears on the two surface elements where Coulomb’s yield condition is met. Gravitation and acceleration terms were neglected in his full field equations. These terms are included in the present treatment, however, since they play an essential role in granular materials flow problems. It is shown that the field equations remain hyperbolic, and the characteristic equations are derived. In addition, a streamline equation, similar to Bernoulli’s classical equation for fluid flow, is derived and used together with the continuity equation to obtain one-dimensional approximate solutions to some typical hopper and chute problems. A solution is obtained for the nonsteady flow from a wedge-shaped hopper when the gate is suddenly opened, including an equation for the minimum slope necessary to prevent arching. Another solution is obtained for the profile of a hopper which has constant wall pressure. Still another solution is obtained for the relationship between the height of a chute and its exit velocity, and it suggests that the maximum trajectory usually is obtained with a horizontal exit since an upward-sloping exit requires a velocity jump at the minimum point in the chute, similar to a hydraulic jump in fluid flow. All of these solutions are ideal in the sense that they include no wall resistance to the flow, and therefore represent maximum flow rates.

Low Velocity Transverse Normal Impact of Graphite Epoxy Composite Laminates.

Abstract: A theoretical-experimental investigation has been conducted to deter mine the response of composite structural elements to low speed, trans verse impact. Experimental results of this investigation showed that as the velocity of impact is increased it requires less impactor kinetic energy to produce damage.A structural dynamic response model consisting of a uniform Bernoulli- Euler beam and a rigid impactor mass was employed. Displacement and velocity distributions were assumed as initial conditions and the solution was expanded in the eigenfunctions of the free vibration problem using Galerkins' method.Good correlation of experimental results with the results of this analyt ical structural dynamic model have been obtained over the velocity range from 0 to 6 m/s.

Relatively finitely determined implies relatively very weak Bernoulli

Abstract: In the isomorphism theory of Bernoulli shifts, the or Hamming distance plays an important role. Two finite state stationary processes are close in the metric if, on the average, strings from one process can be matched with strings from the other process so that their outputs agree with high frequency. Since such agreement must hold in the arbitrarily distant future, it might at first seem difficult to control the pairing.

Hemodynamic Phenomena in Fusiform Aneurysms - I: Steady Flow Conditions

Abstract: Pressure and flow distributions are reported at different steady flows for four latex fusiform aneurysm models. The accuracy of results by pitot-static techniques is examined in terms of experimental evidence and theoretical considerations. Results indicate that pressure distributions are associated with the Reynolds number, the degree of wall divergence and the degree of dilation of the sac, and that these parameters are interactive. The Reynolds number was found to have some predictive value for pressure distribution in the aneurysm, but the Bernoulli principle, based on geometrical variation and laminar flow, was usually inapplicable. Also, disturbed flow (eddies, reversal, turbulence) can occur as functions of the interactive parameters.

On endomorphisms of a solenoid

Abstract: Geometrically simple Bernoulli generators are constructed for certain ergodic endomorphisms of solenoids. An arbitrary ergodic solenoidal group automorphism is obtained as the limit of a sequence of such Bernoulli factors and hence, by a theorem of D. S. Ornstein, must be measuretheoretically isomorphic to a Bernoulli shift. In his survey paper [6], B. Weiss stated that, using Y. Katznelson's methods, he can prove that every ergodic automorphism of a solenoid is isomorphic to a Bernoulli shift. The aim of this note is to give an alternative proof of this result, with a partial result in the endomorphism case. The methods used are similar to those of L. M. Abramov [1], who used geometrically simple generating partitions in order to compute the entropy of certain solenoidal automorphisms. A comparison will show that Abramov's generators are refinements of the Bernoulli generators exhibited in ?2 below. The author would like to acknowledge S. M. Rudolfer, who supervised this work, and R. K. Thomas who suggested the problem to us. For brevity, a working knowledge of measure theory, ergodic theory and topological groups is assumed in what follows, apart from the following fundamental definition of a Bernoulli shift. A measure preserving map 4 from a separable measure space (X, ji) to itself will be called a (one-sided) Bernoulli shift if there is a measurable partition P of X (called a Bernoulli generator for 4) such that (i) {(7-iP}, i _ 0, is an independent family of partitions, and (ii) Vi,0=o47P is the point partition of X. If 4 is invertible, (ii) becomes (ii)' V?? 4. P is the point partition of X. 1. Details of solenoids are well documented (see e.g. [1], [3] and [2, Chapter VIII). The following brief characterisation will be subsequently useful. DEFINITION 1.1. Let G be a noncyclic subgroup of the discrete additive group Q of rational numbers. The character group I of G, called a (onedimensional) solenoid, is a compact, separable, commutative topological group. PROPOSITION 1.2. Let a = (a,, a2.... ) be a sequence of integers ai _ 2. Let Ga be the subgroup of Q generated by the elements nn71 1/ai, for n _ 1. Up to isomorphism, every additive subgroup G of Q, as in 1.1, can be represented as a Ga for some such a. If a is the constant sequence on some integer a, then Ga is the group of a-ary rationals, denoted Ga. Received by the editors December 4, 1974. AMS (MOS) subject classifications (1970). Primary 28A65; Secondary 22D40, 22D45. Key words, and phrases. Solenoid, topological group endomorphism, Bernoulli shift, natural extension, generalised Bernoulli shift. ? American Mathematical Society 1976 69 This content downloaded from 157.55.39.112 on Wed, 07 Sep 2016 05:17:32 UTC All use subject to http://about.jstor.org/terms

On the generalized Bernoulli construction. II. Extension to two and three dimensions

Abstract: The generalized Bernoulli construction is used to study the equation of state of rigid particles in two and three dimensions. A system of hard squares is considered in detail, and results for hard cubes, disks, and spheres are also presented. The discussion permits the introduction of the mean‐free‐path problem in the equation of state, and the result of the virial theorem of Clausius is obtained for these systems. This heuristic method can be used to discuss the equation of state at low and high densities without the use of formal theories.