Uniform persistence

One can work a little harder, using the full theory of Appendix D, and conclude uniform persistence. However, uniform persistence will not be needed in this chapter. 

Proposition 6.2. The system (2.4) is uniformly persistent if and only if the rest points E and E2 are unstable. 

Proof. If El and E2 are unstable, then dissipativeness and uniform persistence (previous proposition) yield the existence of an interior rest point for -k(x, t) (Theorem D.3). If E exists then it is unique and has all eigenvalues negative (Lemma 5.2). Suppose that exists and that Ei is asymptotically stable. Then, since asymptotically stable, Theorem E.l contradicts the uniqueness of E. A similar argument applies if E2 is asymptotically stable. Note that the computations leading up to Lemma 5.1 explicitly determine the signs of the eigenvalues for linearization about Ei and E2. ... 

We can now state the main result of this chapter. It provides sufficient conditions for the coexistence of the two populations in the gradostat and ensures that 2 contains a positive rest point . In fact, it guarantees that the two populations are uniformly persistent in the sense of Appendix D. Briefly, Theorem 4.4 states that coexistence holds if each population can successfully invade its competitor s rest point. 

It is important to point out that the box P is a fixed positive distance from the two coordinate axes. Because every orbit of P corresponding to a positive starting point approaches B, it follows that there exists a positive number 6, independent of the positive initial value, such that each coordinate of the corresponding solution of (3.2) exceeds 6 for all sufficiently large t. This fixed distance 6 provides a cushion against the extinction of either population. Later, we will say that the system (3.2) is uniformly persistent when this situation holds. (See Appendix D.)... 

T will be said to be uniformly persistent if there exists an e such that for all xeE,... 

The following theorem provides a criterion for uniform persistence in terms of the flow on the boundary (see [BFW BW HaWJ). 

Theorem D.2. Let x be a semidynamical system defined on a subset E, the closure of an open set, in a locally compact metric space X. Suppose that dE, the boundary of E, is invariant under w. Assume that tt is dissipative and that the boundary flow ttj is isolated and acyclic with acyclic covering M. Then tt is uniformly persistent if and only if... 

The assumption of the invariance of the boundary is stronger than needed, and the assumption that the space is locally compact can be removed at the expense of further assumptions on the dynamical system. The combination of dissipative and uniform persistence allows the use of fixed-point theorems. The following is sufficient for our applications in K". 

BFW] G. Butler, H. 1. Freedman, and P. Waltman (1986), Uniformly persistent systems, Proceedings of the American Mathematical Society 96 425-30. 

P/M Tool Steels. In conventionally produced high alloy tool steels (slowly cooled cast ingots), carbide tends to segregate (48). Segregated clusters of carbide persist even after hot working, and cause undesirable effects on tool fabrication and tool performance. P/M tool steels, on the other hand, provide very fine and uniform carbides in the compact, the final bar stock, and the tools. Several tool steel suppHers consoHdate gas-atomized tool steel powder by HIP to intermediate shapes, which are then hot-worked to final mill shapes. Water-atomized tool steel powder is also available (see also T OOL materials). ... 

All of the structures shown in Figures 7.2 and 7.3 are D-configurations, and the D-forms of monosaccharides predominate in nature, just as L-amino acids do. These preferences, established in apparently random choices early in evolution, persist uniformly in nature because of the stereospecificity of the enzymes that synthesize and metabolize these small molecules. 

But problems persisted. The catalyst, moving at rapid rates, tended to disintegrate as it impacted the inside surface of equipment. Dust particles formed, clogging pipes and transfer lines and disrupting the smooth flow of operation. This difficulty in turn prevented uniform heat distribution through the system and affected the rate and extent of both cracking and regeneration. Both the volume and C uality of fuel produced suffered. 

When a fluid flowing at a uniform velocity enters a pipe, the layers of fluid adjacent to the walls are slowed down as they are on a plane surface and a boundary layer forms at the entrance. This builds up in thickness as the fluid passes into the pipe. At some distance downstream from the entrance, the boundary layer thickness equals the pipe radius, after which conditions remain constant and fully developed flow exists. If the flow in the boundary layers is streamline where they meet, laminar flow exists in the pipe. If the transition has already taken place before they meet, turbulent flow will persist in the... 

When a fluid flowing with a uniform velocity enters a pipe, a boundary layer forms at the walls and gradually thickens with distance from the entry point. Since the fluid in the boundary layer is retarded and the total flow remains constant, the fluid in the central stream is accelerated. At a certain distance from the inlet, the boundary layers, which have formed in contact with the walls, join at the axis of the pipe, and, from that point onwards, occupy the whole cross-section and consequently remain of a constant thickness. Fulty developed flow then exists. If the boundary layers are still streamline when fully developed flow commences, the flow in the pipe remains streamline. On the other hand, if the boundary layers are already turbulent, turbulent flow will persist, as shown in Figure 11.8. 

Disturbances that increase water scarcity promote the physical uniformity of river systems and the decrease of biological diversity in streams and rivers. The structure and functioning of heavily impacted river systems become mutually and strikingly similar, irrespective of the river s origin and the climate. The more intense and persistent the disturbance, the greater is the resemblance. On the other hand, river organisms use resources most efficiently in spatially heterogeneous chaimels, and under moderate disturbance frequencies, rather than in steady conditions, to which they are not adapted. 

TAetection of the highly potent impurity, 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) in the herbicide 2,4,5-trichlorophenoxyacetic acid (2,4,5-T), necessitated an environmental assessment of the impact of this contaminate. Information was rapidly needed on movement, persistence, and plant uptake to determine whether low concentrations reaching plants, soils, and water posed any threat to man and his environment. Because of the extreme toxicity of TCDD, utmost precautions were taken to reduce or minimize the risk of exposure to laboratory personnel. Synthesis of uniformly labeled C-TCDD by Muelder and Shadoff (I) greatly facilitated TCDD detection in soil and plant experiments. For unlabeled experiments it seemed wise to use only small quantities of diluted solutions in situations where decontamination was feasible and to rely on the sensitivity afforded by electron capture gas chromatography... 

Soil properties A Soil texture (sand, silt, clay), organic matter/carbon content, and pH Stones, roots, and hardpans must be largely absent to allow representative sampling of soil profile Soil properties should appear uniform over test site Soil texture data should be available at time of site selection. Soil properties must match study purpose. This can be realistic use conditions, realistic worst-case or worst-case in terms of agrochemical mobility and persistence Must ensure that the majority of samples can be taken from the deepest sampling horizon. Information about sub-soils can be obtained from soil maps, test coring and on-site interviews... 

Dependence of results from the prior-prejudice distribution. Non-uniform prior-prejudice distributions (NUP for short in what follows) were initially introduced by Jauch and Palmer by centering 3D Gaussian functions at the nuclear positions [29]. They found that the low-density regions of the crystal changed significantly upon introduction of the NUP, but the uneven distribution of errors persisted. 

The laws of classical dynamics were first formulated by Newton. The first law states that any particle will persist in its state of uniform unaccelerated motion unless it is acted upon by a force. Using the notation xiy y, z, for the cartesian coordinates of the ith point particle, of mass mi Newton s equations for n point particles are... 

As the persistence length P approaches infinity, the bending relaxation times T all approach zero, and the amplitude of the uniform mode approaches 1.0, in which case F (l) is precisely the result for a rigid (against flexure) rod. 

In all the materials studied,early photographs show a uniform structure for the exploding wire. A uniform illumination persists for most materials until the beginning of post-dwell conduction, although in AI shortly after expln bright spots and non-unifotm regions of illumination occur... 

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