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Unformatted text preview: 1 1.2. Some BRT settings. The BRT settings were defined in Definition 1.11. Most areas of mathematics have a naturally associated family of (multivariate) functions and sets. This usually forms a natural and interesting BRT setting. This book focuses on five basic BRT settings, as noted in section 1.1. These are formally introduced in Chapter 2. In fact, we have only been able to scratch the surface of BRT even on these basic BRT settings. In this section, we survey a huge range of mathematically interesting BRT settings. This will give the reader a sense of the unusual scope of BRT and a glimpse of what can be expected in the future development of BRT. We provide a plausible estimate that at least 1,000,000 of these mathematically interesting BRT settings represent significantly different BRT phenomena. Any substantial probing of BRT on these settings is beyond the scope of this book. In sections 1.3 and 1.4, we will investigate the status of the Complementation Theorem and the Thin Set Theorem in some very modest sampling of the BRT settings presented in this section. This will give a modest indication as to the depth of BRT and its sensitivity to the choice of BRT setting. I. On N. We now consider a number of natural conditions on functions in MF. These conditions are of three kinds. 1. Bounding conditions. 2. Regularity conditions. 3. Choice of norm. We propose the following basic lower bound conditions on f ∈ MF. i. There exist c,d such that c op i, d op’ j, and for all x ∈ dom(f), cx d op’’ f(x). Here op,op’ ∈ {<,>, ≤ , ≥ ,=}, op’’ ∈ {<, ≤ }, i,j ∈ {0,1/2,1,3/2,2}, and   is the l ∞ norm, the l 1 norm, or the l 2 norm. 2 We propose the following basic upper bound conditions on f ∈ MF. ii. There exist c,d such that c op i, d op’ j, such that for all x ∈ dom(f), cx d op’’ f(x). Here op,op’ ∈ {<,>, ≤ , ≥ ,=}, op’’ ∈ {>, ≥ }, i,j ∈ {0,1/2,1,3/2,2}, and   is the l ∞ norm, the l 1 norm, or the l 2 norm. Each of these conditions in i,ii above results from the choice of 6 parameters: op,op',op'',i,j, . Note that some of the choices of parameters result in degenerate conditions. Each of these basic lower bound conditions and basic upper bound conditions can be modified by using “for all but finitely many x” instead of “for all x”. This doubles the number of lower and upper conditions, and the resulting conditions are called the lower bound conditions and the upper bound conditions. The bounding conditions consist of a conjunction of zero or more conditions, each of which is either a basic lower bound condition or a basic upper bound condition. The five basic BRT settings formally introduced in Chapter 2 are examples of classes of functions obeying bounding conditions....
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math, Sets

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