Nested roots

I have always been fascinated by the mathematical concept of nested roots. As background, most readers will probably be familiar with the commonplace square root symbol in mathematics ... 

Nested roots are certainly more fascinating (and typographically attractive) to look at. Here is an example ... 

If mathematics were a beauty contest, then nested roots would win the prize. They are simply square roots of roots of roots in an infinite cascade. Here is an example of an infinitely nested root ... 

When I asked colleagues to determine a value for the infinite set of nested roots in equation 1,1 surely felt that no human could arrive at a solution. After all, a quattuor-decillion is larger than most computers could assimilate. The behavior of the nested root also makes it particularly difficult to decipher. The continued roots seem to make the values smaller, while the Q multipliers continue to make the values larger. Think of two sumo wrestlers competing for domination on a numerical wrestling mat. How could we ever arrive at an amicable solution ... 

It turns out that it was not too difficult to compute the value for the mysterious nested root. The solution is so simple that it astonished most colleagues who expected a noninteger solution. The value of the nested Q system in equation 1 is simply Q+, and this could have been solved decades ago without computers. 

NExOCTANE A process for making isooctane by dimerizing isobutylene over an acidic ion-exchange catalyst. The product is used as a gasoline additive. Developed from 1997 by Neste Engineering (a subsidiary of Fortum Oil Gas, Finland) and Kellogg Brown Root. The first commercial unit started operating in 2002. See also SP-Isoether. 

To handle the case where m > 1, we need the following definition. We say that a set Sx,y is nested within the set Sa,b if the tree node Vx is contained in the subtree rooted at Vb. Note that if two subsets Sa,b... 

S = 1 and there exists a maximal set Sa,b < 2 with respect to the nesting property such that all sets in S are nested within Sa,b- Let W be the subtree rooted at Vb- The idea is to first cover Af (W U W) and then cover the leaves in U U. The first part of the cover can be obtained by applying the lemma recursively with the input S2, and the second part by applying it recursively with total number of 3(m — fc) — l + 3fc — 1 = 3m — 2 sets. 

The succulent roots, tuhers and hulhs preferred hy the vole are usually stored in large quantities in storage chambers. In a dry, well-lined nest the female gives hirth to 2-6 young, 3-4 times a year from March to October. Since the vole is sexually mature after only 2 months, there can easily be a population explosion. Dry weather, in particular, encourages vole reproduction, whereas in cold, wet weather the nests get very wet and many young voles perish. 

Fig. 93 Soil resistance to penetration and fine-root length density in the soil beneath cutter ant nests (Atta sexdens) and beneath neighboring forest without nest influence.

There is also a more general result, that if two matrices share a nested sequence of invariant subspaces, all of their products also share these subspaces, and the nth roots of the eigenvalues associated with the components within it will not exceed the eigenvalues of the originals. The proof is technical and can be found in Appendix 1 Theorem 4. 

Intrathoracic paragangliomas (PGs) may arise in the anterior mediastinal compartment, in association with the aorticopulmonary root, or along the vertebral column. > Some of these lesions may demonstrate a prominently sclerotic stroma, which can obscure their true lineage.Their neural nature is reflected by their lack of labeling for keratin and EMA. They instead express neurofilament protein, with or without vimentin, in likeness to NBL. Stains for S-100 protein can be used to label sustentacular cells that surround cellular nests in PGs (Fig. 11.24), and the tumor cells themselves often... 

The cause of the problem lies in the numbers 3 and 113 used to raise the individual terms of the array to their cube power, and to take their cube root. When we raised the individual cells in the block (or the individual elements in Array) to their cube power, we reduced the results to single precision. To demonstrate this and, at the same time, fix the problem, add a dimension statement to both macros, reading Dim p As Double , and add a line specifying the value of p (e.g., either p = 3 or p =1/3 ). Finally, in the For Each... Next (or in the nested For... Next loop) refer to p rather than to 3 (or 1/3). The macros now should read like the example shown below, where we have also added a title and some comment lines and, as before, boldfaced the most recent changes. 

A few years ago I asked colleagues whether it were possible for humanity to determine the solution to the following nested set of square roots ... 

Many months went by, and not a single reader could determine a solution. The difficulty was the infinite nesting of roots. Ramanujan finally threw up his hands and gave the answer, 3. It turns out that he had generated the problem years before in the form of a more general theorem ... 

Renker, C., Heinrichs, J., Kaldorf, M., Buscot, R, 2003. Combining nested PCR and restriction digest of the internal transcribed spacer region to characterize arbuscular mycorrhizal fungi on roots from the field. Mycorrhiza 13, 191-198. 

Van Tuinen, D., Jacquot, E., Zhao, B., Gollote, A., Gianinazzi-Pearson, V, 1998. Characterisation of root colonization profiles by a microcosm community of arbuscular mycorrhizal fungi using 25S-rDNA targeted nested PCR. Mol. Ecol. 7, 879-887. 

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