Continued Radicals

A nested radical with terms $a_1, a_2, ldotsa_N$ is an expression of form $sqrt{a_N + cdots + sqrt{a_2 + sqrt{a_1}}}$. The limit as $N$ approaches infinity of such an expression, if it exists, is called a continued radical. We consider the set of real numbers $S(M)$ representable as a continued radical whose terms $a_1, a_2, ldots$ are all from a finite set $M$ of nonnegative real numbers. We give conditions on the set $M$ for $S(M)$ to be (a) an interval, and (b) homeomorphic to the Cantor set

A Class of Periodic Continued Radicals

We compute the limits of a class of periodic continued radicals and we establish a connection between them and the fixed points of the Chebycheff polynomials

Continued Radicals and Cantor Sets

We examine the formation of sets homeomorphic to the ternary Cantor set by continued radicals. We determine properties of bridges and gaps and calculate the thickness of the Cantor set. From this we apply information from continued fractions to continued radicals to obtain new results. We also consider the measure of several Cantor sets

Method of cross-linking polyvinyl alcohol and other water soluble resins

A self supporting sheet structure comprising a water soluble, noncrosslinked polymer such as polyvinyl alcohol which is capable of being crosslinked by reaction with hydrogen atom radicals and hydroxyl molecule radicals is contacted with an aqueous solution having a pH of less than 8 and containing a dissolved salt in an amount sufficient to prevent substantial dissolution of the noncrosslinked polymer in the aqueous solution. The aqueous solution is then irradiated with ionizing radiation to form hydrogen atom radicals and hydroxyl molecule radicals and the irradiation is continued for a time sufficient to effect crosslinking of the water soluble polymer to produce a water insoluble polymer sheet structure. The method has particular application in the production of battery separators and electrode envelopes for alkaline batteries

Continued Radicals

If a1, a2, . . . , an are nonnegative real numbers and fj(x) = paj + x, then f1o f2o· · · fn(0) is a nested radical with terms a1, . . . , an. If it exists, the limit as n ! 1 of such an expression is a continued radical. We consider the set of real numbers S(M) representable as an infinite nested radical whose terms a1, a2, . . . are all from a finite set M. We give conditions on the set M for S(M) to be (a) an interval, and (b) homeomorphic to the Cantor set

Review on Multi-Scale Models of Solid-Electrolyte Interphase Formation

Electrolyte reduction products form the solid-electrolyte interphase (SEI) on negative electrodes of lithium-ion batteries. Even though this process practically stabilizes the electrode-electrolyte interface, it results in continued capacity-fade limiting lifetime and safety of lithium-ion batteries. Recent atomistic and continuum theories give new insights into the growth of structures and the transport of ions in the SEI. The diffusion of neutral radicals has emerged as a prominent candidate for the long-term growth mechanism, because it predicts the observed potential dependence of SEI growth.Comment: 8 pages, 4 figure