Characteristics of cosmic-ray hadronic interactions in the l00_1017 eV range are studied by observing a total of 429 cosmic-ray families of visible energy greater than 100 TeV found in emulsion chamber experiments at high mountain altitudes, Chacaltaya (5200 m above sea level) and the Pamirs (4300 m above sea level). Extensive comparisons were made with simulated families based on models so far proposed, concentrating on the relation between the observed family flux and the behaviour of high-energy showers in the families, hadronic and electromagnetic components. It is concluded that there must be global change in characteristics of hadronic interactions at around l0b~eV deviating from those known in the accelerator energy range, specially in the forwardmost angular region of the collision. A detailed study of a new shower phenomenon of small-pT particle emissions, PT being of the order of 10 MeV/c, is carried out and its relation to the origin of huge "halo" phenomena associated with extremely high energy families is discussed as one of the possibilities. General characteristics of such super-families are surveyed.
Let Σ and Δ be two nonempty finite (not necessarily different) sets (alphabets). As usual, by Σ + (by Δ + ) we denote the set of all nonempty words over the alphabet Σ (respectively, Δ). A map h : Δ + → Σ + is called a morphism if h(p)h(q) = h(pq) for any words p, q ∈ Δ + . We say that a word u ∈ Σ + avoids a word p ∈ Δ + if, for any morphism h : Δ + → Σ + , the word h(p) is not a subword of u. A word p is said to be k-avoidable if there exists an infinite sequence of words u i over some k-letter alphabet such that every word u i avoids p. Finally, a word p is called avoidable if it k-avoidable for some number k; otherwise, the word p is said to be unavoidable. The least number k for which a word is k-avoidable is called the avoidability index of this word. The problem of finding the avoidability index of an arbitrary avoidable word has not been solved; even the question on the complexity of this problem remains open (see the survey [1]). However, for words of certain types, various bounds for avoidability index have been obtained. For example, Petrov [2] showed that the avoidability index of any complete word is at most 4 (a word p is said to be complete if every letter of p occurs twice and, whenever p contains different letters x and y, it also contains xy and yx as subwords). In this note, we specify yet another natural class of words for which the avoidability index has an absolute (i.e., not depending on the number of letters) bound.Recall that a word p = a 1 a 2 . . . a n is called a palindrome if it coincides with its mirror image ← − p = a n . . . a 2 a 1 . We prove the following theorem.
Theorem 1. The avoidability index of any avoidable palindrome does not depend on the number of letters in this word and is at most 16.The avoidable palindrome abacbdbcaba does not belong to any of the previously considered classes of avoidable words, and nothing has been known about its avoidability index so far.To prove the theorem, we need additional information about avoidable words.
FREE DELETIONSLet u be a word over an alphabet Σ. A pair of subsets B, C ⊆ Σ is called a fusion in u if we have x ∈ B ⇔ y ∈ C for any two-letter subword xy of u. A set A ⊆ B \ C is said to be free in the word u. We refer to the removal from the word u of all letters belonging to a free set A as a free deletion; we denote this operation by δ A and the result of applying it to u by u A . A sequence δ A 1 , δ A 2 , . . . , δ A k is called a sequence of free deletions if δ A 1 is a free deletion in u, δ A 2 is a free deletion in u A 1 , and so on.The following theorem from [3] describes a relationship between avoidable words and free deletions in words.
Theorem 2. A word u is unavoidable if and only if there exists a sequence of free deletions such that the application of this sequence to the word u yields the empty word.*
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