On universal partial words

A universal word for a finite alphabet $A$ and some integer $n\geq 1$ is a word over $A$ such that every word in $A^n$ appears exactly once as a subword (cyclically or linearly). It is well-known and easy to prove that universal words exist for any $A$ and $n$. In this work we initiate the systematic study of universal partial words. These are words that in addition to the letters from $A$ may contain an arbitrary number of occurrences of a special `joker' symbol $\Diamond\notin A$, which can be substituted by any symbol from $A$. For example, $u=0\Diamond 011100$ is a linear partial word for the binary alphabet $A=\{0,1\}$ and for $n=3$ (e.g., the first three letters of $u$ yield the subwords $000$ and $010$). We present results on the existence and non-existence of linear and cyclic universal partial words in different situations (depending on the number of $\Diamond$s and their positions), including various explicit constructions. We also provide numerous examples of universal partial words that we found with the help of a computer

On universal partial words for word-patterns and set partitions

Universal words are words containing exactly once each element from a given set of combinatorial structures admiting encoding by words. Universal partial words (u-p-words) contain, in addition to the letters from the alphabet in question, any number of occurrences of a special ``joker'' symbol. We initiate the study of u-p-words for word-patterns (essentially, surjective functions) and (2-)set partitions by proving a number of existence/non-existence results and thus extending the results in the literature on u-p-words and u-p-cycles for words and permutations. We apply methods of graph theory and combinatorics on words to obtain our results

Fabrication of strong long-period gratings in hydrogen-free fibers with 157-nm F<inf>2</inf>-laser radiation

Long-period gratings were fabricated in standard telecommunication fiber (Corning SMF-28) by use of what is believed to be record short-wavelength light from a 157-nm F2 laser. Strong loss peaks were formed without the need for enhancement techniques such as hydrogen loading. The magnitude of the attenuation peak was sensitive to the single-pulse laser fluence, decreasing with increasing pulse fluence as a result of nonuniform 157-nm laser interaction with both the fiber cladding and core. The long-period fiber gratings have good wavelength stability (Δλ ∼ 7 nm) under thermal annealing at 150°C. © 2001 Optical Society of America