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تسجيل مستخدم جديدOn universal partial words
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A universal word for a finite alphabet $A$ and some integer $ngeq 1$ 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 $Diamond otin A$, which can be substituted by any symbol from $A$. For example, $u=0Diamond 011100$ 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.
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