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We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category. [ABSTRACT FROM AUTHOR]

Categorification of some integer sequences are obtained by enumerating the number of sections in the Auslander–Reiten quiver of algebras of finite representation type. [ABSTRACT FROM AUTHOR]

We determine the universal deformation rings of string Λ m. N -modules whose corresponding stable endomorphism ring is isomorphic to k , and which lie either in a connected component of the stable Auslander-Reiten quiver of Λ m , N containing a module with endomorphism ring isomorphic to k or in a periodic component containing only string Λ m , N -modules, where m ≥ 3 and N ≥ 1 are integers, and Λ m , N is a self-injective special biserial k -algebra. [ABSTRACT FROM AUTHOR]

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We describe the irreducible morphisms in the category of modules over a repetitive algebra. We find three special canonical forms: The first canonical form happens when all the component morphisms are split monomorphisms, the second when all the component morphisms are split epimorphisms and the third when there is exactly one irreducible component map. Also, we obtain the same result for the irreducible homomorphisms in the stable category of modules over a repetitive algebra. [ABSTRACT FROM AUTHOR]

In [9] Giraldo and Merklen studied irreducible morphisms in the categories 𝒞(𝒜) and D−(Λ), where 𝒞(𝒜) is the category of complexes over an abelian Krull–Schmidt category 𝒜 and D−(Λ) is the derived category of the bounded above complexes of finite generate left modules, over an Artin algebra Λ. In this work, we continue the study of irreducible morphism having one finite irreducible truncation. [ABSTRACT FROM AUTHOR]

In 1836 E. Midy published in France an article where he showed that if p is a prime number, such that the smallest repeating sequence of digits in the decimal expansion of has an even length, when this sequence is broken into two halves of equal length if these parts are added then the result is a string of 9′′s. Later, J. Lewittes and H.W.Martin generalized this statement when the length of the smallest repeating sequence of digits is e == kd and the sequence is broken into d blocks of equal length and the expansion is over any number base; that fact was named Midy's property. We will give necessary and sufficient conditions (that are easy to check) for the integer N to satisfy Midy's property. [ABSTRACT FROM AUTHOR]