The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and Swinnerton-Dyer. One conjectures that this constant is 1; it is known to be an integer. After surveying what is known about the Manin constant, we establish a new sufficient condition that ensures that the Manin constant is an odd integer. Next, we generalize the notion of the Mani...

The quotient of a triangulated category modulo a subcategory was defined by Verdier. Motivated by the failure of the telescope conjecture, we introduce a new type of quotients for any triangulated category which generalizes Verdier’s construction. Slightly simplifying this concept, the cohomological quotients are flat epimorphisms, whereas the Verdier quotients are Ore localizations. For any co...

The quotient complexity of a regular language L, which is the same as its state complexity, is the number of left quotients of L. An atom of a non-empty regular language L with n quotients is a non-empty intersection of the n quotients, which can be uncomplemented or complemented. An NFA is atomic if the right language of every state is a union of atoms. We characterize all reduced atomic NFAs ...

Let G be a group which is the semidirect product of a normal subgroup N and some subgroup T . Let In(G), n ≥ 1, denote the powers of the augmentation ideal I(G) of the group ring Z(G). Using homological methods the groups Qn(G,H) = In−1(G)I(H)/In(G)I(H), H = G,N, T , are functorially expressed in terms of enveloping algebras of certain Lie rings associated with N and T , in the following cases:...

We start this paper by showing that the Leavitt path algebra of a (row-finite) graph is an algebra of quotients of the corresponding path algebra. The path algebra is semiprime if and only if whenever there is a path connecting two vertices, there is another one in the opposite direction. Semiprimeness is studied because, for acyclic graphs, the Leavitt path algebra is a Fountain-Gould algebra ...

A torsion theory is called differential (higher differential) if a derivation (higher derivation) can be extended from any module to the module of quotients corresponding to the torsion theory. We study conditions equivalent to higher differentiability of a torsion theory. It is known that the Lambek, Goldie and any perfect torsion theories are differential. We show that these classes of torsio...

The Picard group is defined as Γ = SL(2,Z[i]); the ring of 2× 2 matrices with Gaussian integer entries and determinant one. We consider certain graphs associated to quotients Γ/Γ(p) where p is a prime congruent to three mod four and Γ(p) is the congruence subgroup of level p. We prove a decomposition theorem on the vertices of these graphs, and use this decomposition to derive upper and lower b...

A hierarchy of six important structures from abstract algebra (groups, rings, fields etc.) is introduced as Gofer class definitions and laws about them. Many instance declarations are provided, explaining the algebraic construction of integers, quotients, adding i, function spaces, polynomials, and matrices. The definitions include generalized implementations of polynomial division and matrix i...