We give an example of a definable quotient in an o-minimal structure which cannot be eliminated over any set of parameters, giving a negative answer to a question of Eleftheriou, Peterzil, and Ramakrishnan. Equivalently, there is an o-minimal structure M whose elementary diagram does not eliminate imaginaries. We also give a positive answer to a related question, showing that any imaginary in a...

The notions of simplicity and extensionality for coalgebras are generalized beyond Set and a study of the internal dynamics of extensional coalgebras yields the notion of unitary coalgebras. In some cases, unitary coalgebras include extensional ones. Under some reasonable circumstances, both notions of simplicity and extensionality agree. Other extensions of simplicity and extensionality are ∗−...

Many pairing-based protocols require the computation of the product and/or of a quotient of n pairings where n > 1 is a natural integer. Zhang et al.[1] recently showed that the Kachisa-Schafer and Scott family of elliptic curves with embedding degree 16 denoted KSS16 at the 192-bit security level is suitable for such protocols comparatively to the BarettoLynn and Scott family of elliptic curve...

Let A be a finitely generated domain of GK dimension less than 3 over a field K and let Q(A) denote the quotient division algebra of A. Using the ideas of Smoktunowicz, we show that if D is a finitely generated division subalgebras of Q(A) of GK dimension at least 2, then Q(A) is finite dimensional as a left D-vector space. We use this to show that if A is a finitely generated domain of GK dime...

The concepts of fuzzy semi-ideals of R with respect to H≤R and generalized fuzzy quotient rings are introduced. Some properties of fuzzy semiideals are discussed. Finally, several isomorphism theorems for generalized fuzzy quotient rings are established.

The eta invariant of the Dirac operator over a noncompact cofinite quotient of PSL(2,R) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally nontrivial spin structure. When the S1-fibers are rescaled, the metric becomes of nonexact fibred-cusp type near the ends. We completely describe the continuous spe...

We extend the notion of exact completion on a category with weak finite limits to Lawvere’s elementary doctrines. We show how any such doctrine admits an elementary quotient completion, which is the universal solution to adding certain quotients. We note that the elementary quotient completion can be obtained as the composite of two other universal constructions: one adds effective quotients, t...

A duality transform for the coalgebra of the free difference quotient derivationmultiplication of an operator with respect to a free algebra of scalars is constructed. The dual object is realized in an algebra of matricial analytic functions endowed with yet another generalization of the difference quotient derivation.

This paper studies the canonical Chow quotient of a smooth projective variety by a reductive algebraic group. The main observation of this paper is that, over the field of complex numbers, the Chow quotient admits symplectic and other topological interpretations, namely, symplectically, the moduli spaces of stable orbits with prescribed momentum charges; and topologically, the moduli space of s...