Categories of partial functions have become increasingly important principally because of their applications in theoretical computer science. In this note we prove that the category of partial bijections between sets as an inverse-Baer*-category with closed projections and in which the idempotent split is an exact category. Finally the Noether isomorphism theorems are given for this exact categ...

We prove in this note that the global geometric theta lifting for the pair (H,G) is compatible with the Whittaker normalization, where (H,G) = (SO2n, Sp2n), (Sp2n, SO2n+2), or (GLn, GLn+1). More precisely, let k be an algebraically closed field of characteristic p > 2. Let X be a smooth projective connected curve over k. For a stack S write D(S) for the derived category of étale constructible Q̄...

Partial exponential sums are a generalization of classical exponential sums which are useful, for instance, for studying the distribution of the rational points along the fibers of a morphism between varieties over a finite field. They share some properties with the classical ones: their values are also governed by a certain L-function, which is rational and whose reciprocal roots are pure Weil...

Let $\mathrm{h}\mathscr{C}$ be the homotopy category of a stable infinity $\mathscr{C}$. Then $\mathrm{h}\mathscr{C}^{\Delta^{1}}$ morphisms in $\mathscr{C}$ is also triangulated. Hence space $\mathsf{Stab}\,{ \mathrm{h}\mathscr{C}^{\Delta^{1}}}$ stability conditions on well-defined though non-emptiness not obvious. Our basic motivation comparison type $\mathsf{Stab}{\mathrm{h}\mathscr{C}}$ and...

We prove that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is still covariantly finite. This extends a result by Sikko and Smalø. We also prove a triangulated version of the result. As applications, we obtain short proofs to a classical result by Ringel and a recent result by Krause and Solberg. 1. Main Theorems Let C b...

The Madsen-Tillmann spectra defined by categories of threeand four-dimensional Spin manifolds have a very rich algebraic structure, whose surface is scratched here. For Michael Atiyah, in deep gratitude. 1. Cobordism categories 1.1 Many variations and generalizations are possible, but to begin, consider the topological two-category DCobord whose objects are oriented smooth closed d-manifolds (D...

To adopt proof assistants based on logic of total functions in education, one major problem is that of representing partial functions. In particular, one wishes to capture undefinedness in a rigorous way, while staying as close as possible to traditional mathematical practice. In this paper, we propose to represent potentially undefined terms with partial setoids on lifted terms, and to underst...

A graph matching algorithm is illustrated and its performance compared with that of a well known algorithm performing the same task. According to the proposed algorithm, the matching process is carried out by using a State Space Representation: a state represents a partial solution of the matching between two graphs, and a transition between states corresponds to the addition of a new pair of m...

Let T be the set of ground terms over a nite ranked alphabet We de ne partial automata on T and prove that the nitely gener ated congruences on T are in one to one correspondence up to iso morphism with the nite partial automata on T with no inaccessible and no inessential states We give an application in term rewriting every ground term rewrite system has a canonical equivalent system that can...