In this paper, we give three functors $mathfrak{P}$, $[cdot]_K$ and $mathfrak{F}$ on the category of C$^ast$-algebras. The functor $mathfrak{P}$ assigns to each C$^ast$-algebra $mathcal{A}$ a pre-C$^ast$-algebra $mathfrak{P}(mathcal{A})$ with completion $[mathcal{A}]_K$. The functor $[cdot]_K$ assigns to each C$^ast$-algebra $mathcal{A}$ the Cauchy extension $[mathcal{A}]_K$ of $mathcal{A}$ by ...

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolu...

Overview 1 1. Mymotivation: K-theory of schemes 2 2. First steps in homological algebra 3 3. The long exact sequence 6 4. Derived functors 8 5. Cohomology of sheaves 10 6. Cohomology of a Noetherian Affine Scheme 12 7. Čech cohomology of sheaves 12 8. The Cohomology of Projective Space 14 9. Sheaf cohomology on P̃2 16 10. Pushing around sheaves, especially by the Frobenius 17 11. A first look at...

This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the first part of this paper, we obtained several universal deformations for Lie-algebra versions of the above representation using the Schlessinger criteria for...

Proof I : Assume that there exist two potentials V (1) ext (~r) and V (2) ext (~r) differing by more than a constant and giving rise to the same ground state density, n(~r). Obviously, V (1) ext (~r) and V (2) ext (~r) belong to distinct Hamiltonians Ĥ (1) ext(~r) and Ĥ (2) ext(~r), which give rise to distinct wavefunctions Ψ (1) ext(~r) and Ψ (2) ext(~r). Because of the variational principle, ...

The authors study investor beliefs—sentiment and disagreement—about stock market returns during the COVID-19 pandemic using a large number of messages, about 3.7 million, on social media investing platform, StockTwits. rich multimodal features StockTwits data allow to explore evolution sentiment disagreement within across investors, sectors, even industries. find that (disagreement) has sharp d...