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Abstraction, Composition and Contracts: A Sheaf Theoretic Approach [article]

Alberto Speranzon and David I. Spivak and Srivatsan Varadarajan
2018 arXiv   pre-print
In this paper we introduce a new framework for abstraction, composition and analysis of SoS that builds on results and methods developed in sheaf theory, category theory and topos theory.  ...  In particular, we will be modeling behaviors of systems using sheaves, leverage category theoretic methods to define wiring diagrams and formalize composition and, by establishing a connection with topos  ...  In what follows, we present an alternative approach for abstraction and composition of such systems and describe how such methods enable us to prove properties (or express contracts/requirements).  ... 
arXiv:1802.03080v1 fatcat:lpggb4lwgncenbgs6nokgf43ju

An Extended Goodwin Model with Horizontal Trade: A Sheaf Theoretic Approach [article]

Philip Coyle
2017 arXiv   pre-print
It is meaningful to conduct this approach to a system of equations given its novelty, its applicability, and its importance for modeling.  ...  New questions regarding local sections of this sheaf and their possible extensions to global sections are considered. These questions lend themselves to unique analysis about the system.  ...  Conclusion and Further Research In this paper we have explored the extended Goodwin model from a sheaf theoretic approach.  ... 
arXiv:1704.05581v1 fatcat:wl2vhako6rcxnp2xwfwgkdn72i

A Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information

Cliff A. Joslyn, Lauren Charles, Chris DePerno, Nicholas Gould, Kathleen Nowak, Brenda Praggastis, Emilie Purvine, Michael Robinson, Jennifer Strules, Paul Whitney
2020 Sensors  
of the sheaf-based approach.  ...  Because different sensors have differing error models, we seek a theoretically justified quantification of the agreement among ensembles of sensors, both overall for a sensor collection, and also at a  ...  The sheaf theoretical approach can represent all these interactions automatically.  ... 
doi:10.3390/s20123418 pmid:32560463 pmcid:PMC7349656 fatcat:4a3bjwyadnbglboqdt7caacnsy

A Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information [article]

Cliff Joslyn, Lauren Charles, Chris DePerno, Nicholas Gould, Kathleen Nowak, Brenda Praggastis, Emilie Purvine, Michael Robinson, Jennifer Strules, Paul Whitney
2019 arXiv   pre-print
of the sheaf-based approach.  ...  A statistical approach was developed for comparison, a dynamic linear model which was estimated using a Kalman filter. This approach also recovered bear and human locations and sensor accuracies.  ...  This work was partially funded by the Pittman Robertson Federal Aid to Wildlife Restoration Grant and was a joint research project between the North Carolina Wildlife Resources Commission (NCWRC) and the  ... 
arXiv:1912.05487v1 fatcat:726whmbqxbfnzexu5kpmic4bky

On Contractions of smooth varieties [article]

Marco Andreatta, Jarosław A. Wiśniewski
1996 arXiv   pre-print
Let : X Z be a proper surjective map from a smooth complex manifold X onto a normal variety Z. If has connected fibers and -K_X is -ample then is called a good contraction.  ...  and the geometry of :X Z nearby such a fiber.  ...  Abstract. Let ϕ : X → Z be a proper surjective map from a smooth complex manifold X onto a normal variety Z. If ϕ has connected fibers and −K X is ϕ-ample then ϕ is called a good contraction.  ... 
arXiv:alg-geom/9605013v1 fatcat:jzpcu6vyg5amhbxwv6tcubujfi

The derived contraction algebra [article]

Matt Booth
2019 arXiv   pre-print
In this thesis, I construct and prove properties of a new invariant, the derived contraction algebra A^der_con, which I define to be Braun-Chuang-Lazarev's derived quotient of A by e.  ...  Donovan and Wemyss define the contraction algebra A_con to be the quotient of A by e; it is a finite-dimensional noncommutative algebra that is conjectured to completely recover the geometry of the base  ...  I would also like to thank Michael Wemyss for introducing me to contraction algebras and for his support at every stage; this thesis would not have been possible without either of them.  ... 
arXiv:1911.09626v1 fatcat:7qagcosbpfdfrks3lyek6ilshy

Non-commutative Mori contractions and ^1-bundles [article]

Daniel Chan, Adam Nyman
2009 arXiv   pre-print
With the aid of Artin-Zhang's abstract Hilbert schemes, this is used to construct analogues of the extremal contraction of a K-negative curve with self-intersection zero on a smooth projective surface.  ...  As a non-trivial example of the theory developed, we look at non-commutative ruled surfaces and, more generally, at non-commutative ^1-bundles.  ...  Then i. there is a smooth curve X and morphism f : Y −→ X which contracts C. ii. the morphism f is a P 1 -fibration.  ... 
arXiv:0904.1717v1 fatcat:les2hz2o7jd3zeheusqqmwkurq

A^1-homotopy theory and contractible varieties: a survey [article]

Aravind Asok, Paul Arne Østvær
2019 arXiv   pre-print
Our main goal is to highlight the interplay between A^1-homotopy theory and affine algebraic geometry, focusing on the varieties that are "contractible" from various standpoints.  ...  We survey some topics in A^1-homotopy theory.  ...  Acknowledgements Asok was partially supported by National Science Foundation Awards DMS-1254892 and DMS-1802060.  ... 
arXiv:1903.07851v1 fatcat:kzuixkmtbfhznpvhnmbwhsrerm

Non-commutative Mori contractions and P 1 -bundles

Daniel Chan, Adam Nyman
2013 Advances in Mathematics  
With the aid of Artin-Zhang's abstract Hilbert schemes, this is used to construct analogues of the extremal contraction of a K-negative curve with self-intersection zero on a smooth projective surface.  ...  We then show that non-commutative ruled surfaces give examples of the aforementioned non-commutative Mori contractions.  ...  Then i. there is a smooth curve X and morphism f : Y −→ X which contracts C. ii. the morphism f is a P 1 -fibration.  ... 
doi:10.1016/j.aim.2013.06.018 fatcat:3y2kod5jxjbsva7eq56vlxdm7u

The Deformation of Contract in the Information Society

Margaret Jane Radin
2017 Oxford Journal of Legal Studies  
The contemporary lack of "fit" between certain contemporary phenomena and the traditional theories of contract law calls to mind a situation that arose in the history of astronomy.  ...  But these theoretical presumptions did not fit observed facts; planetary orbits did not follow a pattern that can be interpreted as circular motion.  ...  The idea is that a customer is purchasing a composite: an item (such as a computer) together with the terms that come with it. 43 If we call this composite item-plus-terms 42 If a contract is found to  ... 
doi:10.1093/ojls/gqx001 fatcat:cr7ksnqdubcl3mpis2b7k7ctgu

Deletion-contraction triangles for Hausel-Proudfoot varieties [article]

Zsuzsanna Dancso, Michael McBreen, Vivek Shende
2019 arXiv   pre-print
This polynomial satisfies a deletion-contraction relation, which we lift to a deletion-contraction exact triangle for the cohomology of B. There is a corresponding triangle for D.  ...  To a graph, Hausel and Proudfoot associate two complex manifolds, B and D, which behave, respectively like moduli of local systems on a Riemann surface, and moduli of Higgs bundles.  ...  There is an alternative approach to the above lemma.  ... 
arXiv:1910.00979v1 fatcat:yj7brnhyszg5thy47q627rudwy

The Nyquist Theorem For Cellular Sheaves

Michael Robinson
2013 Zenodo  
Our sheaf-theoretic approach allows sufficient generality to treat sampling on non-Euclidean spaces.  ...  , and • for every composition of inclusions a → b → c, F (b → c) • F (a → b) = F (a → b → c).  ... 
doi:10.5281/zenodo.54405 fatcat:2akk2l3jtjcb3dcvoq7qwhe5dy

The Nyquist theorem for cellular sheaves [article]

Michael Robinson
2013 arXiv   pre-print
We develop a unified sampling theory based on sheaves and show that the Shannon-Nyquist theorem is a cohomological consequence of an exact sequence of sheaves.  ...  Using these obstructions, we also present conditions for perfect reconstruction of piecewise linear functions on graphs, a collection of non-bandlimited functions on topologically nontrivial domains.  ...  Our sheaf-theoretic approach allows sufficient generality to treat sampling on non-Euclidean spaces.  ... 
arXiv:1307.7212v1 fatcat:wu6lewzonzd65esfy3rjvgbxza

Tōhoku

Rick Jardine
2015 Inference: International Review of Science  
Rick Jardine evaluates Alexander Grothendieck's 1957 mathematical masterpiece, Some Aspects of Homological Algebra, a turning point in homological algebra, algebraic topology and algebraic geometry, and  ...  for modern mathematics as a whole.  ...  , analytic geometry, and theoretical physics.  ... 
doi:10.37282/991819.15.13 fatcat:3pmvwf64pzc5vddgj5t7zrstfu

A sheaf-theoretic perspective on sampling [article]

Michael Robinson
2014 arXiv   pre-print
It presents sheaf-theoretic approach by solving several different sampling problems involving non-bandlimited functions.  ...  The correct topological tool for modeling all of these situations is the sheaf; a tool which allows local structure and consistency to derive global inferences.  ...  Acknowledgements This work was partly supported under Federal Contract No. FA9550-09-1-0643. The author also wishes to thank the editor for the invitation to write this chapter.  ... 
arXiv:1405.0324v1 fatcat:b4nynhisq5cxvl33ingwelthz4
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