NDLI: Difference Forms

Content Provider	Springer Nature Link
Author	Hydon, Peter E. Mansfield, Elizabeth L.
Copyright Year	2007
Abstract	Currently, there is much interest in the development of geometric integrators, which retain analogues of geometric properties of an approximated system. This paper provides a means of ensuring that finite difference schemes accurately mirror global properties of approximated systems. To this end, we introduce a cohomology theory for lattice varieties, on which finite difference schemes and other difference equations are defined. We do not assume that there is any continuous space, or that a scheme or difference equation has a continuum limit. This distinguishes our approach from theories of “discrete differential forms” built on simplicial approximations and Whitney forms, and from cohomology theories built on cubical complexes. Indeed, whereas cochains on cubical complexes can be mapped injectively to our difference forms, a bijection may not exist. Thus our approach generalizes what can be achieved with cubical cohomology. The fundamental property that we use to prove our results is the natural ordering on the integers. We show that our cohomology can be calculated from a good cover, just as de Rham cohomology can. We postulate that the dimension of solution space of a globally defined linear recurrence relation equals the analogue of the Euler characteristic for the lattice variety. Most of our exposition deals with forward differences, but little modification is needed to treat other finite difference schemes, including Gauss-Legendre and Preissmann schemes.
Ending Page	467
Page Count	41
Starting Page	427
File Format	PDF
ISSN	16153375
e-ISSN	16153383
Journal	Foundations of Computational Mathematics
Issue Number	4
Volume Number	8
Language	English
Publisher	Springer-Verlag
Publisher Date	2007-11-29
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Local exactness de Rham cohomology Exterior algebra, Grassmann algebras Linear and Multilinear Algebras, Matrix Theory Cohomology Difference chains Numerical Analysis Computer Science Difference forms ÄŒech types Discrete version of topics in analysis Applications of Mathematics Math Applications in Computer Science Lattice variety Local difference potentials
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

Sl.	Authority	Responsibilities	Communication Details
1	Ministry of Education (GoI), Department of Higher Education	Sanctioning Authority	https://www.education.gov.in/ict-initiatives
2	Indian Institute of Technology Kharagpur	Host Institute of the Project: The host institute of the project is responsible for providing infrastructure support and hosting the project	https://www.iitkgp.ac.in
3	National Digital Library of India Office, Indian Institute of Technology Kharagpur	The administrative and infrastructural headquarters of the project	Dr. B. Sutradhar bsutra@ndl.gov.in
4	Project PI / Joint PI	Principal Investigator and Joint Principal Investigators of the project	Dr. B. Sutradhar bsutra@ndl.gov.in Prof. Saswat Chakrabarti will be added soon
5	Website/Portal (Helpdesk)	Queries regarding NDLI and its services	support@ndl.gov.in
6	Contents and Copyright Issues	Queries related to content curation and copyright issues	content@ndl.gov.in
7	National Digital Library of India Club (NDLI Club)	Queries related to NDLI Club formation, support, user awareness program, seminar/symposium, collaboration, social media, promotion, and outreach	clubsupport@ndl.gov.in
8	Digital Preservation Centre (DPC)	Assistance with digitizing and archiving copyright-free printed books	dpc@ndl.gov.in
9	IDR Setup or Support	Queries related to establishment and support of Institutional Digital Repository (IDR) and IDR workshops	idr@ndl.gov.in

Digital atlases and difference forms

Exterior powers of F-zips

Castelnuovo–Mumford Regularity and Computing the de Rham Cohomology of Smooth Projective Varieties

Discrete Morse Theory for Computing Cellular Sheaf Cohomology

A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

Typical Real Ranks of Binary Forms

Erratum to: Discrete Lie Advection of Differential Forms

Point Counting in Families of Hyperelliptic Curves

Geometric Variational Crimes: Hilbert Complexes, Finite Element Exterior Calculus, and Problems on Hypersurfaces

Difference Forms

Similar Documents

Digital atlases and difference forms

Exterior powers of F-zips

Castelnuovo–Mumford Regularity and Computing the de Rham Cohomology of Smooth Projective Varieties

Discrete Morse Theory for Computing Cellular Sheaf Cohomology

A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

Typical Real Ranks of Binary Forms

Erratum to: Discrete Lie Advection of Differential Forms

Point Counting in Families of Hyperelliptic Curves

Geometric Variational Crimes: Hilbert Complexes, Finite Element Exterior Calculus, and Problems on Hypersurfaces

Difference Forms