Differential Elimination Theory (DET) View Homepage


Ontology type: schema:MonetaryGrant     


Grant Info

YEARS

2003-2007

FUNDING AMOUNT

N/A

ABSTRACT

Differential equations allow us to model a great variety of phenomena arising in physics, engineering, biology,financial analysis etc. Such ordinary and partial differential equations (ODEs and PDEs) have been studied for along time, and there are both symbolic and numerical approaches to the analysis and solution of differentialequations.In this project we are mainly concerned with the symbolic treatment of ODEs and PDEs, i.e. reductions ofdifferential problems to algebraic ones and algebraic methods for these derived problems.This idea of an algebraic approach to differential equations has a long history. In the 19th century S. Lie initiatedthe investigation of transformations, which leave a given differential equation invariant. Such transformations arecommonly known as Lie symmetries. They form a group, a so-called Lie group. The basic idea here is to find agroup of symmetries of the differential equations and then use this group to reduce the order or the number ofvariables appearing in the equation. At the beginning of the 20th century Riquier and Janet introduced the conceptof involutivity for systems of algebraic differential equations. Their theory leads to canonical systems of generatorsfor differential ideals, and the algorithm for generating such canonical Janet bases is strikingly similar to themethod of Gröbner bases for generating canonical systems for algebraic ideals devoloped by Buchberger. Anotherapproach to symbolic solution of differential equations has been initiated by J. Liouville for the study of formalintegration, and Liouville's method has been extended to an algorithm for solving linear differential equations.There are several advantages of an algebraic analysis of differential equations: sometimes we are actually able togo the whole way and find a symbolic solution to the given DE;if we can derive symmetries of a DE, then this helps a lot in verifying numerical schemes for approximation ofsolutions; we can decide whether a system of ODEs or PDEs admits a solution; if there are solutions, then we canderive differential systems in triangular form such that the solutions of the original system are the (non-singular)solutions of the output system; we can get a complete overview of the algebraic relations satisfied by the solutionsof a given system of differential algebraic equations (DAEs).An effective treatment of these problems depends crucially on algorithms in differential elimination theory, i.e. thetheory of differential ideals in a ring of differential polynomials. The algebraic theory of elimination is welldeveloped, and we can answer the main questions such as consistency of ideals, equality of ideals, the membershipproblem, the radical membership problem, the arithmetic of ideals, etc. constructively. The methods for solvingthese problems are resultants, Gröbner bases, and algebraic involutive bases. For differential ideals there are stillmany open problems. For instance, the membership problem or the ideal inclusion problems for finitely generateddifferential ideals are still not solved.The analysis and improvement of existing algorithms as well as the development of new algorithmic methods indifferential elimination theory is the major goal of the project.Additionally, we will also consider the important application area of control theory. More... »

URL

http://pf.fwf.ac.at/en/research-in-practice/project-finder/11495

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/2201", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/2201", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/2201", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "type": "DefinedTerm"
      }
    ], 
    "description": "Differential equations allow us to model a great variety of phenomena arising in physics, engineering, biology,financial analysis etc. Such ordinary and partial differential equations (ODEs and PDEs) have been studied for along time, and there are both symbolic and numerical approaches to the analysis and solution of differentialequations.In this project we are mainly concerned with the symbolic treatment of ODEs and PDEs, i.e. reductions ofdifferential problems to algebraic ones and algebraic methods for these derived problems.This idea of an algebraic approach to differential equations has a long history. In the 19th century S. Lie initiatedthe investigation of transformations, which leave a given differential equation invariant. Such transformations arecommonly known as Lie symmetries. They form a group, a so-called Lie group. The basic idea here is to find agroup of symmetries of the differential equations and then use this group to reduce the order or the number ofvariables appearing in the equation. At the beginning of the 20th century Riquier and Janet introduced the conceptof involutivity for systems of algebraic differential equations. Their theory leads to canonical systems of generatorsfor differential ideals, and the algorithm for generating such canonical Janet bases is strikingly similar to themethod of Gr\u00f6bner bases for generating canonical systems for algebraic ideals devoloped by Buchberger. Anotherapproach to symbolic solution of differential equations has been initiated by J. Liouville for the study of formalintegration, and Liouville's method has been extended to an algorithm for solving linear differential equations.There are several advantages of an algebraic analysis of differential equations: sometimes we are actually able togo the whole way and find a symbolic solution to the given DE;if we can derive symmetries of a DE, then this helps a lot in verifying numerical schemes for approximation ofsolutions; we can decide whether a system of ODEs or PDEs admits a solution; if there are solutions, then we canderive differential systems in triangular form such that the solutions of the original system are the (non-singular)solutions of the output system; we can get a complete overview of the algebraic relations satisfied by the solutionsof a given system of differential algebraic equations (DAEs).An effective treatment of these problems depends crucially on algorithms in differential elimination theory, i.e. thetheory of differential ideals in a ring of differential polynomials. The algebraic theory of elimination is welldeveloped, and we can answer the main questions such as consistency of ideals, equality of ideals, the membershipproblem, the radical membership problem, the arithmetic of ideals, etc. constructively. The methods for solvingthese problems are resultants, Gr\u00f6bner bases, and algebraic involutive bases. For differential ideals there are stillmany open problems. For instance, the membership problem or the ideal inclusion problems for finitely generateddifferential ideals are still not solved.The analysis and improvement of existing algorithms as well as the development of new algorithmic methods indifferential elimination theory is the major goal of the project.Additionally, we will also consider the important application area of control theory.", 
    "endDate": "2007-04-30T00:00:00Z", 
    "funder": {
      "id": "https://www.grid.ac/institutes/grid.25111.36", 
      "type": "Organization"
    }, 
    "id": "sg:grant.6188012", 
    "identifier": [
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "6188012"
        ]
      }, 
      {
        "name": "fwf_id", 
        "type": "PropertyValue", 
        "value": [
          "P 16357"
        ]
      }
    ], 
    "inLanguage": [
      "en"
    ], 
    "keywords": [
      "equality", 
      "PDE", 
      "major goal", 
      "arithmetic", 
      "Janet", 
      "solution", 
      "important application area", 
      "Lie symmetries", 
      "ideal", 
      "agroup", 
      "open problem", 
      "differential ideals", 
      "control theory", 
      "20th century Riquier", 
      "complete overview", 
      "approximation ofsolutions", 
      "linear differential equations", 
      "improvement", 
      "numerical approach", 
      "differentialequation", 
      "problem", 
      "algebraic approach", 
      "differential system", 
      "algebraic relations", 
      "algebraic theory", 
      "ofdifferential problems", 
      "algebraic differential equations", 
      "idea", 
      "membershipproblem", 
      "basic idea", 
      "beginning", 
      "initiatedthe investigation", 
      "analysis", 
      "solvingthese problems", 
      "time", 
      "system", 
      "symbolic treatment", 
      "formalintegration", 
      "themethod", 
      "solutionsof", 
      "differential algebraic equations", 
      "biology", 
      "differential equation invariant", 
      "canonical system", 
      "algebraic method", 
      "transformation", 
      "DAEs).An effective treatment", 
      "differential elimination theory", 
      "theory", 
      "METHODS", 
      "equations", 
      "partial differential equations", 
      "differential equations", 
      "such transformations", 
      "physics", 
      "such canonical Janet bases", 
      "algebraic involutive bases", 
      "lot", 
      "ideal inclusion problems", 
      "numerical scheme", 
      "instance", 
      "development", 
      "able togo", 
      "Liouville method", 
      "algebraic ideal", 
      "order", 
      "ring", 
      "symbolic solution", 
      "triangular form", 
      "output system", 
      "resultants", 
      "elimination", 
      "ODE", 
      "algorithm", 
      "thetheory", 
      "Buchberger", 
      "groups", 
      "stillmany", 
      "engineering", 
      "algorithms", 
      "Gr\u00f6bner bases", 
      "whole way", 
      "19th century S. Lie", 
      "new algorithmic method", 
      "consistency", 
      "indifferential elimination theory", 
      "project", 
      "reduction", 
      "differential polynomials", 
      "study", 
      "financial analysis", 
      "great variety", 
      "main questions", 
      "conceptof involutivity", 
      "Lie groups", 
      "DE;if", 
      "J. Liouville", 
      "generatorsfor differential ideals", 
      "original system", 
      "generateddifferential ideals", 
      "algebraic analysis", 
      "de", 
      "radical membership problem", 
      "phenomenon", 
      "symmetry", 
      "number ofvariables", 
      "Anotherapproach", 
      "several advantages", 
      "long history", 
      "membership problem", 
      "algebraic one"
    ], 
    "name": "Differential Elimination Theory (DET)", 
    "recipient": [
      {
        "id": "https://www.grid.ac/institutes/grid.9970.7", 
        "type": "Organization"
      }, 
      {
        "affiliation": {
          "id": "https://www.grid.ac/institutes/grid.9970.7", 
          "name": "Universit\u00e4t Linz", 
          "type": "Organization"
        }, 
        "familyName": "WINKLER", 
        "givenName": "Franz", 
        "id": "sg:person.011460276755.42", 
        "type": "Person"
      }, 
      {
        "member": "sg:person.011460276755.42", 
        "roleName": "PI", 
        "type": "Role"
      }
    ], 
    "sameAs": [
      "https://app.dimensions.ai/details/grant/grant.6188012"
    ], 
    "sdDataset": "grants", 
    "sdDatePublished": "2019-03-07T11:31", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com.uberresearch.data.processor/core_data/20181219_192338/projects/base/fwf_projects.xml.gz", 
    "startDate": "2003-09-01T00:00:00Z", 
    "type": "MonetaryGrant", 
    "url": "http://pf.fwf.ac.at/en/research-in-practice/project-finder/11495"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/grant.6188012'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/grant.6188012'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/grant.6188012'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/grant.6188012'


 

This table displays all metadata directly associated to this object as RDF triples.

151 TRIPLES      18 PREDICATES      132 URIs      124 LITERALS      4 BLANK NODES

Subject Predicate Object
1 sg:grant.6188012 schema:about anzsrc-for:2201
2 schema:description Differential equations allow us to model a great variety of phenomena arising in physics, engineering, biology,financial analysis etc. Such ordinary and partial differential equations (ODEs and PDEs) have been studied for along time, and there are both symbolic and numerical approaches to the analysis and solution of differentialequations.In this project we are mainly concerned with the symbolic treatment of ODEs and PDEs, i.e. reductions ofdifferential problems to algebraic ones and algebraic methods for these derived problems.This idea of an algebraic approach to differential equations has a long history. In the 19th century S. Lie initiatedthe investigation of transformations, which leave a given differential equation invariant. Such transformations arecommonly known as Lie symmetries. They form a group, a so-called Lie group. The basic idea here is to find agroup of symmetries of the differential equations and then use this group to reduce the order or the number ofvariables appearing in the equation. At the beginning of the 20th century Riquier and Janet introduced the conceptof involutivity for systems of algebraic differential equations. Their theory leads to canonical systems of generatorsfor differential ideals, and the algorithm for generating such canonical Janet bases is strikingly similar to themethod of Gröbner bases for generating canonical systems for algebraic ideals devoloped by Buchberger. Anotherapproach to symbolic solution of differential equations has been initiated by J. Liouville for the study of formalintegration, and Liouville's method has been extended to an algorithm for solving linear differential equations.There are several advantages of an algebraic analysis of differential equations: sometimes we are actually able togo the whole way and find a symbolic solution to the given DE;if we can derive symmetries of a DE, then this helps a lot in verifying numerical schemes for approximation ofsolutions; we can decide whether a system of ODEs or PDEs admits a solution; if there are solutions, then we canderive differential systems in triangular form such that the solutions of the original system are the (non-singular)solutions of the output system; we can get a complete overview of the algebraic relations satisfied by the solutionsof a given system of differential algebraic equations (DAEs).An effective treatment of these problems depends crucially on algorithms in differential elimination theory, i.e. thetheory of differential ideals in a ring of differential polynomials. The algebraic theory of elimination is welldeveloped, and we can answer the main questions such as consistency of ideals, equality of ideals, the membershipproblem, the radical membership problem, the arithmetic of ideals, etc. constructively. The methods for solvingthese problems are resultants, Gröbner bases, and algebraic involutive bases. For differential ideals there are stillmany open problems. For instance, the membership problem or the ideal inclusion problems for finitely generateddifferential ideals are still not solved.The analysis and improvement of existing algorithms as well as the development of new algorithmic methods indifferential elimination theory is the major goal of the project.Additionally, we will also consider the important application area of control theory.
3 schema:endDate 2007-04-30T00:00:00Z
4 schema:funder https://www.grid.ac/institutes/grid.25111.36
5 schema:identifier Naf237bdbff39412fa85639347a5090ae
6 Nf428072d4dc845328a494905dbb5c663
7 schema:inLanguage en
8 schema:keywords 19th century S. Lie
9 20th century Riquier
10 Anotherapproach
11 Buchberger
12 DAEs).An effective treatment
13 DE;if
14 Gröbner bases
15 J. Liouville
16 Janet
17 Lie groups
18 Lie symmetries
19 Liouville method
20 METHODS
21 ODE
22 PDE
23 able togo
24 agroup
25 algebraic analysis
26 algebraic approach
27 algebraic differential equations
28 algebraic ideal
29 algebraic involutive bases
30 algebraic method
31 algebraic one
32 algebraic relations
33 algebraic theory
34 algorithm
35 algorithms
36 analysis
37 approximation ofsolutions
38 arithmetic
39 basic idea
40 beginning
41 biology
42 canonical system
43 complete overview
44 conceptof involutivity
45 consistency
46 control theory
47 de
48 development
49 differential algebraic equations
50 differential elimination theory
51 differential equation invariant
52 differential equations
53 differential ideals
54 differential polynomials
55 differential system
56 differentialequation
57 elimination
58 engineering
59 equality
60 equations
61 financial analysis
62 formalintegration
63 generateddifferential ideals
64 generatorsfor differential ideals
65 great variety
66 groups
67 idea
68 ideal
69 ideal inclusion problems
70 important application area
71 improvement
72 indifferential elimination theory
73 initiatedthe investigation
74 instance
75 linear differential equations
76 long history
77 lot
78 main questions
79 major goal
80 membership problem
81 membershipproblem
82 new algorithmic method
83 number ofvariables
84 numerical approach
85 numerical scheme
86 ofdifferential problems
87 open problem
88 order
89 original system
90 output system
91 partial differential equations
92 phenomenon
93 physics
94 problem
95 project
96 radical membership problem
97 reduction
98 resultants
99 ring
100 several advantages
101 solution
102 solutionsof
103 solvingthese problems
104 stillmany
105 study
106 such canonical Janet bases
107 such transformations
108 symbolic solution
109 symbolic treatment
110 symmetry
111 system
112 themethod
113 theory
114 thetheory
115 time
116 transformation
117 triangular form
118 whole way
119 schema:name Differential Elimination Theory (DET)
120 schema:recipient N1e57863bcbe340049208ee0f5145e327
121 sg:person.011460276755.42
122 https://www.grid.ac/institutes/grid.9970.7
123 schema:sameAs https://app.dimensions.ai/details/grant/grant.6188012
124 schema:sdDatePublished 2019-03-07T11:31
125 schema:sdLicense https://scigraph.springernature.com/explorer/license/
126 schema:sdPublisher N2b7583f43897496fbaf32a343e034c4f
127 schema:startDate 2003-09-01T00:00:00Z
128 schema:url http://pf.fwf.ac.at/en/research-in-practice/project-finder/11495
129 sgo:license sg:explorer/license/
130 sgo:sdDataset grants
131 rdf:type schema:MonetaryGrant
132 N1e57863bcbe340049208ee0f5145e327 schema:member sg:person.011460276755.42
133 schema:roleName PI
134 rdf:type schema:Role
135 N2b7583f43897496fbaf32a343e034c4f schema:name Springer Nature - SN SciGraph project
136 rdf:type schema:Organization
137 Naf237bdbff39412fa85639347a5090ae schema:name fwf_id
138 schema:value P 16357
139 rdf:type schema:PropertyValue
140 Nf428072d4dc845328a494905dbb5c663 schema:name dimensions_id
141 schema:value 6188012
142 rdf:type schema:PropertyValue
143 anzsrc-for:2201 schema:inDefinedTermSet anzsrc-for:
144 rdf:type schema:DefinedTerm
145 sg:person.011460276755.42 schema:affiliation https://www.grid.ac/institutes/grid.9970.7
146 schema:familyName WINKLER
147 schema:givenName Franz
148 rdf:type schema:Person
149 https://www.grid.ac/institutes/grid.25111.36 schema:Organization
150 https://www.grid.ac/institutes/grid.9970.7 schema:name Universität Linz
151 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...