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The Resource Geometry of Subanalytic and Semialgebraic Sets, by Masahiro Shiota, (electronic resource)
Geometry of Subanalytic and Semialgebraic Sets, by Masahiro Shiota, (electronic resource)
Resource Information
The item Geometry of Subanalytic and Semialgebraic Sets, by Masahiro Shiota, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Oklahoma Libraries.This item is available to borrow from all library branches.
Resource Information
The item Geometry of Subanalytic and Semialgebraic Sets, by Masahiro Shiota, (electronic resource) represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in University of Oklahoma Libraries.
This item is available to borrow from all library branches.
 Summary
 Real analytic sets in Euclidean space (Le. , sets defined locally at each point of Euclidean space by the vanishing of an analytic function) were first investigated in the 1950's by H. Cartan [Car], H. Whitney [WI3], F. Bruhat [WB] and others. Their approach was to derive information about real analytic sets from properties of their complexifications. After some basic geometrical and topological facts were established, however, the study of real analytic sets stagnated. This contrasted the rapid develop ment of complex analytic geometry which followed the groundbreaking work of the early 1950's. Certain pathologies in the real case contributed to this failure to progress. For example, the closure of or the connected components ofa constructible set (Le. , a locally finite union of differ ences of real analytic sets) need not be constructible (e. g. , R  {O} and 3 2 2 { (x, y, z) E R : x = zy2, x + y2 =I O}, respectively). Responding to this in the 1960's, R. Thorn [Thl], S. Lojasiewicz [LI,2] and others undertook the study of a larger class of sets, the semianalytic sets, which are the sets defined locally at each point of Euclidean space by a finite number of ana lytic function equalities and inequalities. They established that semianalytic sets admit Whitney stratifications and triangulations, and using these tools they clarified the local topological structure of these sets. For example, they showed that the closure and the connected components of a semianalytic set are semianalytic
 Language

 eng
 eng
 Edition
 First edition.
 Extent
 1 online resource (XII, 434 p.)
 Note
 Bibliographic Level Mode of Issuance: Monograph
 Contents

 I. Preliminaries
 §1.1. Whitney stratifications
 §1.2. Subanalytic sets and semialgebraic sets
 §1.3. PL topology and C? triangulations
 II. XSets
 §11.1. Xsets
 §11.2. Triangulations of Xsets
 §11.3. Triangulations of X functions
 §11.4. Triangulations of semialgebraic and X0 sets and functions
 §11.5. Cr Xmanifolds
 §11.6. Xtriviality of Xmaps
 §11.7. Xsingularity theory
 III. Hauptvermutung For Polyhedra
 §III.1. Certain conditions for two polyhedra to be PL homeomorphic
 §III.2. Proofs of Theorems III.1.1 and III.1.2
 IV. Triangulations of XMaps
 §IV.l. Conditions for Xmaps to be triangulable
 §IV.2. Proofs of Theorems IV.1.1, IV.1.2, IV.1.2? and IV.1.2?
 §IV.3. Local and global Xtriangulations and uniqueness
 §IV.4. Proofs of Theorems IV.1.10, IV.1.13 and IV.1.13?
 V. DSets
 §V.1. Case where any Dset is locally semilinear
 §V.2. Case where there exists a Dset which is not locally semilinear
 List of Notation
 Isbn
 9781461220084
 Label
 Geometry of Subanalytic and Semialgebraic Sets
 Title
 Geometry of Subanalytic and Semialgebraic Sets
 Statement of responsibility
 by Masahiro Shiota
 Language

 eng
 eng
 Summary
 Real analytic sets in Euclidean space (Le. , sets defined locally at each point of Euclidean space by the vanishing of an analytic function) were first investigated in the 1950's by H. Cartan [Car], H. Whitney [WI3], F. Bruhat [WB] and others. Their approach was to derive information about real analytic sets from properties of their complexifications. After some basic geometrical and topological facts were established, however, the study of real analytic sets stagnated. This contrasted the rapid develop ment of complex analytic geometry which followed the groundbreaking work of the early 1950's. Certain pathologies in the real case contributed to this failure to progress. For example, the closure of or the connected components ofa constructible set (Le. , a locally finite union of differ ences of real analytic sets) need not be constructible (e. g. , R  {O} and 3 2 2 { (x, y, z) E R : x = zy2, x + y2 =I O}, respectively). Responding to this in the 1960's, R. Thorn [Thl], S. Lojasiewicz [LI,2] and others undertook the study of a larger class of sets, the semianalytic sets, which are the sets defined locally at each point of Euclidean space by a finite number of ana lytic function equalities and inequalities. They established that semianalytic sets admit Whitney stratifications and triangulations, and using these tools they clarified the local topological structure of these sets. For example, they showed that the closure and the connected components of a semianalytic set are semianalytic
 http://library.link/vocab/creatorName
 Shiota, Masahiro
 Dewey number
 514
 http://bibfra.me/vocab/relation/httpidlocgovvocabularyrelatorsaut
 f8hyGBFpJd4
 Image bit depth
 0
 Language note
 English
 LC call number
 QA611614.97
 Literary form
 non fiction
 Nature of contents
 dictionaries
 Series statement
 Progress in Mathematics,
 Series volume
 150
 http://library.link/vocab/subjectName

 Topology
 Geometry, algebraic
 Algebraic topology
 Logic, Symbolic and mathematical
 Geometry
 Topology
 Algebraic Geometry
 Algebraic Topology
 Mathematical Logic and Foundations
 Geometry
 Label
 Geometry of Subanalytic and Semialgebraic Sets, by Masahiro Shiota, (electronic resource)
 Note
 Bibliographic Level Mode of Issuance: Monograph
 Antecedent source
 mixed
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Color
 not applicable
 Content category
 text
 Content type code

 txt
 Contents
 I. Preliminaries  §1.1. Whitney stratifications  §1.2. Subanalytic sets and semialgebraic sets  §1.3. PL topology and C? triangulations  II. XSets  §11.1. Xsets  §11.2. Triangulations of Xsets  §11.3. Triangulations of X functions  §11.4. Triangulations of semialgebraic and X0 sets and functions  §11.5. Cr Xmanifolds  §11.6. Xtriviality of Xmaps  §11.7. Xsingularity theory  III. Hauptvermutung For Polyhedra  §III.1. Certain conditions for two polyhedra to be PL homeomorphic  §III.2. Proofs of Theorems III.1.1 and III.1.2  IV. Triangulations of XMaps  §IV.l. Conditions for Xmaps to be triangulable  §IV.2. Proofs of Theorems IV.1.1, IV.1.2, IV.1.2? and IV.1.2?  §IV.3. Local and global Xtriangulations and uniqueness  §IV.4. Proofs of Theorems IV.1.10, IV.1.13 and IV.1.13?  V. DSets  §V.1. Case where any Dset is locally semilinear  §V.2. Case where there exists a Dset which is not locally semilinear  List of Notation
 Dimensions
 unknown
 Edition
 First edition.
 Extent
 1 online resource (XII, 434 p.)
 File format
 multiple file formats
 Form of item
 online
 Isbn
 9781461220084
 Level of compression
 uncompressed
 Media category
 computer
 Media type code

 c
 Other control number
 10.1007/9781461220084
 Quality assurance targets
 absent
 Reformatting quality
 access
 Specific material designation
 remote
 System control number

 (CKB)3400000000089797
 (SSID)ssj0001296929
 (PQKBManifestationID)11886709
 (PQKBTitleCode)TC0001296929
 (PQKBWorkID)11353920
 (PQKB)10928391
 (DEHe213)9781461220084
 (MiAaPQ)EBC3076328
 (EXLCZ)993400000000089797
 Label
 Geometry of Subanalytic and Semialgebraic Sets, by Masahiro Shiota, (electronic resource)
 Note
 Bibliographic Level Mode of Issuance: Monograph
 Antecedent source
 mixed
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code

 cr
 Color
 not applicable
 Content category
 text
 Content type code

 txt
 Contents
 I. Preliminaries  §1.1. Whitney stratifications  §1.2. Subanalytic sets and semialgebraic sets  §1.3. PL topology and C? triangulations  II. XSets  §11.1. Xsets  §11.2. Triangulations of Xsets  §11.3. Triangulations of X functions  §11.4. Triangulations of semialgebraic and X0 sets and functions  §11.5. Cr Xmanifolds  §11.6. Xtriviality of Xmaps  §11.7. Xsingularity theory  III. Hauptvermutung For Polyhedra  §III.1. Certain conditions for two polyhedra to be PL homeomorphic  §III.2. Proofs of Theorems III.1.1 and III.1.2  IV. Triangulations of XMaps  §IV.l. Conditions for Xmaps to be triangulable  §IV.2. Proofs of Theorems IV.1.1, IV.1.2, IV.1.2? and IV.1.2?  §IV.3. Local and global Xtriangulations and uniqueness  §IV.4. Proofs of Theorems IV.1.10, IV.1.13 and IV.1.13?  V. DSets  §V.1. Case where any Dset is locally semilinear  §V.2. Case where there exists a Dset which is not locally semilinear  List of Notation
 Dimensions
 unknown
 Edition
 First edition.
 Extent
 1 online resource (XII, 434 p.)
 File format
 multiple file formats
 Form of item
 online
 Isbn
 9781461220084
 Level of compression
 uncompressed
 Media category
 computer
 Media type code

 c
 Other control number
 10.1007/9781461220084
 Quality assurance targets
 absent
 Reformatting quality
 access
 Specific material designation
 remote
 System control number

 (CKB)3400000000089797
 (SSID)ssj0001296929
 (PQKBManifestationID)11886709
 (PQKBTitleCode)TC0001296929
 (PQKBWorkID)11353920
 (PQKB)10928391
 (DEHe213)9781461220084
 (MiAaPQ)EBC3076328
 (EXLCZ)993400000000089797
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<div class="citation" vocab="http://schema.org/"><i class="fa faexternallinksquare fafw"></i> Data from <span resource="http://link.libraries.ou.edu/portal/GeometryofSubanalyticandSemialgebraicSets/FsPB1ukPgU/" typeof="Book http://bibfra.me/vocab/lite/Item"><span property="name http://bibfra.me/vocab/lite/label"><a href="http://link.libraries.ou.edu/portal/GeometryofSubanalyticandSemialgebraicSets/FsPB1ukPgU/">Geometry of Subanalytic and Semialgebraic Sets, by Masahiro Shiota, (electronic resource)</a></span>  <span property="potentialAction" typeOf="OrganizeAction"><span property="agent" typeof="LibrarySystem http://library.link/vocab/LibrarySystem" resource="http://link.libraries.ou.edu/"><span property="name http://bibfra.me/vocab/lite/label"><a property="url" href="http://link.libraries.ou.edu/">University of Oklahoma Libraries</a></span></span></span></span></div>