Deza E. Generalizations of finite metrics and cuts  (New Jersey, 2016). - ОГЛАВЛЕНИЕ / CONTENTS
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ОбложкаDeza E. Generalizations of finite metrics and cuts / E.Deza, M.Deza, M.D.Sikiric. - New Jersey: World scientific, 2016. - xiii, 303 p.: ill., tab. - Bibliogr.: p.295-303. - ISBN 978-981-4740-39-5
Шифр: (И/ В18-D51) 02
 

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Оглавление / Contents
 
Introduction ................................................... ix

I  Preliminaries ................................................ 1

1  Short preview of the book .................................... 3
   1.1  Outline of Part I. Preliminaries ........................ 4
   1.2  Outline of Part II. Main notions and examples ........... 4
   1.3  Outline of Part III. Cuts, hypermetrics and their
        generalizations ......................................... 6
   1.4  Outline of Part IV. Cones and polytopes of generalized 
        finite metrics .......................................... 6
   1.5  Outline of Part V. Important cases of polyhedra of
        generalized finite semimetrics .......................... 7
2  Main definitions ............................................. 9
   2.1  Graphs .................................................. 9
   2.2  Vector spaces .......................................... 16
   2.3  Matrices ............................................... 19
   2.4  Cones and polytopes .................................... 20
   
II Main notions and examples ................................... 29
3  Non-oriented case: metrics .................................. 31
   3.1  Preliminaries .......................................... 31
   3.2  Definitions ............................................ 32
   3.3  Examples ............................................... 33
4  Oriented case: quasi-metrics ................................ 51
   4.1  Preliminaries .......................................... 51
   4.2  Definitions ............................................ 51
   4.3  Examples ............................................... 53
5  Multidimensional case: m-metrics ............................ 61
   5.1  Preliminaries .......................................... 61
   5.2  Definitions ............................................ 61
   5.3  Examples ............................................... 65
6  Important special case: partial metrics and weightable 
   quasi-metrics ............................................... 83
   6.1  Preliminaries .......................................... 83
   6.2  Definitions ............................................ 83
   6.3  Examples ............................................... 87
   
III  Cuts, hypermetrics and their generalizations .............. 93

7  Cuts and their generalizations .............................. 95
   7.1  Preliminaries .......................................... 95
   7.2  Classical non-oriented case ............................ 95
   7.3  Oriented case .......................................... 99
   7.4  Multidimensional case ................................. 102
   7.5  Important special cases of cut-related constructions .. 104
8  Hypermetrics and their generalizations ..................... 107
   8.1  Preliminaries ......................................... 107
   8.2  Hypermetric and negative type inequalities ............ 107
   8.3  Hypermetrics and distances of negative type ........... 110
   8.4  Some generalizations of hypermetrics .................. 112

IV  Cones and polytopes of generalized finite semimetrics ..... 117

9  Non-oriented case: semimetrics and cuts .................... 119
   9.1  Preliminaries ......................................... 119
   9.2  Cones and polytopes of semimetrics and cuts ........... 120
   9.3  Small cones and polytopes of semimetrics and cuts ..... 124
   9.4  Theorems and conjectures for general case ............. 127
10 Oriented case: quasi-semimetrics and oriented cuts ......... 129
   10.1 Preliminaries ......................................... 129
   10.2 Cones and polytopes of quasi-semimetrics and 
        oriented multicuts .................................... 130
   10.3 Small cones and polytopes of quasi-semimetrics and
        oriented multicuts .................................... 135
   10.4 Theorems and conjectures for general case ............. 149
   10.5 Other constructions of quasi-semimetric polyhedra ..... 156
11 Multidimensional case: ш-hemimetrics ....................... 157
   11.1 Preliminaries ......................................... 157
   11.2 Cones and polytopes of m-hemimetrics and (m, s)-
        supermetrics .......................................... 158
   11.3 Small cones of m-hemimetrics and (m, s)-supermetrics .. 162
   11.4 Some special cases of parameters ...................... 178
   11.5 Theorems and conjectures for general case ............. 182
   
V  Important cases of polyhedra of generalized finite 
   semimetrics ................................................ 187

12 Cones of partial semimetrics and weightable quasi-
   semimetrics ................................................ 189
   12.1 Preliminaries ......................................... 189
   12.2 Polyhedra of partial semimetrics and weightable 
        quasi-semimetrics ..................................... 191
   12.3 Maps P, Q and connections between considered 
        polyhedra ............................................. 195
   12.4 Small polyhedra of partial semimetrics and
        weightable quasi-semimetrics .......................... 198
   12.5 Theorems and conjectures for general case ............. 205
13 Cones of hypermetrics ...................................... 219
   13.1 Preliminaries ......................................... 219
   13.2 Non-oriented case ..................................... 221
   13.3 Partial and weighted hypermetric cones ................ 234
   13.4 Quasi-hypermetric cones ............................... 237
14 Cuts over general graphs ................................... 243
   14.1 Preliminaries ......................................... 243
   14.2 Metric and cut polyhedra over graphs .................. 244
   14.3 Cut polytopes over some graphs ........................ 249
   14.4 Some general results about metric and cut polyhedra
        over graphs ........................................... 255
15 Connections between generalized metrics polyhedra .......... 261
   15.1 Preliminaries ......................................... 261
   15.2 Decomposition of real vector spaces ................... 262
   15.3 Construction of projections of cones on n + 1 points .. 269
   15.4 Projections of METn+1 and CUTn+1 ....................... 277
   15.5 Cases 3 ≤ n ≤ 6 ....................................... 284

Appendixes .................................................... 287
Bibliography .................................................. 295

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