О некоторых многозначных соответствиях в проективной геометрии
В настоящей статье дается критический обзор работ по проективной геометрии К. А. Андреева, А. К. Власова и А. А. Глаголева, относящихся к многозначным соответствиям.
Збережено в:
| Дата: | 1953 |
|---|---|
| Автор: | |
| Формат: | Стаття |
| Мова: | Російська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1953
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/7757 |
| Теги: |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: | |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512747381325824 |
|---|---|
| author | Мандзюк, А. И. Мандзюк, А. И. |
| author_facet | Мандзюк, А. И. Мандзюк, А. И. |
| author_sort | Мандзюк, А. И. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-08-15T10:11:31Z |
| description | В настоящей статье дается критический обзор работ по проективной геометрии К. А. Андреева, А. К. Власова и А. А. Глаголева, относящихся к многозначным соответствиям. |
| first_indexed | 2026-03-24T03:33:42Z |
| format | Article |
| fulltext |
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5. L. Po m c y, Journal de L'Ecole Polytcchnique, Paris. 1923. 1937.
G. R. Stu r 111, Die Lchrc von den geometrischen \'erwandtschaftcn, Bel. 1. II, III
Leipzig und Berlin, 1908.
DoJiytteHa 6 ~rapTa 1953 r.
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| id | umjimathkievua-article-7757 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus |
| last_indexed | 2026-03-24T03:33:42Z |
| publishDate | 1953 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6d/b5f1c42bd12421146ec71d0e5d3d4d6d.pdf |
| spelling | umjimathkievua-article-77572023-08-15T10:11:31Z О некоторых многозначных соответствиях в проективной геометрии О некоторых многозначных соответствиях в проективной геометрии Мандзюк, А. И. Мандзюк, А. И. В настоящей статье дается критический обзор работ по проективной геометрии К. А. Андреева, А. К. Власова и А. А. Глаголева, относящихся к многозначным соответствиям. В настоящей статье дается критический обзор работ по проективной геометрии К. А. Андреева, А. К. Власова и А. А. Глаголева, относящихся к многозначным соответствиям. Institute of Mathematics, NAS of Ukraine 1953-11-09 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7757 Ukrains’kyi Matematychnyi Zhurnal; Vol. 5 No. 4 (1953); 439-452 Український математичний журнал; Том 5 № 4 (1953); 439-452 1027-3190 rus https://umj.imath.kiev.ua/index.php/umj/article/view/7757/9445 Copyright (c) 1953 А. И. Мандзюк |
| spellingShingle | Мандзюк, А. И. Мандзюк, А. И. О некоторых многозначных соответствиях в проективной геометрии |
| title | О некоторых многозначных соответствиях в проективной геометрии |
| title_alt | О некоторых многозначных соответствиях в проективной геометрии |
| title_full | О некоторых многозначных соответствиях в проективной геометрии |
| title_fullStr | О некоторых многозначных соответствиях в проективной геометрии |
| title_full_unstemmed | О некоторых многозначных соответствиях в проективной геометрии |
| title_short | О некоторых многозначных соответствиях в проективной геометрии |
| title_sort | о некоторых многозначных соответствиях в проективной геометрии |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7757 |
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