{
"schema_version": 1,
"kind": "ConceptView",
"producer": "concept-view-1",
"concept_id": "urn:uuid:a548ce48-3340-4d0d-a18e-ca75f604678e",
"labels": {
"zh-CN": "循环",
"en": "cycle"
},
"alternative_labels": [],
"scope_note": "",
"identity_status": "active",
"identity_revision": "2cd2f44e8319443af0c54db7adef6e6ab4a01d6b59921be834e000a5e0c68a35",
"occurrences": [
{
"concept_id": "urn:uuid:a548ce48-3340-4d0d-a18e-ca75f604678e",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "zh-CN",
"route": "zh/chapter-02/02-transformation-and-permutation-groups",
"anchor": "concept-2-cycle",
"chapter": "2",
"section": "2.2",
"offset": 4524,
"range": [
9206,
9420
],
"source_revision": "799abb1b61f01e9b2d7152841faf2f190564a4f0ab51672f131c33ec84fc0875",
"source_category": "translation",
"review_status": "unreviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "循环(cycle)",
"context": {
"range": [
9051,
9649
],
"quote": "一种更简洁的排列符号是循环符号。从任意一个要被置换的元素开始，例如 $a_1$。让 $a_2$ 成为对 $a_1$ 应用 $\\pi$ 变化的结果，让 $a_3$ 成为对 $a_2$ 应用 $\\pi$ 变化的结果，等等。最终，第一个元素 $a_1$ 必须再次出现，比如 $a_{m+1}=a_1$。这就定义了一个循环(cycle)，写成 $\\left(a_1 a_2\\ldots a_m\\right)$。如果 $m=n$，那么 $\\pi$ 就被称为循环排列。如果 $m边界(boundary)，即$\\partial C = 0 .$，则称其为循环(cycle)。若存在一个$(p+1)$-链$C^{\\prime}$使得$C = \\partial C ^ { \\prime }$，则称其为边界。显然，每个边界都是$p$-循环，因为$\\partial C = \\partial ^ { 2 } C ^ { \\prime } = 0$，但反之未必成立。",
"quote_sha256": "87ab1013623f7b1d866eec5fd7e2263c4feb3674f6500fd026699ca6e6470ef7",
"environment": null,
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "en/chapter-17/04-homology-and-cohomology",
"lang": "en",
"source_category": "original_transcription",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-40ff56a73f5de38da30764ac"
},
{
"concept_id": "urn:uuid:a548ce48-3340-4d0d-a18e-ca75f604678e",
"book_id": "szekeres",
"edition_id": "source-snapshot-bc3404156f6e",
"lang": "en",
"route": "en/chapter-17/04-homology-and-cohomology",
"anchor": "first-20322b3f8009b543e4b2-en",
"chapter": "17",
"section": "17.4",
"offset": 9172,
"range": [
12996,
13220
],
"source_revision": "bcdaffd9c93fa5d4dbb16ac427fef782512ff58c5d5472ea282c7769a2173a3d",
"source_category": "original_transcription",
"review_status": "unreviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "cycle",
"context": {
"range": [
12964,
13730
],
"quote": "A $p$-chain $C$ is said to be a cycle if it has no boundary, $\\partial C = 0 .$ . It is said to be a boundary if there exists a $(p+1)$-chain $C^{\\prime}$ such that $C = \\partial C ^ { \\prime }$ . Clearly every boundary is a $p$-cycle since $\\partial C = \\partial ^ { 2 } C ^ { \\prime } = 0$ , but the converse need not be true.",
"quote_sha256": "c4d921ec9c18b90d439dd1bfbace10c116e4b33dd2c52e2985526ab0ff0c0fbb",
"environment": null,
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "zh/chapter-17/04-homology-and-cohomology",
"lang": "zh-CN",
"source_category": "translation",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-f8781a28782bb76b79660b1e"
}
],
"registered_positions": [
{
"route": "zh/chapter-02/02-transformation-and-permutation-groups",
"lang": "zh-CN",
"section": "2.2",
"signal": "term-mark",
"label": "循环",
"english": "cycle",
"offset": 3900
},
{
"route": "en/chapter-17/04-homology-and-cohomology",
"lang": "en",
"section": "17.4",
"signal": "term-mark",
"label": "cycle",
"english": "cycle",
"offset": 8217
},
{
"route": "zh/chapter-17/04-homology-and-cohomology",
"lang": "zh-CN",
"section": "17.4",
"signal": "term-mark",
"label": "循环",
"english": "cycle",
"offset": 6549
}
],
"statements": [],
"resolved_references": [],
"reading_order": [
"occ-c1c06aa5661ec17f964925f6",
"occ-40ff56a73f5de38da30764ac",
"occ-f8781a28782bb76b79660b1e"
],
"recorded_dependencies": [],
"reading_paths": [],
"coverage": {
"registered_positions": 3,
"located_occurrences": 3,
"whole_book_search_complete": false,
"registered_positions_truncated": false,
"independent_human_review_implied": false
},
"unresolved": [],
"base_revision": "14fc3d07069123c241ed4ef5a4a8bff4e7f2eb140c121304669058dee63c8647",
"optional_enrichments": []
}