{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:2MQ3CONEAT3SLCTNMPOUCFB6IV","short_pith_number":"pith:2MQ3CONE","schema_version":"1.0","canonical_sha256":"d321b139a404f7258a6d63dd41143e45597c54e740b6f4a040ad3a274ca667c5","source":{"kind":"arxiv","id":"1707.05764","version":2},"attestation_state":"computed","paper":{"title":"On the distance between homotopy classes in $W^{1/p,p}({\\mathbb S}^1;{\\mathbb S}^1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Itai Shafrir","submitted_at":"2017-07-18T17:47:38Z","abstract_excerpt":"For every $p\\in(1,\\infty)$ there is a natural notion of topological degree for maps in $W^{1/p,p}({\\mathbb S}^1;{\\mathbb S}^1)$ which allows us to write that space as a disjoint union of classes, $W^{1/p,p}({\\mathbb S}^1;{\\mathbb S}^1)=\\bigcup_{d\\in{\\mathbb Z}}\\mathcal{E}_d$. For every pair $d_1,d_2\\in {\\mathbb Z}$, we show that the distance $\\text{Dist}_{W^{1/p,p}}({\\mathcal\n  E}_{d_1}, {\\mathcal E}_{d_2}):=\\sup_{f\\in{\\mathcal E}_{d_1}}\\ \\inf_{g\\in{\\mathcal E}_{d_2}}\\ d_{W^{1/p,p}}(f, g)$ equals the minimal $W^{1/p,p}$-energy in $\\mathcal{E}_{d_1-d_2}$. In the special case $p=2$ we deduce fro"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1707.05764","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-07-18T17:47:38Z","cross_cats_sorted":[],"title_canon_sha256":"ef32ca04cbcc43930654972a5d515f437be4a75fc78bff5bc866015bbe8a28e7","abstract_canon_sha256":"2db5c39efe8468daf852576a5c0a2ef7c5bb6917709f7a3cc2a6430f1360158c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:00.403473Z","signature_b64":"9KdAZAN5e+dIjidehMBD4ti/dgIz9XCMm54S94TIlts1twQav+qiRZJSD2NJoTQqRIlMd0T2zTzze3D/zcPFDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d321b139a404f7258a6d63dd41143e45597c54e740b6f4a040ad3a274ca667c5","last_reissued_at":"2026-05-18T00:27:00.402745Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:00.402745Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the distance between homotopy classes in $W^{1/p,p}({\\mathbb S}^1;{\\mathbb S}^1)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Itai Shafrir","submitted_at":"2017-07-18T17:47:38Z","abstract_excerpt":"For every $p\\in(1,\\infty)$ there is a natural notion of topological degree for maps in $W^{1/p,p}({\\mathbb S}^1;{\\mathbb S}^1)$ which allows us to write that space as a disjoint union of classes, $W^{1/p,p}({\\mathbb S}^1;{\\mathbb S}^1)=\\bigcup_{d\\in{\\mathbb Z}}\\mathcal{E}_d$. For every pair $d_1,d_2\\in {\\mathbb Z}$, we show that the distance $\\text{Dist}_{W^{1/p,p}}({\\mathcal\n  E}_{d_1}, {\\mathcal E}_{d_2}):=\\sup_{f\\in{\\mathcal E}_{d_1}}\\ \\inf_{g\\in{\\mathcal E}_{d_2}}\\ d_{W^{1/p,p}}(f, g)$ equals the minimal $W^{1/p,p}$-energy in $\\mathcal{E}_{d_1-d_2}$. In the special case $p=2$ we deduce fro"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.05764","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1707.05764","created_at":"2026-05-18T00:27:00.402856+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.05764v2","created_at":"2026-05-18T00:27:00.402856+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.05764","created_at":"2026-05-18T00:27:00.402856+00:00"},{"alias_kind":"pith_short_12","alias_value":"2MQ3CONEAT3S","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_16","alias_value":"2MQ3CONEAT3SLCTN","created_at":"2026-05-18T12:30:55.937587+00:00"},{"alias_kind":"pith_short_8","alias_value":"2MQ3CONE","created_at":"2026-05-18T12:30:55.937587+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2MQ3CONEAT3SLCTNMPOUCFB6IV","json":"https://pith.science/pith/2MQ3CONEAT3SLCTNMPOUCFB6IV.json","graph_json":"https://pith.science/api/pith-number/2MQ3CONEAT3SLCTNMPOUCFB6IV/graph.json","events_json":"https://pith.science/api/pith-number/2MQ3CONEAT3SLCTNMPOUCFB6IV/events.json","paper":"https://pith.science/paper/2MQ3CONE"},"agent_actions":{"view_html":"https://pith.science/pith/2MQ3CONEAT3SLCTNMPOUCFB6IV","download_json":"https://pith.science/pith/2MQ3CONEAT3SLCTNMPOUCFB6IV.json","view_paper":"https://pith.science/paper/2MQ3CONE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.05764&json=true","fetch_graph":"https://pith.science/api/pith-number/2MQ3CONEAT3SLCTNMPOUCFB6IV/graph.json","fetch_events":"https://pith.science/api/pith-number/2MQ3CONEAT3SLCTNMPOUCFB6IV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2MQ3CONEAT3SLCTNMPOUCFB6IV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2MQ3CONEAT3SLCTNMPOUCFB6IV/action/storage_attestation","attest_author":"https://pith.science/pith/2MQ3CONEAT3SLCTNMPOUCFB6IV/action/author_attestation","sign_citation":"https://pith.science/pith/2MQ3CONEAT3SLCTNMPOUCFB6IV/action/citation_signature","submit_replication":"https://pith.science/pith/2MQ3CONEAT3SLCTNMPOUCFB6IV/action/replication_record"}},"created_at":"2026-05-18T00:27:00.402856+00:00","updated_at":"2026-05-18T00:27:00.402856+00:00"}