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We associate to each point x\\in M, a Fr\\'{e}chet space H_x(defined in section 3). We prove that if H_x are locally constant, then with certain smoothness and boundedness condition, there exists Darboux chart for the weak symplectic structure."},"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":"1309.1693","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SG","submitted_at":"2013-09-06T16:29:38Z","cross_cats_sorted":["math-ph","math.DG","math.MP"],"title_canon_sha256":"856b24bf814fe229016007a28e33bb84d0889905e236619b032070dc829a14c7","abstract_canon_sha256":"ee9f4f2746c8317de76b050279ea1df11fbc1de4b40712ed989a5a56ff605b48"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:01:58.163028Z","signature_b64":"eX8Ix3afaVKZWG9fxkhxQ82aOekL1NuASaBirAPYswf+4ehOhtJzok+GscX+/RxPtx1amHZeyf8Msk3feHZqCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5de0d065c37c8c82137e904f140f98ca8d840a784c70bc26900b8b5427ca01a6","last_reissued_at":"2026-05-18T03:01:58.162480Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:01:58.162480Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Darboux chart on Projective limit of weak symplectic Banach manifold","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.DG","math.MP"],"primary_cat":"math.SG","authors_text":"Pradip Kumar","submitted_at":"2013-09-06T16:29:38Z","abstract_excerpt":"Suppose M be the projective limit of weak symplectic Banach manifolds \\{(M_i,\\phi_{ij})\\}_{i,j\\in\\mathbb N}, where M_i are modeled over reflexive Banach space and \\sigma is compatible with the inverse system(defined in the article). We associate to each point x\\in M, a Fr\\'{e}chet space H_x(defined in section 3). 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