{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:F3LJDD7X6NQHGJKUTJZQZ5GJRV","short_pith_number":"pith:F3LJDD7X","canonical_record":{"source":{"id":"1311.4282","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-18T07:13:52Z","cross_cats_sorted":[],"title_canon_sha256":"d6e32f23ad60294a821beab966d9b317663d7d6ccb2e960b0ad3315df84f5eb1","abstract_canon_sha256":"1d711419c0de060be8d40313f933db1f3660d56adbd8a2f8665c490228132545"},"schema_version":"1.0"},"canonical_sha256":"2ed6918ff7f3607325549a730cf4c98d59620356c0fcbf89067ebe21041a1483","source":{"kind":"arxiv","id":"1311.4282","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.4282","created_at":"2026-05-18T03:03:46Z"},{"alias_kind":"arxiv_version","alias_value":"1311.4282v3","created_at":"2026-05-18T03:03:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4282","created_at":"2026-05-18T03:03:46Z"},{"alias_kind":"pith_short_12","alias_value":"F3LJDD7X6NQH","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"F3LJDD7X6NQHGJKU","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"F3LJDD7X","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:F3LJDD7X6NQHGJKUTJZQZ5GJRV","target":"record","payload":{"canonical_record":{"source":{"id":"1311.4282","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-18T07:13:52Z","cross_cats_sorted":[],"title_canon_sha256":"d6e32f23ad60294a821beab966d9b317663d7d6ccb2e960b0ad3315df84f5eb1","abstract_canon_sha256":"1d711419c0de060be8d40313f933db1f3660d56adbd8a2f8665c490228132545"},"schema_version":"1.0"},"canonical_sha256":"2ed6918ff7f3607325549a730cf4c98d59620356c0fcbf89067ebe21041a1483","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:03:46.771671Z","signature_b64":"MrLfPiRUMN0knTmERG0qHlcPfKL4pG6qu8oScxhs1lwxdJ1vnL04W/cl6QZyDgMiz1OhWZ4AdsRLhK6JPMNMBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ed6918ff7f3607325549a730cf4c98d59620356c0fcbf89067ebe21041a1483","last_reissued_at":"2026-05-18T03:03:46.770997Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:03:46.770997Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.4282","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:03:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C0v3kPw8/9/KJVe/Gtgi4vQjSBIFFCpK7U37W+uZs2NIo/rI1yHp0Igkk0pxMwJpjfopShNmJHyJ4R+eAWHhBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T05:33:36.022160Z"},"content_sha256":"ed090fc4f6865ba69e007fdaeafbca698c5bdfada0e2d404eb9bdb9e9d5d721c","schema_version":"1.0","event_id":"sha256:ed090fc4f6865ba69e007fdaeafbca698c5bdfada0e2d404eb9bdb9e9d5d721c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:F3LJDD7X6NQHGJKUTJZQZ5GJRV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Uniform Positivity and Continuity of Lyapunov Exponents for a Class of $C^2$ Quasiperiodic Schr\\\"odinger Cocycles ","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Yiqian Wang, Zhenghe Zhang","submitted_at":"2013-11-18T07:13:52Z","abstract_excerpt":"We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the Lyapunov exponents of the corresponding Schr\\\"odinger cocycles are uniformly positive and weak H\\\"older continuous as function of energies. As a corollary, we also obtain that the corresponding integrated density of states (IDS) is weak H\\\"older continous. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to more general $\\mathrm{SL}(2,\\mathbb R)$ cocycles, which in turn can be applied to get uniform pos"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4282","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:03:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NH/GaOVqSjo9ceVCtHD1mgtvGzId+u3etD56AjALa8xRaupppU3Wkn7cXE1yVP8ifWdKqkOwA6oKz02qrimMCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T05:33:36.022526Z"},"content_sha256":"57b6ce796f1b35cb0140e77ed616cd7e180a37140af32d6ce490200051103db6","schema_version":"1.0","event_id":"sha256:57b6ce796f1b35cb0140e77ed616cd7e180a37140af32d6ce490200051103db6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F3LJDD7X6NQHGJKUTJZQZ5GJRV/bundle.json","state_url":"https://pith.science/pith/F3LJDD7X6NQHGJKUTJZQZ5GJRV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F3LJDD7X6NQHGJKUTJZQZ5GJRV/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-23T05:33:36Z","links":{"resolver":"https://pith.science/pith/F3LJDD7X6NQHGJKUTJZQZ5GJRV","bundle":"https://pith.science/pith/F3LJDD7X6NQHGJKUTJZQZ5GJRV/bundle.json","state":"https://pith.science/pith/F3LJDD7X6NQHGJKUTJZQZ5GJRV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F3LJDD7X6NQHGJKUTJZQZ5GJRV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:F3LJDD7X6NQHGJKUTJZQZ5GJRV","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1d711419c0de060be8d40313f933db1f3660d56adbd8a2f8665c490228132545","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-18T07:13:52Z","title_canon_sha256":"d6e32f23ad60294a821beab966d9b317663d7d6ccb2e960b0ad3315df84f5eb1"},"schema_version":"1.0","source":{"id":"1311.4282","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.4282","created_at":"2026-05-18T03:03:46Z"},{"alias_kind":"arxiv_version","alias_value":"1311.4282v3","created_at":"2026-05-18T03:03:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4282","created_at":"2026-05-18T03:03:46Z"},{"alias_kind":"pith_short_12","alias_value":"F3LJDD7X6NQH","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"F3LJDD7X6NQHGJKU","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"F3LJDD7X","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:57b6ce796f1b35cb0140e77ed616cd7e180a37140af32d6ce490200051103db6","target":"graph","created_at":"2026-05-18T03:03:46Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We show that for a class of $C^2$ quasiperiodic potentials and for any Diophantine frequency, the Lyapunov exponents of the corresponding Schr\\\"odinger cocycles are uniformly positive and weak H\\\"older continuous as function of energies. As a corollary, we also obtain that the corresponding integrated density of states (IDS) is weak H\\\"older continous. Our approach is of purely dynamical systems, which depends on a detailed analysis of asymptotic stable and unstable directions. We also apply it to more general $\\mathrm{SL}(2,\\mathbb R)$ cocycles, which in turn can be applied to get uniform pos","authors_text":"Yiqian Wang, Zhenghe Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-18T07:13:52Z","title":"Uniform Positivity and Continuity of Lyapunov Exponents for a Class of $C^2$ Quasiperiodic Schr\\\"odinger Cocycles "},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4282","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ed090fc4f6865ba69e007fdaeafbca698c5bdfada0e2d404eb9bdb9e9d5d721c","target":"record","created_at":"2026-05-18T03:03:46Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1d711419c0de060be8d40313f933db1f3660d56adbd8a2f8665c490228132545","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-11-18T07:13:52Z","title_canon_sha256":"d6e32f23ad60294a821beab966d9b317663d7d6ccb2e960b0ad3315df84f5eb1"},"schema_version":"1.0","source":{"id":"1311.4282","kind":"arxiv","version":3}},"canonical_sha256":"2ed6918ff7f3607325549a730cf4c98d59620356c0fcbf89067ebe21041a1483","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2ed6918ff7f3607325549a730cf4c98d59620356c0fcbf89067ebe21041a1483","first_computed_at":"2026-05-18T03:03:46.770997Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:03:46.770997Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MrLfPiRUMN0knTmERG0qHlcPfKL4pG6qu8oScxhs1lwxdJ1vnL04W/cl6QZyDgMiz1OhWZ4AdsRLhK6JPMNMBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:03:46.771671Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.4282","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ed090fc4f6865ba69e007fdaeafbca698c5bdfada0e2d404eb9bdb9e9d5d721c","sha256:57b6ce796f1b35cb0140e77ed616cd7e180a37140af32d6ce490200051103db6"],"state_sha256":"8822a004de964ce66603a7c7f1b7fd3a27ef00d07cdb3641a94bc315bc18d2d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3Y86qkd8yxqAvcqxC+jV1ifqW2VG6DW1XID/moY0v+LCb5ZZBjBkQFXPSkborj/2bvT4QoMh9ZeyYYPkfuT7CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T05:33:36.024425Z","bundle_sha256":"e55794ff2e781e83eaa92ab325600d51adcb3b92501995664b1ce9c3ce0cf6ff"}}