{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:QGNY6KQGKVFY3SVZK6QEIMJM6V","short_pith_number":"pith:QGNY6KQG","canonical_record":{"source":{"id":"0906.2902","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-06-16T11:46:03Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"b5ed3388074455bf04915783982a57add6f230ed9cf4d03235b0c17f2b96f21d","abstract_canon_sha256":"ea6a6a177ff2800ca1e2b86c7fd88c79e90d6bc12f663241645370a8d85937ce"},"schema_version":"1.0"},"canonical_sha256":"819b8f2a06554b8dcab957a044312cf562e445f64893f160329cb817d1055c31","source":{"kind":"arxiv","id":"0906.2902","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.2902","created_at":"2026-05-18T04:29:42Z"},{"alias_kind":"arxiv_version","alias_value":"0906.2902v1","created_at":"2026-05-18T04:29:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.2902","created_at":"2026-05-18T04:29:42Z"},{"alias_kind":"pith_short_12","alias_value":"QGNY6KQGKVFY","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"QGNY6KQGKVFY3SVZ","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"QGNY6KQG","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:QGNY6KQGKVFY3SVZK6QEIMJM6V","target":"record","payload":{"canonical_record":{"source":{"id":"0906.2902","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-06-16T11:46:03Z","cross_cats_sorted":["math.SP"],"title_canon_sha256":"b5ed3388074455bf04915783982a57add6f230ed9cf4d03235b0c17f2b96f21d","abstract_canon_sha256":"ea6a6a177ff2800ca1e2b86c7fd88c79e90d6bc12f663241645370a8d85937ce"},"schema_version":"1.0"},"canonical_sha256":"819b8f2a06554b8dcab957a044312cf562e445f64893f160329cb817d1055c31","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:29:42.722638Z","signature_b64":"z9qk9vTCixTak0ezUjIVF1XhMjZ//H2d4Xslr9q2LSLXg0vaAEiPyG01xyWw+gXJOdPgvR1qI8KpfItYD3J5BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"819b8f2a06554b8dcab957a044312cf562e445f64893f160329cb817d1055c31","last_reissued_at":"2026-05-18T04:29:42.722222Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:29:42.722222Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0906.2902","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:29:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8hlCb2+qyV+upkvdAUxu3xEnyuLx15ogBqqwN40vqdcp8nih/n1dUu1j49BcHSEBMCiXAkUBOYPJake8+rz5DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T09:20:55.556523Z"},"content_sha256":"d3b97b95726cd347411987c1d7c4c74c71112c164c325dbb9dd10c0eb249a5c9","schema_version":"1.0","event_id":"sha256:d3b97b95726cd347411987c1d7c4c74c71112c164c325dbb9dd10c0eb249a5c9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:QGNY6KQGKVFY3SVZK6QEIMJM6V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Spectral density and Sobolev inequalities for pure and mixed states","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.SP"],"primary_cat":"math.FA","authors_text":"Michel Rumin","submitted_at":"2009-06-16T11:46:03Z","abstract_excerpt":"We prove some general Sobolev-type and related inequalities for positive operators A of given ultracontractive spectral decay, without assuming e^{-tA} is submarkovian. These inequalities hold on functions, or pure states, as usual, but also on mixed states, or density operators in the quantum mechanical sense. This provides universal bounds of Faber-Krahn type on domains, that apply to their whole Dirichlet spectrum distribution, not only the first eigenvalue. Another application is given to relate the Novikov-Shubin numbers of coverings of finite simplicial complexes to the vanishing of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.2902","kind":"arxiv","version":1},"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":"2026-05-18T04:29:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Pfw3qK6IyLL7/k4E78zvO396DOdrOuLxsVEu1SKUHCcvpFTvRi36vYEQ2iLfSMrL+bt3jLZOAS456MHLXAKKCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T09:20:55.556865Z"},"content_sha256":"417e00a431766558f7074c7e3bf105a01e78f754b11dd82595a609f939ddf81c","schema_version":"1.0","event_id":"sha256:417e00a431766558f7074c7e3bf105a01e78f754b11dd82595a609f939ddf81c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QGNY6KQGKVFY3SVZK6QEIMJM6V/bundle.json","state_url":"https://pith.science/pith/QGNY6KQGKVFY3SVZK6QEIMJM6V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QGNY6KQGKVFY3SVZK6QEIMJM6V/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":"2026-06-23T09:20:55Z","links":{"resolver":"https://pith.science/pith/QGNY6KQGKVFY3SVZK6QEIMJM6V","bundle":"https://pith.science/pith/QGNY6KQGKVFY3SVZK6QEIMJM6V/bundle.json","state":"https://pith.science/pith/QGNY6KQGKVFY3SVZK6QEIMJM6V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QGNY6KQGKVFY3SVZK6QEIMJM6V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:QGNY6KQGKVFY3SVZK6QEIMJM6V","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":"ea6a6a177ff2800ca1e2b86c7fd88c79e90d6bc12f663241645370a8d85937ce","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-06-16T11:46:03Z","title_canon_sha256":"b5ed3388074455bf04915783982a57add6f230ed9cf4d03235b0c17f2b96f21d"},"schema_version":"1.0","source":{"id":"0906.2902","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0906.2902","created_at":"2026-05-18T04:29:42Z"},{"alias_kind":"arxiv_version","alias_value":"0906.2902v1","created_at":"2026-05-18T04:29:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0906.2902","created_at":"2026-05-18T04:29:42Z"},{"alias_kind":"pith_short_12","alias_value":"QGNY6KQGKVFY","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"QGNY6KQGKVFY3SVZ","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"QGNY6KQG","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:417e00a431766558f7074c7e3bf105a01e78f754b11dd82595a609f939ddf81c","target":"graph","created_at":"2026-05-18T04:29:42Z","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 prove some general Sobolev-type and related inequalities for positive operators A of given ultracontractive spectral decay, without assuming e^{-tA} is submarkovian. These inequalities hold on functions, or pure states, as usual, but also on mixed states, or density operators in the quantum mechanical sense. This provides universal bounds of Faber-Krahn type on domains, that apply to their whole Dirichlet spectrum distribution, not only the first eigenvalue. Another application is given to relate the Novikov-Shubin numbers of coverings of finite simplicial complexes to the vanishing of the ","authors_text":"Michel Rumin","cross_cats":["math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-06-16T11:46:03Z","title":"Spectral density and Sobolev inequalities for pure and mixed states"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.2902","kind":"arxiv","version":1},"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:d3b97b95726cd347411987c1d7c4c74c71112c164c325dbb9dd10c0eb249a5c9","target":"record","created_at":"2026-05-18T04:29:42Z","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":"ea6a6a177ff2800ca1e2b86c7fd88c79e90d6bc12f663241645370a8d85937ce","cross_cats_sorted":["math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-06-16T11:46:03Z","title_canon_sha256":"b5ed3388074455bf04915783982a57add6f230ed9cf4d03235b0c17f2b96f21d"},"schema_version":"1.0","source":{"id":"0906.2902","kind":"arxiv","version":1}},"canonical_sha256":"819b8f2a06554b8dcab957a044312cf562e445f64893f160329cb817d1055c31","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"819b8f2a06554b8dcab957a044312cf562e445f64893f160329cb817d1055c31","first_computed_at":"2026-05-18T04:29:42.722222Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:29:42.722222Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"z9qk9vTCixTak0ezUjIVF1XhMjZ//H2d4Xslr9q2LSLXg0vaAEiPyG01xyWw+gXJOdPgvR1qI8KpfItYD3J5BA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:29:42.722638Z","signed_message":"canonical_sha256_bytes"},"source_id":"0906.2902","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d3b97b95726cd347411987c1d7c4c74c71112c164c325dbb9dd10c0eb249a5c9","sha256:417e00a431766558f7074c7e3bf105a01e78f754b11dd82595a609f939ddf81c"],"state_sha256":"e3d9af2cf50646017229a4a4c4f7b319b4cb4570c619c8d2dc2a3ff093abbec8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"p3E/9zexDVMdf4TX7PRn+d98ofodI8j7kq5dWFBlrfbKVO5408O0ppn+yvZO4ZCe3/huTQz7KJFUjjCMgGwKBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T09:20:55.559588Z","bundle_sha256":"9f73ccc45d9d04fc0178bf58c6d754b461a9a49dcc7ee37d369cd5ce36ab2b52"}}