{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:F3E7UMR37JUT4XOEK777RMPQOH","short_pith_number":"pith:F3E7UMR3","canonical_record":{"source":{"id":"1103.3695","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-03-18T19:54:37Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"6ac7787d85115dc4bb31d06cd71005414df5a3cb1af1ed1d1db284cb9787ec51","abstract_canon_sha256":"115b09f10aa2983aa6ef1ca990f413da8ffb72610c4599a5a52ff0cebf21aaab"},"schema_version":"1.0"},"canonical_sha256":"2ec9fa323bfa693e5dc457fff8b1f071ff5489c6453e0133c541a2151436ce54","source":{"kind":"arxiv","id":"1103.3695","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.3695","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"arxiv_version","alias_value":"1103.3695v2","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.3695","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"pith_short_12","alias_value":"F3E7UMR37JUT","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"F3E7UMR37JUT4XOE","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"F3E7UMR3","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:F3E7UMR37JUT4XOEK777RMPQOH","target":"record","payload":{"canonical_record":{"source":{"id":"1103.3695","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-03-18T19:54:37Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"6ac7787d85115dc4bb31d06cd71005414df5a3cb1af1ed1d1db284cb9787ec51","abstract_canon_sha256":"115b09f10aa2983aa6ef1ca990f413da8ffb72610c4599a5a52ff0cebf21aaab"},"schema_version":"1.0"},"canonical_sha256":"2ec9fa323bfa693e5dc457fff8b1f071ff5489c6453e0133c541a2151436ce54","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:26:32.710035Z","signature_b64":"VL+A/DARL6vpxLw8c/YGztp7fh8e/J11cIGG6MjtOn2jMMRu3JppsIn589SxxlWBUjACnjV4ghHgWhb6i9J4CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2ec9fa323bfa693e5dc457fff8b1f071ff5489c6453e0133c541a2151436ce54","last_reissued_at":"2026-05-18T03:26:32.709622Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:26:32.709622Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1103.3695","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:26:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"opEVk6/+RzoK0VrbCLTu3Z20oRtPsycDRkupJxx6fTJGVo8el4Gp1WyGOV9BNduEvQ1F9LM+2ZCTf8zUDZnCBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T19:08:22.267191Z"},"content_sha256":"6d68fc10596ea86520f014cacc498b09f02726eae89c4657ca60529acb22cd05","schema_version":"1.0","event_id":"sha256:6d68fc10596ea86520f014cacc498b09f02726eae89c4657ca60529acb22cd05"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:F3E7UMR37JUT4XOEK777RMPQOH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.FA","authors_text":"Daniel Lenz, Matthias Keller, Rados{\\l}aw Wojciechowski, Sebastian Haeseler","submitted_at":"2011-03-18T19:54:37Z","abstract_excerpt":"We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard formal Laplacian and we can characterize the Dirichlet and Neumann Laplacians as the largest and smallest Markovian restrictions of the standard formal Laplacian. In the case of general graphs, this class contains the Dirichlet and Neumann Laplacians and we describe how these may differ from each other, characterize when they agree, and study connections to e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.3695","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:26:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YHMqwuqqsyiP3IXbZsM9OapBvCeTr40lWLFlpKSzpwtNcOUTqzpQIE7p4ihIMlvt1BS5M7AIyF2He3v5Sdn/DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T19:08:22.267564Z"},"content_sha256":"64bf8c821fa572028db0d66246a25daafea6e859444c296557af47177248ed89","schema_version":"1.0","event_id":"sha256:64bf8c821fa572028db0d66246a25daafea6e859444c296557af47177248ed89"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F3E7UMR37JUT4XOEK777RMPQOH/bundle.json","state_url":"https://pith.science/pith/F3E7UMR37JUT4XOEK777RMPQOH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F3E7UMR37JUT4XOEK777RMPQOH/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":"2026-06-01T19:08:22Z","links":{"resolver":"https://pith.science/pith/F3E7UMR37JUT4XOEK777RMPQOH","bundle":"https://pith.science/pith/F3E7UMR37JUT4XOEK777RMPQOH/bundle.json","state":"https://pith.science/pith/F3E7UMR37JUT4XOEK777RMPQOH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F3E7UMR37JUT4XOEK777RMPQOH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:F3E7UMR37JUT4XOEK777RMPQOH","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":"115b09f10aa2983aa6ef1ca990f413da8ffb72610c4599a5a52ff0cebf21aaab","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-03-18T19:54:37Z","title_canon_sha256":"6ac7787d85115dc4bb31d06cd71005414df5a3cb1af1ed1d1db284cb9787ec51"},"schema_version":"1.0","source":{"id":"1103.3695","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1103.3695","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"arxiv_version","alias_value":"1103.3695v2","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1103.3695","created_at":"2026-05-18T03:26:32Z"},{"alias_kind":"pith_short_12","alias_value":"F3E7UMR37JUT","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"F3E7UMR37JUT4XOE","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"F3E7UMR3","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:64bf8c821fa572028db0d66246a25daafea6e859444c296557af47177248ed89","target":"graph","created_at":"2026-05-18T03:26:32Z","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 study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard formal Laplacian and we can characterize the Dirichlet and Neumann Laplacians as the largest and smallest Markovian restrictions of the standard formal Laplacian. In the case of general graphs, this class contains the Dirichlet and Neumann Laplacians and we describe how these may differ from each other, characterize when they agree, and study connections to e","authors_text":"Daniel Lenz, Matthias Keller, Rados{\\l}aw Wojciechowski, Sebastian Haeseler","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-03-18T19:54:37Z","title":"Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.3695","kind":"arxiv","version":2},"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":"sha256:6d68fc10596ea86520f014cacc498b09f02726eae89c4657ca60529acb22cd05","target":"record","created_at":"2026-05-18T03:26:32Z","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":"115b09f10aa2983aa6ef1ca990f413da8ffb72610c4599a5a52ff0cebf21aaab","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2011-03-18T19:54:37Z","title_canon_sha256":"6ac7787d85115dc4bb31d06cd71005414df5a3cb1af1ed1d1db284cb9787ec51"},"schema_version":"1.0","source":{"id":"1103.3695","kind":"arxiv","version":2}},"canonical_sha256":"2ec9fa323bfa693e5dc457fff8b1f071ff5489c6453e0133c541a2151436ce54","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2ec9fa323bfa693e5dc457fff8b1f071ff5489c6453e0133c541a2151436ce54","first_computed_at":"2026-05-18T03:26:32.709622Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:26:32.709622Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VL+A/DARL6vpxLw8c/YGztp7fh8e/J11cIGG6MjtOn2jMMRu3JppsIn589SxxlWBUjACnjV4ghHgWhb6i9J4CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:26:32.710035Z","signed_message":"canonical_sha256_bytes"},"source_id":"1103.3695","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6d68fc10596ea86520f014cacc498b09f02726eae89c4657ca60529acb22cd05","sha256:64bf8c821fa572028db0d66246a25daafea6e859444c296557af47177248ed89"],"state_sha256":"9bd4405cba68d91d6e90dcbb911dc1ea3062ef69b8031b79bae6ba514775c3d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BT5tE2OXO9q1rfpCJQeJRhta0LWpZsP74YstyvEmhh/3tPrN+KrJJBthJ7Ml9ouUj8beIq2MzIXKWF9WcVJPAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T19:08:22.269532Z","bundle_sha256":"c664919fdad0f7a4c901698cadfff9fab615f6bcd8f9fbdd319eeff85e863ca0"}}