{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:2OQ2FV53GRTDOHUWI3URMCYYSU","short_pith_number":"pith:2OQ2FV53","canonical_record":{"source":{"id":"2004.10441","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2020-04-22T08:31:08Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"2c0e5593d413c4c5653e2928ed76b32b735f31462777675d69f2fdfbe8b7cf31","abstract_canon_sha256":"50379314e01214a82c1057576924d432fe9052fd3583a738acf86bbdf0cee8ee"},"schema_version":"1.0"},"canonical_sha256":"d3a1a2d7bb3466371e9646e9160b1895248aa7f7f0c767acdace323aa90ee771","source":{"kind":"arxiv","id":"2004.10441","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2004.10441","created_at":"2026-07-05T02:55:06Z"},{"alias_kind":"arxiv_version","alias_value":"2004.10441v1","created_at":"2026-07-05T02:55:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2004.10441","created_at":"2026-07-05T02:55:06Z"},{"alias_kind":"pith_short_12","alias_value":"2OQ2FV53GRTD","created_at":"2026-07-05T02:55:06Z"},{"alias_kind":"pith_short_16","alias_value":"2OQ2FV53GRTDOHUW","created_at":"2026-07-05T02:55:06Z"},{"alias_kind":"pith_short_8","alias_value":"2OQ2FV53","created_at":"2026-07-05T02:55:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:2OQ2FV53GRTDOHUWI3URMCYYSU","target":"record","payload":{"canonical_record":{"source":{"id":"2004.10441","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2020-04-22T08:31:08Z","cross_cats_sorted":["math.MP"],"title_canon_sha256":"2c0e5593d413c4c5653e2928ed76b32b735f31462777675d69f2fdfbe8b7cf31","abstract_canon_sha256":"50379314e01214a82c1057576924d432fe9052fd3583a738acf86bbdf0cee8ee"},"schema_version":"1.0"},"canonical_sha256":"d3a1a2d7bb3466371e9646e9160b1895248aa7f7f0c767acdace323aa90ee771","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:55:06.097198Z","signature_b64":"2pwn7C624RkYKzJ1Jn6EUWkqfR9uf6CBd1LwhclCNdzPpE3fAXG5NWEYa5TIgDx0x9nQef2LSXRO+MJYdr56AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d3a1a2d7bb3466371e9646e9160b1895248aa7f7f0c767acdace323aa90ee771","last_reissued_at":"2026-07-05T02:55:06.096797Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:55:06.096797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2004.10441","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":"2026-07-05T02:55:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9n+tSu/bAXqklF4UzK1VdLdKVG26/q64e4n8XENQPsZWc1jmnA3p/DS5dQbNkTJyoiWtiVoVKziwsKxDC3NeAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T09:07:59.637158Z"},"content_sha256":"53bf87a75d5ff7f13892627c7ed68bd723f191f4cd12bf141c18d79230852579","schema_version":"1.0","event_id":"sha256:53bf87a75d5ff7f13892627c7ed68bd723f191f4cd12bf141c18d79230852579"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:2OQ2FV53GRTDOHUWI3URMCYYSU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A modified Euler-Maclaurin formula in 1D and 2D with applications in statistical physics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Jihong Guo, Yunpeng Liu","submitted_at":"2020-04-22T08:31:08Z","abstract_excerpt":"The Euler-Maclaurin summation formula is generalized to a modified form by expanding the periodic Bernoulli polynomials as its Fourier series and taking cuts, which includes both the Euler-Maclaurin summation formula and the Poission summation formula as special cases. By making use of the modified formula, a numerical summation method is obtained and the error can be controlled. The modified formula is also generalized from one dimention to two dimentions. Examples of its applications in statistical physics are also discussed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2004.10441","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2004.10441/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T02:55:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iPgMPTxUM4BcP7oBApjzYlY0IyhhuE6xDTUjV8CfzltB4xf0u/nyNm3tNjdtoujfkVYpZ5STOBqupHjpsxUBCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-05T09:07:59.637550Z"},"content_sha256":"59ffcce1f9b5ae34a1013274a918cb7540292ff8620660d6f4e9132590fc99ab","schema_version":"1.0","event_id":"sha256:59ffcce1f9b5ae34a1013274a918cb7540292ff8620660d6f4e9132590fc99ab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2OQ2FV53GRTDOHUWI3URMCYYSU/bundle.json","state_url":"https://pith.science/pith/2OQ2FV53GRTDOHUWI3URMCYYSU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2OQ2FV53GRTDOHUWI3URMCYYSU/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":"2026-07-05T09:07:59Z","links":{"resolver":"https://pith.science/pith/2OQ2FV53GRTDOHUWI3URMCYYSU","bundle":"https://pith.science/pith/2OQ2FV53GRTDOHUWI3URMCYYSU/bundle.json","state":"https://pith.science/pith/2OQ2FV53GRTDOHUWI3URMCYYSU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2OQ2FV53GRTDOHUWI3URMCYYSU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:2OQ2FV53GRTDOHUWI3URMCYYSU","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":"50379314e01214a82c1057576924d432fe9052fd3583a738acf86bbdf0cee8ee","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2020-04-22T08:31:08Z","title_canon_sha256":"2c0e5593d413c4c5653e2928ed76b32b735f31462777675d69f2fdfbe8b7cf31"},"schema_version":"1.0","source":{"id":"2004.10441","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2004.10441","created_at":"2026-07-05T02:55:06Z"},{"alias_kind":"arxiv_version","alias_value":"2004.10441v1","created_at":"2026-07-05T02:55:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2004.10441","created_at":"2026-07-05T02:55:06Z"},{"alias_kind":"pith_short_12","alias_value":"2OQ2FV53GRTD","created_at":"2026-07-05T02:55:06Z"},{"alias_kind":"pith_short_16","alias_value":"2OQ2FV53GRTDOHUW","created_at":"2026-07-05T02:55:06Z"},{"alias_kind":"pith_short_8","alias_value":"2OQ2FV53","created_at":"2026-07-05T02:55:06Z"}],"graph_snapshots":[{"event_id":"sha256:59ffcce1f9b5ae34a1013274a918cb7540292ff8620660d6f4e9132590fc99ab","target":"graph","created_at":"2026-07-05T02:55:06Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2004.10441/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Euler-Maclaurin summation formula is generalized to a modified form by expanding the periodic Bernoulli polynomials as its Fourier series and taking cuts, which includes both the Euler-Maclaurin summation formula and the Poission summation formula as special cases. By making use of the modified formula, a numerical summation method is obtained and the error can be controlled. The modified formula is also generalized from one dimention to two dimentions. Examples of its applications in statistical physics are also discussed.","authors_text":"Jihong Guo, Yunpeng Liu","cross_cats":["math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2020-04-22T08:31:08Z","title":"A modified Euler-Maclaurin formula in 1D and 2D with applications in statistical physics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2004.10441","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":"sha256:53bf87a75d5ff7f13892627c7ed68bd723f191f4cd12bf141c18d79230852579","target":"record","created_at":"2026-07-05T02:55:06Z","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":"50379314e01214a82c1057576924d432fe9052fd3583a738acf86bbdf0cee8ee","cross_cats_sorted":["math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2020-04-22T08:31:08Z","title_canon_sha256":"2c0e5593d413c4c5653e2928ed76b32b735f31462777675d69f2fdfbe8b7cf31"},"schema_version":"1.0","source":{"id":"2004.10441","kind":"arxiv","version":1}},"canonical_sha256":"d3a1a2d7bb3466371e9646e9160b1895248aa7f7f0c767acdace323aa90ee771","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d3a1a2d7bb3466371e9646e9160b1895248aa7f7f0c767acdace323aa90ee771","first_computed_at":"2026-07-05T02:55:06.096797Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:55:06.096797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"2pwn7C624RkYKzJ1Jn6EUWkqfR9uf6CBd1LwhclCNdzPpE3fAXG5NWEYa5TIgDx0x9nQef2LSXRO+MJYdr56AA==","signature_status":"signed_v1","signed_at":"2026-07-05T02:55:06.097198Z","signed_message":"canonical_sha256_bytes"},"source_id":"2004.10441","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53bf87a75d5ff7f13892627c7ed68bd723f191f4cd12bf141c18d79230852579","sha256:59ffcce1f9b5ae34a1013274a918cb7540292ff8620660d6f4e9132590fc99ab"],"state_sha256":"6e09c5dea0f14b61af8d4364de2b0062b25654299ec74f02c10fcbfd4007a9d7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DhBQekiIB40C5XOKqK8Ew+v0kjmZfT8NpKfgJMVSM3O3WB3HrYuxXzmgAwGonxWp5GpLC0KfLEJLBNJwzwbvBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-05T09:07:59.639555Z","bundle_sha256":"112d2a71917eaa00f8ed527bf7aeab569bd837aa69db2b24d8806046c5965d24"}}