{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2007:K6MMHOVLWH3BEXPJTGBCCAGUEN","short_pith_number":"pith:K6MMHOVL","canonical_record":{"source":{"id":"0711.4117","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2007-11-26T21:03:40Z","cross_cats_sorted":[],"title_canon_sha256":"71b66b69015abdb0d83e34322fea9303f949663bd17af13edc70ba9d132933c6","abstract_canon_sha256":"c5a38bd58c146556185f46fdfdbd12aa803b408486c50c38882618f108a90d55"},"schema_version":"1.0"},"canonical_sha256":"5798c3baabb1f6125de999822100d42367478ac6e73ac48155bf9e52e8148123","source":{"kind":"arxiv","id":"0711.4117","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0711.4117","created_at":"2026-05-18T03:49:09Z"},{"alias_kind":"arxiv_version","alias_value":"0711.4117v1","created_at":"2026-05-18T03:49:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0711.4117","created_at":"2026-05-18T03:49:09Z"},{"alias_kind":"pith_short_12","alias_value":"K6MMHOVLWH3B","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"K6MMHOVLWH3BEXPJ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"K6MMHOVL","created_at":"2026-05-18T12:25:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2007:K6MMHOVLWH3BEXPJTGBCCAGUEN","target":"record","payload":{"canonical_record":{"source":{"id":"0711.4117","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AP","submitted_at":"2007-11-26T21:03:40Z","cross_cats_sorted":[],"title_canon_sha256":"71b66b69015abdb0d83e34322fea9303f949663bd17af13edc70ba9d132933c6","abstract_canon_sha256":"c5a38bd58c146556185f46fdfdbd12aa803b408486c50c38882618f108a90d55"},"schema_version":"1.0"},"canonical_sha256":"5798c3baabb1f6125de999822100d42367478ac6e73ac48155bf9e52e8148123","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:09.500702Z","signature_b64":"JhMvGsqDD43Uus7BQEYj6KPlQUVowyEuwUbHSgt7dC0tUsnLYR2bLp7oiWch2YNIgHZIppTx7qPuM6181tvEBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5798c3baabb1f6125de999822100d42367478ac6e73ac48155bf9e52e8148123","last_reissued_at":"2026-05-18T03:49:09.500254Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:09.500254Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0711.4117","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":"2026-05-18T03:49:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4hWC42TGhN/IMcf81CDb2k/V2og0ofCNRTOycq8L/+njId6/QneDe4kEFNsGOpuTJSfpnPVQKDUCcYBH55dEAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T06:21:15.866390Z"},"content_sha256":"b4488c6eff01a1068309718105c0c48859207b8cb4ff386c4479e3348814e2e9","schema_version":"1.0","event_id":"sha256:b4488c6eff01a1068309718105c0c48859207b8cb4ff386c4479e3348814e2e9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2007:K6MMHOVLWH3BEXPJTGBCCAGUEN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Simplified Calculation for the Fundamental Solution to the Heat Equation on the Heisenberg Group","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Albert Boggess, Andrew Raich","submitted_at":"2007-11-26T21:03:40Z","abstract_excerpt":"Let $L = -1/4 (\\sum_{j=1}^n(X_j^2+Y_j^2)+i\\gamma T)$ where $\\gamma$ is a complex number, $X_j$, $Y_j$, and $T$ are the left invariant vector fields of the Heisenberg group structure for $R^n \\times R^n \\times R$. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the Heat Equation $\\partial_s\\rho = -L\\rho$. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the $\\Box_b$-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted dbar-operat"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.4117","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":"2026-05-18T03:49:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xGaEyprQ1RMr+Hnfb4mngjoKSfWmV1hUHX1ed4MKMnFRKBlkHna1NiiA8m6Op6JToYwYYreUKCRk0JkPnXUVCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T06:21:15.866811Z"},"content_sha256":"a476903fe7afea9bb6e7a476040acdd66455ae9739d6f28b44b9dc4904fe2b0b","schema_version":"1.0","event_id":"sha256:a476903fe7afea9bb6e7a476040acdd66455ae9739d6f28b44b9dc4904fe2b0b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K6MMHOVLWH3BEXPJTGBCCAGUEN/bundle.json","state_url":"https://pith.science/pith/K6MMHOVLWH3BEXPJTGBCCAGUEN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K6MMHOVLWH3BEXPJTGBCCAGUEN/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":"2026-06-26T06:21:15Z","links":{"resolver":"https://pith.science/pith/K6MMHOVLWH3BEXPJTGBCCAGUEN","bundle":"https://pith.science/pith/K6MMHOVLWH3BEXPJTGBCCAGUEN/bundle.json","state":"https://pith.science/pith/K6MMHOVLWH3BEXPJTGBCCAGUEN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K6MMHOVLWH3BEXPJTGBCCAGUEN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2007:K6MMHOVLWH3BEXPJTGBCCAGUEN","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":"c5a38bd58c146556185f46fdfdbd12aa803b408486c50c38882618f108a90d55","cross_cats_sorted":[],"license":"","primary_cat":"math.AP","submitted_at":"2007-11-26T21:03:40Z","title_canon_sha256":"71b66b69015abdb0d83e34322fea9303f949663bd17af13edc70ba9d132933c6"},"schema_version":"1.0","source":{"id":"0711.4117","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0711.4117","created_at":"2026-05-18T03:49:09Z"},{"alias_kind":"arxiv_version","alias_value":"0711.4117v1","created_at":"2026-05-18T03:49:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0711.4117","created_at":"2026-05-18T03:49:09Z"},{"alias_kind":"pith_short_12","alias_value":"K6MMHOVLWH3B","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_16","alias_value":"K6MMHOVLWH3BEXPJ","created_at":"2026-05-18T12:25:55Z"},{"alias_kind":"pith_short_8","alias_value":"K6MMHOVL","created_at":"2026-05-18T12:25:55Z"}],"graph_snapshots":[{"event_id":"sha256:a476903fe7afea9bb6e7a476040acdd66455ae9739d6f28b44b9dc4904fe2b0b","target":"graph","created_at":"2026-05-18T03:49:09Z","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":"Let $L = -1/4 (\\sum_{j=1}^n(X_j^2+Y_j^2)+i\\gamma T)$ where $\\gamma$ is a complex number, $X_j$, $Y_j$, and $T$ are the left invariant vector fields of the Heisenberg group structure for $R^n \\times R^n \\times R$. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the Heat Equation $\\partial_s\\rho = -L\\rho$. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the $\\Box_b$-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted dbar-operat","authors_text":"Albert Boggess, Andrew Raich","cross_cats":[],"headline":"","license":"","primary_cat":"math.AP","submitted_at":"2007-11-26T21:03:40Z","title":"A Simplified Calculation for the Fundamental Solution to the Heat Equation on the Heisenberg Group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0711.4117","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:b4488c6eff01a1068309718105c0c48859207b8cb4ff386c4479e3348814e2e9","target":"record","created_at":"2026-05-18T03:49:09Z","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":"c5a38bd58c146556185f46fdfdbd12aa803b408486c50c38882618f108a90d55","cross_cats_sorted":[],"license":"","primary_cat":"math.AP","submitted_at":"2007-11-26T21:03:40Z","title_canon_sha256":"71b66b69015abdb0d83e34322fea9303f949663bd17af13edc70ba9d132933c6"},"schema_version":"1.0","source":{"id":"0711.4117","kind":"arxiv","version":1}},"canonical_sha256":"5798c3baabb1f6125de999822100d42367478ac6e73ac48155bf9e52e8148123","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5798c3baabb1f6125de999822100d42367478ac6e73ac48155bf9e52e8148123","first_computed_at":"2026-05-18T03:49:09.500254Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:49:09.500254Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"JhMvGsqDD43Uus7BQEYj6KPlQUVowyEuwUbHSgt7dC0tUsnLYR2bLp7oiWch2YNIgHZIppTx7qPuM6181tvEBw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:49:09.500702Z","signed_message":"canonical_sha256_bytes"},"source_id":"0711.4117","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b4488c6eff01a1068309718105c0c48859207b8cb4ff386c4479e3348814e2e9","sha256:a476903fe7afea9bb6e7a476040acdd66455ae9739d6f28b44b9dc4904fe2b0b"],"state_sha256":"3225517aa58c59e795c816137bb3d6ec2ff5b7eb2d55acee12abc9e70992b3fb"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tDjVtpftfhBlGIwykH2kjwi4a01yE9PuYb8bm5mTWx1R5kHt+4PUrKsjbfe6iYkrSGzM4sDaGmIJtuuUIIdPDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T06:21:15.868818Z","bundle_sha256":"42edc7aee009a8816299cc642dd21a8437d566229807031c6014bb429980a5ad"}}