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In our problem, the source $f = f(x,y,t)$ and the final data $h(x,y)$ are unknown. We only know random noise data $g_{ij}(t)$ and $d_{ij}$ satisfying the regression models g_{ij}(t) &=& f(x_i,y_j,t) + \\vartheta\\xi_{ij}(t), d_{ij} &=& h(x_i,y_j) + \\s"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1606.05463","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-06-17T09:29:53Z","cross_cats_sorted":[],"title_canon_sha256":"0d2906af419eb76401047eaaca83b45c6936021ba9b4d3e3aa78fa140e7cbe62","abstract_canon_sha256":"4895f4f6be9cd721c2426d1758bb18b98b2b83d656bc52395eeaa711bfa5a8af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:19.520027Z","signature_b64":"/el50DEBx7DeqLAmhB3x3iv1m1ANzpA4q7N+ym6ydvwicBJ6eT3LGT7UAQyx9KV10vvP5LRzBM204mThNj23Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d47e2f7860bcf03b5c25079ad9562f4b6120c33b697b1a586928280f01015ebc","last_reissued_at":"2026-05-18T01:12:19.519699Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:19.519699Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Two Dimensional Backward Heat Problem With Statistical Discrete Data","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Dang Duc Trong, Nguyen Dang Minh, Nguyen Huy Tuan, To Duc Khanh","submitted_at":"2016-06-17T09:29:53Z","abstract_excerpt":"In this paper, we focus on the backward heat problem of finding the function $\\theta(x,y)=u(x,y,0)$ such that \\[ {l l l} u_t - a(t)(u_{xx} + u_{yy}) & = f(x,y,t), & \\qquad (x,y,t) \\in \\Omega\\times (0,T), u(x,y,T) & = h(x,y), & \\qquad (x,y) \\in\\bar{\\Omega}. \\] where $\\Omega = (0,\\pi) \\times (0,\\pi)$ and the heat transfer coefficient $a(t)$ is known. In our problem, the source $f = f(x,y,t)$ and the final data $h(x,y)$ are unknown. We only know random noise data $g_{ij}(t)$ and $d_{ij}$ satisfying the regression models g_{ij}(t) &=& f(x_i,y_j,t) + \\vartheta\\xi_{ij}(t), d_{ij} &=& h(x_i,y_j) + \\s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05463","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1606.05463","created_at":"2026-05-18T01:12:19.519750+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.05463v1","created_at":"2026-05-18T01:12:19.519750+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05463","created_at":"2026-05-18T01:12:19.519750+00:00"},{"alias_kind":"pith_short_12","alias_value":"2R7C66DAXTYD","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_16","alias_value":"2R7C66DAXTYDWXBF","created_at":"2026-05-18T12:29:55.572404+00:00"},{"alias_kind":"pith_short_8","alias_value":"2R7C66DA","created_at":"2026-05-18T12:29:55.572404+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/2R7C66DAXTYDWXBFA6NNSVRPJN","json":"https://pith.science/pith/2R7C66DAXTYDWXBFA6NNSVRPJN.json","graph_json":"https://pith.science/api/pith-number/2R7C66DAXTYDWXBFA6NNSVRPJN/graph.json","events_json":"https://pith.science/api/pith-number/2R7C66DAXTYDWXBFA6NNSVRPJN/events.json","paper":"https://pith.science/paper/2R7C66DA"},"agent_actions":{"view_html":"https://pith.science/pith/2R7C66DAXTYDWXBFA6NNSVRPJN","download_json":"https://pith.science/pith/2R7C66DAXTYDWXBFA6NNSVRPJN.json","view_paper":"https://pith.science/paper/2R7C66DA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.05463&json=true","fetch_graph":"https://pith.science/api/pith-number/2R7C66DAXTYDWXBFA6NNSVRPJN/graph.json","fetch_events":"https://pith.science/api/pith-number/2R7C66DAXTYDWXBFA6NNSVRPJN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/2R7C66DAXTYDWXBFA6NNSVRPJN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/2R7C66DAXTYDWXBFA6NNSVRPJN/action/storage_attestation","attest_author":"https://pith.science/pith/2R7C66DAXTYDWXBFA6NNSVRPJN/action/author_attestation","sign_citation":"https://pith.science/pith/2R7C66DAXTYDWXBFA6NNSVRPJN/action/citation_signature","submit_replication":"https://pith.science/pith/2R7C66DAXTYDWXBFA6NNSVRPJN/action/replication_record"}},"created_at":"2026-05-18T01:12:19.519750+00:00","updated_at":"2026-05-18T01:12:19.519750+00:00"}