{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:AGZ5MDPCOI5BZ2LFVWOWJUWMPQ","short_pith_number":"pith:AGZ5MDPC","canonical_record":{"source":{"id":"1504.07386","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-28T09:13:35Z","cross_cats_sorted":[],"title_canon_sha256":"5386c12a71978eb7d12df769878434c0e28088b3f334abf67e8a1257ff733e43","abstract_canon_sha256":"94e7df159350cac82a756a70e837ac9acdcadb8a2222a909c9ece1b61f83acfb"},"schema_version":"1.0"},"canonical_sha256":"01b3d60de2723a1ce965ad9d64d2cc7c3e7480a70b5678c6cf35b924ab0e0c00","source":{"kind":"arxiv","id":"1504.07386","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.07386","created_at":"2026-05-18T02:16:38Z"},{"alias_kind":"arxiv_version","alias_value":"1504.07386v4","created_at":"2026-05-18T02:16:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07386","created_at":"2026-05-18T02:16:38Z"},{"alias_kind":"pith_short_12","alias_value":"AGZ5MDPCOI5B","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"AGZ5MDPCOI5BZ2LF","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"AGZ5MDPC","created_at":"2026-05-18T12:29:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:AGZ5MDPCOI5BZ2LFVWOWJUWMPQ","target":"record","payload":{"canonical_record":{"source":{"id":"1504.07386","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-28T09:13:35Z","cross_cats_sorted":[],"title_canon_sha256":"5386c12a71978eb7d12df769878434c0e28088b3f334abf67e8a1257ff733e43","abstract_canon_sha256":"94e7df159350cac82a756a70e837ac9acdcadb8a2222a909c9ece1b61f83acfb"},"schema_version":"1.0"},"canonical_sha256":"01b3d60de2723a1ce965ad9d64d2cc7c3e7480a70b5678c6cf35b924ab0e0c00","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:38.823648Z","signature_b64":"0z6SwF4/KCGpHeh3AlYTqxjDU6+RV1NhHMcvBeOli2mMo6XruEk4JHVPdUcnqLQYIv5fcafChQNJ5487ObwJCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"01b3d60de2723a1ce965ad9d64d2cc7c3e7480a70b5678c6cf35b924ab0e0c00","last_reissued_at":"2026-05-18T02:16:38.823154Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:38.823154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1504.07386","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:16:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V+h41ADC8/A8Lj2/+zAqCADzXDLY1zyu/poWOlqswzJIfQ66Z4cgMIcScO+q0H4SzFmEHKM5zlxswD58d+S7BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T08:07:38.220744Z"},"content_sha256":"594506eb27fa226b1cbe8cea492ba1f6eaa4606ffcea9afa347e2372af737c78","schema_version":"1.0","event_id":"sha256:594506eb27fa226b1cbe8cea492ba1f6eaa4606ffcea9afa347e2372af737c78"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:AGZ5MDPCOI5BZ2LFVWOWJUWMPQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kyeong-Hun Kim, Sungbin Lim","submitted_at":"2015-04-28T09:13:35Z","abstract_excerpt":"Let $p(t,x)$ be the fundamental solution to the problem $$ \\partial_{t}^{\\alpha}u=-(-\\Delta)^{\\beta}u, \\quad \\alpha\\in (0,2), \\, \\beta\\in (0,\\infty). $$ In this paper we provide the asymptotic behaviors and sharp upper bounds of $p(t,x)$ and its space and time fractional derivatives $$ D_{x}^{n}(-\\Delta_x)^{\\gamma}D_{t}^{\\sigma}I_{t}^{\\delta}p(t,x), \\quad \\forall\\,\\, n\\in\\mathbb{Z}_{+}, \\,\\, \\gamma\\in[0,\\beta],\\,\\, \\sigma, \\delta \\in[0,\\infty), $$ where $D_{x}^n$ is a partial derivative of order $n$ with respect to $x$, $(-\\Delta_x)^{\\gamma}$ is a fractional Laplace operator and $D_{t}^{\\sigma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07386","kind":"arxiv","version":4},"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":"2026-05-18T02:16:38Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HsRqf/2pK4sUQ8cttILJ/D3Og3DvnLCmBGSx1PdR/NglA7iLMUTW2J2v4RjwIcxHtp+CWrGe1cLczBOR4K/dBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T08:07:38.221237Z"},"content_sha256":"6c6c68dd6e8050ba32d5231d0fa80bb6d8c30b92d99f34ff3716ce48857b5f32","schema_version":"1.0","event_id":"sha256:6c6c68dd6e8050ba32d5231d0fa80bb6d8c30b92d99f34ff3716ce48857b5f32"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AGZ5MDPCOI5BZ2LFVWOWJUWMPQ/bundle.json","state_url":"https://pith.science/pith/AGZ5MDPCOI5BZ2LFVWOWJUWMPQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AGZ5MDPCOI5BZ2LFVWOWJUWMPQ/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":"2026-05-27T08:07:38Z","links":{"resolver":"https://pith.science/pith/AGZ5MDPCOI5BZ2LFVWOWJUWMPQ","bundle":"https://pith.science/pith/AGZ5MDPCOI5BZ2LFVWOWJUWMPQ/bundle.json","state":"https://pith.science/pith/AGZ5MDPCOI5BZ2LFVWOWJUWMPQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AGZ5MDPCOI5BZ2LFVWOWJUWMPQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:AGZ5MDPCOI5BZ2LFVWOWJUWMPQ","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":"94e7df159350cac82a756a70e837ac9acdcadb8a2222a909c9ece1b61f83acfb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-28T09:13:35Z","title_canon_sha256":"5386c12a71978eb7d12df769878434c0e28088b3f334abf67e8a1257ff733e43"},"schema_version":"1.0","source":{"id":"1504.07386","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.07386","created_at":"2026-05-18T02:16:38Z"},{"alias_kind":"arxiv_version","alias_value":"1504.07386v4","created_at":"2026-05-18T02:16:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.07386","created_at":"2026-05-18T02:16:38Z"},{"alias_kind":"pith_short_12","alias_value":"AGZ5MDPCOI5B","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_16","alias_value":"AGZ5MDPCOI5BZ2LF","created_at":"2026-05-18T12:29:10Z"},{"alias_kind":"pith_short_8","alias_value":"AGZ5MDPC","created_at":"2026-05-18T12:29:10Z"}],"graph_snapshots":[{"event_id":"sha256:6c6c68dd6e8050ba32d5231d0fa80bb6d8c30b92d99f34ff3716ce48857b5f32","target":"graph","created_at":"2026-05-18T02:16:38Z","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 $p(t,x)$ be the fundamental solution to the problem $$ \\partial_{t}^{\\alpha}u=-(-\\Delta)^{\\beta}u, \\quad \\alpha\\in (0,2), \\, \\beta\\in (0,\\infty). $$ In this paper we provide the asymptotic behaviors and sharp upper bounds of $p(t,x)$ and its space and time fractional derivatives $$ D_{x}^{n}(-\\Delta_x)^{\\gamma}D_{t}^{\\sigma}I_{t}^{\\delta}p(t,x), \\quad \\forall\\,\\, n\\in\\mathbb{Z}_{+}, \\,\\, \\gamma\\in[0,\\beta],\\,\\, \\sigma, \\delta \\in[0,\\infty), $$ where $D_{x}^n$ is a partial derivative of order $n$ with respect to $x$, $(-\\Delta_x)^{\\gamma}$ is a fractional Laplace operator and $D_{t}^{\\sigma","authors_text":"Kyeong-Hun Kim, Sungbin Lim","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-28T09:13:35Z","title":"Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07386","kind":"arxiv","version":4},"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":"sha256:594506eb27fa226b1cbe8cea492ba1f6eaa4606ffcea9afa347e2372af737c78","target":"record","created_at":"2026-05-18T02:16:38Z","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":"94e7df159350cac82a756a70e837ac9acdcadb8a2222a909c9ece1b61f83acfb","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-28T09:13:35Z","title_canon_sha256":"5386c12a71978eb7d12df769878434c0e28088b3f334abf67e8a1257ff733e43"},"schema_version":"1.0","source":{"id":"1504.07386","kind":"arxiv","version":4}},"canonical_sha256":"01b3d60de2723a1ce965ad9d64d2cc7c3e7480a70b5678c6cf35b924ab0e0c00","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"01b3d60de2723a1ce965ad9d64d2cc7c3e7480a70b5678c6cf35b924ab0e0c00","first_computed_at":"2026-05-18T02:16:38.823154Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:16:38.823154Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0z6SwF4/KCGpHeh3AlYTqxjDU6+RV1NhHMcvBeOli2mMo6XruEk4JHVPdUcnqLQYIv5fcafChQNJ5487ObwJCw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:16:38.823648Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.07386","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:594506eb27fa226b1cbe8cea492ba1f6eaa4606ffcea9afa347e2372af737c78","sha256:6c6c68dd6e8050ba32d5231d0fa80bb6d8c30b92d99f34ff3716ce48857b5f32"],"state_sha256":"45339cf8809ffe78eecd819427b017b85bd68b1a5ca269362550226e20695186"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RyajHsCw+b09Nc55jt+JC5KdOcg04wl/rZKaeSJaT6ZBwzcHWFPbTtXt5u6Yv1GE1POMynkVsnQY+5QwURWMBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T08:07:38.224603Z","bundle_sha256":"9a2dcc4ac99575ec1d27a5b8b6485753b00c06e4421b032fb699cd4c997277ce"}}