{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:NNE7IXRZ3FPX55XPVICKIJVDSZ","short_pith_number":"pith:NNE7IXRZ","canonical_record":{"source":{"id":"1805.00253","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-05-01T09:24:08Z","cross_cats_sorted":[],"title_canon_sha256":"eeb08025e96596a8dbc6110241362e68b7132bf34ad7aede81489125b5c4fe92","abstract_canon_sha256":"a58acdb580f28f9371fea96f82b5ebd7ac017f8d77e47a67a1041acfbd3618ce"},"schema_version":"1.0"},"canonical_sha256":"6b49f45e39d95f7ef6efaa04a426a396643659c61233512193f8b1c73e11c3fb","source":{"kind":"arxiv","id":"1805.00253","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.00253","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"arxiv_version","alias_value":"1805.00253v2","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.00253","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"pith_short_12","alias_value":"NNE7IXRZ3FPX","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NNE7IXRZ3FPX55XP","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NNE7IXRZ","created_at":"2026-05-18T12:32:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:NNE7IXRZ3FPX55XPVICKIJVDSZ","target":"record","payload":{"canonical_record":{"source":{"id":"1805.00253","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-05-01T09:24:08Z","cross_cats_sorted":[],"title_canon_sha256":"eeb08025e96596a8dbc6110241362e68b7132bf34ad7aede81489125b5c4fe92","abstract_canon_sha256":"a58acdb580f28f9371fea96f82b5ebd7ac017f8d77e47a67a1041acfbd3618ce"},"schema_version":"1.0"},"canonical_sha256":"6b49f45e39d95f7ef6efaa04a426a396643659c61233512193f8b1c73e11c3fb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:44.320331Z","signature_b64":"rzxNFeHhosEvPmucLKSRuWSAbbHOGENEGXxhcswWKvQ/3uNu/vxXPILIO15KWzfiB3Miw3128EkmZfJtQPloAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b49f45e39d95f7ef6efaa04a426a396643659c61233512193f8b1c73e11c3fb","last_reissued_at":"2026-05-18T00:13:44.319677Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:44.319677Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.00253","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":"2026-05-18T00:13:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GmyfrhOrYUpw/O6FnXAx9BDZO1LhqCR/UzhzUNu687fL6+Lr4MlO8ujgvC9Ow6qYUpRKfwVncCvRKncxX9p0BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:23:22.939939Z"},"content_sha256":"eef7dd788ab2b012a900585ec4dcf4191388d0553791bea40ab14a855ec49767","schema_version":"1.0","event_id":"sha256:eef7dd788ab2b012a900585ec4dcf4191388d0553791bea40ab14a855ec49767"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:NNE7IXRZ3FPX55XPVICKIJVDSZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Singularity of the $n$-th eigenvalue of high dimensional Sturm-Liouville problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.SP","authors_text":"Hao Zhu, Lei Liu, Li Wu, Xijun Hu","submitted_at":"2018-05-01T09:24:08Z","abstract_excerpt":"It is natural to consider continuous dependence of the $n$-th eigenvalue on $d$-dimensional ($d\\geq2$) Sturm-Liouville problems after the results on $1$-dimensional case by Kong, Wu and Zettl [14]. In this paper, we find all the boundary conditions such that the $n$-th eigenvalue is not continuous, and give complete characterization of asymptotic behavior of the $n$-th eigenvalue. This renders a precise description of the jump phenomena of the $n$-th eigenvalue near such a boundary condition. Furthermore, we divide the space of boundary conditions into $2d+1$ layers and show that the $n$-th ei"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.00253","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":"2026-05-18T00:13:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sPKHMyyqV5Vu8ccRN2A5q2L7JGsSBfQ1y6hZhbMxTofCP3fQZb8VL/75Jj+BEqNiN6Ss+aMku4lmj191ULcPCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T09:23:22.940589Z"},"content_sha256":"fc2744ed258b721aaf30e3acf405b3ac2bde9bc9286c78f9a090414decec99e6","schema_version":"1.0","event_id":"sha256:fc2744ed258b721aaf30e3acf405b3ac2bde9bc9286c78f9a090414decec99e6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NNE7IXRZ3FPX55XPVICKIJVDSZ/bundle.json","state_url":"https://pith.science/pith/NNE7IXRZ3FPX55XPVICKIJVDSZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NNE7IXRZ3FPX55XPVICKIJVDSZ/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":"2026-05-27T09:23:22Z","links":{"resolver":"https://pith.science/pith/NNE7IXRZ3FPX55XPVICKIJVDSZ","bundle":"https://pith.science/pith/NNE7IXRZ3FPX55XPVICKIJVDSZ/bundle.json","state":"https://pith.science/pith/NNE7IXRZ3FPX55XPVICKIJVDSZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NNE7IXRZ3FPX55XPVICKIJVDSZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:NNE7IXRZ3FPX55XPVICKIJVDSZ","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":"a58acdb580f28f9371fea96f82b5ebd7ac017f8d77e47a67a1041acfbd3618ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-05-01T09:24:08Z","title_canon_sha256":"eeb08025e96596a8dbc6110241362e68b7132bf34ad7aede81489125b5c4fe92"},"schema_version":"1.0","source":{"id":"1805.00253","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.00253","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"arxiv_version","alias_value":"1805.00253v2","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.00253","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"pith_short_12","alias_value":"NNE7IXRZ3FPX","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NNE7IXRZ3FPX55XP","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NNE7IXRZ","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:fc2744ed258b721aaf30e3acf405b3ac2bde9bc9286c78f9a090414decec99e6","target":"graph","created_at":"2026-05-18T00:13:44Z","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":"It is natural to consider continuous dependence of the $n$-th eigenvalue on $d$-dimensional ($d\\geq2$) Sturm-Liouville problems after the results on $1$-dimensional case by Kong, Wu and Zettl [14]. In this paper, we find all the boundary conditions such that the $n$-th eigenvalue is not continuous, and give complete characterization of asymptotic behavior of the $n$-th eigenvalue. This renders a precise description of the jump phenomena of the $n$-th eigenvalue near such a boundary condition. Furthermore, we divide the space of boundary conditions into $2d+1$ layers and show that the $n$-th ei","authors_text":"Hao Zhu, Lei Liu, Li Wu, Xijun Hu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-05-01T09:24:08Z","title":"Singularity of the $n$-th eigenvalue of high dimensional Sturm-Liouville problems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.00253","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:eef7dd788ab2b012a900585ec4dcf4191388d0553791bea40ab14a855ec49767","target":"record","created_at":"2026-05-18T00:13:44Z","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":"a58acdb580f28f9371fea96f82b5ebd7ac017f8d77e47a67a1041acfbd3618ce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2018-05-01T09:24:08Z","title_canon_sha256":"eeb08025e96596a8dbc6110241362e68b7132bf34ad7aede81489125b5c4fe92"},"schema_version":"1.0","source":{"id":"1805.00253","kind":"arxiv","version":2}},"canonical_sha256":"6b49f45e39d95f7ef6efaa04a426a396643659c61233512193f8b1c73e11c3fb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6b49f45e39d95f7ef6efaa04a426a396643659c61233512193f8b1c73e11c3fb","first_computed_at":"2026-05-18T00:13:44.319677Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:13:44.319677Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rzxNFeHhosEvPmucLKSRuWSAbbHOGENEGXxhcswWKvQ/3uNu/vxXPILIO15KWzfiB3Miw3128EkmZfJtQPloAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:13:44.320331Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.00253","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:eef7dd788ab2b012a900585ec4dcf4191388d0553791bea40ab14a855ec49767","sha256:fc2744ed258b721aaf30e3acf405b3ac2bde9bc9286c78f9a090414decec99e6"],"state_sha256":"c86c3feccf9c9c11a047163e236eb141c2ab78854e77282e1595fe6d3f7a2d0c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gp2rQuwR/fTrDQ7bJszyfYjx80URbh8xFTXhmiIIm3Qux5BurhdUTaDn0i4D/RTsYHXV4Pmu3wzU7nk6RqNDDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T09:23:22.944152Z","bundle_sha256":"4a84a0d17e503798205db50a7c44bc2b5c5f9b56306f9e44755dfcce77378b3e"}}