{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:3EX7VVURANNP7ZXYWOFCI2N2PR","short_pith_number":"pith:3EX7VVUR","canonical_record":{"source":{"id":"1904.02538","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-04-03T02:28:36Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"dfbbf8ee5fbe467c80e5e4cb2d19a3886dc2591e8baf0ca0064d7f19a409a939","abstract_canon_sha256":"565830739997b313d9ee383c37963d5655abd68dcad3a1a0c47522e41eb7f64e"},"schema_version":"1.0"},"canonical_sha256":"d92ffad691035affe6f8b38a2469ba7c733169f764ca4824b3c57b1c54d2662e","source":{"kind":"arxiv","id":"1904.02538","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.02538","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"arxiv_version","alias_value":"1904.02538v1","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.02538","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"pith_short_12","alias_value":"3EX7VVURANNP","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"3EX7VVURANNP7ZXY","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"3EX7VVUR","created_at":"2026-05-18T12:33:07Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:3EX7VVURANNP7ZXYWOFCI2N2PR","target":"record","payload":{"canonical_record":{"source":{"id":"1904.02538","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-04-03T02:28:36Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"dfbbf8ee5fbe467c80e5e4cb2d19a3886dc2591e8baf0ca0064d7f19a409a939","abstract_canon_sha256":"565830739997b313d9ee383c37963d5655abd68dcad3a1a0c47522e41eb7f64e"},"schema_version":"1.0"},"canonical_sha256":"d92ffad691035affe6f8b38a2469ba7c733169f764ca4824b3c57b1c54d2662e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:23.356764Z","signature_b64":"tG77+Qmcfy+7E/cszPsLYA3QeMz9KhtSC+UGy4IeyCO86iJnnO/2fcF59a+2q1a+wyq++SRFC7cuOqDtm8DzDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d92ffad691035affe6f8b38a2469ba7c733169f764ca4824b3c57b1c54d2662e","last_reissued_at":"2026-05-17T23:49:23.356185Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:23.356185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.02538","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":"2026-05-17T23:49:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BODAH7c1L21sexByuLznmwAyw2UvNZM2i7SulYAOVm7brPliFRD5j6S8L/TY0SqOc9/Qnkm7e+HscByXESIxAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T16:42:18.465118Z"},"content_sha256":"1e47674457db3752462cb87c3ee797b7bbeb72da9b4033139c5422abee6c8764","schema_version":"1.0","event_id":"sha256:1e47674457db3752462cb87c3ee797b7bbeb72da9b4033139c5422abee6c8764"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:3EX7VVURANNP7ZXYWOFCI2N2PR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalizations of Schoenberg's theorem on positive definite kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Juan C. Vera, Olga Kuryatnikova","submitted_at":"2019-04-03T02:28:36Z","abstract_excerpt":"The seminal theorem of I.J. Schoenberg characterizes positive definite (p.d.) kernels on the unit sphere $S^{n-1}$ invariant under the automorphisms of the sphere. We obtain two generalizations of this theorem for p.d. kernels on fiber bundles. Our first theorem characterizes invariant p.d. kernels on bundles whose fiber is a product of a compact set and the unit sphere. This result implies, in particular, a characterization of invariant under the automorphisms of the sphere, p.d. kernels on a product of $S^{n-1}$ and a compact set. Our second result characterizes invariant p.d. kernels on the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.02538","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":"2026-05-17T23:49:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BTRJawxnSPjyZL8KXg7YJdxhxcYTNap4IDd4uRsEzL9KTlSHSducZe9GK/j2axsJXHVz9jHNCNIcjaIuqXXbCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T16:42:18.465859Z"},"content_sha256":"01be42e8a4dfec702b9df64c1cc5b5870dacc71247e1928af2783ce628921048","schema_version":"1.0","event_id":"sha256:01be42e8a4dfec702b9df64c1cc5b5870dacc71247e1928af2783ce628921048"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3EX7VVURANNP7ZXYWOFCI2N2PR/bundle.json","state_url":"https://pith.science/pith/3EX7VVURANNP7ZXYWOFCI2N2PR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3EX7VVURANNP7ZXYWOFCI2N2PR/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":"2026-06-08T16:42:18Z","links":{"resolver":"https://pith.science/pith/3EX7VVURANNP7ZXYWOFCI2N2PR","bundle":"https://pith.science/pith/3EX7VVURANNP7ZXYWOFCI2N2PR/bundle.json","state":"https://pith.science/pith/3EX7VVURANNP7ZXYWOFCI2N2PR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3EX7VVURANNP7ZXYWOFCI2N2PR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:3EX7VVURANNP7ZXYWOFCI2N2PR","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":"565830739997b313d9ee383c37963d5655abd68dcad3a1a0c47522e41eb7f64e","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-04-03T02:28:36Z","title_canon_sha256":"dfbbf8ee5fbe467c80e5e4cb2d19a3886dc2591e8baf0ca0064d7f19a409a939"},"schema_version":"1.0","source":{"id":"1904.02538","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.02538","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"arxiv_version","alias_value":"1904.02538v1","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.02538","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"pith_short_12","alias_value":"3EX7VVURANNP","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_16","alias_value":"3EX7VVURANNP7ZXY","created_at":"2026-05-18T12:33:07Z"},{"alias_kind":"pith_short_8","alias_value":"3EX7VVUR","created_at":"2026-05-18T12:33:07Z"}],"graph_snapshots":[{"event_id":"sha256:01be42e8a4dfec702b9df64c1cc5b5870dacc71247e1928af2783ce628921048","target":"graph","created_at":"2026-05-17T23:49:23Z","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":"The seminal theorem of I.J. Schoenberg characterizes positive definite (p.d.) kernels on the unit sphere $S^{n-1}$ invariant under the automorphisms of the sphere. We obtain two generalizations of this theorem for p.d. kernels on fiber bundles. Our first theorem characterizes invariant p.d. kernels on bundles whose fiber is a product of a compact set and the unit sphere. This result implies, in particular, a characterization of invariant under the automorphisms of the sphere, p.d. kernels on a product of $S^{n-1}$ and a compact set. Our second result characterizes invariant p.d. kernels on the","authors_text":"Juan C. Vera, Olga Kuryatnikova","cross_cats":["math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-04-03T02:28:36Z","title":"Generalizations of Schoenberg's theorem on positive definite kernels"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.02538","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":"sha256:1e47674457db3752462cb87c3ee797b7bbeb72da9b4033139c5422abee6c8764","target":"record","created_at":"2026-05-17T23:49:23Z","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":"565830739997b313d9ee383c37963d5655abd68dcad3a1a0c47522e41eb7f64e","cross_cats_sorted":["math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2019-04-03T02:28:36Z","title_canon_sha256":"dfbbf8ee5fbe467c80e5e4cb2d19a3886dc2591e8baf0ca0064d7f19a409a939"},"schema_version":"1.0","source":{"id":"1904.02538","kind":"arxiv","version":1}},"canonical_sha256":"d92ffad691035affe6f8b38a2469ba7c733169f764ca4824b3c57b1c54d2662e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d92ffad691035affe6f8b38a2469ba7c733169f764ca4824b3c57b1c54d2662e","first_computed_at":"2026-05-17T23:49:23.356185Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:49:23.356185Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tG77+Qmcfy+7E/cszPsLYA3QeMz9KhtSC+UGy4IeyCO86iJnnO/2fcF59a+2q1a+wyq++SRFC7cuOqDtm8DzDg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:49:23.356764Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.02538","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1e47674457db3752462cb87c3ee797b7bbeb72da9b4033139c5422abee6c8764","sha256:01be42e8a4dfec702b9df64c1cc5b5870dacc71247e1928af2783ce628921048"],"state_sha256":"19e32b2d66c7c455985ed1d2846f829c1230175ca42fdfd6a078f33f16ccc9d8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q+IovEOyBgnOXh0xfik3PIjOIfgONWxx5uq0+zNBXM5XgPBqM4gZPBAnd1lGpxzNDkbGnhR6jHo9PfUcNSicCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T16:42:18.469822Z","bundle_sha256":"b92139ebb1ff475a7e06e3fb86611e7ccbe4d28a5f7f7b7661c6434194468936"}}