{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2000:2D7YBW4UT44ILNULVGP2M5MFZD","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":"6b43af4980643103d1c361feab4364e0ee68ad11e92cf7aeb09f2794bcc16d86","cross_cats_sorted":["math-ph","math.MP"],"license":"","primary_cat":"math.SP","submitted_at":"2000-08-11T20:45:10Z","title_canon_sha256":"2791ba79a13504035f74e4834fa926e3cc47ee93325659f730c2d7590ecad809"},"schema_version":"1.0","source":{"id":"math/0008088","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0008088","created_at":"2026-05-18T01:05:38Z"},{"alias_kind":"arxiv_version","alias_value":"math/0008088v1","created_at":"2026-05-18T01:05:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0008088","created_at":"2026-05-18T01:05:38Z"},{"alias_kind":"pith_short_12","alias_value":"2D7YBW4UT44I","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_16","alias_value":"2D7YBW4UT44ILNUL","created_at":"2026-05-18T12:25:49Z"},{"alias_kind":"pith_short_8","alias_value":"2D7YBW4U","created_at":"2026-05-18T12:25:49Z"}],"graph_snapshots":[{"event_id":"sha256:edf0c1ad5f985ddb29b892624d472280d724e63ccf850188941b35d4d866b196","target":"graph","created_at":"2026-05-18T01:05:38Z","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":"For a domain $\\Omega$ contained in a hemisphere of the $n$-dimensional sphere $\\SS^n$ we prove the optimal result $\\lambda_2/\\lambda_1(\\Omega) \\le \\lambda_2/\\lambda_1(\\Omega^{\\star})$ for the ratio of its first two Dirichlet eigenvalues where $\\Omega^{\\star}$, the symmetric rearrangement of $\\Omega$ in $\\SS^n$, is a geodesic ball in $\\SS^n$ having the same $n$-volume as $\\Omega$. We also show that $\\lambda_2/\\lambda_1$ for geodesic balls of geodesic radius $\\theta_1$ less than or equal to $\\pi/2$ is an increasing function of $\\theta_1$ which runs between the value $(j_{n/2,1}/j_{n/2-1,1})^2$ f","authors_text":"Mark S. Ashbaugh, Rafael D. Benguria","cross_cats":["math-ph","math.MP"],"headline":"","license":"","primary_cat":"math.SP","submitted_at":"2000-08-11T20:45:10Z","title":"A Sharp Bound for the Ratio of the First Two Dirichlet Eigenvalues of a Domain in a Hemisphere of S^n"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0008088","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:b9ff879e660e334056997d59b1172e45e571f5190fa91192ed07f609159e3514","target":"record","created_at":"2026-05-18T01:05:38Z","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":"6b43af4980643103d1c361feab4364e0ee68ad11e92cf7aeb09f2794bcc16d86","cross_cats_sorted":["math-ph","math.MP"],"license":"","primary_cat":"math.SP","submitted_at":"2000-08-11T20:45:10Z","title_canon_sha256":"2791ba79a13504035f74e4834fa926e3cc47ee93325659f730c2d7590ecad809"},"schema_version":"1.0","source":{"id":"math/0008088","kind":"arxiv","version":1}},"canonical_sha256":"d0ff80db949f3885b68ba99fa67585c8d55051fb5b7911dd4c1518149f06f6ae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d0ff80db949f3885b68ba99fa67585c8d55051fb5b7911dd4c1518149f06f6ae","first_computed_at":"2026-05-18T01:05:38.389784Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:38.389784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HfD/api1kAmujnugsNhappUieZqJtVDCh3lrtazyXZraq0EhiOvAZYgrqwr/jUHrequk4ICtL5a/ZWUmyruOBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:38.390278Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0008088","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b9ff879e660e334056997d59b1172e45e571f5190fa91192ed07f609159e3514","sha256:edf0c1ad5f985ddb29b892624d472280d724e63ccf850188941b35d4d866b196"],"state_sha256":"ec8199df36628a860f7078ebd75d8aab04ca8bd57e85cfe70ac616a63a2fd30d"}