{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:7RYMUMK6JUUKLJAIMCWAKD3K67","short_pith_number":"pith:7RYMUMK6","canonical_record":{"source":{"id":"1310.5571","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-21T14:38:59Z","cross_cats_sorted":["math-ph","math.AP","math.DG","math.MP"],"title_canon_sha256":"cb87d6f101ca51eb8d5d0b098afebc5214c07f3aa39bf7f0704a3b59b1fd0b26","abstract_canon_sha256":"5f91f40414b0e8a8e157fa8ba0a913e0633240c204b0a9e5b18268e61f416436"},"schema_version":"1.0"},"canonical_sha256":"fc70ca315e4d28a5a40860ac050f6af7d2e5b07b260a45da93b5ba766c38f9c4","source":{"kind":"arxiv","id":"1310.5571","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.5571","created_at":"2026-05-18T01:57:23Z"},{"alias_kind":"arxiv_version","alias_value":"1310.5571v2","created_at":"2026-05-18T01:57:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.5571","created_at":"2026-05-18T01:57:23Z"},{"alias_kind":"pith_short_12","alias_value":"7RYMUMK6JUUK","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7RYMUMK6JUUKLJAI","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7RYMUMK6","created_at":"2026-05-18T12:27:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:7RYMUMK6JUUKLJAIMCWAKD3K67","target":"record","payload":{"canonical_record":{"source":{"id":"1310.5571","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-21T14:38:59Z","cross_cats_sorted":["math-ph","math.AP","math.DG","math.MP"],"title_canon_sha256":"cb87d6f101ca51eb8d5d0b098afebc5214c07f3aa39bf7f0704a3b59b1fd0b26","abstract_canon_sha256":"5f91f40414b0e8a8e157fa8ba0a913e0633240c204b0a9e5b18268e61f416436"},"schema_version":"1.0"},"canonical_sha256":"fc70ca315e4d28a5a40860ac050f6af7d2e5b07b260a45da93b5ba766c38f9c4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:57:23.690298Z","signature_b64":"kIqqO++gGKhXjErWbg4Ir39urZPIdHA4KtKmvp9eUAa9kn/PycMgZM5yRl2Kr1vGO5RpFvsSHlD5K1FKXIhQDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc70ca315e4d28a5a40860ac050f6af7d2e5b07b260a45da93b5ba766c38f9c4","last_reissued_at":"2026-05-18T01:57:23.689798Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:57:23.689798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1310.5571","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":"2026-05-18T01:57:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gzYkEeTac++dyrfxHy7FtOrzav25XhdHxtk3WDwDFcTK+mmDNOgN2kLcF8ayvnjGdnoO8AZmcSCKB9c6TXLHCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T07:51:24.961447Z"},"content_sha256":"d87ef1765694c883193c178fee3866ac53fcb8a52ea50e01e3e1fa5dc260b283","schema_version":"1.0","event_id":"sha256:d87ef1765694c883193c178fee3866ac53fcb8a52ea50e01e3e1fa5dc260b283"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:7RYMUMK6JUUKLJAIMCWAKD3K67","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Critical points of multidimensional random Fourier series: variance estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.AP","math.DG","math.MP"],"primary_cat":"math.PR","authors_text":"Liviu I. Nicolaescu","submitted_at":"2013-10-21T14:38:59Z","abstract_excerpt":"To any positive number $\\varepsilon$ and any nonnegative even Schwartz function $w:\\mathbb{R}\\to\\mathbb{R}$ we associate the random function $u^\\varepsilon$ on the $m$-torus $T^m_\\varepsilon:=\\mathbb{R}^m/(\\varepsilon^{-1}\\mathbb{Z})^m$ defined as the real part of the random Fourier series $$ \\sum_{\\nu\\in\\mathbb{Z}^m} X_{\\nu,\\varepsilon} \\exp\\bigl(\\; 2\\pi \\varepsilon \\sqrt{-1} \\;(\\nu\\cdot \\theta)\\;\\bigr),$$ where $X_{\\nu,\\varepsilon}$ are complex independent Gaussian random variables with variance $w(\\varepsilon|\\nu|)$. Let $N^\\varepsilon$ denote the number of critical points of $u^\\varepsilon"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.5571","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":"2026-05-18T01:57:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0ipvDNV2RaCvw3mfE72B8GWNbuD4JcW4y5X18FrKcxHNAlAXdcAUKPQD2ooHfItDTDfXs5fWSIYTlDvHXH2pDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T07:51:24.961802Z"},"content_sha256":"8fb32757aab8a312f61dfdf5aadf53672e4ecd559807c4ef840f60cb80c33720","schema_version":"1.0","event_id":"sha256:8fb32757aab8a312f61dfdf5aadf53672e4ecd559807c4ef840f60cb80c33720"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7RYMUMK6JUUKLJAIMCWAKD3K67/bundle.json","state_url":"https://pith.science/pith/7RYMUMK6JUUKLJAIMCWAKD3K67/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7RYMUMK6JUUKLJAIMCWAKD3K67/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":"2026-06-23T07:51:24Z","links":{"resolver":"https://pith.science/pith/7RYMUMK6JUUKLJAIMCWAKD3K67","bundle":"https://pith.science/pith/7RYMUMK6JUUKLJAIMCWAKD3K67/bundle.json","state":"https://pith.science/pith/7RYMUMK6JUUKLJAIMCWAKD3K67/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7RYMUMK6JUUKLJAIMCWAKD3K67/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:7RYMUMK6JUUKLJAIMCWAKD3K67","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":"5f91f40414b0e8a8e157fa8ba0a913e0633240c204b0a9e5b18268e61f416436","cross_cats_sorted":["math-ph","math.AP","math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-21T14:38:59Z","title_canon_sha256":"cb87d6f101ca51eb8d5d0b098afebc5214c07f3aa39bf7f0704a3b59b1fd0b26"},"schema_version":"1.0","source":{"id":"1310.5571","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.5571","created_at":"2026-05-18T01:57:23Z"},{"alias_kind":"arxiv_version","alias_value":"1310.5571v2","created_at":"2026-05-18T01:57:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.5571","created_at":"2026-05-18T01:57:23Z"},{"alias_kind":"pith_short_12","alias_value":"7RYMUMK6JUUK","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_16","alias_value":"7RYMUMK6JUUKLJAI","created_at":"2026-05-18T12:27:36Z"},{"alias_kind":"pith_short_8","alias_value":"7RYMUMK6","created_at":"2026-05-18T12:27:36Z"}],"graph_snapshots":[{"event_id":"sha256:8fb32757aab8a312f61dfdf5aadf53672e4ecd559807c4ef840f60cb80c33720","target":"graph","created_at":"2026-05-18T01:57:23Z","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":"To any positive number $\\varepsilon$ and any nonnegative even Schwartz function $w:\\mathbb{R}\\to\\mathbb{R}$ we associate the random function $u^\\varepsilon$ on the $m$-torus $T^m_\\varepsilon:=\\mathbb{R}^m/(\\varepsilon^{-1}\\mathbb{Z})^m$ defined as the real part of the random Fourier series $$ \\sum_{\\nu\\in\\mathbb{Z}^m} X_{\\nu,\\varepsilon} \\exp\\bigl(\\; 2\\pi \\varepsilon \\sqrt{-1} \\;(\\nu\\cdot \\theta)\\;\\bigr),$$ where $X_{\\nu,\\varepsilon}$ are complex independent Gaussian random variables with variance $w(\\varepsilon|\\nu|)$. Let $N^\\varepsilon$ denote the number of critical points of $u^\\varepsilon","authors_text":"Liviu I. Nicolaescu","cross_cats":["math-ph","math.AP","math.DG","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-21T14:38:59Z","title":"Critical points of multidimensional random Fourier series: variance estimates"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.5571","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:d87ef1765694c883193c178fee3866ac53fcb8a52ea50e01e3e1fa5dc260b283","target":"record","created_at":"2026-05-18T01:57:23Z","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":"5f91f40414b0e8a8e157fa8ba0a913e0633240c204b0a9e5b18268e61f416436","cross_cats_sorted":["math-ph","math.AP","math.DG","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-10-21T14:38:59Z","title_canon_sha256":"cb87d6f101ca51eb8d5d0b098afebc5214c07f3aa39bf7f0704a3b59b1fd0b26"},"schema_version":"1.0","source":{"id":"1310.5571","kind":"arxiv","version":2}},"canonical_sha256":"fc70ca315e4d28a5a40860ac050f6af7d2e5b07b260a45da93b5ba766c38f9c4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc70ca315e4d28a5a40860ac050f6af7d2e5b07b260a45da93b5ba766c38f9c4","first_computed_at":"2026-05-18T01:57:23.689798Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:57:23.689798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kIqqO++gGKhXjErWbg4Ir39urZPIdHA4KtKmvp9eUAa9kn/PycMgZM5yRl2Kr1vGO5RpFvsSHlD5K1FKXIhQDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:57:23.690298Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.5571","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d87ef1765694c883193c178fee3866ac53fcb8a52ea50e01e3e1fa5dc260b283","sha256:8fb32757aab8a312f61dfdf5aadf53672e4ecd559807c4ef840f60cb80c33720"],"state_sha256":"825d1788a2b53e08b010663acd86294c4934fbe9783981cb18812d5e76d7a929"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uPU/CyJSrnntLzWfUV//VHGQuSDgFbcpHDe4DfJuTTcFaCaRqBX75qKnrgMpamJ6aKCIlsPdtwf4z+2WTmg5DQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T07:51:24.963766Z","bundle_sha256":"413728daca0c03ca944c79dbe1e78579f8c157df38d8425021912cfe186617d7"}}