{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:4R6QRTZ2JBHEBBJWPKWNNCLJTP","short_pith_number":"pith:4R6QRTZ2","canonical_record":{"source":{"id":"1901.01676","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-01-07T06:35:18Z","cross_cats_sorted":[],"title_canon_sha256":"ce39f788f3e17f29b8ec2727d7dec7f839952c427defffcdf0f90425882ddf65","abstract_canon_sha256":"d2ba4e0bb708adfc7ea7b7d5235c14e401ca87a915cd010edb7aa29b525e7ff4"},"schema_version":"1.0"},"canonical_sha256":"e47d08cf3a484e4085367aacd689699be9a4160de513fc8fb24ad28ca399fd32","source":{"kind":"arxiv","id":"1901.01676","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.01676","created_at":"2026-05-17T23:56:51Z"},{"alias_kind":"arxiv_version","alias_value":"1901.01676v1","created_at":"2026-05-17T23:56:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.01676","created_at":"2026-05-17T23:56:51Z"},{"alias_kind":"pith_short_12","alias_value":"4R6QRTZ2JBHE","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"4R6QRTZ2JBHEBBJW","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"4R6QRTZ2","created_at":"2026-05-18T12:33:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:4R6QRTZ2JBHEBBJWPKWNNCLJTP","target":"record","payload":{"canonical_record":{"source":{"id":"1901.01676","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-01-07T06:35:18Z","cross_cats_sorted":[],"title_canon_sha256":"ce39f788f3e17f29b8ec2727d7dec7f839952c427defffcdf0f90425882ddf65","abstract_canon_sha256":"d2ba4e0bb708adfc7ea7b7d5235c14e401ca87a915cd010edb7aa29b525e7ff4"},"schema_version":"1.0"},"canonical_sha256":"e47d08cf3a484e4085367aacd689699be9a4160de513fc8fb24ad28ca399fd32","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:51.479860Z","signature_b64":"tCGUDw6kmVCr3SIH1bfqE3KCwxvMe6BgWPna8ZsyoDxxY5dsvguFllqH7igPvFbR3dCthPEiv+w/McCdvF0ZDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e47d08cf3a484e4085367aacd689699be9a4160de513fc8fb24ad28ca399fd32","last_reissued_at":"2026-05-17T23:56:51.479441Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:51.479441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1901.01676","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":"2026-05-17T23:56:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HW60+eMpLzeAgMgVRj4i1//FIrTyQAargvpuKdpxr9R8w0XZGv9vZiE8EusQMEEhujA6hwFGy6vogclJCKTABg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T00:12:32.446189Z"},"content_sha256":"a7b30a01224e2747cf0c805b3da8fdd94e6973dc08b108882e84186a158354a5","schema_version":"1.0","event_id":"sha256:a7b30a01224e2747cf0c805b3da8fdd94e6973dc08b108882e84186a158354a5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:4R6QRTZ2JBHEBBJWPKWNNCLJTP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Uncertainty principles on nilpotent Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"Ajay Kumar, Jyoti Sharma","submitted_at":"2019-01-07T06:35:18Z","abstract_excerpt":"Hardy's type uncertainty principle on connected nilpotent Lie groups for the Fourier transform is proved. An analogue of Hardy's theorem for Gabor transform has been established for connected and simply connected nilpotent Lie groups. Finally Beurling's theorem for Gabor transform is discussed for groups of the form $\\mathbb{R}_n \\times K$, where $K$ is a compact group"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01676","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":"2026-05-17T23:56:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Pz4Af492FtXrVEWvLapI86CM+nnCCUOLQH+TMg5+8PXpmuC9bwm6KQSCdMt5Vk1pRokCSEeG0kB53mNDRcyGBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T00:12:32.446550Z"},"content_sha256":"cdb25a9dc98a37edc92b2736d2a717bbdcbaef696640202345236262688fc4f1","schema_version":"1.0","event_id":"sha256:cdb25a9dc98a37edc92b2736d2a717bbdcbaef696640202345236262688fc4f1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4R6QRTZ2JBHEBBJWPKWNNCLJTP/bundle.json","state_url":"https://pith.science/pith/4R6QRTZ2JBHEBBJWPKWNNCLJTP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4R6QRTZ2JBHEBBJWPKWNNCLJTP/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":"2026-06-26T00:12:32Z","links":{"resolver":"https://pith.science/pith/4R6QRTZ2JBHEBBJWPKWNNCLJTP","bundle":"https://pith.science/pith/4R6QRTZ2JBHEBBJWPKWNNCLJTP/bundle.json","state":"https://pith.science/pith/4R6QRTZ2JBHEBBJWPKWNNCLJTP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4R6QRTZ2JBHEBBJWPKWNNCLJTP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:4R6QRTZ2JBHEBBJWPKWNNCLJTP","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":"d2ba4e0bb708adfc7ea7b7d5235c14e401ca87a915cd010edb7aa29b525e7ff4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-01-07T06:35:18Z","title_canon_sha256":"ce39f788f3e17f29b8ec2727d7dec7f839952c427defffcdf0f90425882ddf65"},"schema_version":"1.0","source":{"id":"1901.01676","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.01676","created_at":"2026-05-17T23:56:51Z"},{"alias_kind":"arxiv_version","alias_value":"1901.01676v1","created_at":"2026-05-17T23:56:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.01676","created_at":"2026-05-17T23:56:51Z"},{"alias_kind":"pith_short_12","alias_value":"4R6QRTZ2JBHE","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_16","alias_value":"4R6QRTZ2JBHEBBJW","created_at":"2026-05-18T12:33:10Z"},{"alias_kind":"pith_short_8","alias_value":"4R6QRTZ2","created_at":"2026-05-18T12:33:10Z"}],"graph_snapshots":[{"event_id":"sha256:cdb25a9dc98a37edc92b2736d2a717bbdcbaef696640202345236262688fc4f1","target":"graph","created_at":"2026-05-17T23:56:51Z","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":"Hardy's type uncertainty principle on connected nilpotent Lie groups for the Fourier transform is proved. An analogue of Hardy's theorem for Gabor transform has been established for connected and simply connected nilpotent Lie groups. Finally Beurling's theorem for Gabor transform is discussed for groups of the form $\\mathbb{R}_n \\times K$, where $K$ is a compact group","authors_text":"Ajay Kumar, Jyoti Sharma","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-01-07T06:35:18Z","title":"Uncertainty principles on nilpotent Lie groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.01676","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:a7b30a01224e2747cf0c805b3da8fdd94e6973dc08b108882e84186a158354a5","target":"record","created_at":"2026-05-17T23:56:51Z","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":"d2ba4e0bb708adfc7ea7b7d5235c14e401ca87a915cd010edb7aa29b525e7ff4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2019-01-07T06:35:18Z","title_canon_sha256":"ce39f788f3e17f29b8ec2727d7dec7f839952c427defffcdf0f90425882ddf65"},"schema_version":"1.0","source":{"id":"1901.01676","kind":"arxiv","version":1}},"canonical_sha256":"e47d08cf3a484e4085367aacd689699be9a4160de513fc8fb24ad28ca399fd32","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e47d08cf3a484e4085367aacd689699be9a4160de513fc8fb24ad28ca399fd32","first_computed_at":"2026-05-17T23:56:51.479441Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:51.479441Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tCGUDw6kmVCr3SIH1bfqE3KCwxvMe6BgWPna8ZsyoDxxY5dsvguFllqH7igPvFbR3dCthPEiv+w/McCdvF0ZDQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:51.479860Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.01676","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a7b30a01224e2747cf0c805b3da8fdd94e6973dc08b108882e84186a158354a5","sha256:cdb25a9dc98a37edc92b2736d2a717bbdcbaef696640202345236262688fc4f1"],"state_sha256":"0384fc8a644dd8e9f111b5e2b013112c245808b060844c52e03890d71ae2d295"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nPPZAVHukqLi/LQD3vS65vZ4NZsz/4tSt+bSO+aFqjtf8bl62RHwsiISYwDScQlftqTUP4YUY4jsX0GrlFHrAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T00:12:32.448430Z","bundle_sha256":"6d0b379c8b49915b7078cd591909f77521f898f814d416f2cab17785fb4e634f"}}