{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:YSXPCYWEKBTYOKR66QVMNGOIXH","short_pith_number":"pith:YSXPCYWE","canonical_record":{"source":{"id":"2109.04586","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-09-09T23:20:15Z","cross_cats_sorted":[],"title_canon_sha256":"1dc1cf3b456863b8872369f74e60d7fcc9c5b3e559f95f8c4dd644733998b87a","abstract_canon_sha256":"fc7a809e0badb681788f5a1e147b4b2441afa6e4514202c5fe388ef9d9c05b89"},"schema_version":"1.0"},"canonical_sha256":"c4aef162c45067872a3ef42ac699c8b9df0359050f8ffdaf02f36b4eaa539863","source":{"kind":"arxiv","id":"2109.04586","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.04586","created_at":"2026-06-23T02:13:11Z"},{"alias_kind":"arxiv_version","alias_value":"2109.04586v1","created_at":"2026-06-23T02:13:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.04586","created_at":"2026-06-23T02:13:11Z"},{"alias_kind":"pith_short_12","alias_value":"YSXPCYWEKBTY","created_at":"2026-06-23T02:13:11Z"},{"alias_kind":"pith_short_16","alias_value":"YSXPCYWEKBTYOKR6","created_at":"2026-06-23T02:13:11Z"},{"alias_kind":"pith_short_8","alias_value":"YSXPCYWE","created_at":"2026-06-23T02:13:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:YSXPCYWEKBTYOKR66QVMNGOIXH","target":"record","payload":{"canonical_record":{"source":{"id":"2109.04586","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-09-09T23:20:15Z","cross_cats_sorted":[],"title_canon_sha256":"1dc1cf3b456863b8872369f74e60d7fcc9c5b3e559f95f8c4dd644733998b87a","abstract_canon_sha256":"fc7a809e0badb681788f5a1e147b4b2441afa6e4514202c5fe388ef9d9c05b89"},"schema_version":"1.0"},"canonical_sha256":"c4aef162c45067872a3ef42ac699c8b9df0359050f8ffdaf02f36b4eaa539863","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T02:13:11.475915Z","signature_b64":"kPI0GrhXxzkNfnZYTXio0lq71bHL6NLCkDBs3ooi8LBz4EJEO/kazGdxqai0NX1XcYqNScNg0dYMrPLVyZU1DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c4aef162c45067872a3ef42ac699c8b9df0359050f8ffdaf02f36b4eaa539863","last_reissued_at":"2026-06-23T02:13:11.475527Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T02:13:11.475527Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2109.04586","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":"2026-06-23T02:13:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aFTaKkqLDlYKia0EkgLKE/A/d9caG0N4uqqUgtYlq5nsa0TqqYl+XeL2OE51H1QuvOcPhHY6uon3HSZrX/xYBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T00:27:27.883327Z"},"content_sha256":"a8b4301463d7b3149ede6861f656d04f9c30f9146cd59c3236a670bc0b4cf23f","schema_version":"1.0","event_id":"sha256:a8b4301463d7b3149ede6861f656d04f9c30f9146cd59c3236a670bc0b4cf23f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:YSXPCYWEKBTYOKR66QVMNGOIXH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The critical point and the $p$-norm of $A_s$ and $C$-matrices","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Javad Mashreghi, Ludovick Bouthat","submitted_at":"2021-09-09T23:20:15Z","abstract_excerpt":"The $L$-matrix $A_s=[1/(n+s)]$ was introduced in \\cite{MRtmp}. As a surprising property, we showed that its 2-norm is constant for $s \\geq s_0$, where the critical point $s_0$ is unknown but relies in the interval $(1/4,1/2)$. In this note, using some delicate calculations we sharpen this result by improving the upper and lower bounds of the interval surrounding $s_0$. Moreover, we show that the same property persists for the $p$-norm of $A_s$ matrices. We also obtain the 2-norm of a family of $C$-matrices with lacunary sequences."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.04586","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.04586/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-06-23T02:13:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oy+roDlMyMsyvHHhfl22Hk5JwhOILnLdK8poqwk8RVUj5siNu723lAmzoRKNhlpIT5GT7eUp7/KBhInkqQ/1DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T00:27:27.883749Z"},"content_sha256":"7d83fb2d6190e9ddaf77f64437602893bd0745c55affe6c9b8bd6e1d09ed6930","schema_version":"1.0","event_id":"sha256:7d83fb2d6190e9ddaf77f64437602893bd0745c55affe6c9b8bd6e1d09ed6930"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/YSXPCYWEKBTYOKR66QVMNGOIXH/bundle.json","state_url":"https://pith.science/pith/YSXPCYWEKBTYOKR66QVMNGOIXH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/YSXPCYWEKBTYOKR66QVMNGOIXH/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":"2026-06-29T00:27:27Z","links":{"resolver":"https://pith.science/pith/YSXPCYWEKBTYOKR66QVMNGOIXH","bundle":"https://pith.science/pith/YSXPCYWEKBTYOKR66QVMNGOIXH/bundle.json","state":"https://pith.science/pith/YSXPCYWEKBTYOKR66QVMNGOIXH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/YSXPCYWEKBTYOKR66QVMNGOIXH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:YSXPCYWEKBTYOKR66QVMNGOIXH","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":"fc7a809e0badb681788f5a1e147b4b2441afa6e4514202c5fe388ef9d9c05b89","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-09-09T23:20:15Z","title_canon_sha256":"1dc1cf3b456863b8872369f74e60d7fcc9c5b3e559f95f8c4dd644733998b87a"},"schema_version":"1.0","source":{"id":"2109.04586","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2109.04586","created_at":"2026-06-23T02:13:11Z"},{"alias_kind":"arxiv_version","alias_value":"2109.04586v1","created_at":"2026-06-23T02:13:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2109.04586","created_at":"2026-06-23T02:13:11Z"},{"alias_kind":"pith_short_12","alias_value":"YSXPCYWEKBTY","created_at":"2026-06-23T02:13:11Z"},{"alias_kind":"pith_short_16","alias_value":"YSXPCYWEKBTYOKR6","created_at":"2026-06-23T02:13:11Z"},{"alias_kind":"pith_short_8","alias_value":"YSXPCYWE","created_at":"2026-06-23T02:13:11Z"}],"graph_snapshots":[{"event_id":"sha256:7d83fb2d6190e9ddaf77f64437602893bd0745c55affe6c9b8bd6e1d09ed6930","target":"graph","created_at":"2026-06-23T02:13:11Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2109.04586/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The $L$-matrix $A_s=[1/(n+s)]$ was introduced in \\cite{MRtmp}. As a surprising property, we showed that its 2-norm is constant for $s \\geq s_0$, where the critical point $s_0$ is unknown but relies in the interval $(1/4,1/2)$. In this note, using some delicate calculations we sharpen this result by improving the upper and lower bounds of the interval surrounding $s_0$. Moreover, we show that the same property persists for the $p$-norm of $A_s$ matrices. We also obtain the 2-norm of a family of $C$-matrices with lacunary sequences.","authors_text":"Javad Mashreghi, Ludovick Bouthat","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-09-09T23:20:15Z","title":"The critical point and the $p$-norm of $A_s$ and $C$-matrices"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.04586","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:a8b4301463d7b3149ede6861f656d04f9c30f9146cd59c3236a670bc0b4cf23f","target":"record","created_at":"2026-06-23T02:13:11Z","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":"fc7a809e0badb681788f5a1e147b4b2441afa6e4514202c5fe388ef9d9c05b89","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2021-09-09T23:20:15Z","title_canon_sha256":"1dc1cf3b456863b8872369f74e60d7fcc9c5b3e559f95f8c4dd644733998b87a"},"schema_version":"1.0","source":{"id":"2109.04586","kind":"arxiv","version":1}},"canonical_sha256":"c4aef162c45067872a3ef42ac699c8b9df0359050f8ffdaf02f36b4eaa539863","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c4aef162c45067872a3ef42ac699c8b9df0359050f8ffdaf02f36b4eaa539863","first_computed_at":"2026-06-23T02:13:11.475527Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-23T02:13:11.475527Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kPI0GrhXxzkNfnZYTXio0lq71bHL6NLCkDBs3ooi8LBz4EJEO/kazGdxqai0NX1XcYqNScNg0dYMrPLVyZU1DQ==","signature_status":"signed_v1","signed_at":"2026-06-23T02:13:11.475915Z","signed_message":"canonical_sha256_bytes"},"source_id":"2109.04586","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a8b4301463d7b3149ede6861f656d04f9c30f9146cd59c3236a670bc0b4cf23f","sha256:7d83fb2d6190e9ddaf77f64437602893bd0745c55affe6c9b8bd6e1d09ed6930"],"state_sha256":"9bb9b19d746200029b5ba03d494e60950d7a5ffede327bbcf2f133ab3b0b48df"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jN3OndZUIlt7/B0EsvrdSi8iczXiCqhB1gOThysl7RcFaY/7x5qigai+ISZDILFFKfG48zaDYFvtXAxUJEi7Bg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T00:27:27.885663Z","bundle_sha256":"6fa59cac783a3e42ba43e56408a05c52fcb10ede4882ecc5008106b0fe0455ff"}}