{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:T6XO5OAZRZE3AUGIA6SLZOYUZ7","short_pith_number":"pith:T6XO5OAZ","canonical_record":{"source":{"id":"1405.3619","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-05-14T18:52:22Z","cross_cats_sorted":[],"title_canon_sha256":"5806f5f1538b44b42c2b38a56e20d0b39e7031483e18b1e371d96d6c93543320","abstract_canon_sha256":"fb23756031c53d8412388a891007c96c89e08b2dfef1a20af49e00cf2d8a2630"},"schema_version":"1.0"},"canonical_sha256":"9faeeeb8198e49b050c807a4bcbb14cfd6b74e7aa057b9178ccbe30ae6ec594f","source":{"kind":"arxiv","id":"1405.3619","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.3619","created_at":"2026-05-18T02:51:50Z"},{"alias_kind":"arxiv_version","alias_value":"1405.3619v1","created_at":"2026-05-18T02:51:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.3619","created_at":"2026-05-18T02:51:50Z"},{"alias_kind":"pith_short_12","alias_value":"T6XO5OAZRZE3","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"T6XO5OAZRZE3AUGI","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"T6XO5OAZ","created_at":"2026-05-18T12:28:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:T6XO5OAZRZE3AUGIA6SLZOYUZ7","target":"record","payload":{"canonical_record":{"source":{"id":"1405.3619","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-05-14T18:52:22Z","cross_cats_sorted":[],"title_canon_sha256":"5806f5f1538b44b42c2b38a56e20d0b39e7031483e18b1e371d96d6c93543320","abstract_canon_sha256":"fb23756031c53d8412388a891007c96c89e08b2dfef1a20af49e00cf2d8a2630"},"schema_version":"1.0"},"canonical_sha256":"9faeeeb8198e49b050c807a4bcbb14cfd6b74e7aa057b9178ccbe30ae6ec594f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:51:50.435399Z","signature_b64":"g6IHuFZAzB7gbA7RSyUCfYjor212m6TAa9or1DI6rWlT0fIQtXvM/oFrdUFGXhz3gUvjzsf5Qu28OUJOcSRDBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9faeeeb8198e49b050c807a4bcbb14cfd6b74e7aa057b9178ccbe30ae6ec594f","last_reissued_at":"2026-05-18T02:51:50.434877Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:51:50.434877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1405.3619","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":"2026-05-18T02:51:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ehFuZd9WVtFgLdR66ER/8a3RFRaNFYYwA7OzzjD3zTBn1N3d3+oLnmHU06DExs6z3vzVLPvhrDx2RckWRkxBAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T10:39:22.068454Z"},"content_sha256":"6c9e08488254c8f21a8af9ee3f04e7a6c302c97a57101ba7028cfa7ac815461e","schema_version":"1.0","event_id":"sha256:6c9e08488254c8f21a8af9ee3f04e7a6c302c97a57101ba7028cfa7ac815461e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:T6XO5OAZRZE3AUGIA6SLZOYUZ7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lacunary ideal convergence in probabilistic normed spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ayhan Esi, Bipan Hazarika","submitted_at":"2014-05-14T18:52:22Z","abstract_excerpt":"An ideal $I$ is a family of subsets of positive integers $\\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_k)$ of real numbers is said to be lacunary $I$-convergent to a real number $\\ell$, if for each $ \\varepsilon> 0$ the set $$\\left\\{r\\in \\mathbb{N}:\\frac{1}{h_r}\\sum_{k\\in J_r} |x_{k}-\\ell|\\geq \\varepsilon\\right\\}$$ belongs to $I.$ The aim of this paper is to study the notion of lacunary $I$-convergence in probabilistic normed spaces as a variant of the notion of ideal convergence. Also lacunary $I$-limit points and lacunary $I$-cluster poin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.3619","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":"2026-05-18T02:51:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rso78cL/1jKBn6nvsE1mwqS1H12jz5pk/aGNRkCjHtGCupJQTspZWpDgB3a3rxgrq2ERJomlXrJ5Tn3mt1MdDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T10:39:22.068811Z"},"content_sha256":"02a483ccea406113e3bb358ca5145c3999c11325faf9a3d8563e5ee48f17f420","schema_version":"1.0","event_id":"sha256:02a483ccea406113e3bb358ca5145c3999c11325faf9a3d8563e5ee48f17f420"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/T6XO5OAZRZE3AUGIA6SLZOYUZ7/bundle.json","state_url":"https://pith.science/pith/T6XO5OAZRZE3AUGIA6SLZOYUZ7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/T6XO5OAZRZE3AUGIA6SLZOYUZ7/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":"2026-06-03T10:39:22Z","links":{"resolver":"https://pith.science/pith/T6XO5OAZRZE3AUGIA6SLZOYUZ7","bundle":"https://pith.science/pith/T6XO5OAZRZE3AUGIA6SLZOYUZ7/bundle.json","state":"https://pith.science/pith/T6XO5OAZRZE3AUGIA6SLZOYUZ7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/T6XO5OAZRZE3AUGIA6SLZOYUZ7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:T6XO5OAZRZE3AUGIA6SLZOYUZ7","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":"fb23756031c53d8412388a891007c96c89e08b2dfef1a20af49e00cf2d8a2630","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-05-14T18:52:22Z","title_canon_sha256":"5806f5f1538b44b42c2b38a56e20d0b39e7031483e18b1e371d96d6c93543320"},"schema_version":"1.0","source":{"id":"1405.3619","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1405.3619","created_at":"2026-05-18T02:51:50Z"},{"alias_kind":"arxiv_version","alias_value":"1405.3619v1","created_at":"2026-05-18T02:51:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.3619","created_at":"2026-05-18T02:51:50Z"},{"alias_kind":"pith_short_12","alias_value":"T6XO5OAZRZE3","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"T6XO5OAZRZE3AUGI","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"T6XO5OAZ","created_at":"2026-05-18T12:28:49Z"}],"graph_snapshots":[{"event_id":"sha256:02a483ccea406113e3bb358ca5145c3999c11325faf9a3d8563e5ee48f17f420","target":"graph","created_at":"2026-05-18T02:51:50Z","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":"An ideal $I$ is a family of subsets of positive integers $\\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_k)$ of real numbers is said to be lacunary $I$-convergent to a real number $\\ell$, if for each $ \\varepsilon> 0$ the set $$\\left\\{r\\in \\mathbb{N}:\\frac{1}{h_r}\\sum_{k\\in J_r} |x_{k}-\\ell|\\geq \\varepsilon\\right\\}$$ belongs to $I.$ The aim of this paper is to study the notion of lacunary $I$-convergence in probabilistic normed spaces as a variant of the notion of ideal convergence. Also lacunary $I$-limit points and lacunary $I$-cluster poin","authors_text":"Ayhan Esi, Bipan Hazarika","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-05-14T18:52:22Z","title":"Lacunary ideal convergence in probabilistic normed spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.3619","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":"sha256:6c9e08488254c8f21a8af9ee3f04e7a6c302c97a57101ba7028cfa7ac815461e","target":"record","created_at":"2026-05-18T02:51:50Z","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":"fb23756031c53d8412388a891007c96c89e08b2dfef1a20af49e00cf2d8a2630","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2014-05-14T18:52:22Z","title_canon_sha256":"5806f5f1538b44b42c2b38a56e20d0b39e7031483e18b1e371d96d6c93543320"},"schema_version":"1.0","source":{"id":"1405.3619","kind":"arxiv","version":1}},"canonical_sha256":"9faeeeb8198e49b050c807a4bcbb14cfd6b74e7aa057b9178ccbe30ae6ec594f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9faeeeb8198e49b050c807a4bcbb14cfd6b74e7aa057b9178ccbe30ae6ec594f","first_computed_at":"2026-05-18T02:51:50.434877Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:51:50.434877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g6IHuFZAzB7gbA7RSyUCfYjor212m6TAa9or1DI6rWlT0fIQtXvM/oFrdUFGXhz3gUvjzsf5Qu28OUJOcSRDBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:51:50.435399Z","signed_message":"canonical_sha256_bytes"},"source_id":"1405.3619","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6c9e08488254c8f21a8af9ee3f04e7a6c302c97a57101ba7028cfa7ac815461e","sha256:02a483ccea406113e3bb358ca5145c3999c11325faf9a3d8563e5ee48f17f420"],"state_sha256":"088e56fbad60b23048aff029a5766ce5f67995cf9ffb501a2723ac36b4858772"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LKSlNMK7aMEFuFcYEaz3de2KaYj8Hbutb+6X2IE7L0O3WSxEYndNmHyjjZm3bByxYluSVBajgrQ/4m/ct0LNBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T10:39:22.070741Z","bundle_sha256":"7387d4d4ef7619595595d7dddca679852201fe9cbbdb1c9216dcda5c559277c9"}}