{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:6Q7MQDPDPKFVZMQ3WFU2EAV22M","short_pith_number":"pith:6Q7MQDPD","canonical_record":{"source":{"id":"1211.3285","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-14T12:06:25Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"80c81d315a704aa7f7c48bbb75e36add089cacb06ab7912aa9e2a29195c4ba4b","abstract_canon_sha256":"c3a798ffc6e995c4efa5c5dd79c9b7ab5be92464af56ace89fa4187036113be1"},"schema_version":"1.0"},"canonical_sha256":"f43ec80de37a8b5cb21bb169a202bad32ece40c843087c4fcacc0aa436d96349","source":{"kind":"arxiv","id":"1211.3285","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.3285","created_at":"2026-05-18T02:50:13Z"},{"alias_kind":"arxiv_version","alias_value":"1211.3285v2","created_at":"2026-05-18T02:50:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.3285","created_at":"2026-05-18T02:50:13Z"},{"alias_kind":"pith_short_12","alias_value":"6Q7MQDPDPKFV","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"6Q7MQDPDPKFVZMQ3","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"6Q7MQDPD","created_at":"2026-05-18T12:26:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:6Q7MQDPDPKFVZMQ3WFU2EAV22M","target":"record","payload":{"canonical_record":{"source":{"id":"1211.3285","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-14T12:06:25Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"80c81d315a704aa7f7c48bbb75e36add089cacb06ab7912aa9e2a29195c4ba4b","abstract_canon_sha256":"c3a798ffc6e995c4efa5c5dd79c9b7ab5be92464af56ace89fa4187036113be1"},"schema_version":"1.0"},"canonical_sha256":"f43ec80de37a8b5cb21bb169a202bad32ece40c843087c4fcacc0aa436d96349","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:13.619304Z","signature_b64":"yDUZTWdUUXtKeY1d0LZ0XApuvRbWKt9BV4lYmQAmJD9n6SfaNnz6ZKIKBNXcP1jYN8oUISC0TpZFmy413nnDBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f43ec80de37a8b5cb21bb169a202bad32ece40c843087c4fcacc0aa436d96349","last_reissued_at":"2026-05-18T02:50:13.618801Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:13.618801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.3285","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":"2026-05-18T02:50:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dijLXAsS7gSSc9vDk2KIjK5pGTHrFw+3U21vBabsPi3qqIFPXBuCBp7EtYlDwV3Pg584j/JdqWRN4/1/AkUtDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T18:38:40.594972Z"},"content_sha256":"added0f18ff2f9b60dcec9cf84ae17be31add4aadc0dbc362c07903424a78080","schema_version":"1.0","event_id":"sha256:added0f18ff2f9b60dcec9cf84ae17be31add4aadc0dbc362c07903424a78080"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:6Q7MQDPDPKFVZMQ3WFU2EAV22M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Cram\\'er transform and t-entropy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.PR","authors_text":"Krzysztof Zajkowski, Urszula Ostaszewska","submitted_at":"2012-11-14T12:06:25Z","abstract_excerpt":"t-entropy is the convex conjugate of the logarithm of the spectral radius of a weighted composition operator (WCO). Let $X$ be a nonnegative random variable. We show how the Cram\\'er transform with respect to the spectral radius of WCO is expressed by the t-entropy and the Cram\\'er transform of the given random variable X."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3285","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":"2026-05-18T02:50:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6rrv07in6TkxbEDoZBwYk/4kb0LM0l1oTLXlhIm8i3jl/9YWeJ4ZlEdq4AYWRIduKTJ5qi0Z52vdsMGoATsHCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T18:38:40.595345Z"},"content_sha256":"caae4a905a2bffcb8995ccbd0deb17ff347ab7bf11d423158ff3737a8e9cb53e","schema_version":"1.0","event_id":"sha256:caae4a905a2bffcb8995ccbd0deb17ff347ab7bf11d423158ff3737a8e9cb53e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6Q7MQDPDPKFVZMQ3WFU2EAV22M/bundle.json","state_url":"https://pith.science/pith/6Q7MQDPDPKFVZMQ3WFU2EAV22M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6Q7MQDPDPKFVZMQ3WFU2EAV22M/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":"2026-06-27T18:38:40Z","links":{"resolver":"https://pith.science/pith/6Q7MQDPDPKFVZMQ3WFU2EAV22M","bundle":"https://pith.science/pith/6Q7MQDPDPKFVZMQ3WFU2EAV22M/bundle.json","state":"https://pith.science/pith/6Q7MQDPDPKFVZMQ3WFU2EAV22M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6Q7MQDPDPKFVZMQ3WFU2EAV22M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:6Q7MQDPDPKFVZMQ3WFU2EAV22M","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":"c3a798ffc6e995c4efa5c5dd79c9b7ab5be92464af56ace89fa4187036113be1","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-14T12:06:25Z","title_canon_sha256":"80c81d315a704aa7f7c48bbb75e36add089cacb06ab7912aa9e2a29195c4ba4b"},"schema_version":"1.0","source":{"id":"1211.3285","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.3285","created_at":"2026-05-18T02:50:13Z"},{"alias_kind":"arxiv_version","alias_value":"1211.3285v2","created_at":"2026-05-18T02:50:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.3285","created_at":"2026-05-18T02:50:13Z"},{"alias_kind":"pith_short_12","alias_value":"6Q7MQDPDPKFV","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_16","alias_value":"6Q7MQDPDPKFVZMQ3","created_at":"2026-05-18T12:26:56Z"},{"alias_kind":"pith_short_8","alias_value":"6Q7MQDPD","created_at":"2026-05-18T12:26:56Z"}],"graph_snapshots":[{"event_id":"sha256:caae4a905a2bffcb8995ccbd0deb17ff347ab7bf11d423158ff3737a8e9cb53e","target":"graph","created_at":"2026-05-18T02:50:13Z","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":"t-entropy is the convex conjugate of the logarithm of the spectral radius of a weighted composition operator (WCO). Let $X$ be a nonnegative random variable. We show how the Cram\\'er transform with respect to the spectral radius of WCO is expressed by the t-entropy and the Cram\\'er transform of the given random variable X.","authors_text":"Krzysztof Zajkowski, Urszula Ostaszewska","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-14T12:06:25Z","title":"Cram\\'er transform and t-entropy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3285","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:added0f18ff2f9b60dcec9cf84ae17be31add4aadc0dbc362c07903424a78080","target":"record","created_at":"2026-05-18T02:50:13Z","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":"c3a798ffc6e995c4efa5c5dd79c9b7ab5be92464af56ace89fa4187036113be1","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-11-14T12:06:25Z","title_canon_sha256":"80c81d315a704aa7f7c48bbb75e36add089cacb06ab7912aa9e2a29195c4ba4b"},"schema_version":"1.0","source":{"id":"1211.3285","kind":"arxiv","version":2}},"canonical_sha256":"f43ec80de37a8b5cb21bb169a202bad32ece40c843087c4fcacc0aa436d96349","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f43ec80de37a8b5cb21bb169a202bad32ece40c843087c4fcacc0aa436d96349","first_computed_at":"2026-05-18T02:50:13.618801Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:50:13.618801Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yDUZTWdUUXtKeY1d0LZ0XApuvRbWKt9BV4lYmQAmJD9n6SfaNnz6ZKIKBNXcP1jYN8oUISC0TpZFmy413nnDBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:50:13.619304Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.3285","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:added0f18ff2f9b60dcec9cf84ae17be31add4aadc0dbc362c07903424a78080","sha256:caae4a905a2bffcb8995ccbd0deb17ff347ab7bf11d423158ff3737a8e9cb53e"],"state_sha256":"7e0284d8922ab7ac5391c6d2c9ca52c1b9ed38dd103c104854880160497a5271"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zlRzlgGg88kCWhYOBVvxpwTkbSGoSKvRXT1AwUFEb56a/NM3kPqs7TXWN4JOzunOxtfir4pxHbfjR44EXUmTDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T18:38:40.597212Z","bundle_sha256":"d917bfeb93e55827b8c2bb62440f6cde377769089f3053404ec52110bc4d1be5"}}