{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:K5M6BJYV3VCKU6ILLJGODCXS3A","short_pith_number":"pith:K5M6BJYV","canonical_record":{"source":{"id":"1810.07995","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-18T11:36:07Z","cross_cats_sorted":[],"title_canon_sha256":"0f625302184ea2e72e581a69d7b0d70de688c2c0ec830a7a98cb9c60d5807974","abstract_canon_sha256":"a315329dd0658402021d10beb5f59d30853c57de0786b92506e736e286f2387c"},"schema_version":"1.0"},"canonical_sha256":"5759e0a715dd44aa790b5a4ce18af2d82863cdbbd86bb2ff4eddba5dc25842dc","source":{"kind":"arxiv","id":"1810.07995","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.07995","created_at":"2026-05-18T00:02:52Z"},{"alias_kind":"arxiv_version","alias_value":"1810.07995v1","created_at":"2026-05-18T00:02:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.07995","created_at":"2026-05-18T00:02:52Z"},{"alias_kind":"pith_short_12","alias_value":"K5M6BJYV3VCK","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_16","alias_value":"K5M6BJYV3VCKU6IL","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_8","alias_value":"K5M6BJYV","created_at":"2026-05-18T12:32:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:K5M6BJYV3VCKU6ILLJGODCXS3A","target":"record","payload":{"canonical_record":{"source":{"id":"1810.07995","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-18T11:36:07Z","cross_cats_sorted":[],"title_canon_sha256":"0f625302184ea2e72e581a69d7b0d70de688c2c0ec830a7a98cb9c60d5807974","abstract_canon_sha256":"a315329dd0658402021d10beb5f59d30853c57de0786b92506e736e286f2387c"},"schema_version":"1.0"},"canonical_sha256":"5759e0a715dd44aa790b5a4ce18af2d82863cdbbd86bb2ff4eddba5dc25842dc","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:52.866929Z","signature_b64":"spTbRuSwNhZiDU4qjsZEIXnCxiP8CRDtuymEzt6/EWC1nd6gGrojo/QdmY60xulf6s3pD1cXYvlKdIQzyRiABw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5759e0a715dd44aa790b5a4ce18af2d82863cdbbd86bb2ff4eddba5dc25842dc","last_reissued_at":"2026-05-18T00:02:52.866244Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:52.866244Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1810.07995","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":"2026-05-18T00:02:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kvlauKVRHWbvUqDFskj53jNuG7/8jbZtpQ3a184EqBeuWxAXh5k7hxOO8R+5/s7hK9q8pDDRHTjSE1GS6z3dAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T08:41:57.210864Z"},"content_sha256":"59ab43d49393491b979964d5aa92c44c8a9413f98cb833af3f0affdf3c9058dc","schema_version":"1.0","event_id":"sha256:59ab43d49393491b979964d5aa92c44c8a9413f98cb833af3f0affdf3c9058dc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:K5M6BJYV3VCKU6ILLJGODCXS3A","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Double phase problems with variable growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Du\\v{s}an D. Repov\\v{s}, Matija Cencelj, Vicen\\c{t}iu D. R\\u{a}dulescu","submitted_at":"2018-10-18T11:36:07Z","abstract_excerpt":"We consider a class of double phase variational integrals driven by nonhomogeneous potentials. We study the associated Euler equation and we highlight the existence of two different Rayleigh quotients. One of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues. The notion of eigenvalue is understood in the sense of pairs of nonlinear operators, as introduced by Fu\\v{c}ik, Ne\\v{c}as, Sou\\v{c}ek, and Sou\\v{c}ek. The analysis developed in this paper extends the abstract framework corresponding to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07995","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":"2026-05-18T00:02:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DQhCaDX6hwYF8f+QpNaXYq1a9i1b1I/kTuOD+V+W6jc/mTDbdb8SdjE98DXmhVKpVQMWd0gs8RT2mlHZhKKpAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T08:41:57.211219Z"},"content_sha256":"9d3fba8591d069ce45e71d179a2a625405ce5772463983de66cde53776f7d6f6","schema_version":"1.0","event_id":"sha256:9d3fba8591d069ce45e71d179a2a625405ce5772463983de66cde53776f7d6f6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K5M6BJYV3VCKU6ILLJGODCXS3A/bundle.json","state_url":"https://pith.science/pith/K5M6BJYV3VCKU6ILLJGODCXS3A/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K5M6BJYV3VCKU6ILLJGODCXS3A/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":"2026-05-31T08:41:57Z","links":{"resolver":"https://pith.science/pith/K5M6BJYV3VCKU6ILLJGODCXS3A","bundle":"https://pith.science/pith/K5M6BJYV3VCKU6ILLJGODCXS3A/bundle.json","state":"https://pith.science/pith/K5M6BJYV3VCKU6ILLJGODCXS3A/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K5M6BJYV3VCKU6ILLJGODCXS3A/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:K5M6BJYV3VCKU6ILLJGODCXS3A","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":"a315329dd0658402021d10beb5f59d30853c57de0786b92506e736e286f2387c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-18T11:36:07Z","title_canon_sha256":"0f625302184ea2e72e581a69d7b0d70de688c2c0ec830a7a98cb9c60d5807974"},"schema_version":"1.0","source":{"id":"1810.07995","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.07995","created_at":"2026-05-18T00:02:52Z"},{"alias_kind":"arxiv_version","alias_value":"1810.07995v1","created_at":"2026-05-18T00:02:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.07995","created_at":"2026-05-18T00:02:52Z"},{"alias_kind":"pith_short_12","alias_value":"K5M6BJYV3VCK","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_16","alias_value":"K5M6BJYV3VCKU6IL","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_8","alias_value":"K5M6BJYV","created_at":"2026-05-18T12:32:33Z"}],"graph_snapshots":[{"event_id":"sha256:9d3fba8591d069ce45e71d179a2a625405ce5772463983de66cde53776f7d6f6","target":"graph","created_at":"2026-05-18T00:02:52Z","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":"We consider a class of double phase variational integrals driven by nonhomogeneous potentials. We study the associated Euler equation and we highlight the existence of two different Rayleigh quotients. One of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues. The notion of eigenvalue is understood in the sense of pairs of nonlinear operators, as introduced by Fu\\v{c}ik, Ne\\v{c}as, Sou\\v{c}ek, and Sou\\v{c}ek. The analysis developed in this paper extends the abstract framework corresponding to","authors_text":"Du\\v{s}an D. Repov\\v{s}, Matija Cencelj, Vicen\\c{t}iu D. R\\u{a}dulescu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-18T11:36:07Z","title":"Double phase problems with variable growth"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07995","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":"sha256:59ab43d49393491b979964d5aa92c44c8a9413f98cb833af3f0affdf3c9058dc","target":"record","created_at":"2026-05-18T00:02:52Z","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":"a315329dd0658402021d10beb5f59d30853c57de0786b92506e736e286f2387c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-10-18T11:36:07Z","title_canon_sha256":"0f625302184ea2e72e581a69d7b0d70de688c2c0ec830a7a98cb9c60d5807974"},"schema_version":"1.0","source":{"id":"1810.07995","kind":"arxiv","version":1}},"canonical_sha256":"5759e0a715dd44aa790b5a4ce18af2d82863cdbbd86bb2ff4eddba5dc25842dc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5759e0a715dd44aa790b5a4ce18af2d82863cdbbd86bb2ff4eddba5dc25842dc","first_computed_at":"2026-05-18T00:02:52.866244Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:02:52.866244Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"spTbRuSwNhZiDU4qjsZEIXnCxiP8CRDtuymEzt6/EWC1nd6gGrojo/QdmY60xulf6s3pD1cXYvlKdIQzyRiABw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:02:52.866929Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.07995","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:59ab43d49393491b979964d5aa92c44c8a9413f98cb833af3f0affdf3c9058dc","sha256:9d3fba8591d069ce45e71d179a2a625405ce5772463983de66cde53776f7d6f6"],"state_sha256":"201602c3c84a1df8d0d657633d52479e4304ce35fd98ab8bb1cf222338fe94a9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BNNl2zopgtkLAJkWkT09jW3xpTdv+aScuW7ZKWnAS7h3CBt4oK5zpE06SqzTWtdppN59veHtJeKp4jqU8q3QAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T08:41:57.213386Z","bundle_sha256":"b61af6bc617874ece958930865e691270c01894fd41a17d0932b0c5b5c886349"}}