{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:XFX2RLR5PQTX75IC2EOTD2SSCS","short_pith_number":"pith:XFX2RLR5","canonical_record":{"source":{"id":"1409.3346","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-11T08:07:07Z","cross_cats_sorted":[],"title_canon_sha256":"22a169a7c99db73ab50d74bd87c855c9abb150b15bbfa242d7d6c7ac36d254a0","abstract_canon_sha256":"95c9fd5db44de6529545ce7d86435dcbbba0f828a9c1c10484889e40bdf06773"},"schema_version":"1.0"},"canonical_sha256":"b96fa8ae3d7c277ff502d11d31ea52148b61704e3eb1744de638b8cebdc4bee7","source":{"kind":"arxiv","id":"1409.3346","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.3346","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"arxiv_version","alias_value":"1409.3346v1","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.3346","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"pith_short_12","alias_value":"XFX2RLR5PQTX","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"XFX2RLR5PQTX75IC","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"XFX2RLR5","created_at":"2026-05-18T12:28:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:XFX2RLR5PQTX75IC2EOTD2SSCS","target":"record","payload":{"canonical_record":{"source":{"id":"1409.3346","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-11T08:07:07Z","cross_cats_sorted":[],"title_canon_sha256":"22a169a7c99db73ab50d74bd87c855c9abb150b15bbfa242d7d6c7ac36d254a0","abstract_canon_sha256":"95c9fd5db44de6529545ce7d86435dcbbba0f828a9c1c10484889e40bdf06773"},"schema_version":"1.0"},"canonical_sha256":"b96fa8ae3d7c277ff502d11d31ea52148b61704e3eb1744de638b8cebdc4bee7","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:43:01.582655Z","signature_b64":"kp9Jh0YsMYfRjP3C3SHmWTS/Ri6CXQWpywt+r+1MkJZfKEZtTnYeu4fZyQpEhGCIUnCLjSvWhMN0ClOXPbJcDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b96fa8ae3d7c277ff502d11d31ea52148b61704e3eb1744de638b8cebdc4bee7","last_reissued_at":"2026-05-18T02:43:01.582202Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:43:01.582202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.3346","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":"2026-05-18T02:43:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+LpjCfAWYVQnHkBcEjBsTrptAzrI28AlSydLu9D/m2noHOQEASHeU0GMAdpjuc8/I71JYTBoTSPNJyeV6QL+DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T05:47:58.709268Z"},"content_sha256":"c1b4fa09745b7b1d67138db2ebf3d7ccb54eace88948143267fd05d1b6291c9e","schema_version":"1.0","event_id":"sha256:c1b4fa09745b7b1d67138db2ebf3d7ccb54eace88948143267fd05d1b6291c9e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:XFX2RLR5PQTX75IC2EOTD2SSCS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Criticality theory of half-linear equations with the (p,A)-Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Netanel Regev, Yehuda Pinchover","submitted_at":"2014-09-11T08:07:07Z","abstract_excerpt":"We study positive solutions of half-linear second-order elliptic equations of the form $$Q_{A,V}(u):= -\\mathrm{div} (|\\nabla u|_{A}^{p-2}A(x)\\nabla u)+ V(x)|u|^{p-2}u=0 \\quad \\mbox{in }\\Omega,$$ where $1<p<\\infty$, $\\Omega$ is a domain in $\\mathbb{R}^{n}$, $n\\geq 2$, $V\\in L_{\\mathrm{loc}}^{\\infty}(\\Omega)$, $A=\\big(a_{ij}\\big)\\in L_{\\mathrm{loc}}^{\\infty}(\\Omega,\\mathbb{R}^{n^2})$ is a symmetric and locally uniformly positive definite matrix in $\\Omega$, and $$|\\xi|_{A}^{2}:=\\left\\langle A(x)\\xi,\\xi\\right\\rangle=\\sum_{i,j=1}^n a_{ij}(x)\\xi_i\\xi_j \\qquad x\\in \\Omega, \\xi=(\\xi_1,\\ldots,\\xi_n)\\i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.3346","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":"2026-05-18T02:43:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rNiDT07gSm6TfPgbAKmm00+nWOCMsGh4hwIPb6qIMr4s7AdUwNwOcdEGumaS8Glz8IpZfyNBPuWKA6ABRoxIDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T05:47:58.709632Z"},"content_sha256":"3a0d7bee4a82051154b7792110fee5a315d4f8b042759c16b25ba0d69dc2831e","schema_version":"1.0","event_id":"sha256:3a0d7bee4a82051154b7792110fee5a315d4f8b042759c16b25ba0d69dc2831e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XFX2RLR5PQTX75IC2EOTD2SSCS/bundle.json","state_url":"https://pith.science/pith/XFX2RLR5PQTX75IC2EOTD2SSCS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XFX2RLR5PQTX75IC2EOTD2SSCS/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":"2026-07-02T05:47:58Z","links":{"resolver":"https://pith.science/pith/XFX2RLR5PQTX75IC2EOTD2SSCS","bundle":"https://pith.science/pith/XFX2RLR5PQTX75IC2EOTD2SSCS/bundle.json","state":"https://pith.science/pith/XFX2RLR5PQTX75IC2EOTD2SSCS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XFX2RLR5PQTX75IC2EOTD2SSCS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:XFX2RLR5PQTX75IC2EOTD2SSCS","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":"95c9fd5db44de6529545ce7d86435dcbbba0f828a9c1c10484889e40bdf06773","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-11T08:07:07Z","title_canon_sha256":"22a169a7c99db73ab50d74bd87c855c9abb150b15bbfa242d7d6c7ac36d254a0"},"schema_version":"1.0","source":{"id":"1409.3346","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.3346","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"arxiv_version","alias_value":"1409.3346v1","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.3346","created_at":"2026-05-18T02:43:01Z"},{"alias_kind":"pith_short_12","alias_value":"XFX2RLR5PQTX","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_16","alias_value":"XFX2RLR5PQTX75IC","created_at":"2026-05-18T12:28:57Z"},{"alias_kind":"pith_short_8","alias_value":"XFX2RLR5","created_at":"2026-05-18T12:28:57Z"}],"graph_snapshots":[{"event_id":"sha256:3a0d7bee4a82051154b7792110fee5a315d4f8b042759c16b25ba0d69dc2831e","target":"graph","created_at":"2026-05-18T02:43:01Z","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 study positive solutions of half-linear second-order elliptic equations of the form $$Q_{A,V}(u):= -\\mathrm{div} (|\\nabla u|_{A}^{p-2}A(x)\\nabla u)+ V(x)|u|^{p-2}u=0 \\quad \\mbox{in }\\Omega,$$ where $1<p<\\infty$, $\\Omega$ is a domain in $\\mathbb{R}^{n}$, $n\\geq 2$, $V\\in L_{\\mathrm{loc}}^{\\infty}(\\Omega)$, $A=\\big(a_{ij}\\big)\\in L_{\\mathrm{loc}}^{\\infty}(\\Omega,\\mathbb{R}^{n^2})$ is a symmetric and locally uniformly positive definite matrix in $\\Omega$, and $$|\\xi|_{A}^{2}:=\\left\\langle A(x)\\xi,\\xi\\right\\rangle=\\sum_{i,j=1}^n a_{ij}(x)\\xi_i\\xi_j \\qquad x\\in \\Omega, \\xi=(\\xi_1,\\ldots,\\xi_n)\\i","authors_text":"Netanel Regev, Yehuda Pinchover","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-11T08:07:07Z","title":"Criticality theory of half-linear equations with the (p,A)-Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.3346","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:c1b4fa09745b7b1d67138db2ebf3d7ccb54eace88948143267fd05d1b6291c9e","target":"record","created_at":"2026-05-18T02:43:01Z","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":"95c9fd5db44de6529545ce7d86435dcbbba0f828a9c1c10484889e40bdf06773","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-11T08:07:07Z","title_canon_sha256":"22a169a7c99db73ab50d74bd87c855c9abb150b15bbfa242d7d6c7ac36d254a0"},"schema_version":"1.0","source":{"id":"1409.3346","kind":"arxiv","version":1}},"canonical_sha256":"b96fa8ae3d7c277ff502d11d31ea52148b61704e3eb1744de638b8cebdc4bee7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b96fa8ae3d7c277ff502d11d31ea52148b61704e3eb1744de638b8cebdc4bee7","first_computed_at":"2026-05-18T02:43:01.582202Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:43:01.582202Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kp9Jh0YsMYfRjP3C3SHmWTS/Ri6CXQWpywt+r+1MkJZfKEZtTnYeu4fZyQpEhGCIUnCLjSvWhMN0ClOXPbJcDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:43:01.582655Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.3346","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c1b4fa09745b7b1d67138db2ebf3d7ccb54eace88948143267fd05d1b6291c9e","sha256:3a0d7bee4a82051154b7792110fee5a315d4f8b042759c16b25ba0d69dc2831e"],"state_sha256":"a5574b62a0035a2c9b2fc77f7292d8c2a5eaaa246900cc49894e1fe5130f5972"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fecujMG5RQ3b/ow5UdR313Bnyu4nSHHDrdjktG8v/MF6GvjZ0ydQ6oCbx1bXZi8g9XyR+f67xBXyb04wZU3dCw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T05:47:58.711615Z","bundle_sha256":"14657944b6b4c498db2bfdbb56734443f47ede7fd8d0f9b1026ebb84ff769b58"}}