{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:NMCXACN4QRAMHUEW6A4QBYSIQB","short_pith_number":"pith:NMCXACN4","canonical_record":{"source":{"id":"1401.2338","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-10T14:13:50Z","cross_cats_sorted":[],"title_canon_sha256":"cf2c8b598f0a39160a65facd4828d2bfefae8330546e01bf3887416a7ff974f7","abstract_canon_sha256":"17c3a365a86915969f7ca901bc5a89686abf3b15b94ea3d82e76d3d6336335f7"},"schema_version":"1.0"},"canonical_sha256":"6b057009bc8440c3d096f03900e248807be8c11d263a5c7f952fc09c5e4d1d4a","source":{"kind":"arxiv","id":"1401.2338","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.2338","created_at":"2026-05-18T03:02:45Z"},{"alias_kind":"arxiv_version","alias_value":"1401.2338v1","created_at":"2026-05-18T03:02:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.2338","created_at":"2026-05-18T03:02:45Z"},{"alias_kind":"pith_short_12","alias_value":"NMCXACN4QRAM","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"NMCXACN4QRAMHUEW","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"NMCXACN4","created_at":"2026-05-18T12:28:41Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:NMCXACN4QRAMHUEW6A4QBYSIQB","target":"record","payload":{"canonical_record":{"source":{"id":"1401.2338","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-10T14:13:50Z","cross_cats_sorted":[],"title_canon_sha256":"cf2c8b598f0a39160a65facd4828d2bfefae8330546e01bf3887416a7ff974f7","abstract_canon_sha256":"17c3a365a86915969f7ca901bc5a89686abf3b15b94ea3d82e76d3d6336335f7"},"schema_version":"1.0"},"canonical_sha256":"6b057009bc8440c3d096f03900e248807be8c11d263a5c7f952fc09c5e4d1d4a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:45.545231Z","signature_b64":"KtQtNGPVT6zjQ740tZmCWp8j9z3MZVF1wdvg9EV1viJeyQhbKfv7xyEj+1rgWAA3Dun6ZT3Ajf5ICQ7pGesKBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6b057009bc8440c3d096f03900e248807be8c11d263a5c7f952fc09c5e4d1d4a","last_reissued_at":"2026-05-18T03:02:45.544565Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:45.544565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1401.2338","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":"2026-05-18T03:02:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J0Y5nX+hgjC+TyNub/ADwjfWkCx0H0OeSKFZTvdmGZW6k1wxguPNOl008T4R2IKTHIxVdfvLhpb7BeFk/YlNDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T01:11:40.199816Z"},"content_sha256":"abbb312e721205e27b6fb0b5056072e041d7a3fcc09a89da6090fa144910db05","schema_version":"1.0","event_id":"sha256:abbb312e721205e27b6fb0b5056072e041d7a3fcc09a89da6090fa144910db05"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:NMCXACN4QRAMHUEW6A4QBYSIQB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotic Behavior of Gradient Flows Driven by Nonlocal Power Repulsion and Attraction Potentials in One Dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daniel Matthes, Jan-Christian H\\\"utter, Marco Di Francesco, Massimo Fornasier","submitted_at":"2014-01-10T14:13:50Z","abstract_excerpt":"We study the long time behavior of the Wasserstein gradient flow for an energy functional consisting of two components: particles are attracted to a fixed profile $\\omega$ by means of an interaction kernel $\\psi_a(z)=|z|^{q_a}$,and they repel each other by means of another kernel $\\psi_r(z)=|z|^{q_r}$. We focus on the case of one space dimension and assume that $1\\le q_r\\le q_a\\le 2$.\n  Our main result is that the flow converges to an equilibrium if either $q_r<q_a$ or $1\\le q_r=q_a\\le4/3$,and if the solution has the same (conserved) mass as the reference state $\\omega$. In the cases $q_r=1$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.2338","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":"2026-05-18T03:02:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qKi0Zk6ZXiWC4U/nRyd47/lum6hT27axCNNLzF1SoaJHhsEy0piMLJKIZWhfR4peXsye4Na7e8B20b03FfO8DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T01:11:40.200170Z"},"content_sha256":"8bdf7700661d7d1816138f024a4b44a7583b069a225701851ac5a859a725a882","schema_version":"1.0","event_id":"sha256:8bdf7700661d7d1816138f024a4b44a7583b069a225701851ac5a859a725a882"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NMCXACN4QRAMHUEW6A4QBYSIQB/bundle.json","state_url":"https://pith.science/pith/NMCXACN4QRAMHUEW6A4QBYSIQB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NMCXACN4QRAMHUEW6A4QBYSIQB/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":"2026-06-04T01:11:40Z","links":{"resolver":"https://pith.science/pith/NMCXACN4QRAMHUEW6A4QBYSIQB","bundle":"https://pith.science/pith/NMCXACN4QRAMHUEW6A4QBYSIQB/bundle.json","state":"https://pith.science/pith/NMCXACN4QRAMHUEW6A4QBYSIQB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NMCXACN4QRAMHUEW6A4QBYSIQB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:NMCXACN4QRAMHUEW6A4QBYSIQB","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":"17c3a365a86915969f7ca901bc5a89686abf3b15b94ea3d82e76d3d6336335f7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-10T14:13:50Z","title_canon_sha256":"cf2c8b598f0a39160a65facd4828d2bfefae8330546e01bf3887416a7ff974f7"},"schema_version":"1.0","source":{"id":"1401.2338","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.2338","created_at":"2026-05-18T03:02:45Z"},{"alias_kind":"arxiv_version","alias_value":"1401.2338v1","created_at":"2026-05-18T03:02:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.2338","created_at":"2026-05-18T03:02:45Z"},{"alias_kind":"pith_short_12","alias_value":"NMCXACN4QRAM","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_16","alias_value":"NMCXACN4QRAMHUEW","created_at":"2026-05-18T12:28:41Z"},{"alias_kind":"pith_short_8","alias_value":"NMCXACN4","created_at":"2026-05-18T12:28:41Z"}],"graph_snapshots":[{"event_id":"sha256:8bdf7700661d7d1816138f024a4b44a7583b069a225701851ac5a859a725a882","target":"graph","created_at":"2026-05-18T03:02:45Z","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 the long time behavior of the Wasserstein gradient flow for an energy functional consisting of two components: particles are attracted to a fixed profile $\\omega$ by means of an interaction kernel $\\psi_a(z)=|z|^{q_a}$,and they repel each other by means of another kernel $\\psi_r(z)=|z|^{q_r}$. We focus on the case of one space dimension and assume that $1\\le q_r\\le q_a\\le 2$.\n  Our main result is that the flow converges to an equilibrium if either $q_r<q_a$ or $1\\le q_r=q_a\\le4/3$,and if the solution has the same (conserved) mass as the reference state $\\omega$. In the cases $q_r=1$ a","authors_text":"Daniel Matthes, Jan-Christian H\\\"utter, Marco Di Francesco, Massimo Fornasier","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-10T14:13:50Z","title":"Asymptotic Behavior of Gradient Flows Driven by Nonlocal Power Repulsion and Attraction Potentials in One Dimension"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.2338","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:abbb312e721205e27b6fb0b5056072e041d7a3fcc09a89da6090fa144910db05","target":"record","created_at":"2026-05-18T03:02:45Z","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":"17c3a365a86915969f7ca901bc5a89686abf3b15b94ea3d82e76d3d6336335f7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-01-10T14:13:50Z","title_canon_sha256":"cf2c8b598f0a39160a65facd4828d2bfefae8330546e01bf3887416a7ff974f7"},"schema_version":"1.0","source":{"id":"1401.2338","kind":"arxiv","version":1}},"canonical_sha256":"6b057009bc8440c3d096f03900e248807be8c11d263a5c7f952fc09c5e4d1d4a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6b057009bc8440c3d096f03900e248807be8c11d263a5c7f952fc09c5e4d1d4a","first_computed_at":"2026-05-18T03:02:45.544565Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:02:45.544565Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"KtQtNGPVT6zjQ740tZmCWp8j9z3MZVF1wdvg9EV1viJeyQhbKfv7xyEj+1rgWAA3Dun6ZT3Ajf5ICQ7pGesKBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:02:45.545231Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.2338","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:abbb312e721205e27b6fb0b5056072e041d7a3fcc09a89da6090fa144910db05","sha256:8bdf7700661d7d1816138f024a4b44a7583b069a225701851ac5a859a725a882"],"state_sha256":"5312849e486c83c6f7cb016cd19fea08a230e17f086ee3f939b7684ddc7ee928"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q0yKw6oaOpfS/8d7ZGoTiFSN+Kp6eZGQeCNF3npe0LUQO+cMXjnzpsl2LWkD9GPhTU4HC9TV/uHuU2zuFUt3Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T01:11:40.202106Z","bundle_sha256":"b76d5d2d046888219a5a71916572f89f9117a29f1dbdcce1b65de9b020963a43"}}