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For any given finite measure $\\mu$ on $V$, define functionals \\begin{eqnarray*}\n  J_{ \\beta }(u)&=&\\frac{1}{2}\\int_{V}|\\nabla u|^{2}d \\mu -\\beta \\log\\int_{V}he^{u}d \\mu,\n  J_{ \\alpha ,\\beta }(u)&=&\\frac{1}{2}\\int_{V}\\left(|\\nabla u|^{2}- \\alpha u^{2}\\right) d \\mu -\\beta \\log\\int_{V}he^{u}d \\mu \\end{eqnarray*} on the functional space $$ {\\bf H}= \\left\\{ u\\in{\\bf W}^{1,2}(V) \\Bigg| \\int_{V}u\\!\\ d\\mu =0 \\right\\}. $$\n  For any $\\beta \\in \\mathbb{R}$, we"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2308.10002","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2023-08-19T12:43:24Z","cross_cats_sorted":[],"title_canon_sha256":"8d08b7a447f74e992a551bce20259064c53fecef5a166ebd2013232f5740202c","abstract_canon_sha256":"f96b95c25145b9d92763bf42dedbe5bc8c0ac68a052479dac19f6b07198cbf89"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:42:49.846218Z","signature_b64":"PuFMuiUDA6c4aN9nWj6DtA1CfCRVTZn2kfgO+SLPMSzoLPX2g7mzruSOBxAvyboOaZU4vYAZJbuLZVZGXK76Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c08dae367bbc384eb33fc0b587a59141a7f9112d3c14426e2bef503e4dca9719","last_reissued_at":"2026-07-05T06:42:49.845752Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:42:49.845752Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence of Solutions to a Class of Kazdan-Warner Equations on Finite Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Qianwei Zhang, Yi Li","submitted_at":"2023-08-19T12:43:24Z","abstract_excerpt":"Let $G=(V, E)$ be a connected finite graph, $h$ be a positive function on $V$ and $\\lambda _{1}(V)$ be the first non-zero eigenvalue of $-\\Delta$. For any given finite measure $\\mu$ on $V$, define functionals \\begin{eqnarray*}\n  J_{ \\beta }(u)&=&\\frac{1}{2}\\int_{V}|\\nabla u|^{2}d \\mu -\\beta \\log\\int_{V}he^{u}d \\mu,\n  J_{ \\alpha ,\\beta }(u)&=&\\frac{1}{2}\\int_{V}\\left(|\\nabla u|^{2}- \\alpha u^{2}\\right) d \\mu -\\beta \\log\\int_{V}he^{u}d \\mu \\end{eqnarray*} on the functional space $$ {\\bf H}= \\left\\{ u\\in{\\bf W}^{1,2}(V) \\Bigg| \\int_{V}u\\!\\ d\\mu =0 \\right\\}. $$\n  For any $\\beta \\in \\mathbb{R}$, we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2308.10002","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2308.10002/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2308.10002","created_at":"2026-07-05T06:42:49.845807+00:00"},{"alias_kind":"arxiv_version","alias_value":"2308.10002v1","created_at":"2026-07-05T06:42:49.845807+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2308.10002","created_at":"2026-07-05T06:42:49.845807+00:00"},{"alias_kind":"pith_short_12","alias_value":"YCG24NT3XQ4E","created_at":"2026-07-05T06:42:49.845807+00:00"},{"alias_kind":"pith_short_16","alias_value":"YCG24NT3XQ4E5MZ7","created_at":"2026-07-05T06:42:49.845807+00:00"},{"alias_kind":"pith_short_8","alias_value":"YCG24NT3","created_at":"2026-07-05T06:42:49.845807+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2509.01155","citing_title":"On finite-energy solutions of Kazan-Warner equations on the lattice graph","ref_index":49,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YCG24NT3XQ4E5MZ7YC2YPJMRIG","json":"https://pith.science/pith/YCG24NT3XQ4E5MZ7YC2YPJMRIG.json","graph_json":"https://pith.science/api/pith-number/YCG24NT3XQ4E5MZ7YC2YPJMRIG/graph.json","events_json":"https://pith.science/api/pith-number/YCG24NT3XQ4E5MZ7YC2YPJMRIG/events.json","paper":"https://pith.science/paper/YCG24NT3"},"agent_actions":{"view_html":"https://pith.science/pith/YCG24NT3XQ4E5MZ7YC2YPJMRIG","download_json":"https://pith.science/pith/YCG24NT3XQ4E5MZ7YC2YPJMRIG.json","view_paper":"https://pith.science/paper/YCG24NT3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2308.10002&json=true","fetch_graph":"https://pith.science/api/pith-number/YCG24NT3XQ4E5MZ7YC2YPJMRIG/graph.json","fetch_events":"https://pith.science/api/pith-number/YCG24NT3XQ4E5MZ7YC2YPJMRIG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YCG24NT3XQ4E5MZ7YC2YPJMRIG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YCG24NT3XQ4E5MZ7YC2YPJMRIG/action/storage_attestation","attest_author":"https://pith.science/pith/YCG24NT3XQ4E5MZ7YC2YPJMRIG/action/author_attestation","sign_citation":"https://pith.science/pith/YCG24NT3XQ4E5MZ7YC2YPJMRIG/action/citation_signature","submit_replication":"https://pith.science/pith/YCG24NT3XQ4E5MZ7YC2YPJMRIG/action/replication_record"}},"created_at":"2026-07-05T06:42:49.845807+00:00","updated_at":"2026-07-05T06:42:49.845807+00:00"}