{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:2F4RHR4V7LMCB2IFNOGXVJ4DTH","short_pith_number":"pith:2F4RHR4V","canonical_record":{"source":{"id":"1703.10874","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-31T12:10:30Z","cross_cats_sorted":[],"title_canon_sha256":"b959b99b7393fca455dd35c26992152c37b6dcbf50dd6526eb0d2e4da62e4f89","abstract_canon_sha256":"b60abb34332a2a478ef7802383fa7335551dfb8ee2117fc161e540b9570f0db4"},"schema_version":"1.0"},"canonical_sha256":"d17913c795fad820e9056b8d7aa78399ed13c155a354f21eb7fbc9eefc4f5d0f","source":{"kind":"arxiv","id":"1703.10874","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.10874","created_at":"2026-05-18T00:47:32Z"},{"alias_kind":"arxiv_version","alias_value":"1703.10874v1","created_at":"2026-05-18T00:47:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.10874","created_at":"2026-05-18T00:47:32Z"},{"alias_kind":"pith_short_12","alias_value":"2F4RHR4V7LMC","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"2F4RHR4V7LMCB2IF","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"2F4RHR4V","created_at":"2026-05-18T12:30:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:2F4RHR4V7LMCB2IFNOGXVJ4DTH","target":"record","payload":{"canonical_record":{"source":{"id":"1703.10874","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-31T12:10:30Z","cross_cats_sorted":[],"title_canon_sha256":"b959b99b7393fca455dd35c26992152c37b6dcbf50dd6526eb0d2e4da62e4f89","abstract_canon_sha256":"b60abb34332a2a478ef7802383fa7335551dfb8ee2117fc161e540b9570f0db4"},"schema_version":"1.0"},"canonical_sha256":"d17913c795fad820e9056b8d7aa78399ed13c155a354f21eb7fbc9eefc4f5d0f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:32.437628Z","signature_b64":"YlYzHIy2n2ffYuCftZverMV6IeHZ3zTQqv/ylZ+DiaPTksXqxx8WySBKXSZQ+XSg8Q4+892hWhNrak/+muhbCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d17913c795fad820e9056b8d7aa78399ed13c155a354f21eb7fbc9eefc4f5d0f","last_reissued_at":"2026-05-18T00:47:32.436997Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:32.436997Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1703.10874","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":"2026-05-18T00:47:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B7ANtEcg9VgoeVF2UJH/ip52ApqGsPOifKJswQthIOweGokntliMLZD0/tAY/zCB6z5LQ26HaAKCFA3vCqlMDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:03:12.236945Z"},"content_sha256":"d2d487fe9a680d11c6a7eedcecc5fd10e2764c645276c3fd7f06ef77803cf56e","schema_version":"1.0","event_id":"sha256:d2d487fe9a680d11c6a7eedcecc5fd10e2764c645276c3fd7f06ef77803cf56e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:2F4RHR4V7LMCB2IFNOGXVJ4DTH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Nicolas Fournier","submitted_at":"2017-03-31T12:10:30Z","abstract_excerpt":"We consider the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution $(f_t)_{t\\geq 0}$, once the initial condition $f_0$ with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a $(0,\\infty)\\times\\mathbb{R}^3$ random variable $(M_t,V_t)$ such that $E[M_t {\\bf 1}_{\\{V_t \\in \\cdot\\}}]=f_t$. We also write down a series expansion of $f_t$. Although both the algorithm and the series expansion might be theoretically interesting in that they expl"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.10874","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":"2026-05-18T00:47:32Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6pb5XK5WfaNZpNBeaRM5BvphsKKRQ4KNUmiQrG1YQR3MDt3y7dxL/KLlJXkigc8cUByvry1xVqqtHETukrM6Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:03:12.237627Z"},"content_sha256":"122b5027b2f6f7b9dcd5ab2d563ecca94c010f4475cec7b7b7a1b52607d63c4b","schema_version":"1.0","event_id":"sha256:122b5027b2f6f7b9dcd5ab2d563ecca94c010f4475cec7b7b7a1b52607d63c4b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/2F4RHR4V7LMCB2IFNOGXVJ4DTH/bundle.json","state_url":"https://pith.science/pith/2F4RHR4V7LMCB2IFNOGXVJ4DTH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/2F4RHR4V7LMCB2IFNOGXVJ4DTH/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":"2026-05-27T14:03:12Z","links":{"resolver":"https://pith.science/pith/2F4RHR4V7LMCB2IFNOGXVJ4DTH","bundle":"https://pith.science/pith/2F4RHR4V7LMCB2IFNOGXVJ4DTH/bundle.json","state":"https://pith.science/pith/2F4RHR4V7LMCB2IFNOGXVJ4DTH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/2F4RHR4V7LMCB2IFNOGXVJ4DTH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:2F4RHR4V7LMCB2IFNOGXVJ4DTH","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":"b60abb34332a2a478ef7802383fa7335551dfb8ee2117fc161e540b9570f0db4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-31T12:10:30Z","title_canon_sha256":"b959b99b7393fca455dd35c26992152c37b6dcbf50dd6526eb0d2e4da62e4f89"},"schema_version":"1.0","source":{"id":"1703.10874","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.10874","created_at":"2026-05-18T00:47:32Z"},{"alias_kind":"arxiv_version","alias_value":"1703.10874v1","created_at":"2026-05-18T00:47:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.10874","created_at":"2026-05-18T00:47:32Z"},{"alias_kind":"pith_short_12","alias_value":"2F4RHR4V7LMC","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"2F4RHR4V7LMCB2IF","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"2F4RHR4V","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:122b5027b2f6f7b9dcd5ab2d563ecca94c010f4475cec7b7b7a1b52607d63c4b","target":"graph","created_at":"2026-05-18T00:47:32Z","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 the spatially homogeneous Boltzmann equation for hard potentials with angular cutoff. This equation has a unique conservative weak solution $(f_t)_{t\\geq 0}$, once the initial condition $f_0$ with finite mass and energy is fixed. Taking advantage of the energy conservation, we propose a recursive algorithm that produces a $(0,\\infty)\\times\\mathbb{R}^3$ random variable $(M_t,V_t)$ such that $E[M_t {\\bf 1}_{\\{V_t \\in \\cdot\\}}]=f_t$. We also write down a series expansion of $f_t$. Although both the algorithm and the series expansion might be theoretically interesting in that they expl","authors_text":"Nicolas Fournier","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-31T12:10:30Z","title":"A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.10874","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:d2d487fe9a680d11c6a7eedcecc5fd10e2764c645276c3fd7f06ef77803cf56e","target":"record","created_at":"2026-05-18T00:47:32Z","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":"b60abb34332a2a478ef7802383fa7335551dfb8ee2117fc161e540b9570f0db4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-03-31T12:10:30Z","title_canon_sha256":"b959b99b7393fca455dd35c26992152c37b6dcbf50dd6526eb0d2e4da62e4f89"},"schema_version":"1.0","source":{"id":"1703.10874","kind":"arxiv","version":1}},"canonical_sha256":"d17913c795fad820e9056b8d7aa78399ed13c155a354f21eb7fbc9eefc4f5d0f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d17913c795fad820e9056b8d7aa78399ed13c155a354f21eb7fbc9eefc4f5d0f","first_computed_at":"2026-05-18T00:47:32.436997Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:47:32.436997Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YlYzHIy2n2ffYuCftZverMV6IeHZ3zTQqv/ylZ+DiaPTksXqxx8WySBKXSZQ+XSg8Q4+892hWhNrak/+muhbCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:47:32.437628Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.10874","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d2d487fe9a680d11c6a7eedcecc5fd10e2764c645276c3fd7f06ef77803cf56e","sha256:122b5027b2f6f7b9dcd5ab2d563ecca94c010f4475cec7b7b7a1b52607d63c4b"],"state_sha256":"39f2c8f9f86164d2e95ba8eb03dae66ca42e3a7b7043731091cf97f1b6488662"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uFthJMzWvrOkiRGRlToAZN5olTgVhbDBFyeXplxMRlLbARU+aMfwgHpXkQfgA7fqZN1N/qZp1Wtg2p901KxSBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T14:03:12.241147Z","bundle_sha256":"a795883cda55cbc56f2bf2d3f0fcf67f252b349efeeb9d9df993efad43ac22b4"}}