Paper Citation Record · LEDGER

Exact hopping and collision times for two hard discs in a box

As of 16 August 2026, this Paper Citation Record lists 32 of 32 outbound references and 0 inbound Pith citation observations for arXiv:1908.04749.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
1908.04749 v1

Coverage vector

measured 32 of 32 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-14T13:48:04.847527Z

measured 32 of 32 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-16T06:30:59.297886+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

32 of 32 outbound references displayed

verified exact0
verified fuzzy19
unresolved13
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation 1debe9f5-1835-4508-8c21-b5d4c6408135 · outbound

This paper cites In the above for- mulation we deﬁne ghop(x1,x2,y1,y2) := y1− y2, so that ∇ghop(x) = (0,0,1,−1) and‖∇ghop(x)‖ = √.

Exact hopping and collision times for two hard discs in a box In the above for- mulation we deﬁne ghop(x1,x2,y1,y2) := y1− y2, so that ∇ghop(x) = (0,0,1,−1) and‖∇ghop(x)‖ = √

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.330155Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation c8f67211-79a5-4a2f-b5ae-475c8f83d75d · outbound

This paper cites diagonal.

Exact hopping and collision times for two hard discs in a box diagonal

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.344620Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 92063533-97b1-4618-b49a-1183b6897d3a · outbound

This paper cites This results in the following expression: Ahop = x1,x2=a y1,y2=b∫∫∫∫ x1,x2=−a y1,y2=−b dx 1dx 2dy 1dy 2 1{(x1−x2)2+(y1−y2)2≥(2r)2} δ (y1− y2√ 2 ).

Exact hopping and collision times for two hard discs in a box This results in the following expression: Ahop = x1,x2=a y1,y2=b∫∫∫∫ x1,x2=−a y1,y2=−b dx 1dx 2dy 1dy 2 1{(x1−x2)2+(y1−y2)2≥(2r)2} δ (y1− y2√ 2 )

Reference 3

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.315014Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation e2f5beb2-e74c-4068-91f1-23fdd30d0c25 · outbound

This paper cites For r < w/4, we ﬁnd for the corresponding area Acoll = √ 2(16πabr− 32(a + b)r2 + 16r3).

Exact hopping and collision times for two hard discs in a box For r < w/4, we ﬁnd for the corresponding area Acoll = √ 2(16πabr− 32(a + b)r2 + 16r3)

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.300278Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation efb8b9a5-d266-4bc5-aab9-f6ab80f562e9 · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 5

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:05.285253Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 8d935ff9-246e-4c01-988e-e84ae56312d0 · outbound

This paper cites If w,h > 4r, then the limits of integration are unaffected by the radius of the circles.

Exact hopping and collision times for two hard discs in a box If w,h > 4r, then the limits of integration are unaffected by the radius of the circles

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.270746Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation eedea004-89a2-4fff-a9c9-0db0836efa88 · outbound

This paper cites We use a suitable Dirac delta to represent each codimension-1 collision event (contact of a disc with a wall or another disc).

Exact hopping and collision times for two hard discs in a box We use a suitable Dirac delta to represent each codimension-1 collision event (contact of a disc with a wall or another disc)

Reference 7

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.255569Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 1f8d3544-583a-4eff-9035-53caae7ea898 · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 8

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:05.240106Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation fa19e0a5-4e56-4ba9-86cd-40089a888b09 · outbound

This paper cites Boltzmann, Sitz.

Exact hopping and collision times for two hard discs in a box Boltzmann, Sitz

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.225463Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 48e3a838-7d59-4865-9385-85a34f7dec73 · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 10

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:05.211404Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 932ed4e8-59ba-4377-9378-653140afefb8 · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 11

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:05.197073Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 3dfda6a8-aae1-4f0c-81c3-33aa123f7171 · outbound

This paper cites Szasz, ed., Hard Ball Systems and the Lorentz Gas , Enc.

Exact hopping and collision times for two hard discs in a box Szasz, ed., Hard Ball Systems and the Lorentz Gas , Enc

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.181122Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 51e58134-ae09-43fe-9cda-0b9fbf1edf24 · outbound

This paper cites Bowles, K.

Exact hopping and collision times for two hard discs in a box Bowles, K

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.165520Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation b359a311-0538-41f9-8ac9-950975041379 · outbound

This paper cites Awazu, Phys.

Exact hopping and collision times for two hard discs in a box Awazu, Phys

Reference 14

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.149406Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 843240c9-65af-41e4-9500-4c8822cfb748 · outbound

This paper cites Munakata and G.

Exact hopping and collision times for two hard discs in a box Munakata and G

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.133550Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation b19f9d0f-adca-4a77-9426-0093e765768f · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 16

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:05.117791Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 285f2b9c-f392-454d-9036-85354f46628f · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 17

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:05.102404Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 6bb72f33-3238-494e-871e-0ecc50d2e559 · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 18

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:05.085156Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 0133a6db-285f-4d9e-b5e1-8b020517f567 · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 19

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:05.070303Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation e8357355-7405-4cbc-a0dc-8679033701b7 · outbound

This paper cites Uraganase and T.

Exact hopping and collision times for two hard discs in a box Uraganase and T

Reference 20

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.055474Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 5f12db9d-40d1-4070-85a8-31ed623d0174 · outbound

This paper cites Simanyi, Ergodic Theory and Dynamical Systems19 (1999).

Exact hopping and collision times for two hard discs in a box Simanyi, Ergodic Theory and Dynamical Systems19 (1999)

Reference 21

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.041167Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 6708c4c3-e5e3-4f45-832c-89ef336aeb35 · outbound

This paper cites Chernov, J.

Exact hopping and collision times for two hard discs in a box Chernov, J

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:05.027022Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 53e6ae95-2d0b-4fda-a6f3-2bc208a8aa3f · outbound

This paper cites Bezanson, A.

Exact hopping and collision times for two hard discs in a box Bezanson, A

Reference 23

Resolution: unresolved; no resolver link, observed 2026-08-14T13:48:04.808509Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 21d70d5a-587c-4775-96d0-040125297e31 · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 24

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:05.001435Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 27004bac-65c7-4885-a65a-00e9cd5332d5 · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 25

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:04.988004Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation d9d8e28d-86b6-42e3-ba87-84be64d02e6f · outbound

This paper cites Chernov and R.

Exact hopping and collision times for two hard discs in a box Chernov and R

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:04.974352Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation e6e187d4-1fc2-492b-8e5e-31fc2da20e6d · outbound

This paper cites Machta and R.

Exact hopping and collision times for two hard discs in a box Machta and R

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:04.959688Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation f0bed0bd-cdee-4584-a377-e5d853460f10 · outbound

This paper cites H ¨ormander, The Analysis of Linear Partial Differential Op- erators I (Springer, 1983).

Exact hopping and collision times for two hard discs in a box H ¨ormander, The Analysis of Linear Partial Differential Op- erators I (Springer, 1983)

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:04.943378Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation c114ff39-d894-43bc-84e3-e104aea5fca2 · outbound

This paper cites Zappa, M.

Exact hopping and collision times for two hard discs in a box Zappa, M

Reference 29

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:04.928113Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation a9ce2586-51ff-40c4-91f6-ffc9d36545d6 · outbound

This paper cites Hirata, B.

Exact hopping and collision times for two hard discs in a box Hirata, B

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-14T13:48:04.912637Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 59214eec-4c0d-4fa9-8e9b-683e85f544f1 · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 31

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:04.898130Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Observation 486faf64-3dd5-442c-992a-b0cb188b648d · outbound

This paper cites an unresolved cited work.

Exact hopping and collision times for two hard discs in a box Unresolved cited work

Reference 32

Resolution: unresolved; raw_fallback, observed 2026-08-14T13:48:04.882989Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-16T06:30:59.297886+00:00.

Pith citing papers

No inbound Pith citation observations are available.