A Lower Bound for the Number of Elastic Collisions

Burdzy, Krzysztof; Duarte, Mauricio

doi:10.1007/s00220-019-03399-3

Cited by 10 publications

(12 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It has been proved in [9] that if the balls have equal radii and masses then, K(n, d) K(n, 2) > n 3 /27 for n 3, d 2.…”

Section: Review Of Existing Resultsmentioning

confidence: 99%

“…It has been proved in that if the balls have equal radii and masses then,

\begin{matrix} true \\ right K (n, d) ⩾ K (n, 2) > n^{3} / 27 for n ⩾ 3, d ⩾ 2 . \end{matrix}

Another lower bound has been recently announced in for

d ⩾ 3

\begin{matrix} true \\ right K (n, d) ⩾ K (n, 3) ⩾ 2^{⌊ n / 2 ⌋} for n ⩾ 3, d ⩾ 3 . \end{matrix}

Note that gives a larger number of collisions than only for values of

n

larger than or equal to 14 and only for

d > 2

.…”

Section: Introductionmentioning

confidence: 97%

“…Let

K (n, d)

be the maximum number of elastic collisions that

n

balls in

R^{d}

, of equal radii and masses, can undergo. It is easy to see that

K (n, 1) = n (n - 1) / 2

for

n ⩾ 2

, and that

K (n, d)

is a non‐decreasing function of

d

(see ). Hence,

K (n, d) ⩾ n (n - 1) / 2

for all

n ⩾ 2

and

d ⩾ 1

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the number of hard ball collisions

Burdzy

Duarte

2019

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

We give a new and elementary proof that the number of elastic collisions of a finite number of balls in the Euclidean space is finite. We show that if there are n balls of equal masses and radii 1, and at the time of a collision between any two balls the distance between any other pair of balls is greater than n−n, then the total number of collisions is bounded by nfalse(5/2+εfalse)n, for any fixed ε>0 and large n. We also show that if there is a number of collisions larger than ncn for an appropriate c>0, then a large number of these collisions occur within a subfamily of balls that form a very tight configuration.

show abstract

“…It has been proved in [9] that if the balls have equal radii and masses then, K(n, d) K(n, 2) > n 3 /27 for n 3, d 2.…”

Section: Review Of Existing Resultsmentioning

confidence: 99%

“…It has been proved in that if the balls have equal radii and masses then,

\begin{matrix} true \\ right K (n, d) ⩾ K (n, 2) > n^{3} / 27 for n ⩾ 3, d ⩾ 2 . \end{matrix}

Another lower bound has been recently announced in for

d ⩾ 3

\begin{matrix} true \\ right K (n, d) ⩾ K (n, 3) ⩾ 2^{⌊ n / 2 ⌋} for n ⩾ 3, d ⩾ 3 . \end{matrix}

Note that gives a larger number of collisions than only for values of

n

larger than or equal to 14 and only for

d > 2

.…”

Section: Introductionmentioning

confidence: 97%

See 1 more Smart Citation

On the number of hard ball collisions

Burdzy

Duarte

2019

J. London Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…It has been proved in [10] by example that the number of elastic collisions of n balls in d-dimensional space is greater than n 3 /27 for n ≥ 3 and d ≥ 2, for some initial conditions. The previously known lower bound was of order n 2 (that bound was for balls in dimension 1 and was totally elementary).…”

Section: 1mentioning

confidence: 99%

Protecting Billiard Balls From Collisions

Athreya

Burdzy

2020

Arnold Math J.

Self Cite

View full text Add to dashboard Cite

We present a game inspired by research on the possible number of billiard ball collisions in the whole Euclidean space. One player tries to place n static "balls" with zero radius (i.e., points) in a way that will minimize the total number of possible collisions caused by the cue ball. The other player tries to find initial conditions for the cue ball to maximize the number of collisions. The value of the game is √ n (up to constants). The lower bound is based on the Erdős-Szekeres Theorem. The upper bound may be considered a generalization of the Erdős-Szekeres Theorem.

show abstract

“…After the (23) collision at B, particle 3 has to "maneuver around" particle 2 to collide with particle 1 in the fourth collision. See [8] for an alternate viewpoint on this sequence.…”

Section: A Detour: the Foch Sequencementioning

confidence: 99%