John Golden · @mathhombre
142 followers · 59 posts · Server mathstodon.xyz

Mathematicians Christian JƤkel and Lennart Van Hirtum et al. simultaneously discover the 42-digit Dedekind number after 32 years of trying.

The exact values of the Dedekind numbers are known for \(0\leq n\leq9\):
\(2,3,6,20,168,7581,7828354,2414682040998,\)
\(56130437228687557907788,\)
\(286386577668298411128469151667598498812366\)
(sequence A000372 in the OEIS)

šŸ”— scitechdaily.com/elusive-ninth

šŸ”— sciencealert.com/mathematician

Summation formulašŸ‘‡
Kisielewicz (1988) rewrote the logical definition of antichains into the following arithmetic formula for the Dedekind numbers:
\[\displaystyle M(n)=\sum_{k=1}^{2^{2^n}} \prod_{j=1}^{2^n-1} \prod_{i=0}^{j-1} \left(1-b_i^k b_j^k\prod_{m=0}^{\log_2 i} (1-b_m^i+b_m^i b_m^j)\right)\]

where \(b_i^k\) is the \(i\)th bit of the number \(k\), which can be written using the floor function as
\[\displaystyle b_i^k=\left\lfloor\frac{k}{2^i}\right\rfloor - 2\left\lfloor\frac{k}{2^{i+1}}\right\rfloor.\]

However, this formula is not helpful for computing the values of \(M(n)\) for large \(n\) due to the large number of terms in the summation.

Asymptotics:
The logarithm of the Dedekind numbers can be estimated accurately via the bounds
\[\displaystyle{n\choose \lfloor n/2\rfloor}\le \log_2 M(n)\le {n\choose \lfloor n/2\rfloor}\left(1+O\left(\frac{\log n}{n}\right)\right).\]

Here the left inequality counts the number of antichains in which each set has exactly \(\lfloor n/2\rfloor\) elements, and the right inequality was proven by Kleitman & Markowsky (1975).

#pustamraut #egr #challengingproblem #pustam #mathhistory #difficultproblem #richarddedekind #challenging #mathematicians #discovery #sequence #mathematics #numbertheory #dedekind #dedekindnumber

Last updated 1 year ago

Mathematicians Christian JƤkel and Lennart Van Hirtum et al. simultaneously discover the 42-digit Dedekind number after 32 years of trying.

The exact values of the Dedekind numbers are known for \(0\leq n\leq9\):
\(2,3,6,20,168,7581,7828354,2414682040998,56130437228687557907788,286386577668298411128469151667598498812366\)
(sequence A000372 in the OEIS)

šŸ”— scitechdaily.com/elusive-ninth

šŸ”— sciencealert.com/mathematician

Summation formulašŸ‘‡
Kisielewicz (1988) rewrote the logical definition of antichains into the following arithmetic formula for the Dedekind numbers:
\[\displaystyle M(n)=\sum_{k=1}^{2^{2^n}} \prod_{j=1}^{2^n-1} \prod_{i=0}^{j-1} \left(1-b_i^k b_j^k\prod_{m=0}^{\log_2 i} (1-b_m^i+b_m^i b_m^j)\right)\]

where \(b_i^k\) is the \(i\)th bit of the number \(k\), which can be written using the floor function as
\[\displaystyle b_i^k=\left\lfloor\frac{k}{2^i}\right\rfloor - 2\left\lfloor\frac{k}{2^{i+1}}\right\rfloor.\]

However, this formula is not helpful for computing the values of \(M(n)\) for large \(n\) due to the large number of terms in the summation.

Asymptotics:
The logarithm of the Dedekind numbers can be estimated accurately via the bounds
\[\displaystyle{n\choose \lfloor n/2\rfloor}\le \log_2 M(n)\le {n\choose \lfloor n/2\rfloor}\left(1+O\left(\frac{\log n}{n}\right)\right).\]

Here the left inequality counts the number of antichains in which each set has exactly \(\lfloor n/2\rfloor\) elements, and the right inequality was proven by Kleitman & Markowsky (1975).

#pustamraut #egr #challengingproblem #pustam #mathhistory #difficultproblem #richarddedekind #challenging #mathematicians #discovery #sequence #mathematics #numbertheory #dedekind #dedekindnumber

Last updated 1 year ago

Kati Kish Bar-On · @kati_kish
24 followers · 25 posts · Server mathstodon.xyz

šŸŽ‰The third (and final!) paper from my Ph.D. dissertation is now out at @PhilSciJournal as šŸ‘€ First view article šŸ‘€
šŸ§Interested in how mathematical communities react to new ideas?
Read on:
(if you don't have šŸ”, please pm me!) cambridge.org/core/journals/ph

#mathhistory #histofscience #philsci

Last updated 1 year ago

Charlotte Kirchhoff-Lukat · @charlottekl
237 followers · 379 posts · Server mathstodon.xyz

@rrogers šŸ˜Æ
Do you think that's still at play, though? After all, the UK has had mathematical excellence at the top of its university system for centuries now. I had the privilege of studying at in , where you trip over mathematical history constantly. (Great opportunity for learning some history of maths for me!)

#mathhistory #education #maths #cambridge #trinity

Last updated 2 years ago

HannahCrazyhawk · @HannahCrazyhawk
159 followers · 203 posts · Server universeodon.com
Kati Kish Bar-On · @kati_kish
6 followers · 1 posts · Server mathstodon.xyz

So I have been told that I should introduce myself in this platform, and what a better way to do that then to share a new publication!
Excited to share that my paper ā€œFrom philosophical traditions to scientific developments: reconsidering the response to Brouwerā€™s intuitionismā€ is now available in Synthese: doi.org/10.1007/s11229-022-039 Interested in reading and donā€™t have access? PM me for a copy šŸ“„

#mathhistory #philsci #introduction

Last updated 2 years ago

Michael Evans · @MEvans
17 followers · 40 posts · Server mathstodon.xyz




December 1st, 1792.
Nikolay Ivanovich Lobachevsky is born.

And Tom Lehrer sings about it!

youtu.be/_lf1mB8OZeg

#matheducation #stem #mathhistory #math

Last updated 2 years ago

Infrapink (he/his/him) · @Infrapink
18 followers · 248 posts · Server mastodon.ie

The 's methods for calculating the positions of the sun and moon at times specify that the absolute value of the jya of an arc must be added to some value in some cases, and subtracted in others.

Since the jya is the predecessor to the modern sine, in this we see the earliest inklings of negative values of trigonometric functions. It's interesting how ideas develop.

#suryasiddhanta #jyotisha #trigonometry #maths #historyofmaths #mathhistory

Last updated 2 years ago