Mathematicians Christian Jäkel 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)

🔗 https://scitechdaily.com/elusive-ninth-dedekind-number-discovered-unlocking-a-decades-old-mystery-of-mathematics/?expand_article=1

🔗 https://www.sciencealert.com/mathematicians-discover-the-ninth-dedekind-number-after-32-years-of-searching

Summation formula👇
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).

#DedekindNumber #Dedekind #NumberTheory #Mathematics #Sequence #Discovery #Mathematicians #Challenging #RichardDedekind #DifficultProblem #MathHistory #Pustam #ChallengingProblem #EGR #PustamRaut

Mathematicians Christian Jäkel 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)

🔗 https://scitechdaily.com/elusive-ninth-dedekind-number-discovered-unlocking-a-decades-old-mystery-of-mathematics/?expand_article=1

🔗 https://www.sciencealert.com/mathematicians-discover-the-ninth-dedekind-number-after-32-years-of-searching

Summation formula👇
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).

#DedekindNumber #Dedekind #NumberTheory #Mathematics #Sequence #Discovery #Mathematicians #Challenging #RichardDedekind #DifficultProblem #MathHistory #Pustam #ChallengingProblem #EGR #PustamRaut

Dear diary,

Collect car ... into #Tewkesbury for coffee ... start engine an hour later and red light is back on 🙃

Booked in again for replacement replacement #EGR.

I mean, 3 miles up the road and the newly fitted replacement exhaust gas recirculation valve fails.

I, er, ...

The #EGR valve system in your car is designed to reduce emissions. Under certain driving conditions, such as high load or idle, the engine will automatically turn on the EGR valve to allow exhaust gas to re-enter the combustion chamber and be burned with new fuel. The EGR valve is designed to close when not needed so that no excessive pressure builds up in the system.
https://www.shareaword.com.au/egr-valves-what-they-are-the-parts-that-comprise-them-and-replacing-egr-valves/

Autoon tarjolla n. 600€ korjaus. Liittyy pakokaasujen polton optimointiin päästöjen osalta.

* missäs niitä sähköautoja oli myytävänä? *

#autoilu #EGR