The CAG had initially estimated that the coal scam caused a massive loss of Rs 10.6 lakh crore to the exchequer, but its final report tabled in Parliament put the figure at Rs 1.86 lakh crore.
#CoalScam #HC #CBI #Respond #Ex- #Secy #Challenging
https://www.news18.com/india/coal-scam-hc-asks-cbi-to-respond-to-appeal-by-ex-coal-secy-former-public-servant-challenging-three-year-jail-8538985.html

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

Dark Souls is a challenge for a vTuber like me ! This ledge is quite difficult in particular. Classical videogame moment.

You can find the full video here: https://www.youtube.com/live/KSO-KOwQr9k?feature=share

#DarkSouls #VideoGames #Challenging #vTuber #Difficult #Ledge #Clips

In this picture, a large mountain is visible in the background, with a series of trees and paths leading up to it. The mountain is rugged and towering, with its imposing size and majesty dominating the landscape. The sky above is clear and blue, with a few fluffy clouds visible on the horizon, creating a sense of tranquility and calmness.

#Mountain #Nature #Hiking #Landscape #Path #Trees #Summertime #Rocky #Terrain #Challenging #Difficult #Explore #Climbing #Peak #Majestic

A visit to a potential garden design client…she showed me the backyard. In a rather perverse way, I liked the image…though I wouldn’t be the one having to clear and weed it!

#gardening #weeds #GardenDesign #UrbanGardens #challenging #rejuvenation

In this picture, a large mountain is visible in the background, with a series of trees and paths leading up to it. The mountain is rugged and towering, with its imposing size and majesty dominating the landscape. The sky above is clear and blue, with a few fluffy clouds visible on the horizon, creating a sense of tranquility and calmness.

#Mountain #Nature #Hiking #Landscape #Path #Trees #Summertime #Rocky #Terrain #Challenging #Difficult #Explore #Climbing #Peak #Majestic

Top 5 VR puzzle games - VR puzzle games provide a unique and immersive gameplay experienc... - https://cointelegraph.com/news/top-5-vr-puzzle-games #problem-solvingskills #immersiveexperience #virtualreality #vrpuzzlegames #captivating #challenging #puzzles

On the back of a suggestion from my #RSE #mentor I've decided to setup a blog on Design Patterns in Python where I'll learn and document how to use #DesignPatterns
Starting with the low hanging fruit of #FactoryMethods

https://ns-rse.github.io/design-patterns/

I've also setup an adjacent repository and I'm putting all the code in its own repo (as the blog is written in #quarto )

https://github.com/ns-rse/python-design-patterns

Lets see if I can learn a design pattern once a month #challenging

I'm trying to get better at asking for what I need. I have been going through a few #challenging situations very recently, and I could use a boost. No, not that kind of boost. An #EmotionalBoost. I know few of you know me, but if you have any sort of #support to throw my way, even as a stranger, I'll take it.

Attached is a #ComicStrip I created on stripcreator.com many years ago. I felt it was kind of relevant to the situation.

Thanks in advance for the #whack.

#MentalHealth

New of my Benedictus recording.

Finished all the choir. Tracks (esp. Sopranos) will need some post-processing… The days I was a soprano myself are more than thirty years gone…

When all that is done, I can correctly record the soloist and wrap up the mix.

#Ardour #HomeStudio #Choir #Recording #Challenging

I intentionally poke out of my comfort zone constantly, I love growing, I love being challenged, I love expanding what I think I'm capable of handling or appreciating, I love widening my comfort zone to include more and more

#growing #changing #comfortzone #challenge #challenging #exploring #exploration

I'm trying to get better at asking for what I need. I have been going through a few #challenging situations very recently, and I could use a boost. No, not that kind of boost. An #EmotionalBoost. I know few of you know me, but if you have any sort of #support to throw my way, even as a stranger, I'll take it.

Attached is a #ComicStrip I created on stripcreator.com many years ago. I felt it was kind of relevant to the situation.

Thanks in advance for the #whack.

#MentalHealth

Die Tür vom Gym ist offen..... und es duftet hervorragend nach frisch gemachten Süßkram von einem der Restaurants. #challenging

#Canadian #history from #stories of 1st #Chinese women to come to #Canada & subsequent #generations of #ChineseCanadian women. Amazing tale of #courageous #women who left behind their families, knowing they would never see them again & of girls who were shipped off to #NewWorld to marry men they'd never met. These are the women who fought against the many forms of #racism they faced in Canada while, at the same time, #challenging #sexism within their own #communities .
https://www.nfb.ca/film/under-the-willow-tree-pioneer-chinese-women/

what was your most #challenging environment where you took #photos or #videos?

A topic I will talk about in one of the next #Podcast Episodes. How to prepare your gear and yourself for situations outside of the comfort zone.

Would love to hear your story!

Which day will we be living in precedented, certain, unchallenging times? Asking for a planet...

Just wondering... I was gonna set an alarm...

#California #Chico #Redding #NorCal #unprecedented #uncertain #challenging #precedented #certain #unchallenging #alarm #justwondering #askingforafriend #askingforaplanet #mastodon #mastodonfollow

Am I #challenging to #friends? #goofy

It drives me insane when I can’t find something, astrology makes it easier to understand difficult times. This is definitely the energy of Mercury opposite Mars. Very scattered energy. Luckily Mercury is sextile Saturn offering us a more grounded & realistic approach to any ideas or actions we take. If you’re feeling anxious, frustrated or irrational this week you are feeling this intense energy #astrology #transits #scattered #grounded #mercury #opposite #mars #saturn #challenging

Sky Swing Fling. Made this to become a pop up or folding piece but it hasn't happened yet and so why not share it as is.

There's a theme of floating, dangling, drifting, contorting, constriction and freedom coming through work this year.

#Hope #Hopeful #Challenges #Challenging #HardTimes #Floating #Flying #Imagination
#Art #Artist #NaiveArt