co.combinatorics – Happy ants never leave compact domain?

I am curious if the following seemingly simple question has an easy answer?

Consider an ant population of $N$ ants that lives in $mathbb R^2$. Each ant can be considered to be a disk or radius $1.$ Ants like to be close to their peers but also not too close. The optimal distance between the center of two ants is $5^{2/3}$. So given two coordinates of centers of ants $x_i,x_j in mathbb R^2$. Their happiness $H$ is $H(x_i,x_j):=-vert vert x_i-x_j vert^{3/2} -5 vert.$ Distances $vert x_i-x_jvert le 1$ are not allowed since ants cannot be on top of each other.

Now consider the total happiness $H_N:=sum_{i<j} H(x_i,x_j).$ The question is the following:

Consider any maximiser $x=(x_1,..,x_N)$ of happiness $H_N.$ Can we show that any optimizer is always contained in a ball $B(y(x),rsqrt{N})$ where $r$ is a universal positive constant independent of $N$ and the maximiser and $y$ is allowed to depend on $x$. That means, all $x_i$ of the optimizer are in $B(y(x),rsqrt{N}).$

The conjecture of $B(y(x),rsqrt{N})$ is due to the fact that order $N$ lattice particles would fit into a ball of radius $B(y(x),rsqrt{N}).$

The question sounds almost like a school problem, but I just cannot exclude weird configurations.

Happy New Year…Given Covid-19, What Inspires You To Create A Life Worth Living?

Hello, I’m curious to learn about how people make meaning in their lives and what measurement tools are used to monitor progress toward fulfillment.

So, thank you in advance for your reflective thoughts…

What is most meaningful to you and what does it inspire you to create?

How do you measure your progress toward fulfillment? Is it by money, job titles, number of children, number of vacations, number of posts (lol) or something else and how does it support your progress?

Looking forward with glee to hearing all your answers…


с Новым Годом From Parkinhost! (That’s Happy New Year for a Russian VPS Starting at $4.90/month!)

From plague to riots to wildfires to elections to murder hornets…2020 was quite a year!  Of course there were some good things in 2020, like some of our great offers.  And now we’ll close out with the last offer of 2020 from ParkinHost.

They debuted during our Black Friday/Cyber Monday season and now they’re back with another offer in Russia.  With ten years in the business and 4000+ customers, these folks look like they’re in the business for the long haul.

Here’s what they had to say about themselves:

ParkInHost is an entity of DiggDigital known for its low-cost high-quality Unmanaged VPS and Dedicated hosting services in European DC’s. 

The company was founded by 2 friends while studying there Engineering in 2010. It took nearly 4 years to understand the markets, marketing, and streamline operations with DC’s and our partners. Its been 10 years since we are in a business serving more than 4000+ new clients each year.

Our motto was to provide quick support and assistance when clients require us. We support Via Live Chat, Skype, and Support ticket 24/7 and 365 days a year.

As of today we still have a small team of 14 members only working in rotational shifts to give the best to clients.”

Their WHOIS is public and they are registered in India (PAN AAKFD2487J). You can find their TOS on their website and you can pay with Paypal, Bitcoin, Alipay, Perfectmoney, Webmoney, Ideal, Dotpay, and Sofort Banking.

Remember that the LEB community would love to hear about your experience in the comments section below!

Now read more to see the offers!

Russia VPS – HDD

  • 1GB RAM

  • x vCPU

  • 50GB HDD space

  • Unlimited

  • 200 Mbps

  • 1x IPv4

  • KVM

  • Coupon: PARKINHOST50

  • $4.9/month

  • $52.20/year

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Russia VPS – SSD

  • 1GB RAM

  • 1x vCPU
  • 25GB SSD space

  • Unlimited

  • 200 Mbps

  • 1x IPv4

  • KVM

  • Coupon: PARKINHOST50

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Additionally, ParkinHost has special offers on powerful Russian Dedicated Servers.


  • Intel Xeon E5-2680v3 x2

  • 128GB RAM

  • 2x 960GB SSDs

  • Hardware RAID10

  • 1Gbps uplink

Please let us know if you have any questions/comments and enjoy!


I’m Andrew, techno polymath and long-time LowEndTalk community Moderator. My technical interests include all things Unix, perl, python, shell scripting, and relational database systems. I enjoy writing technical articles here on LowEndBox to help people get more out of their VPSes.

Happy Holidays, dear friends! | Proxies-free

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