More information: Dhagash Mehta et al. “Kinetic Transition Networks for the Thomson Problem and Smale’s Seventh Problem.” Physical Review Letters. DOI: 10.1103/PhysRevLett.117.028301 , Also at arXiv:1605.08459 [cond-mat.soft] This puzzle is the essence of Thomson’s problem, which asks how to arrange equal charges (such as electrons) on the surface of a sphere in a way that minimizes their electrostatic potential energy—the energy caused by all of the electrons repelling each other. According to Coulomb’s law, the total energy is inversely related to the sum of the distances between all possible pairs of charges, so the goal is to spread the charges as far apart as possible.This task is more difficult than it sounds—Thomson’s problem has been rigorously solved only for numbers of 2, 3, 4, 6, and 12 charges. In 1998, mathematician Steven Smale identified the problem of how to choose starting points close to the lowest-energy state (which makes it easier to solve Thomson’s problem) as the seventh problem on his list of 18 unsolved problems for the 21st century. Part of the reason why Thomson’s problem is so important is because its applications are so far-reaching. In 1904, J.J. Thomson originally proposed the model of charges on a sphere to describe the structure of an atom. Even though experiments disproved this “plum pudding model” long ago, the Thomson problem still has notable applications in chemistry (for understanding how electrons fill electron shells in atoms), biology (for determining the arrangements of proteins on the shells of spherical viruses), as well as in physics, computer science, and such practical applications as determining the optimal placement of communication satellites around the Earth. Spheres on treesNow in a new paper published in Physical Review Letters, a team of mathematicians, engineers, and scientists from the US, the UK, and Australia has taken a new approach to Thomson’s problem that makes it much easier to determine the lowest-energy configuration. For seven numbers of charges (every third number from 132 to 150), they have constructed tree-shaped disconnectivity graphs, where the vertical axis or “trunk” corresponds to the energy of a particular charge arrangement. Each “branch” terminates at a local minimum, which are the states that have lower energies than all of their neighboring states, and so they are candidates for the ultimate lowest energy state, the global minimum. By visualizing the problem in this way, the researchers noticed that these particular graphs don’t have lots of branches extending from lots of other branches. Instead, every branch connects to only a few other branches and then to the trunk at regularly spaced energy thresholds, so that the graph resembles a palm tree or single funnel structure.The researchers found that this “funneled potential energy landscape” is characteristic of a highly ordered structure and displays characteristics of a small-world network. As a result, it provides an important clue for finding the global minimum. It tells the researchers to start their optimization algorithms using the local minima because, in these networks, it turns out that every local minimum is always within 5-7 steps (branches) of the global minimum. This is true even for local minima that have much higher energies than the global minimum, and even when the total number of local minima is very large. (Phys.org)—How do you arrange a group of points on the surface of a sphere so that all the points are as far apart from each other as possible? With two points, the answer is easy: place them on opposite sides of the sphere, as if they are endpoints of the diameter. With three points, make them the vertices of an equilateral triangle, and so on. But as the number of points increases, so does the difficulty of the problem. By visualizing the problem from a new perspective, the researchers found that lower-energy configurations have more connections than higher-energy configurations do. Credit: Mehta et al. ©2016 American Physical Society Journal information: Physical Review Letters This document is subject to copyright. Apart from any fair dealing for the purpose of private study or research, no part may be reproduced without the written permission. The content is provided for information purposes only. Spin glass physics with trapped ions © 2016 Phys.org The disconnectivity graph for 147 charges on a sphere has a structure-seeking “palm tree” organization. The five lowest minimum energy configurations are shown. Credit: Mehta et al. ©2016 American Physical Society Citation: Researchers chip away at Smale’s 7th unsolved problem in mathematics (2016, July 15) retrieved 18 August 2019 from https://phys.org/news/2016-07-chip-smale-7th-unsolved-problem.html The disconnectivity graph reveals other information, such as that lower-energy local minima have more connections to other states than do higher-energy local minima. The researchers also discovered that the global minimum is always the most highly connected node in the entire network, making it the network’s central node. Implications for Thomson’s and Smale’s problemsUsing this insight from the network’s single-funneled, small-world structure, finding the global minimum for Thomson’s problem becomes much easier than before.”Our work looks at Smale’s seventh problem from a completely different perspective and sheds novel light on it,” coauthor Dhagash Mehta, at the University of Notre Dame and the University of Adelaide, told Phys.org. “In this work, methods developed by the theoretical chemistry community have helped understand a deep mathematical problem. Often it is the other way around.”As the researchers explain, it’s easier to solve Thomson’s problem in these particular cases than it is to solve Smale’s problem (of choosing good starting points). So although the results will likely be useful, they do not go very far toward solving Smale’s seventh problem.”I think ‘chip away’ is about right,” said coauthor David Wales at University Chemical Laboratories in Cambridge, UK. “There is no rigorous mathematical progress on the problem from an analytic point of view.”In the future, the researchers plan to extend this approach to larger numbers of charges. From earlier work, they expect that landscapes with more than 400 charges will start to display multiple funnels, so the small-world structure may disappear. “While we have only shown data for seven numbers, we have strong reasons to believe that the single funnel is a feature for numbers less than 150,” said coauthor Halim Kusumaatmaja at Durham University in Durham, UK. “For larger numbers, there will likely be multiple funnels. Nonetheless, the network analysis could still be exploited to help us quickly identify candidates for the global minimum.”Other lines of work include exploiting the small-world properties discovered here to improve other optimization algorithms and develop novel algorithms, as well as to incorporate weight and direction into these networks, which may provide additional insight into the Thomson problem.”The social network analogy for networks of minima of the Thomson problem will go further when we analyze other network properties of these networks of minima,” Mehta said. “Our results will also help in constructing novel algorithms to find the global minimum more efficiently by exploiting these network properties.” Explore further
Chief Minister Nitish Kumar’s tweet likening himself to ‘chandan’ (sandalwood), which remains unaffected despite snakes wrapping themselves around it, has created a flutter in poll-bound Bihar as it is being seen as directed against his ally Rashtriya Janata Dal (RJD) chief Lalu Prasad Yadav.One @SunilVChandak posed a question to Kumar on his programme “Ask Nitish” on Twitter on July 20 – “#AskNitish if you win with Laluji and the numbers are significant, how you will be able to give good development-ordinated government?” Also Read – Need to understand why law graduate’s natural choice is not legal profession: CJIIn reply, Kumar had tweeted, “Bihar’s development is my sole agenda. Jo Rahim
Kolkata: BJP should take the blame for the “shabby arrangement” at Prime Minister Narendra Modi’s rally at Midnapore, where 90 people were injured due to collapse of a canopy, Trinamool Congress leader Saugata Roy said in the Lok Sabha.”The Prime Minister addressed the Kisan Kalyan rally in Midnapore. People were brought in from Jharkhand, Odisha. During the rally, a tent collapsed, leaving 90 people injured, of which 50 were women. BJP had organised the rally. They must take blame for the shabby arrangement,” Roy said during Zero Hour. Also Read – Rain batters Kolkata, cripples normal lifeA makeshift tent had collapsed during Modi’s rally in Midnapore on July 16, following which the Centre had sought a report from the Bengal government.The tent was erected next to the main entrance of the rally venue to shelter people from the rain and the incident had happened when the Prime Minister was mid-way through his speech.Meanwhile, a team comprising senior officials of the Ministry of Home Affairs visited the spot in connection with the incident. Also Read – Speeding Jaguar crashes into Mercedes car in Kolkata, 2 pedestrians killedJoint Secretary Arti Bhatnagar and two senior SPG officials visited the spot after holding a detailed discussion with senior officials of the district administration and police.The police have also been looking into the role of the decorators that had erected the pandal. It may be mentioned that no fault of the Public Works Department was found with regard to the collapse, according to sources in Nabanna.It has been stated that the area was under the SPG from seven days ahead of the rally. So, the state police does not have much to do in this connection.
Kolkata: The 10th edition of Apeejay Kolkata Literary Festival will see the participation of writers, poets and dignitaries from China, Italy, France, Australia, Singapore, USA, UK and Pakistan, alongside noted names from Indian literary circles. Scheduled to take place from January 18 to 20 next year, it will feature Andrew Sean Greer, Pulitzer winner for Fiction 2018; Ira Mukhoty, known for her feminist writing; Devdutt Pattanaik, mythology expert and author; Ramchandra Guha, historian; Ravinder Singh, Young Adult author; Upamanyu Chatterjee, celebrated writer; Ravish Kumar, journalist, Poet and author; along with veteran Bollywood actor Naseeruddin Shah. Also Read – Rain batters Kolkata, cripples normal life “Apeejay Kolkata Literary Festival (AKLF) has connected, empowered and mobilised new voices and ideas since its inception in 2010. In the last nine years, the festival has contributed greatly to restoring Kolkata’s literary edge, a distinct and valued identity, I felt, at the time, it was losing. “AKLF has developed as the fountainhead of the mission we pursue 365 days at Oxford Bookstores – of books, reading, literature, writers, publishers, building readership, creating awareness and attraction for the world of books and popularising the whole gamut that goes with books and publishing,” said Priti Paul, Director, Apeejay Surrendra Group. Also Read – Speeding Jaguar crashes into Mercedes car in Kolkata, 2 pedestrians killed Novelist of the much acclaimed “All The Lives We Never Lived” Anuradha Roy; former Jammu and Kashmir Chief Minister Farooq Abdullah; and Sahitya Akademi winner Jerry Pinto will also be participating. They will be joined by Ashwin Sanghi, Rajmohan Gandhi, Shashi Tharoor, Shobhaa De, and Ratna Pathak Shah. AKLF 2019 will focus on health, current affairs, women’s issues, and children’s literature among other themes. The 10th edition of the festival will be held in Park Street’s iconic spaces giving everyone an opportunity to (re)visit, admire and enjoy their tangible and intangible heritage.
Free Webinar | Sept 5: Tips and Tools for Making Progress Toward Important Goals Attend this free webinar and learn how you can maximize efficiency while getting the most critical things done right. Brought to you by PCWorld Register Now » 3 min read December 5, 2007 A few years back, when I replaced my aging Athlon XP-based home-built PC with a faster, quieter system, I stored the old one away presuming that someday I’d do something interesting with it. Microsoft’s Windows Home Server proved the perfect excuse to do just that. The end result is a highly useful, though sometimes frustratingly simplistic, addition to my home tech lineup.Windows Home Server is Microsoft’s first stab at a consumer server product that sits at the heart of your network where other PCs can access its content. Available now on hardware such as HP’s MediaSmart Home Server or as a $180 software package from system-builder sites such as Newegg, Windows Home Server lets you store and stream media files, back up multiple PCs, and connect remotely via the Web. Plus, Microsoft says that add-on features such as video recording and home automation are on the way from third-party vendors.Windows Home Server requires an ethernet connection between the server and the network (Microsoft deemed wireless networking too flaky). I installed a prerelease copy of the operating system in about 2 hours; the only snag I hit involved enabling the remote Web access features. My router turned out to be the problem, and one I wasn’t able to resolve with tech support. To connect client PCs to the server, you install a simple console application on each that also lets you tweak the server’s settings.Soon, I was streaming music, photos, and standard-definition video to my 802.11g-enabled notebook, flawlessly. I experienced some stutters with a high-def test video file, but that’s an 802.11g bandwidth issue. Everything streamed cleanly to my ethernet-connected Windows XP Pro PC and my Xbox 360. The 360 connection is great, as it allows me to access media where I most enjoy it: on my couch, in front of my HDTV.Windows Home Server also lets you back up the entire contents of each connected PC to the server’s hard drive, as a compressed file that it updates daily with only the changes that have occurred since the last backup. The backup feature is quite slick, and it illustrates the degree to which Microsoft has successfully simplified an often-complicated process. You can add more hard drives to the server, and even enable data duplication (essentially RAID 1 data mirroring). But the nitty-gritty settings for such features are largely inaccessible, hidden behind basic wizards and check boxes.Simplicity is great, but I think enthusiasts like me will want access to more knobs and switches. Microsoft might envision a home server in every house, but I’m not sure the average PC user would even know what to do with one (yet). After having lived with the server for a few weeks, though, I can’t imagine not having one–a clear sign that it’s filling a need in my geek household. Better yet, it let me pull my old workhorse PC out of retirement.