I have now done it a couple of times, but always have to look it up. Here are the steps to re-install Grub from a live Ubuntu CD, when you have Windows and Linux on your hard disk. You must be root, or run these commands with
mount /dev/sda2 /mnt
mount /dev/sda3 /mnt/boot, if you have a separate boot partition
mount -o bind /dev /mnt/dev
mount -o bind /sys /mnt/sys
mount -t proc /proc /mnt/proc
chroot /mnt /bin/bash
Above commands are from method 3 in GRUB (in German).
Below I copy some paragraphs from a recent article in Polymath 8 – a Success! | Combinatorics and more by Gil Kalai on Zhang’s breakthrough regarding prime numbers:
Twin primes are two primes and . The ancient twin prime conjecture asserts that there are infinitely many twin primes. (This conjecture is still not proved.) The prime number theorem asserts that there are (asymptotically) primes whose value is smaller than a positive integer n, and this implies that we can find arbitrary large pairs of consecutive primes and such that is at most . Until a few years ago nothing asymptotically better was known. Goldston, Pintz, and Yıldırım (GPY), showed in 2005 that there infinitely many pairs of primes and such that is . A crucial idea was to derive information on gaps of primes from the distribution of primes in arithmetic progressions. GPY showed that conditioned on the Elliott-Halberstam conjecture (EHC) there are infinitely many primes of bounded gaps (going all the way to 16, depending on a certain parameter in the conjecture, but not to 2). Yitang Zhang did not prove the EHC but based on further understanding of the situation found a way to shortcut the conjecture and to prove that there are infinitely many primes with bounded gaps unconditionally!
Gil Kalai goes on: Here is a very nice 2007 survey article by Kannan Soundararajan on this general area of research and the GPY breakthrough.
Now some astonishing facts on Yitang Zhang from the English Wikipedia: After graduation, Zhang had a hard time finding an academic position. In a recent article, Zhang’s thesis advisor, Professor Tzuong-Tsieng Moh, recalled that “Sometimes I regretted not ﬁxing him a job” and “He never came back to me requesting recommendation letters.” He managed to find a position as a lecturer after many years. He is still currently a lecturer at the University of New Hampshire, where he was hired by Kenneth Appel back in 1999. Prior to getting back to academia, he worked for several years as an accountant and a delivery worker for a New York City restaurant. He also worked in a motel in Kentucky and in a Subway sandwich shop.
Added 01-Jan-2014: Unheralded Mathematician Bridges the Prime Gap.
Except notably Russia and Africa this blog has been visited by quite a number of countries. I am surprised.
Also see Statistics of this Blog: Top Views by Country.
Below table is from work-with-us (data as of 22-Sep-2013): One of these things is not like the other.
||Monthly Uniques (US)
Georg Hager’s Blog posted an illustrative article on icc versus g++ performance w.r.t. OpenMP. Dr. Georg Hager is one of the authors of Introduction to High Performance Computing for Scientists and Engineers.
double precision, dimension(N) :: a,b,c,d
! initialization etc. omitted
s = walltime()
!$omp parallel private(R,i)
a(i) = b(i) + c(i) * d(i)
!$omp end do
!$omp end parallel
MFlops = R*N/(e-s)/1.e6
JpGraph is a library of PHP code to draw a variety of graphs:
- line+bar+pie charts
- radar+polar+contour graphs
- bar+QR codes
- Gantt charts
- a couple of other chart types
Here are two examples:
Inspired by a discussion with my son regarding movement of a point-charge between two other charges I revisited the definition of the line integral. Wikipedia offers an excellent visualization of the definition of the line integral for a vector field. See animated graphic below:
This is something a book barely can do.
BTW, still today, I find multivariate calculus interesting, see On Differential Forms.
Just read on Torricelli’s trumpet in Wikipedia. This states that there is a body having infinite surface but finite volume! That sounds contradictory at first.
Function in question is
from to .
Above plot is from Maxima using:
plot3d( [x,1/x*sin(v),1/x*cos(v)], [x,1,8], [v,0,2*%pi] );
Brian Koberlein wrote a very good introductory article on curvature of space and time in the vicinity of gravity: see The Attraction of Curves – Brian Koberlein.
Last year I used a drop-in replacement for the ordinary Linux
sort command called
nsort from Ordinal Technology. Ordinal’s
nsort is free but not open-source. One thing is clear, however, it is very fast.
nsort was written by Chris Nyberg.
The motivation for looking for a faster sort was as follows. I had to drop all duplicate records from a single Oracle database table. The table had more than 800 million records. It was later found out, i.e., after I already had the solution, that from the initial number of records only 3% of the records would remain, i.e., 97% of the records were indeed duplicates. The solution basically was to extract all data from the table with a small C program. The extracted data was then sorted (
sort -u), the result then loaded into the database table again.
nsort instead of plain
sort runtime was one-third. In my case overall runtime went down from 60 minutes to 20 minutes.
Nsort user guide is the very readable user’s guide to
nsort can be found at sortbenchmark.org.